Static without plate 258 220 203 168 122 86 58 38 24 PREDIWARE
Static with plate 241 212 197 166 122 86 59 39 24 PREDIWARE
Dynamic with plate 296 262 246 213 169 133 105 82 64 Table 2-4 Comparison between PREDIWARE direct calculation results and Alize ones; deflections [µµµµm]; S2 structure Considered stress σσσσXX AC bottom σσσσZZ AC bottom σσσσXX UGA Top σσσσZZ UGA Top σσσσXX S Top σσσσZZ S Top Alizé LCPC -1,166 0,05 0,003 0,05 0,001 0,028 PREDIWARE
Static without plate -1,159 0,048 0,0033 0,050 0,0014 0,0281 PREDIWARE
Static with plate -1,180 0,046 0,0069 0,048 0,0039 0,0281 PREDIWARE
Dynamic with plate -0,854 0,035 0,0070 0,036 0,0040 0,0253 Table 2-5 Comparison between PREDIWARE direct calculation results and Alize ones; stresses [MPa]; S2
structure Considered strain εεεεXX AC bottom εεεεZZ UGA Top εεεεZZ S Top Alizé LCPC -77,6 240,4 226,9 PREDIWARE
Static without plate -77,0 239,5 226,4
PREDIWARE
Static with plate -78,2 232,6 227,9
PREDIWARE
Dynamic with plate -73,9 250,5 171,5
Table 2-6 Comparison between PREDIWARE direct calculation results and Alize ones; strains [µµµµm/m]; S2
structure
It appears that surface deflections, stresses and strains provided by Alize and PREDIWARE calculation in static mode and without the loading plate are rigorously the same (1 µm difference only for G4). Precision on geophones is of the micron range whereas precisions on
stresses and strains are respectively better than 0,01 MPa and 1 µm/m.
Adding a loading plate is of no effect on outer deflections. Nevertheless some discrepancies are observed in the plate area with regard to the no-plate configuration. The influence on stresses is negligible, except for tangent compression in both unbound materials. Effect on strains is limited.
Comparison between static and dynamic modellings shows that deflections are higher in the dynamic case. This implies that the static modelling overestimates backcalculated moduli with regards to the dynamic one. Let us remind that the tested structure presents shallow bedrock (3 m).
Let us also consider the half-space case.
The latter is dealt with on structure S3 (point Pl1; see appendix 1.1). Moduli of the reference
set are usual moduli. The respective values of 4 700, 9 000, 200, 150 and 120 MPa have been chosen for surface (AC1) and base (AC2) asphalt concrete layers, humidified Unbound Graded
Aggregate (UGA), untreated gravel (G) and Subgrade (S).
Geophone G1 G2 G3 G4 G5 G6 G7 G8 G9 PREDIWARE
Static with plate 415 334 316 276 229 189 160 135 117 PREDIWARE
Dynamic with plate 213 168 159 138 114 92 78 67 59 Table 2-7 Comparison between static and dynamic calculations; S3 structure
This time, the inverse observation is made i.e. static modelling underestimates backcalculated moduli with regard to a dynamic one.
This supports [Mera, 1995] observations (see literature review) on numerical signals, who shows that the static modellings tends to overestimate stiffnesses for shallow bedrocks and underestimate them on half-spaces.
4.2.2Validation of the backcalculation phase
The main principle consists in choosing for a pavement structure a reference moduli data set and determining, using the previously validated PREDIWARE direct calculation option, the related simulated deflection data set.
Then accuracy and robustness of algorithm is tested by performing backcalculations with several seed moduli set, when considering that the previously simulated deflections are the experimental data to be matched. Dispersion on backcalculated moduli is studied.
This work is performed on static and M1 dynamic (both with and without damping)
modellings, using a common seed moduli set. The S3 structure (point Pl1) of appendix 1.1 has
been retained for this study. First, static and dynamic without damping results are compared. Then, the influence of damping is studied.
Note that due to very time-consuming calculations in dynamic mode, the work is performed using a ten (10) seed moduli sets sample (called Reference set (Ref) and SMS 2 to 9 in the following). It could be interesting to reiterate it on larger data set (100 seed moduli data sets for instance.)
Reference moduli set is, as above 4700, 9000, 200, 150 and 120 MPa from highest to lowest layer. Other seed moduli sets have been arbitrary defined using the Random Excel function, when imposing a realistic variation range. Mean, standard deviation (Std. Dev.) and variance (Var.) are collected in Table 2-8 and Table 2-9 (blue part)
The latter collect results of the respective pseudo-static and dynamic without damping
calculations. Note that the quality of fitting, expressed in terms of normalized mean squared error (Norm. MSE) is in all cases excellent.
The comparison of resulting backcalculated mean values with reference values demonstrates that a good accuracy on backcalculated moduli can be reached with only twenty (20)
iterations. The study of variances of results (yellow boxes), compared to initial scatter (blue boxes) shows that robustness is much better in the dynamic case. Such a result can be explained by the high overdetermination of the problem when the whole time-histories are considered instead of the peak values only.
