Bundesanstalt für Materialforschung und -prüfung

International Symposium (NDT-CE 2003)

Non-Destructive Testing in Civil Engineering 2003
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Enhanced Backcalculation Techniques for Assessing Highway Structural Properties

Carl A. Lenngren, Chalmers University of Technology, Göteborg, Sweden
Johan Olsson, Chalmers University of Technology, Göteborg, Sweden


Backcalculation of road layer elastic moduli is an important procedure in determining stresses and strains in road structures due for overlay design. By using the non-destructive device the falling weight deflectometer, it is a proven method of achieving a sound mechanistic design of the new layer thickness. Some assumptions are made in the process. One is introducing full friction in layer interfaces. The present paper shows that full friction may not always be the case in reality. If slippage occurs it leads to an underestimation of unbound base and subbase layers. If not accounted for, some critical strains determining the overlay design can be underestimated as well. However, layer interface slippage can be backcalculated in favor of getting a better fit between measured and backcalculated deflection basins.


The Falling Weight Deflectometer (FWD) is the most commonly used tool for non-destructive testing of roads and airfields in the world today. Data from the tests are used for backcalculating elastic properties of layers in the road structure as well as the subgrade. Thus, it is possible to calculate a response on any given load that could be used in a mechanistic design procedure. Many engineers have developed a confidence in this method and its results are being used in the further reconstruction design of the road. However, sometimes assumptions in the model are violating factual conditions. The present paper looks upon the effects of faulty layer interface assumptions in the elastic layer analysis. A finite element study confirms the predicted results of not dealing with layer slippage.


The derived backcalculated elastic modulus of the unbound base layer is sometimes very low, e.g. 150 MPa or less. In many cases the less than realistic results depend on limitations of the model used. Non-linearity and viscoelastic effects may not be accounted for. However, the FWD is using a nominal load of 50 kN and a nominal load time history to set a standard for engineers to use. It has been found that full friction in the interface between layers is not always occurring in reality. When an approaching load such as from a truck gets to any arbitrary point the underside of the bound layer will be in a compression state, followed by expansion and finally go back to a compression state in the longitudinal direction as the load leaves. This effect may sometimes result in a tiny gap between the interfaces, which will be exaggerated when the underside is cold as is a well-known fact for Portland cement concrete (PCC) slabs, see Figure 1. Even if visco-elastic materials like asphalt concrete relaxes, the effect leaves the interface in a somewhat disturbed condition that may affect the deflections as measured by the FWD.

Fig 1: A wheel load would leave the asphalt layer underside in a compressed state in the longitudinal direction of travel.

In the mid 1990:ies some cases with suspiciously low unbound base moduli were backcalculated with a new approach stemming from the backcalculation of PCC pavements. A method of dealing with this phenomenon by introducing a thin air gap between layers is described in (Lenngren, 2002). Figure 2 shows how the stiffness of an unbound base layer is affected if an air gap is not accounted for.

Fig 2: Air gap influence on a four-layer asphalt concrete system backcalculated without gap.


For the purpose of testing different situations with the two programs an input module was written that could handle a number of constructions and loading cases to be tested for the two different programs and their respective input data formats. Thus, it became easy to identify model situations that the CLEVERCALC backcalculation program was able to cope with correctly and maybe more important identify those situations where it was difficult to retrieve the original E-moduli.

As regarding thin layers of air, it was found that it is computably possible indeed to work with soft layers adjacent to very stiff ones, like concrete to corresponding air pressures, with roughly a factor 400,000. However, the computation times go up considerably if the air layer is thin, something one should be aware of when generating the deflection files in forward calculation, and the backcalculation may suffer if the selected air modulus is too low depending on declaration of variables in the program. The soft layer stiffness did not compromise the calculation for the two programs tested with a factor of roughly 50,000 corresponding to values of 0.1 kPa for asphalt concrete (AC) pavements and 1 kPa for PCC pavements.

Further tests showed that that the backcalculation program was not consistent if the degree of freedom was higher than five, e.g. solving five layers and an air gap. However, much of this problem can be attributed to the number of sensors and their distance from the center of the loading plate. A practical solution to tackle this problem is to first do the backcalculation with fewer layers for solving the subgrade modulus. Thus, in a second step the subgrade can be fixed so that the problem is reduced to a four-layer system et cetera. The following was concluded for AC pavements for small gaps of soft layers backcalculated without the gap.

