Stress-strain Response of High Strength Concrete and Application of the Existing Models

Stress-strain model of concrete is essentially required during design phases of structural members. With the evolution of normal concrete to High Strength Concrete (HSC); various predictive models of stress-strain behavior of High Strength Concrete (HSC) are available in the literature. Such models developed by various researchers are differing to each other, because of the different mix proportions and material properties. This study represents a comparative analysis of available stress-strain models with the experimental results of three different series (100% cement concrete, Silica Fume (SF) concrete and Metakaolin (MK) concrete) of high strength concrete mixes. Compressive strength and stress-strain behavior of 100×200 mm cylinders made of all Prepared mixes was determined at with curing age of 28 days. Compressive strength of all mixes was found in the range of 71-87 MPa. Stress-strain behavior of tested cylinders was found much different from the available predictive models. In view of the dissimilarity occurred between the predictive stress-strain behavior and the experimental data; a new predictive model is proposed, which adequately satisfy the experimental results.


INTRODUCTION
Use of High Strength Concrete (HSC) particularly in mega-construction is becoming more popular, because of its value-added benefits, which resulted in the reduction of structural member sizes, durability and longer life.For concrete as structural material, its compressive strength is an essential parameter requires for ultimate strength design of various structural members.Failure response of a structural member is usually studied using nonlinear analysis, for which the compressive stress-strain curves are used as the main design basis (Lu and Zhao, 2010).Recently, Lu and Zhao (2010) represented a review of the compressive stress-strain models published during last few years.It was observed that most of the available models are not capable to predict the stress-strain response of HSC.Therefore, they proposed a new empirical model, which according to them, is found more versatile and applied on all their experimental results and compared with the previously generated stress-strain curves of Hsu and Hsu (1994), Van Gysel and Taerwe (1996) and Wee et al. (1996).
An interesting question is to find the reasons of invalidation of predictive models to different sets of experimental results.Undoubtedly, the prediction of concrete behavior has been just like obtaining a specific number by rolling a dice.Where, despite using the same concrete mix ingredients, quantities, size of the molds, curing procedure and temperature, there is less probability of obtaining the same compressive stressstrain behavior and strength.At the time when Popovics (1973) and Carreira and Chu (1985) proposed their models for the compressive strength of HSC, the compressive strength of the concrete was not as high as it is today, even the cement composition has been changed and Superplasticizers (SPs) are so advanced, which can better improve the concrete quality.Therefore, there are very less chances of the potential applicability of the predictive models of Popovics (1973) and Carreira and Chu (1985) for the HSCs of today.
Additionally, mineral admixtures or Supplementary Cementitious Materials (SCMs) have now become an essential constituent of the HSC composition, which may possibly influence the stressstrain behavior.Furthermore, the mix design which authors used to propose their predictive models for the compressive stress-strain behavior of HSC were adopted e.g., Popovics (1973), Carreira and Chu (1985), CEB-FIP Model Code 90 (1993), Van Gysel and Taerwe (1996) and Lu and Zhao (2010).Only few researchers used the original mix designs to propose their model e.g., Wang et al. (1978), Hsu and Hsu (1994) and Wee et al. (1996).Upon reviewing these models, it has been found that, while proposing their models, described the deficiencies of the previous predictive models of the stress-strain behavior of HSC; for example, Hsu and Hsu (1994) claimed that, their model is simple and capable of predicting the complete stress-strain curve of HSC of compressive strength Author's remarks Single model capable of predicting stress-strain response from the origin to ultimate (i.e., 0 < ε ≤ ε I ).Sargin and Handa (1969) A, B, C and D are constants which can be estimated by considering the condition A Up to 76 Carreira and Chu (1985)

