Development of statistical models for prediction of mechanical properties of plain concrete

2.1. Materials

The locally available ingredient materials viz., cement, fine aggregates (river sand), and coarse aggregate are used for casting of concrete specimens. Ordinary Portland Cement (OPC) 53 grade, as stipulated in IS 12269 (20(20) IS 12269: 2013, Indian standard specifications for ordinary portland cement, 53 Grade. Bureau of Indian Standards, New Delhi, India.
), is used. The consistency, soundness, and setting time properties of OPC are assessed as per IS 4031 (21(21) IS 4031, Methods of physical tests for hydraulic cement. Bureau of Indian Standards, New Delhi, India.
). Similarly, the physical properties of fine and coarse aggregates such as sieve analysis, water absorption, and specific gravity are determined following IS 2386 (22(22) IS 2386, Methods of test for aggregates for concrete. Bureau of Indian Standards, New Delhi, India.
). Table 3 shows the test results obtained while determining the physical properties of the ingredient materials.

2.2. Specimen

The concrete specimens are prepared following IS 10086 and IS 516 (23(23) IS 10086-1982 (1995). Specification for moulds for use in tests of cement and concrete. Bureau of Indian Standards, New Delhi.
, 24(24) IS 516: 1959 (Reaffirmed 2004). Indian standard methods of tests for strength of concrete. Bureau of Indian Standards, New Delhi.
). The cube specimens (150 mm [5.90 in.]) and cylinder specimens (300 x 150 Φ mm [11.811 x 5.90 in.]) are cast for compressive strength testing. Beam specimens (150 x 150 x 700 mm [5.90 x 5.90 x 27.56 in.]) are prepared for flexural testing and an additional set of cylindrical specimens (300 x 150 Φ mm [11.811 x 5.90 in.]) are cast for determining the modulus of elasticity. All the above specimens are prepared in each of the eight mixes. The specimens after casting are kept for initial setting at room temperature of 28^° C [82.4^° F] and relative humidity of 90%. The specimens are demoulded after 24 hours, and then the specimens are moist cured by ponding in a curing tank for 28 days.

2.3. Testing of specimen

The testing program for all the specimens is conducted as per IS 516 (24(24) IS 516: 1959 (Reaffirmed 2004). Indian standard methods of tests for strength of concrete. Bureau of Indian Standards, New Delhi.
). All the specimens are tested in Multi-function control console (MCC-8) machine for compression, flexure strength, and modulus of elasticity following IS 516 (clause 5) (24(24) IS 516: 1959 (Reaffirmed 2004). Indian standard methods of tests for strength of concrete. Bureau of Indian Standards, New Delhi.
). All the readings of load and deformation are recorded by a calibrated electronic control system attached to a host PC. Digital linear variable displacement transducer (LVDT) and extensometer are used to measure the deformation. The machine is kept in the load control mechanism throughout the testing program. Care is taken that the vertical axis of the machine platen coincides with the axis of the specimen.

The testing surfaces are milled before tests. The specimens are tested in saturated surface dry conditions. The cube specimens used for compressive testing are subjected to load on the adjacent side of casting. However, in the case of cylinder compression testing, the cylinders are necessarily kept vertical. The load rate used is kept at 0.233 MPa/sec (140 kg/sq.cm/min [1991.27 lbs/sq. in/min]) as recommended by IS 516 (24(24) IS 516: 1959 (Reaffirmed 2004). Indian standard methods of tests for strength of concrete. Bureau of Indian Standards, New Delhi.
). The specimens are tested until failure. Four specimens of both cube and cylinders are tested for a period of 28 days, per mix.

In the flexural testing of concrete beams, the four-point loading mechanism is used. Two LVDT’s are used for measuring the deflection of the beams. All the contact surfaces are cleaned to remove loose materials. The supports are 600 mm [23.62 in.] apart, and loading rollers are 200 mm [7.874 in.] apart, thereby dividing the beam into three parts of 200 mm [7.874 in.] each. Figure 1 shows the beam and LVDT arrangement for the flexural test. The load is applied at the rate of 0.0114 MPa/sec (7 kg/sq.cm/mm) as per IS 516 (24(24) IS 516: 1959 (Reaffirmed 2004). Indian standard methods of tests for strength of concrete. Bureau of Indian Standards, New Delhi.
). The load is increased until the failure of specimens. For every mix, three specimens are tested for the period of 28 days.