Three-layer AC systems:

  • AC modulus 0-5 % lower
  • Unbound base modulus, 2 -50 % lower
  • Subgrade modulus, within 1 %

Four-layer AC systems:

  • Asphalt modulus 4 % lower to 1 % one percent higher
  • Thin unbound base modulus, severely underrated, 3 - 50 % lower depending on size of gap. (The size is thus easy to backcalculate).
  • Subbase layer overrated 1- 3 %
  • Subgrade modulus, less than 1 % deviation

All backcalculations were made with a seed moduli routine suggested by Newcomb, (1986). This with the intention of introducing unbiased seeding in the calculations. All basins solved for better than 0.6 root mean square (RMS) error. Some of them much better.


As pointed out by Romanoschi (1999), many linear layer elastic backcalculation programs use full friction between all layer interfaces. As an alternative to the air gap approach, the test bench program was also written to accommodate the effect of slippage between layers.

In the first simulation a number of constructions were backcalculated on forward calculated data where the spring compliance factor was varied from full friction to full slippage in steps of ten percent. It seemed that once the slip exceeds 10-20 percent it has little effect on the moduli, i.e. it does not matter if the slip is 50 or 100 %, as material Poisson's ratio prevents the layer to move horizontally up to a certain degree. The question of interest is more of what happens if the upper layers do slip somewhat, which indeed is very likely.

Fig 3: Asphalt four-layer system with air gap, backcalculated without gap affects the modulus of the unbound base the most. The thinner base layer makes the effect on the base modulus more apparent than for the three-layer system.

A three layer asphalt concrete pavement resulted in somewhat different deflection basins, especially for the inner sensors. The deflection input data set was then used to backcalculate these basins with full friction between layers. This resulted in the backcalculated parameters shown in Table 1. It is interesting to see the large effect this has on the unbound base. The asphalt layer stiffness decreases somewhat for small slips, then increases with the slip. The subgrade is not affected, but the strain on top of the subgrade does increase, which would mean a higher rutting rate. The strain in the asphalt is also higher despite higher backcalculated stiffness.

Slip E(1) E(2) E(3) Backc:d AC strain Actual AC Strain Backcal:d Subgrade Strain Actual Subgrade Strain
0.000 5897 300 75 184 182 -342 -344
0.001 5715 263 75 197 201 -358 -388
0.005 6210 201 76 206 217 -371 -426
0.010 6566 176 75 207 222 -372 -437
0.050 7120 147 75 206 228 -371 -448
0.100 7213 142 74 206 228 -371 -449
Table 1: Results of backcalculated moduli using full friction on deflections generated with slip.

Thus, when the friction between layer one and two decreases, the horizontal strain at the bottom of the asphalt layer and the vertical strain on top of the subgrade increases.

Slip E(1) E(2) E(3) E(4) Backc:d AC strain Actual AC Strain Backcal:d Subgrade Strain Actual Subgrade Strain
0 5882 311 100 75 206 205 -278 -280
0.001 5875 216 103 75 224 227 -291 -311
0.002 5990 173 103 75 231 235 -295 -326
0.005 6071 134 100 75 240 245 -299 -344
0.010 6080 126 94 75 244 250 -300 -355
0.020 6042 129 88 75 246 254 -300 -362
0.050 5988 135 83 75 247 257 -300 -367
0.100 5965 138 81 75 248 258 -301 -368
0.500 5950 140 80 75 248 259 -301 -369
1.000 5948 140 80 75 248 259 -301 -369
Table 2: Results of backcalculated moduli using full friction on deflections generated with slip on a four-layer asphalt concrete pavement.

Again, the model used was the same as for the air gap data described above and shown in Figure 3. The effect on the unbound base is considerable and may explain for a lot of underestimated modulus on base courses. The asphalt layer stiffness is not affected that much but the horizontal strain at the bottom of the layer is somewhat underestimated. As usual, the subgrade modulus is not, and the subbase is only marginally affected. Both asphalt strain and subgrade strain show higher values, but the original model with slip yielded an asphalt strain of 250 microstrains at one percent slip. If backcalculated with full friction 244 microstrain is achieved, so a slight underestimation would be made if a design was conducted on these figures. The underestimation for the subgrade is even higher, 355 microstrains in the original model compared to only 300 in the backcalculated one. Converted into traffic as number of standard axles it is by a factor (355/300)^4 = 1.96. E.g. the rutting would be underestimated by half its real value.


