Empirical models and design codes in prediction of modulus of elasticity of concrete

Abstract

Modulus of Elasticity (MOE) is a key parameter in reinforced concrete design. It represents the stress-strain relationship in the elastic range and is used in the prediction of concrete structures. Out of range estimation of MOE in the existing codes of practice strongly affect the design and performance of the concrete structures. This study includes: (a) evaluation and comparison of the existing analytical models to estimating the MOE in normal strength concrete, and (b) proposing and verifying a new model. In addition, a wide range of experimental databases and empirical models to estimate the MOE from compressive strength and density of concrete are evaluated to verification of the proposed model. The results show underestimation of MOE of conventional concrete in majority of the existing models. Also, considering the consistency between density and mechanical properties of concrete, the predicted MOE in the models including density effect, are more compatible with the experimental results.

Keywords

modulus of elasticity / normal strength normal weight concrete / empirical models / design codes / compressive strength / density

Introduction

Despite innovations and advances in construction materials, Normal Strength Normal Weight Concrete (NSNWC) is still the most widely used composite material in the construction industry. Modulus of Elasticity (MOE) and Compressive Strength (CS) are the most well-known parameters of concrete used in design of concrete structures. Other mechanical properties of concrete can be displayed in terms of CS and not often in terms of MOE [¹].

Accurate prediction of MOE is crucial in the reinforced and prestressed concrete for elastic shortening claculation, shrinkage and creep loss evaluations as well as crack control that is directly related to the durability of concrete structures [²,³]. MOE is also used to calculate the deflection of structures under short-term and long-term loading and serviceability aspects. Its ccurate prediction directly influences the reliable prediction of creep [⁴,⁵]. It is also used in seismic analysis for rational deformation and drift calculations. MOE is an essential parameter in determination of the stress and strain distributions and displacements, especially, when the elasticity considerations govern the concrete structure design.

MOE can be defined both theoretically and experimentally. In the theoretical model concrete behaves in a multi-phase system, and the MOE is obtained as a function of the elastic behavior of its components [³]. The hardened concrete is combination of components including coarse and fine aggregates and cement, however, elasticity of concrete doesn’t follow their elastic behavior [³,⁶]. Fig. 1 compares the stress-strain behavior in the hardened concrete with components of the concrete mixture. The volume fraction and stiffness of the aggregates primarily affect the modulus of elasticity of concrete; however, the effect of aggregate-paste bond is also important [⁷].

The laboratory and in-situ testing are preferred methods to get an accurate value of MOE; however, it is determined from CS in majority of the construction projects. Considering the non-linear and brittle behavior of concrete, CS is not the only key factor in MOE variations. Topcu and Ugurlu (2007) [⁸] explained the other effective parameters on MOE value as shown in Fig. 2.

Geological distribution, density and porosity of aggregate are crucial factors in determination of the modulus of elasticity of concrete. A dense aggregate results in a better modulus of elasticity of concrete [⁹,¹⁰]. In addition, a higher content of coarse aggregate in the concrete mixture exhibit slightly higher modulus of elasticity [¹¹]. Effect of aggregate on the mechanical properties of concrete, especially, the modulus of elasticity of concrete is as important as the water to binder ratio in the concrete mixture [¹²].

The cement content as paste material to bond the aggregates is also an important parameter in the mechanical properties and modulus of elasticity of concrete. Increasing the paste content decrease the void content of the mixture, that in turn, increases the modulus of elasticity of the hardened concrete [¹¹].

Similar to the compressive strength of concrete, the modulus of elasticity also experiences a reduction as the water to binder ration increases [¹³].

Russian Standard (SP 52-101-2003) [¹⁴] defines the coarse aggregate properties, mix design, curing conditions, loading rate, chemical admixtures and mineral admixtures as operative parameters on MOE of the hardened concrete. Local aggregates and paste material, testing conditions, test method and loading rate also defined as effective parameters on MOE. Levtchit et al (2004) [¹⁵] reported that the age-related increment of MOE is more significant than the CS effect; even though, with no increase of CS, the increased MOE value was observed.

There are several ways to determine the MOE of concrete from test specimens in the literature. Nonetheless, the technical instructions in the utilized standards and the applied controls on the compression machine influence the test results [¹⁴]. The instructions in ASTM-C469-02El are globally used in determination of MOE in experimental investigations and the codes of practice. Sometimes, the researchers propose completely different models by using the same testing standards. The difference comes from using different curing conditions, mix designs, local materials and untilizing different testing methods.