Seed moduli sets Backcalculated moduli sets
AC1 AC2 UGA G S AC1 AC2 UGA G S
Norm. MSE Ref 4700 9000 200 150 120 4700 9000 200 150 120 0 SMS2 4476 10957 241 342 138 4657 9211 192 157 119 6.91E-03 SMS3 6572 13368 275 260 108 4677 9115 196 154 119 2.27E-03 SMS4 5054 10458 281 234 74 4654 9229 191 158 119 8.67E-03 SMS5 2741 6691 255 205 38 4640 9292 189 160 118 1.36E-02 SMS6 2645 3635 199 211 49 4837 8467 216 138 123 4.26E-02 SMS7 2014 8236 280 192 79 4681 9080 197 152 120 9.27E-04 SMS8 4848 4366 338 204 146 5154 7446 250 116 129 4.17E-01 SMS9 3443 7908 256 237 82 4682 9083 197 153 120 1.05E-03 SMS10 6332 8377 160 182 41 4672 9148 194 156 119 4.37E-03 Mean 4055 8291 258 226 93 4742 8880 203 149 121 Std Dev 1450 3120 43 53 38 165 589 19 14 3 Var 35.8% 37.6% 16.7% 23.6% 40.8% 3.5% 6.7% 9.5% 9.3% 2.9%
Table 2-8 Robustness of the pseudo-static backcalculation procedure; S3 structure
Seed moduli sets Backcalculated moduli sets
AC1 AC2 UGA G S AC1 AC2 UGA G S
Norm. MSE Ref 4700 9000 200 150 120 4700 9000 200 150 120 0 SMS2 4476 10957 241 342 138 4665 9088 200.04 150.04 120.02 2.7E-03 SMS3 6572 13368 275 260 108 4938 8464 199.31 149.81 119.82 9.9E-02 SMS4 5054 10458 281 234 74 4747 8889 199.90 149.95 119.96 4.8E-03 SMS5 2741 6691 255 205 38 4755 8871 199.87 149.94 119.96 6.2E-03 SMS6 2645 3635 199 211 49 4762 8843 200.01 149.93 119.95 8.5E-03 SMS7 2014 8236 280 192 79 4545 9383 200.43 150.18 120.11 4.2E-02 SMS8 4848 4366 338 204 146 4844 8621 200.23 149.85 119.86 4.5E-02 SMS9 3443 7908 256 237 82 4688 9033 199.99 150.02 120.01 6.4E-04 SMS10 6332 8377 160 182 41 4723 8944 199.97 149.97 119.98 1.5E-03 Mean 4055 8291 258 226 93 4736 8914 200 150 120 Std Dev 1450 3120 43 53 38 105 251 0 0 0 Var 35.8% 37.6% 16.7% 23.6% 40.8% 2.2% 2.8% 0.1% 0.1% 0.1%
Table 2-9 Robustness of the dynamic without damping backcalculation procedure; S3 structure
Note that in the dynamic case, precision found on backcalculated moduli is excellent for deepest layers (subgrade, untreated gravel, and UGA), and slightly less satisfactory for surface layers, which is consistent with the previous sensitivity study conclusions.
Associated results obtained when introducing damping are not presented here. The corresponding precision on backcalculated moduli is once more excellent.
4.2.3Computation times
Static direct calculations are virtually instantaneous (less than 1 s calculation time) Nevertheless, computation times for the dynamic direct calculations relative to the S3
structure (1 time step out of 3, i.e. 80 time increments) was 30 s per calculation, with a ratio CPU time / machine residence time of 0,96.
The structure presents 4 layers, so that each step of the iterative process using the Gauss Newton algorithm requires respectively 6 and 5 direct calculations with and without damping. Considering a mean value of 15 iterations to converge, mean computation time for the whole backcalculation procedure is about 25 to 45 min.
Material configuration was:
- Intel Core 2 Quad CPU Q6600; 4 × 2,4 GHz - 2Go RAM
- OS: Windows XP
A calculation accelerator (option MUL in CESAR) allows significant computation gains (ratio in the range of 2 to 5 according to tested meshs). It is for the moment only available in the development version, also owned by STAC, in a LINUX environment. PREDIWARE works at present only in a Windows environment. It is planned to adapt PREDIWARE to LINUX but this has not been done at the time of working.
Partial conclusion
It has been shown that the accuracy of the direct calculation using PREDIWARE is
sufficiently good, so that the mesh discretization has no discernable influence on the global HWD moduli determination.
Accuracy and robustness of the dynamic numerical backcalculation procedure has also been demonstrated for both pseudo-static and dynamic methods. Normalized mean squared errors are negligible compared with errors generated by imposed parameters (especially layer thicknesses), so that the errors on backcalculated moduli inherent to numerical
backcalculation can be neglected.
Comparison between static and dynamic without damping corresponding results asserts that the dynamic backcalculation method provides more robust convergence than the pseudo-static one. Influence of damping has also been studied. It appears that this parameter can be
integrated in the backcalculation process.
Nevertheless, dynamic calculations present an operational weakness due to their
computational time. It strengthens interest for the self-adjoints theory (see appendix 3.3), whose implementation in PREDIWARE is in progress.