In order to study the structural response of decoupling the layers in a pavement structure a FE-model was created in the general purpose software ABAQUS. This method of simulation offers the possibility to change the contact condition between the different layers in a way that is not too complicated. A two layer structure were chosen consisting of an asphalt layer placed on top of a base layer of sand. The outline of the pavement structure can be found in figure 6 below. Output, i.e. strains, from the simulation were gathered in the center of the structure in the bottom of the asphalt layer. The dimension for the FE-model used in this model is 16 m in length, 5 m height and 4 m wide. In order to minimize the calculation time a symmetry line was used so that only half of the geometry of the real structure had to be created in the FE-geometry description. Even with this restriction 5000 8-node brick elements were used in the simulation to avoid getting distortion in the results originating from the numerical procedures. A moving load equivalent to a super single tire, model 425/65 R22.5 with a contact pressure 0.95 MPa, was placed on the surface making one passage over the output data point shown in Figure 4.

Fig 4: The 3D FE-model used in the numerical experiments.

Two sets of calculations were performed on the geometrical model presented above in Figure 5. The first one labeled FEM-TIED was performed with full interaction between the two layers in the structure. In the second simulation, denoted FEM-FRICTION, the two layers were separated and were only interacting through contact in the vertical direction and a standard friction law in the horizontal direction that states that the friction forces correspond to a friction constant m multiplied with the force normal to the surfaces in contact. The friction constant may vary between m =0 which means that there is no friction between the surfaces to m =1 that means full friction. Even though this type of interaction model is quite fundamental in it's formulation it can still capture very interesting phenomena i.e. it is only effective when the surfaces are pressed together and is totally inactive and allows the surfaces to disconnect when tensional forces are present. The asphalt layer was assumed to be visco-elastic and was modeled by the well-known Burger material model, which has been implemented into ABAQUS through a user routine created by the authors. In both simulations the assumption was made that the sand in the second layer could be described by the use of the standard linear elastic material model.

The first simulation that was performed with total interaction between the two layers shows a good agreement with test results from a test performed on a heavy vehicle simulator in Delft called LINTRACK. The result from this simulation is presented as the curve market FEM-TIED in Figure 5 below.

Fig 5: The horizontal strain measured and calculated in the base of the AC layer.

In the second simulation the layers only interacted through a friction contact law and where the assumption was made that
m =0.8. This value of the friction constant could be discussed but the estimation was based on the idea that the friction between the two layers should be high due to that the bottom surface of the asphalt would be rough. The results from the second simulation indicate a different response from the road structure. A significantly larger amount of permanent strains is left when the tire load has made its passage compared to the completely joined case. The difference in structural response is made even clearer when studying the displacement plots in Figure 6. In this plot the gap forming, which is a result of the stress state shown in Figure 1, during the load passage is clearly visible. The deformation in the figure is magnified but it still shows the difference in physical response.

Fig 6: The gap that occur between asphalt and sand during a load passage.

Even with an elementary road structure, as the one presented in the FE-simulations above, it is clear that the interaction between different layers has a vital impact on the response of a numerical model. It is therefore of utter importance that this phenomena can be taken into consideration when trying to simulate the response of a multi layer structure in the cases where no perfect contact and interaction conditions exists.


By introducing more variables in a backcalculation method one does not always yield anticipated results. With more degrees of freedom the goodness of the fit naturally improves, but some parameters often seem to be out of range, which is why most programs would have an option of restricting the range for e.g. resilient modulus. However, one might not be able to detect flawed layers or other defects by doing so. In order to avoid arbitrary limits one could plot the average root mean square of sensors, i.e. the backcalculated versus the measured basin, as a function of the E-modulus of the layer in question as described by Lenngren, (1994). An example of a "difficult" layer to solve was presented from Minnesota TH19 consisting of a 19 cm Portland cement concrete (PCC) slab over a thin 10 cm thick unbound gravel base. Thus, the PCC-slab is much stiffer than the base layer and the consequence of trying to backcalculate such a structure is that the base stiffness may be assessed any value within a wide range and still the backcalculated versus the measured basin would yield a low RMS. The deflection on the surface is primarily a function of the thickness and the inverse of the stiffness of each layer, so in this case one would need a lot of resolution on the surface being able to backcalculate the thin base layer modulus correctly. In the example above the latitude to be within one percent error per sensor for the three respective layers are given in Table 3.

Layer Minimum Best fit Maximum Max/min
PCC 40000 42000 44000 1.10
Base 5 34 180 36.0
Subgrade 210 230 270 1.29
Table 3: Sensitivity range for solutions within one percent error. Stiffness in MPa.

