Issue 52

M.F. Bouali et alii, Frattura ed Integrità Strutturale, 52 (2020) 82-97; DOI: 10.3221/IGF-ESIS.52.07

0.00 6.93 10.20 11.14 13.25

0.00 -16.40 -23.51 -25.55 -27.05 0.00 -22.37 -33.18 -38.63 -39.97 0.00 -19.13 -24.97 -29.26 -29.90 0.00 -3.74 -9.75 -13.65 -17.36

0.00 14.29 18.75 18.95 20.03 0.00 16.94 22.55 25.19 25.78 0.00 16.85 19.78 22.32 22.42

0.00 4.27 5.56 5.01 6.63 0.00 0.13 2.24 5.16 6.67 0.00 6.49 6.01 8.24 8.74 0.00 2.77 8.27

0.00 3.36 5.56 6.59 9.19

0.00 13.03 16.42 16.50 17.80 0.00 12.55 17.16 20.40 21.59 0.00 15.37 17.20 19.74 20.06 0.00 4.18 10.07 13.80 17.43

0.00 -10.26 -16.31 -19.54 -22.49 0.00 -9.90 -19.54 -27.73 -31.45 0.00 -12.66 -17.27 -23.02 -25.09 0.00 -3.47 -9.44

M10 4/10 550 A

0.00 1.06 2.24 7.22 10.05 0.00 5.58 6.01 9.82 11.34

0.00 3.72 8.48

M10 4/10 430 A

13.21 15.46 0.00 9.18 10.78 14.37 15.45

M10 4/10 520 S

0.00 2.57 8.27

0.00 3.71 9.66

0.00 3.29 9.11

M10 4/10 750 S

12.99 16.77 -13.43 -17.23 Table 10: Error percentages of composite models and experimental results in  9  (%). 13.52 17.22 11.97 15.72 12.24 16.14

The Bache and Nepper-Christensen and Hirsch-Dougill models underestimate the Young’s modulus of LWAC measured in [3]. Bache and Nepper-Christensen model, 43/75 cases give  E  smaller than 10% and  E ranges from 31.45% to 1.62%. For the Hirsch-Dougill model, 29/75 cases give  E  smaller than 10% with 23 cases smaller than 5%. It can be seen by examining Fig. 4 that the most accurate models are those of Maxwell, Counto1 and Hashin-Hansen which give less errors percentages (Fig. 4 and Tab. 10). Statistical analysis In order to confirm what has been announced previously and distinguish the most suitable model for predicting the effective elasticity modulus of the LWAC, a global statistical study was carried out on all the experimental values of the three researchers (119 measures). To this effect, the mean values and standard deviation for all composite models used in this study and experimental data are calculated as seen in Tab. 10.

Bache and Nepper Christensen

Hashin Hansen

Popovics Hirsch- Dougill

Maxwell

Counto1

Counto2

Mean Values Standard deviation

-6.90 8.24

-9.66 11.14

-2.72 5.72

0.29 5.27

-0.23 5.32

-5.94 7.12

-6.42 8.46

Table 10: Mean values and standard deviation of composite models and all experimental data in  7, 8, 9  .

Fig. 5 shows the normal distribution approximation of error percentage for all 07 composite analytical models. Every estimator has a pick on the mean value and a standard deviation presented by a tight or wide curve. As expected, the Maxwell, Counto1 and Hashin-Hansen composite models provide a good prediction of experimental Young’s modulus of all LWAC tested by De Larrard  7  , Yang and Huang  8  and Ke Y et al.  9  (119 values) with a maximum volume fraction of aggregates Vg equal to 49.37%. It is clear from curves of Fig. 5 that the best curves that fit experimental data are respectively Maxwell, Counto1 and Hashin-Hansen models because the mean values are closest to zero than others. It is also important to notice that the standard deviation of both models (Maxwell 5.27, Counto1 5.32 and Hashin-Hansen 5.72) are tight which indicates that there is a high concentration of estimated values around of zero.

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