This study aims to develop new model to predict MOE from CS of concrete. The main objectives of this study are:

a) Comparison and evaluation of the previously developed 35 models in international design codes and 19 empirical models to predict MOE of conventional concrete;

b) Comparison and verification of the predictions of the models by the experimental values of MOE in laboratory conditions from previously conducted investigations of 100 mixtures;

c) Proposing and verifying a new MOE model for normal strength concrete.

Research significance

Accurate assessments of concrete structures need appropriate material models, advanced structural simulation tools and probabilistic approaches to consider unavoidable uncertainties. It is a fact that the mechanical properties of concrete are highly dependent on the types and proportions of binders and aggregates that are not the same, worldwide. Since limited numbers of analytical models are used to predict the MOE values in majority of the countries around the world, it is vital to investigate that whether all the hypotheses are covering all the experimental results by utilizing different materials, test methods and curing conditions.

The utilized empirical data in the proposed model covers a wide range of the experiments including different curing methods, mixture design, local fine and coarse aggregates, testing setup and loading techniques. Therefore, the developed model poses the capability to consider a wide range of unseen parameters to provide an accurate prediction of MOE of concrete.

Experimental database for MOE

Utilizing the results of published researches as database is an effective tool for applicabillty of the various MOE models for normal strength concrete. The experimental database has been drawn mainly from the research articles presented at conferences and published articles on mechanical properties of concrete. Despite the advantages of using experimental results from different investigations, it can be problematic due to deficient information about the exact composition of the concrete mixtures; size of the specimen and curing conditions. In addition, the testing method varies by different researches and in some cases this information is not entirely presented [¹⁶].

This study utilizes the results of previously conducted twenty laboratory investigations [⁸,¹⁷–³⁵] to propose and verification of a new equation to predict MOE from CS of concrete. Predictions of the proposed model are also compared and verified by the existing models in the literature and codes of practice.

Existing models of modulus of elasticity

Models in codes of practice

In absence of the accurate testing system to define the exact value of MOE in the applied concrete in the construction, the existing relationships in codes of practice are frequently used to predict the MOE . However, the problem is that for the same range of CS there are many relationships to predict the MOE. In addition, there are considerable differences in prediction of MOE of the same grade of concrete by different models.

To have a comprehensive documentation of the relationships between MOE and CS, and possibly, to perform the most accurate comparison, the existing models for normal strength concrete in 35 international design codes have been investigated. Table 1 shows the most common models of MOE in the international codes of practice. There are some density and compressive strength limitations in application of some of the models in Table 1.

The proposed relationships in SP52-101-2003 [¹⁴], JSCE-2007 [³⁶] and BS 5400-4 (1990) [³⁷] in Table 1 are results of regression analysis of the corresponding data in Tables 2, 3 and 4, respectively.

Notwithstanding the local environmental conditions, material resources, curing condition and testing methods, some countries apply the technical instructions and models developed in other parts of the world. In this regards, as presented in Table 5, the technical publications of American Concrete Institute (ACI), Canadian Standards Association (CSA) and British Standards Institution (BSI) are basis of the local standards in most parts of the world.

Empirical models

Well-investigated and verified empirical equations are reliable foundations to establish models to predict the mechanical properties of concrete in the design codes. Annually, many experiments are being conducted to make a better estimation of MOE from test results. The considerable point is that these experiments cover a wide variety of aggregates, paste type and mix design, and evaluate the effect of different parameters such as concrete density on the MOE value. This study investigates the proposed empirical models since 1943 to predict the MOE in normal-strength concrete. Table 6 shows the existing empirical models of MOE and the limitations applied on each model. Evidently, some similar empirical models apply different ranges of CS to estimate the MOE of normal strength concrete.

Proposed analytical model

The relationship between CS and MOE is always a matter of interest to select a proper concrete to design the structural reinforced concrete elements. In other words, CS plays a key role in determination of the MOE value in the hardened concrete and majority of the MOE models are founded on CS. To estimate the modulus of elasticity from compressive strength in normal strength concrete at age 28 days, Equation (1) is proposed based on regression analyses of the experimental data. Considering the better computability of the model, the model in ACI-318-08 [³⁸] is selected as the basis for the proposed model.