to 80
Does not fit on current data, due to negative value of β Two models, one for combine prediction of the behavior of rising branch and falling branch up to the limiting strain (i.e., 0 < ε ≤ ε =, C ), while the second model predicts the behavior of the depressing branch from limiting strain up to the ultimate strain (i.e., ε Modified form of Sargin and Handa (1969) Hsu and Hsu (1994 where, The expression is the modified form of Carreira and Chu (1985) Van Gysel and Taerwe (1996) For The expression is the modified form of CEB-FIP Model Code 90 (1993) Lu and Zhao (2010 where, ε 2 is the strain at 0.8f = ′ in the falling branch of stress-strain curve 50-140 For 0 ≤ ε ≤ ε 2 , modified form of Sargin and Handa (1969) and for ε > ε 2 , model of Van Gysel and Taerwe (1996) is used Two models, one for predicting the behavior of the rising branch up to the peak point (i.e., 0 < ε ≤ ε = ′ ), while the second model for predicting the behavior of falling branch from the peak point to the ultimate (i.e., ε = ′ < ε ≤ ε I ).Wee et al. (1996) For where, The maximum compressive strength in MPa; ε = ′ : The corresponding strains; β: The parameter that controls the descending branch; w: The unit weigh of the concrete in kg m % exceeding 69 MPa.However, Carreira and Chu (1985) and in CEB-FIP Model Code 90 (1993), stress-strain models were already proposed for compressive strength exceeding 69 MPa and the model of Carreira and Chu (1985) was simpler.Similarly, the model of CEB-FIP Code 90 (1993) has the assumption of the fixed value of the strain and the steepness of the softening branch, which are not appreciated by the researchers (Lu and Zhao, 2010;Van Gysel and Taerwe, 1996) and therefore, predictive model of CEB-FIP Model Code 90 (1993) is not suitable for HSC.Wee et al. (1996) proposed their model for HSC of compressive strength ranging from 50 to 120 MPa.The appreciable fact is that, they carried out a vast experimental investigation, which was not done earlier by any author who proposed their models for the compressive stress-strain curves.In order to determine influence of different mix compositions on the stress-strain behavior of their specimens; they (Wee et al., 1996) applied four previously proposed models of Hognestad (1951), Wang et al. (1978), Carreira and Chu (1985) and CEB-FIP Model Code 90 (1993) to predict the stress-strain curves and found, the model of Wang et al. (1978) has the best fit.In spite of the closeness to the actual stressstrain behavior, Wee et al. (1996) and Lu and Zhao (2010) did not appreciate the predictive model proposed by Wang et al. (1978) due to its complexity, which is tedious and requires computer aid.Therefore, they (Wee et al., 1996) proposed a new predictive model, based on the model of Carreira and Chu (1985) (Table 1).
According to the authors of the current investigation, none of the predictive models, themselves, are not deficient as they were mainly based on the experimental results of the others rather than their original experimental investigation.Therefore, there is a great need of experimental investigation to determine the fitting of all predictive models as represented in Table 1.Upon the examination of these models, it can be observed that, the predictive models of HSC are based mainly on the two parameters required to generate stress-strain curves including maximum compressive strength and its corresponding strains.Moreover, to date, three categories of these models are proposed, as follow: Type I: Single model capable of predicting stressstrain response from the origin to ultimate (i.e., 0 < ≤ ε I ).Type II: Two models, one for predicting the behavior of the rising branch up to the peak point (i.e., 0 < ≤ ε = ′ ), while the second model for predicting the behavior of falling branch from the peak point to the ultimate (i.e., ε = ′ < ≤ ε I ).Type III: Two models, one for combine prediction of the behavior of rising branch and falling branch up to the limiting strain (i.e., 0 < ≤ ε =, C )., while the second model predicts the behavior of the depressing branch from limiting strain up to the ultimate strain (i.e., ε =, C < ≤ ε I ).
The principal aim of this study is to investigate the stress-strain behavior as obtained by using different predictive models and to identify their deficiencies as highlighted by different researchers.A predictive model is also proposed based on the proposed models of Hsu and Hsu (1994) as well as Lu and Zhao (2010) which is capable of generating stress-strain curves closer to the experimental results.

EXPERIMENTAL PROGRAM
Material properties and mix design: Detail of the material used for the preparation of the HSC and their quantities are presented in Table 2; however, physical and chemical given of cement and mineral admixtures used are listed in Table 3.

Casting and testing of specimens:
For each series (Table 2), two batches of HSC were prepared and from each batch, three cylinders of size 100×200 mm were cast and cured for 28 days.Before pouring the concrete in the molds conforming to specification C470/C470M, slump of the concrete was determined, immediately after mixing the concrete in compliance with ASTM C192/C192M-13a and it was found that, slump is within the limit of 100±10 mm then, the all concrete specimens were demolded and cured.

ANALYSIS OF THE EXPERIMENTAL RESULTS
Compressive strength results consisting of mean strength, standard deviation and sample variance are listed in Table 4. Based on the statistical analysis of the test results, standard deviation was obtained between 1 to 1.75 MPa and the coefficient of variation was found lower than 3%.The consistency of the stress-strain curves is shown in Fig. 1 indicates that, all mixes were well designed and highly cohesive.Highest average compressive strength was obtained in series "M" (Table 4), which was 17.66 and 7.22% higher than series "P" and series "S", respectively.The compressive strength of series "S" was 9.74% higher than of series "P".

