3.1 Comparison of Young’s moduli
Figure 5 shows an example of the stress-strain responses in the direct tension test. In the direct tension test for the tensile Young’s modulus, the load was temporarily removed for measurement of strain at unloading. The residual strain at unloading is inelastic, so the strain including the residual strain is inappropriate for the evaluation of Young’s modulus. The strain shown in Figure 5 was evaluated by subtracting from strain at loading to residual strain at unloading. The stress-strain responses were almost linear up to the maximum point, as shown in Figure 5, so the gradient of the regression line was taken as the tensile Young's modulus.
Figure 6 presents the result of the compression test with the same procedure of the direct tension test. The ft in Figure 6 means the splitting tensile strength at the test age (7 days). One of the compressive Young's modulus in this study (hereinafter referred to as linear modulus) was obtained from the slope of this linear regression line.
The result of the compression test based on JIS is demonstrated in Figure 7. The result in Figure 7 was regressed by the cubic equation shown in Equation (1).
where, σ is stress (MPa), ε is strain (x 10-6), a, b and c are coefficients.
The experimental data was accurately regressed by the cubic curve in Equation (1), and R2 (coefficient of determination) was more than 0.99. The coefficient c in Equation (1) corresponds to the initial tangent modulus. The secant modulus, applied as a general Young’s modulus of concrete in Japan was evaluated from the data at the strain of 50 x 10-6 and the data at the stress of 1/3 of the compressive strength (fc'), as shown in Figure 7.
Figure 8 shows the results of the tensile Young's modulus and the compressive Young’s moduli (the secant modulus, the initial tangent modulus and the linear modulus). The secant modulus was obviously smaller than the tensile Young’s modulus, with the average value being 87% of the tensile Young’s modulus. This result indicates that tensile stresses evaluated using the secant modulus from the compression test might be underestimated. On the other hands, the initial tangent modulus and the linear modulus were nearly equal to the tensile Young’s modulus. This was because the stress – strain relationship shown in Figure 6 was almost linear in the stress range less than the tensile strength and the stiffness was equivalent regardless of tensile stress and compressive stress. However, the stress was not proportional to the strain as the stress increased, particularly beyond 1/3 of the compressive strength, as shown in Figure 7. Therefore, the difference between the tensile Young's modulus and the secant modulus can be caused by the difference in the level of the stress and strain used for the evaluation of the Young's modulus. In other words, the Young's modulus evaluated from the compressive stress-strain curve at the stress range less than the splitting tensile strength may be suitable for crack estimation even if the stress field differs.
In addition, the loading apparatus and dog-bone shaped specimen used in the direct tension test shown in Figure 2 and Figure 3 are not typical, so it is difficult to obtain the development evaluation of Young’s modulus at early ages based on many tests. The compression tests by using a typical cylindrical specimen can be relatively easily performed at the various ages of concrete. The cylindrical specimen for the initial tangent modulus has already fractured after the compression test. The compression test for the initial tangent modulus requires many specimens in order to obtain stiffness development as ages. The loading stresses in the compression test for the linear modulus are below the tensile strength. The cylindrical specimen for the linear modulus hardly has a failure and can be repeatedly used for the compression tests at the various ages, so stiffness development evaluated by the linear modulus includes less errors between specimens.
3.2 Results of the splitting tensile strength and the Young’s modulus
The Young’s modulus (the linear modulus) obtained from the compression test and the splitting tensile strength during the tensile creep test are shown in Figure 9 and Figure 10, respectively. The development of the Young’s modulus and the splitting tensile strength of the fly ash concrete is more gradual than that of the normal concrete. The stiffness of the normal concrete and the fly ash concrete have been developing, so the elastic strain could be decreasing during the tensile creep test with a constant load.
The loading stress determined based on the splitting tensile strength at the loading age is given in Table 4. The stress/strain ratio was set to 30% or 40% at the loading start and was gradually decreased during the tensile creep test due to development of tensile strength as shown in Figure 10.