Standard cylinders (300 x 150Φmm [11.811 x 5.90 in.]) are used for measuring the modulus of elasticity of concrete. The load is continuously applied at the rate of 0.233 MPa/sec (140 kg/sq.cm/mm [1991.27 lbs/sq. in/min]) as per the guidelines of IS 516 until the average stress of (C+5) kg/sq.cm is obtained, where, C is one-third of the average compressive strength. The load is maintained at this stress value for at least a minute and then gradually unloaded to reach an average stress of 1.5 kg/sq.cm [21.94 lbs/sq.in]. The load is again applied for the second time at the same rate until the average stress of (C+1.5) kg/sq.cm is reached, and unloading is carried out in the same way. The load is further applied for the third time and in ten approximately equal increments of stress reaching up to an average stress of (C+1.5) kg/sq.cm. Two extensometers are diagonally attached to the vertical surface of the specimen for measuring the deformation of specimens at every load change. Figure 2 presents the arrangement of extensometers and the specimen arrangement for the determination of modulus of elasticity test. Three specimens are tested for a period of 28 days per mix proportion.

4.1. Prediction of cube compressive strength

In developing a model through SPSS, compressive strength is used as a dependent, and the quantity of cement, fine aggregate, coarse aggregate, and water in 1 cu.m of concrete are used as variables. The model is developed with SPSS software (version 23) using the experimental data generated during testing of cube compressive strength as presented by Eq. 1. Out of the available 32 strength values, 17 have been used to generate the model, while the remaining 15 strength values have been used to validate the proposed model.

where $fck$ is the compressive strength of cube specimen (MPa) at the age of 28 days, C, FA, CA, and W are the quantities of the cement, fine aggregate, coarse aggregate, and water respectively in kg/m³ of concrete mix.

Pearson’s R for the above-proposed model is found as 0.98, indicating the effectiveness of the proposed model. It is worth noting here that the coefficient of water quantity is obtained to be negative, indicating that compressive strength is inversely proportional to the quantity of water added to the concrete mix. The generated predictive models from the available test data, as detailed in the previous section, are validated from the remaining test results. Table 5 presents actual experimental values of the compressive strength of cube that are obtained from a test program conducted and the predicted values that are obtained from the proposed model [1]. The table also depicts the percentage deviation in the predicted values of compressive strength. It can be seen that in all cases, the percentage deviation is less than 5% except for one value, which is deviating by 9.98%, still less than 10%, and hence well acceptable. These show the effectiveness of the proposed model in predicting the cube compressive strength of the concrete. The efficacy of the present model is further checked with the results available in literature (8(8) Anbuvelan, K., Subramanian, K. (2014). An empirical relationship between modulus of elasticity, modulus of rupture and compressive strength of M60 concrete containing metakaolin. Research Journal of Applied Sciences, Engineering, and Technology, 8(11), 1294-1298.
, 25(25) Shelke, N.L., Gadve, S. (2016). Prediction of compressive strength of concrete based on accelerated strength. Structural Engineering and Mechanics: An International Journal, 58(6), 989-999.
). It can be seen that these experimental values are also in good agreement with the results obtained from the models.

Additionally, to assess the efficacy of the proposed model [1], predicted values are plotted against experimental values, as shown in Figure 5, for which the R² value is found to be 0.97. It is observed that linear fit of the predicted values plotted against experimental values obtained from available literature lies within the 95% prediction band. This indicates that predicted values are in consonance with experimental values. All the predicted values obtained from the model lies in the 95% prediction band.

4.2. Prediction of cylinder compressive strength

According to the standard code guidelines of some of the nations such as America (ACI318), New Zealand (NZS3101), and Europe (EN 199211), the compressive strength is measured on cylindrical specimens of standard size, generally 300 mm x 150 mm for the purpose of designing. Hence, a model for prediction of cylinder compressive strength is also developed on the same lines as cube compressive strength as stated in [2].

Where $fck\text{'}$ is the compressive strength of cylinder (MPa) at the age of 28 days.