Comparison of the mix design used in current investigation:
The authors collected the data of the experimental mix designs used to formulate the predictive models for High Strength Concrete (HSC) from 1990 to date to observe the effect of mix design and aggregate size on the stress-strain curves.The experimental investigation carried by the Wee et al. (1996) was the only comprehensive and close to the current experimental investigation; however, in rest of the investigations, previously generated data was used.It was however, mentioned by Popovics (1973) and Carreira and Chu (1985) that, there are several unseen factors which affect the stress-strain behavior.For example stress-strain curve is very much sensitive to the testing conditions (type and stiffness of the compression testing machine, loading rate and duration); specimen's shape and size; position, type and length of the strain gages applied on the specimens, specimen age, concrete mix composition (particularly coarse aggregate size and quantity), etc., (Popovics, 1973;Carreira and Chu, 1985).Carreira and Chu (1985) further added that, the shape of the descending branch is influenced by the stiffness of the specimen versus stiffness of the compression testing machine, development of the micro cracks at the interface of the aggregate and cement matrix.That is the reason; stressstrain relation is strongly affected by the rate of the strains, quality, content and characteristics of the cement matrix and aggregates.
The experimental data of Wee et al. (1996) is given in Table 5 and compared with the experimental data in Fig. 2.
As mentioned in Table 5, for the same quantity of water and the same size of aggregates, Wee et al. (1996) varied their experimental data by increasing the amount of cement paste (cement+water) and lowering of the aggregate content especially fine aggregate.By doing so, higher strength of the concrete was obtained while the post peak branch of the stress strain curve was observed steeper.The increase in compressive strength is mainly due to the increase in the cement content which offered substantial strength to the cement matrix and shifted the failure towards the aggregates and results higher strength properties.Referring to the Fig. 2, it can be seen that the stress-strain curves of current study appropriately close to those of Wee et al. (1996).This shows that, predictive model of Wee et al. (1996) can closely predicts the stress-strain behaviour of the current study.
Application of the existing predictive stress-strain models on the current data: All models, as listed in the Table 1, are applied on the current experimental results given in Table 3.The predicted and experimental stress-strain curves are presented in Fig. 3 to 5. It can be seen that, the models of Wee et al. (1996) and Lu and Zhao (2010) well represent the post peak branch of the stress-strain curves, while the proposed model of Hsu and Hsu (1994) is suitable for the ascending branch.For all models illustrated in Table 1, goodness of fit of the predicted stress-strain curves to the experimental stress-strain curves of series "P" and series "M" is investigated in terms of Root mean square error "˞H˟" and absolute fraction of variance "ˢ", following the procedure in Khan et al. (2013).These parameters were calculated by selecting the stress values obtained by experiment and calculated from the predictive models at the same level of strain values.
"˞H˟" and "ˢ" were calculated manually using Eq. ( 1) and ( 2), which are described as follow (Table 6): (1) where, "t C ", "J " and "J" represent the experimental results (used as target), predicted results (used as output) and number of observations, respectively.As mentioned in Table 5, the model of Wee et al. (1996) and Lu and Zhao (2010) better captured the shape of the descending branch of the current experimental results.However, as mentioned by Lu and Zhao (2010), the drawback of the model of Wee et al. (1996) is the discontinuity of the curve at maximum compressive stress.Therefore, the proposed model of Lu and Zhao (2010) is selected as the suitable model for the descending branch and the model of Hsu and Hsu (1994) is selected to predict the rising branch of all stress-strain curves shown in Fig. 3 to 5.

Newly proposed compressive stress-strain curves of HSC:
As mentioned in the previous section, the models of the Hsu and Hsu (1994) and Lu and Zhao (2010) are the better representative models of the rising and falling branches of the stress-strain curves, respectively.Therefore, two equations are proposed for the current experimental results.The first equation was proposed for 0 < ε ≤ ε =, C , where ε =, C is the strain corresponding to the limiting stress (f =, C ) level of 0.96f = ′ in the falling branch, while second equation was proposed for f = > f =, C or f = > .96f = ′ , whereas, Lu and Zhao (2010) suggested the limiting stress (f =, C ) level of 0.96 f = ′ is the value of ε =, C corresponding to 0.8 f =

′
(Table 1).The reason for selecting the limiting stress (f =, C ) level of 0.96 f = ′ in the falling branch is the appearance of discontinuity when the model was applied to the current experimental data.
The description of the proposed equations by the authors of the current study is as follows: For 0 < ≤ , : where,

= Ә
'.$% ә % + 2.59 and J = 3 For > , : In the following Eq.( 1) and ( 2), the term "J" is the parameter, which contributes in the toughness of the curve, while "β " and exponential function "1 + 0.1 (J − 1)" decides the shape of the curve in the     Lu and Zhao (2010) ascending and descending branches of the stress-strain curves.
Using Eq. ( 1) and ( 2), new stress-strain curves are generated and shown in Fig. 6 (for series "P"), Fig. 7 (for Series "S") and Fig. 8 (for Series "M").In the following figures, the predicted stress-strain curves are also compared with the experimental stress-strain curves and those generated by using the model of Lu and Zhao (2010).It can be observed that, the proposed model for the prediction of stress-strain curves can better predicts the stress-strain curves as compared to the model of Lu and Zhao (2010).
The fit goodness of the predicted stress-strain curves to the experimental stress-strain curves is shown in Table 5, which confirms that the proposed model predicts the stress-strain curves better than the predictive model of Lu and Zhao (2010).
It is worth mentioning that, the predictive model proposed by the authors is investigated for the current mix design (Table 2) and the testing conditions mentioned in this study.In case of different mix design and testing conditions, predictive results and ˦ , may tend to vary (Table 7).

Fig. 7 :Fig. 8 :
Fig.7: Comparison of the proposed model with the experimental data of series "S" and the proposed model ofLu and Zhao (2010)

Table 1 :
Existing stress-strain models for HSC

Table 2 :
Mix materials and quantities for HSC preparation

Table 4 :
Compression test results at curing age of 28 days

Table 6 :
"RMS" and "V" of the all predictive models