3.3 Creep behavior
Figure 11 shows an example of the strain behavior of the loaded specimens and the non-loaded specimens. The data between the loading strain and non-loading strains are the strains at temporary unloading in order to measure the elastic strain during the tensile creep test. Figure 12 shows the creep strain and the specific creep of the normal concrete (N3-30). The creep strains were evaluated by subtracting the non-loading strain and the elastic strain from the loading strain, assuming that the elastic strain was constant at the start of the loading. The specific creep obtained by dividing the creep strain by the loading stress. The loading stresses of N3-30 differed from 0.366 MPa to 0.714 MPa as shown in Figure 12 a), so the creep strain was greater at higher loading stresses. The behaviors of the specific creep shown in Figure 12 b) were relatively similar even at the different loading stresses. In Figure 12 b), the average of the 3 results is also demonstrated. The average specific creep was in the range of ± 6 x 10-6/MPa. Figure 13 presents the creep strain and the specific creep of the fly ash concrete (FA3-30) The creep strain behaviors of the fly ash concrete shown in Figure 13 a) were not as clearly different as that of the normal concrete, because the loading stresses for each creep test with fly ash concrete were equivalent. Though the experiment in the present paper cannot indicate a definite factor, the average specific creep of FA3-30 shown in Figure 13 b) had larger errors than that of the normal concrete and was in the range of ± 10 x 10-6/MPa. It should be noted that the elastic strain at early age decreases as the concrete age increases because of the Young’s modulus development during the tensile creep test as shown in Figure 9.
In this study, two sets of elastic strains for tensile creep behavior were evaluated, one was from the measurement during the tensile creep test, and the other was estimated by dividing the loading stress to the Young’ modulus based on the Hooke’s Law, as shown in Figure 14 and Figure 15. The estimated strains at the start of the loading were not equal to the elastic strains from the measurement, so the modified strain was evaluated by multiplying the ratio α of the measured strain to the estimated strain at the loading. Both the measured strains and the modified strains were decreased with increasing ages. This result indicates that the creep determined from assuming a constant elastic strain has been underestimated. The modified strain of N7-30 was equivalent to the measured strain at the end of loading. However, the modified strains such as N3-30b had a different behavior with the measured strains during the tensile creep test. The elastic strains, shown as red lines in Figure 14 and Figure 15, for evaluating the creep in this study were therefore determined with reference to the measured strains rather than the modified strains. It can be seen that the applied strain of N3-30b decreased in 2 days, while the elastic strains of N7-30 decreased more gradually.
Figure 16 shows the specific creep based on the decreasing elastic strains. The dotted lines in Figure 16 represent the results when using the constant elastic strain. In the case when the load started at the age of 3 days, the difference in the specific creep caused by considering the development of the Young’s modulus expanded with decreasing the elastic strain. In this case, the maximum increase of the specific creep of the normal concrete was 10.1 x 10-6/MPa, and was 8.0 x 10-6/MPa on average. The fly ash concrete also had the specific creep’s maximum increase of 11.7 x 10-6/MPa, and 8.4 x 10-6/MPa on average. Such increase was larger than the difference in the specific creep caused by mixing fly ash into the normal concrete. In other words, the decrease of the elastic strain contributes more in the evaluation of the specific creep than the use of the fly ash mixing. This result indicates that decrease in elastic strain due to stiffness development is more major factor than the effect of fly ash in creep evaluation of fly ash concrete at early age. On the other hands, the specific creep when starting the load at the age of 7 days as shown in Figure 17 was less affected by the fly ash mixing and the decreasing elastic strain than by starting the load at the age of 3 days.
Figure 18 presents the specific creep of N3-40 with a stress/strength ratio of 40% at load-starting. One was relatively similar to N3-30, and the others were approximately 2-3 times than N3-30. Figure 19 shows the results of FA3-40. The two specific creep behaviors of FA3-40 until 6 days were almost equal to that of FA3-30, and the other one was around 1.8 times of FA3-30 at 14 days. The actual stress/strength ratio based on the splitting tensile strength is also shown in Figure 18 and Figure 19. The stress/strength ratio was decreased from 40% to less than 30% in 4 days (age of 7 days). However, the specific creep behavior of N3-40 and FA3-40 after 4 days differed from that of N3-30 and FA3-30. These results indicate that creep strain with the stress/strain ratio of 40% might be nonlinear to loading stress and the specific creep of concrete with a stress history which is nonlinear to creep strain might be greater even if a stress/strength ratio was decreased to less than 30%. In addition, the specific creep of N3-30 and N3-40 at 14 days almost converged, and the specific creep of FA3-30 and FA3-40 was increasing at 14 days yet.