(1){{\displaystyle E}}_c=\mathrm{9.19}{({{\displaystyle f}}_c^{\text{'}})}^{\mathrm{0.354}}

where, E_c (GPa) is modulus of elasticity and {{\displaystyle f}}_c^{\text{'}} (MPa) is compressive strength of concrete. This equation gives a matching estimation of MOE in the CS ranging from 20 to 50 MP.

According to Fig. 3, predictions of the proposed model of MOE from CS in Eq. (1), are in good agreement with the experimental results and the correlation factor between the experimental and predicted values is 0.78.

Fig. 4 compares the predictions of the models in design codes presented in Table 1, with predictions of the proposed model in Eq. (1). Fig. 5 compares the predictions of empirical models presented in Table 2 with predictions of the proposed model in Eq. (1). The comparison in both figures is performed in the CS ranging 20 to 50 MPa.

According to Fig. 4, the models in design codes produces considerable under and overestimated MOE to use for concrete design purposes, generally. However, some models such as ACI-318(08) [³⁸], CSA-A23.03-M94 [³⁹] and VBC-95 [⁴⁰] give reasonable estimation of the MOE in agreement with the experimental data. Fig. 6 compares the predicted MOE by the best matching models in codes of practice and the proposed model with the experimental MOE data.

According to Figs. 4 and 5, in general, the empirical models give better estimation of the MOE for normal strength concrete. However, some considerable far estimated values of MOE, in comparison with the predictions of the proposed model are obvious. Also, some empirical models such as the models developed by Pauw (1960) [⁴¹], Leemann and Hoffmann (2005) [²⁶] and Haranki (2009) [⁵] give realistic estimation of MOE, in agreement with the experimental data. Fig. 7 compares the predicted MOE by the best matching empirical models and the proposed model with the experimental MOE values.

Results and discussions

As shown in Table 7, the proposed model in this study provides a better prediction of MOE with a coefficient of correlation factor (R²) 0.78, compared to correlation factors 0.83 for the ACI 318-08 [³⁸], 0.71 for the CSA.A23-M94 [³⁹] and 0.63 for the VBC-95 [⁴⁰] models, respectively.

Table 8 shows the correlation factor between the experimental data and predictions of the empirical models. According to Table 8, the proposed model provides a better estimation of MOE for the specified ranges of CS. The correlation factor for proposed model is 0.86, while, it ranges from 0.82 for Pauw (1960) [⁴¹], 0.81 for Haranki (2009) [⁵] and 0.75 for Leemann and Hoffmann (2005) [²⁶] models, respectively.

This study investigates the relationship between the compressive strength in the range of 20-50 MPa and the modulus of elasticity of conventional concrete. According to Figs. 3, 4, 5 , the MOE is well predicted by the proposed model and the best fitting models, however in the CS ranging 27–37 MPa, the predicted MOE values are more compatible with the experimental data. This is the most common range of CS of the conventional concrete used in the common construction projects, in which, the proposed equation gives more realistic estimations of MOE than other existing models.

Along with the effect of compressive strength, effect of the concrete density is also included in 22% and 37% of the models in design codes in Table 1 and the empirical models in Table 6, respectively. According to Figs. 5 and 6, in general, the estimated MOE by these equations including density effect are more compatible with the experimental data.

There is a power type nonlinear relationship between the modulus of elasticity and compressive strength of conventional concrete in almost all the existing models. However, the predicted values of MOE for the same range of CS are completely different from the experimented data. The models in some globally referred codes of practice give significantly under or over estimations from the experimental MOE, which in turn, can affect the performance, economy and safety of the designed concrete structure.

The proposed model gives the most compatible results with the collected experimental data. Considering the position of this model in Fig. 4, majority of the existing models in design codes underestimate the MOE for normal strength concrete. Among the classified models in Table 5, the first class (ACI-318-08) [³⁸] gives the most compatible predictions of MOE with the experimental values. The second class (ACI-312-92) [⁸] has the highest overestimation of MOE in comparison with other models. The classes 4 and 5 also overestimate the MOE values; however, they are not as far as predictions of the second class. Finally, the class 4 has the lowest estimation of MOE among all groups.