For the above model, the correlating coefficient is obtained as 0.98, which alludes to the effectiveness of the model. The comparison of the compressive strength of the cylinder across different mixes is as shown in Table 6. From the testing program, experimental values are obtained and the proposed model [2] is used to predict the corresponding values. The percentage deviation in predicted value from experimental value is also shown in Table 6. In no case, the percentage deviation is more than 5% except in one instance where it is 9.09%. These figures clearly indicate the closeness of the proposed model to the experimental results. The efficiency of the developed model is further checked with the results available in literature (8(8) Anbuvelan, K., Subramanian, K. (2014). An empirical relationship between modulus of elasticity, modulus of rupture and compressive strength of M60 concrete containing metakaolin. Research Journal of Applied Sciences, Engineering, and Technology, 8(11), 1294-1298.
, 16(16) Ahmed, M., Mallick, J., Hasan, M. (2016). A study of factors affecting the flexural tensile strength of concrete. Journal of King Saud University Engineering Sciences, 28(2), 147-156.
, 26(26) Ashour, S.A. (2000). Effect of compressive strength and tensile reinforcement ratio on flexural behavior of high-strength concrete beams. Engineering Structures, 22(5), 413-423.
). These experimental values imply to be in good agreement with the results obtained from the models.

Moreover, when plotted, the R² value for predicted values against experimental values is 0.97. This indicates the higher efficacy of the prediction model. Also, as seen in Figure 6, almost all values lie in the 95% prediction band. The linear fit of the predicted values plotted against experimental values obtained from available literature lies close to prediction band. This signifies the effectiveness of the proposed model.

4.3. Prediction of flexural strength

The tensile strength of concrete is most commonly attributed to flexural strength (modulus of rupture). Almost all structural components are subjected to significant flexure stresses. Several structural components viz., slabs, beams, road pavements are designed basically for the flexural loads. In the investigation of flexural strength according to standard codes, it is revealed that the process involves tedious experimentation. Therefore, the standard guidelines of different countries have given an empirical equation for estimating the flexural strength of concrete correlating to compressive strength. However, compressive strength also needs aout 28 days of experimentation. Hence, for estimating the flexural strength at an early age with accuracy, a predictive model is developed as given in [3].

where, $fcr$ is the flexural tensile strength of concrete (MPa).

Pearson’s R, obtained after multiple regression analyses for the above mentioned model, is 0.96. Table 7 shows the deviation of experimental values acquired from the test program for flexural tensile strength and predicted values obtained from the proposed model [3]. It is witnessed that the predicted values have a percentage deviation of less than 9%. These indicate that the proposed model gives value in the vicinity of experimental values. The effectiveness of the developed model is further validated by comparing it to reported results (27(27) Khayat, K.H., Bickley, J.A., Hooton, R.D. (1995). High-strength concrete properties derived from compressive strength values. Cement, Concrete and Aggregates, 17(2), 126-133.
). The results obtained from the models are consistent with these experimental values.

Figure 7 shows the curve fit obtained from the linear regression analysis. The R² value or coefficient of determination is obtained as 0.94. In addition to this, it can be seen from the curve that all experimental values are in the 95% prediction band. The predicted values plotted against experimental values obtained from available literature lies in to prediction band. This indicates the efficiency of the proposed model.

4.4. Prediction of modulus of elasticity

The modulus of elasticity is the valuable mechanical property used in the analysis and designing of concrete structures. The modulus of elasticity is the measure of flexibility of the material. The deflection pattern of materials is governed by its modulus of elasticity and hence defines the serviceability of the material. The existing standard guidelines give the correlation of modulus of elasticity with compressive strength. In addition to this, standard experimental procedures are also prescribed for its determination. However, this process requires a longer duration of time. Hence for prompt analysis of elastic property of the designed concrete, the predictive model is articulated. [4] gives the prediction model for modulus of elasticity of concrete.

where, Ec is the modulus of elasticity of concrete (MPa).

Also, after performing statistical analysis, Pearson’s R-value obtained is 0.976. The predicted values obtained from the proposed model [4] are compared with the experimental values of modulus of elasticity. It is apparent that none of the predicted values have a percentage deviation of more than 4.5%, as shown in Table 8. This indicates that the proposed model give values, which are very close to the experimental values. The effectiveness of the developed model is further examined with the results available in literature (9(9) Vijayalaxmi, B., Ajay, H., Ranjith, A., Sandya, D. (2015). Experimental investigation on elastic properties of concrete incorporating GGBFS. International Journal for Research in Applied Science & Engineering Technology, 3(5), 487-496.
, 26(26) Ashour, S.A. (2000). Effect of compressive strength and tensile reinforcement ratio on flexural behavior of high-strength concrete beams. Engineering Structures, 22(5), 413-423.
). It is apparent that these experimental values are also in good agreement with the results obtained from the models.

Additionally, a curve fit with a 95% prediction band is also developed, as shown in Figure 8. The efficacy of the above model can be seen from the R² value, which is obtained as 0.98. The predicted values plotted against experimental values obtained from available literature fall within the prediction band. This demonstrate the efficacy of the proposed model.