The quote between maximum and minimum "acceptable" solution illustrates how difficult it is to determine the modulus for the given input data. It is clearly difficult to improve the solution of the base layer, so in order to check if sensor spacing or other means of input data could improve the backcalculation, the benchmark program was used for this purpose. The forward input data were changed slightly to 40000 MPa for the PCC slab. A common value of 300 MPa was used for the gravel base and a rather soft 75 MPa subgrade was assigned for the subgrade. It turned out that CLEVERCALC consistently returned too low base layer values and too high subgrade values. Tweaking the input with a thin air layer did not improve the results and changing the friction between layers produced random variability. Concentrating the sensors around 100 - 450 mm from the center of the loading plate did improve the result but not convincingly so, and it was felt that more than ten sensors (the maximum allowed in CLEVERCALC) would be needed. Clearly, in this case the low backcalculated base modulus was more an artifact of the slight differences in forward and backward calling routines. Finally, an ELSYM5 produced deflection basin was used as an input. As seen in Table 4 it is slightly different than the test bench program. The table also shows the deflections the program backcalculated.

  D0 D200 D300 D450 D600 D900 D1200
Test bench Basin 256 246 238 225 211 182 156
Backcalculated test bench 259 243 238 226 211 182 155
ELSYM5 Basin 255 239 235 222 208 180 153
Backcalculated ELSYM5 255 239 235 222 208 180 153
Table 4: Two differently generated forward basins and backcalculated deflections. Sensor spacing in mm. Deflections in micrometers.

Regarding the moduli Table 5 shows the result and the low modulus based on test bench basin can now be attributed to artifacts of the program. This is not saying that the program is bad in backcalculating the right moduli; years of use has validated it. Rather, it is showing that there are situations were the information in the input data just is not there and that it is very important to know when this occurs. This is such an example. The difference in moduli from the ELSYM5 basin could be attributed to rounding errors of the deflections as they should solve perfectly.

  PCC Modulus Percent Change Base Modulus Percent Change Subgrade Modulus Percent Change
Original 40000 - 300 - 75 -
Test bench B/C 39235 -1.9 213 -29.0 76 +1.3
ELSYM B/C 40378 +0.9 154 -48.6 77 +2.6
Table 5: Two differently generated forward basins and backcalculated layer moduli in MPa.

However, when the layers in the model are rather easy to solve, i.e. when they are thick enough to contribute to the response on the surface, unanticipated behavior could be attributed to faulty interface treatment between layers. As for backcalculation optimization techniques, one could try using a number of options. First one could try incrementing the spring compliance from full friction to full slippage and then just see what solution has the best fit. If a high slippage turns out to be the best fit one might consider introducing a thin layer of air or water for that matter to see if the fit improves further. Another approach could be assuming full or near full friction when the layer moduli are close and then progressively assume more slip if the two respective layers are far apart in stiffness. However, the latter case may impose horizontal movements between layers in turn creating friction between the two, so linear elastic modeling may not suffice, describing the mode. Finite element models as described by Romanoschi, (1999) may indicate whether simple friction models can be used or if there is need for shear and viscous modeling at the interfaces.


There is good reason to cope with interface conditions when trying to backcalculate elastic moduli. Special care should be taken when the underside of the top layer is cold and contracted. Pavements under live traffic are also more sensitive as traffic creates a compressed state in the longitudinal direction. As more sophisticated programs are appearing there are of course better ways to deal with non-linearity and other non-elastic properties. However, the speed and straight-forwardness of the linear elastic programs make them simple to use and most likely they will be used for years to come. Coping with the interface conditions can be regarded as tweaking the backcalculation process to further enhance the data used for an overlay design. This also means that it should not be used at all times, but only when it really seems to improve the results. Thus, any method should not be used as an all time remedy or quick fix, but only when really needed. Software that creates data files for both backcalculation and forward calculation and checks the results by sensitivity analyses is really helpful in such cases and is truly recommended.


  1. Lenngren C.A., Non-Destructive Testing Utilizing Controlled Variable Rise Time, Proceedings Fourth International Conference on Bearing Capacity of Roads and Airfields, Vol. 1, pp 467-490, 1994.
  2. Lenngren C.A., Backcalculation of Thin Air - A Way to Cope with Slipping and Sliding Layer Interfaces, Proceedings Six International Conference on Bearing Capacity of Roads, Railways and Airfields, Lisbon, Portugal A.A. Balkema Publishers. Vol. 1, pp 203-212, 2002.
  3. Newcomb D.E., Development and Evaluation of Regression Method to Interpret Dynamic Pavement Deflections, PhD. Dissertation, Department of Civil Engineering, University of Washington, Seattle, Washington, 1986
  4. Romanoschi S.A., Characterization of Pavement Layer Interfaces. Ph.D. Dissertation, Louisiana State University, Baton Rouge, 1999
  5. Washington State Transportation Center University of Washington, Washington State Department of Evercalc User's Guide Version 1.1. April 1987
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