The proposed model based on a wide range of experimental data covers the effects of different environmental conditions and mixture design on conventional concrete. Considerable the number of effective parameters to estimate the modulus of elasticity of conventional concrete. Considering the number of experimental data for each range of the compressive strength in the experimental data, the proposed model poses a better consistency of the relationship between input data and the estimated parameter.

The models in FHWA (2000) [⁴²] and NCHRP (2003) [⁴³] significantly underestimate the MOE values, sigmificantly while, CEB-FIP (2003) [⁴⁴], AASHTO-LRFD [⁴⁵] and OHBDC (1983) [⁴⁶] models have the highest predictions among the models. The difference between the predicted values of these codes, especially for CS under 40 MPa is about 80%. By increasing the CS, the difference in the predicted values decreases to about 50% in the CS around 50 MPa. The model in NCHRP (2003) [⁴³] gives about 500% lower MOE for CS under 43 MPa. While, in the range of CS between 43 MPa–50 MPa, the estimated values are about 50% above the experimental MOE.

All the models evaluated in Fig. 4 show a non-linear pattern between CS and MOE by similar trends, in general. . However, the model in NCHRP (2003) [⁴³] gives substantially different predictions and estimates significantly lower and higher MOE for CS around 20 and 50 MPa, respectively.

According to Fig. 4, the models in Japanese code (AIJ-1985) [⁴⁷], Canadian code (CSA.A23.04) [⁴⁸], American code (FHWA-2000) [⁴²], New Zealander code (GDC-2000) [¹], South African code (SABS0100-92) [⁴], Peruvian code (NTE-E.060-2009) [⁴⁹] and British code (BS8110-97) [⁵⁰] underestimate the MOE. While, the models proposed in Australian code (AS3600-09) [⁵¹], Japanese code (JSCE-2007) [³⁶], Indian code (IS-4562000) [⁵²], Dutch code (VBC-1995) [⁴⁰] and Finnish code (RakMK-2012) [⁵³] predict matching MOE values with the proposed model.. In other words, the latter group ofcodes are more compatible with the proposed equation and experimental data.

Majority of the models in Tables 1 and 5 are modification of the preliminary primary developed models in the literature. The experimental MOE data for normal-strength concrete in this study is taken from a wide range of experiments including different types of aggregates, environmental conditions, curing methods , testing equipment and assessment methods. According to Tables 7 and 8, the initial models such as Pauw (1960) [⁴¹] predict more compatible values of MOE with the experimental results for normal-strength concrete.

Concluding remarks

Rational and simplified analytical model to predict the modulus of elasticity of convention concrete has been proposed and compared with the existing empirical models and the models in a wide range of international design codes and experimental data. The following conclusions can be drawn from the study:

- The proposed MOE model based on the model in ACI-318(08) gives reasonable estimation of the experimental results for conventional concrete in the range of compressive strength between 20–50 MPa;

- Existing MOE models in 35 international design codes and 19 empirical models are investigated. Unexpectedly, majority of the models give out of range predictions; generally, lower than the experimental values of MOE. Some models underestimate the MOE in lower CS range and overestimate it in upper levels of CS. In the worst case, the ratio of overestimated to underestimated MOE is bout 5;

- Application of some national models in other regions of the world should be verified by the environmental conditions, local aggregates and testing equipment and methods;

- The models including the density effect give better estimation of MOE from compressive strength;

- The primary models such as Pauw (1960) , are more compatible with the randomly selected experimental results with a wide variety of aggregates, curing condition and testing methods and standards;

- Based on the similar basis and compatible predictions of MOE, international design codes are classified in 5 main groups. ACI-318-08 and ACI-312-92 as representatives of groups 1 and 2 give the most compatible predictions and the highest overestimation of MOE, respectively.

- BS 8110 (1997) and CSA-A23.3 as representatives of groups 4 and 5 also give MOE higher than the experimental data.MOE is significantly underestimated in FHWA (2000) and NCHRP (2003). While, CEB-FIP (2003), AASHTO-LRFD and OHBDC (1983) models have the highest predictions of MOE.

- AIJ-1985, CSA.A23.04, FHWA-2000, GDC-2000, SABS0100-92, NTE-E.060-2009 and BS-8110-97 underestimate the MOE. While, the models proposed in AS-3600-09, JSCE-2007, IS-4562000, VBC-1995 and RakMK-2012 predict matching MOE values with the proposed model.

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