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Federal Highway Administration Research and Technology
Coordinating, Developing, and Delivering Highway Transportation Innovations

This report is an archived publication and may contain dated technical, contact, and link information
Publication Number: FHWA-HRT-05-152
Date: February 2006

Guidelines for Review and Evaluation of Backcalculation Results

Chapter 3. Forwardcalculation Spreadsheets


Using the forward calculation principles outlined in the preceding sections, four generic forward calculation spreadsheets are provided, as follows:

  • Forwardcalculation for flexible pavement sections using U.S. Customary units.
  • Forwardcalculation for flexible pavement sections using SI units.
  • Forwardcalculation for rigid pavement sections using U.S. Customary units.
  • Forwardcalculation for rigid pavement sections using SI units.

Please note, however, that there are several constraints to using these spreadsheets. One constraint is that they are presently formatted for seven deflection readings. Accordingly, if your FWD generates more than seven deflection readings, the analyst should only paste in data for the seven most appropriate sensor positions obtained, depending on the units and type of pavement, etc.

Before using the provided forward calculation spreadsheets, the following constraints should be noted:

Notes on the FWD Deflection Data

For flexible pavements, three of the seven chosen deflection readings must be positioned either at 0, 8, and 12 inches or 0, 200, and 300 mm. Further, these three must be ordered as the first three of the seven deflection positions selected. For rigid pavements, four of the chosen seven deflection readings must be positioned either at 0, 12, 24, and 36 inches or at 0, 300, 600, and 900 mm. Furthermore, these four must be the first, third, fifth, and sixth of the seven deflection positions selected. The use of these critical positions makes it possible to calculate either the AREA12 term or the AREA36 term for AC or PCC pavements, respectively. The appropriate AREA term is, in turn, used in the calculation of the bound surface course stiffness. The remaining sensor positions should be chosen such that they span the region in the deflection basin where one of these is approximately one-half of the center deflection reading, for all lines of FWD input data. This enables proper use of the Hogg model for forward calculating the effective subgrade modulus and the depth to the effective hard layer.

Although any drop-by-drop FWD data may be used as input, to improve the random accuracy of the deflection readings, it is recommended that averages of multiple drops are used, especially in the case of rigid pavements or flexible pavements where the asphalt temperature is very low. This is important because the basin will be very flat, and the method is highly sensitive to very small errors in any of the AREA terms used for forward calculation of the bound surface course.

Determine the Pavement Layer Structures for Forwardcalculation

When using the forward calculation techniques, the operator must forward calculate elastic moduli for up to three layers for each deflection basin. Given that requirement, the pavement system being analyzed must be divided or combined into a two- or three-layer structure, as follows:

  1. Surface (bound) layer (AC or PCC).
  2. Base layer (unbound or granular for flexible pavements, unbound or treated for rigid pavements).
  3. Subgrade layer; depth to apparent stiff layer calculated from the deflection basin.

For rigid pavements, the uppermost base layer below the PCC slab was considered the base layer.

Using the Forwardcalculation Spreadsheets

The four forward calculation spreadsheets provided are self-explanatory in almost all circumstances. For first-time users, the following steps should be carried out when using any of these spreadsheets:

  1. Load the Microsoft® EXCEL spreadsheet of choice (flexible or rigid pavement type, U.S. Customary or SI units). If the necessary options and/or add-ins are installed in your version of EXCEL, the spreadsheet will ask whether you want to load the macros or not. To this you should click the icon reading, "Enable Macros." If they are not installed, please install any necessary options or add-ins.
  2. Once the spreadsheet is loaded, save it under a new name so the template is not lost-forever!
  3. All gray shaded areas in the spreadsheet must be filled in with correct input data. If these data are pasted in from another worksheet, to retain the gray shading the input data should also be shaded gray.
  4. In cell I-2, type in a name or identifier for the project.
  5. In cells C-8 through R-8, and down to a maximum of about 1,000 lines of load-deflection data, paste in the FWD data you wish to process through forward calculation. Use the format and (especially) the units shown in the example provided. (Note: Columns C through J of input data are for identification purposes only, and are not used for any forward calculation purposes-you may change the headings or choose to not use, as appropriate.)
  6. In cell U-5, fill in the plate radius. Please note, as indicated in cell U-6, that there are only two possible radii: 300 mm or 12 inches (or equivalent in opposite units).
  7. In cells W-5 through AB-5, fill in the sensor positions used, in the appropriate units. Note that, in cell V-5 (= 0), the center deflection is required and that some of the cells between W-5 and AA-5 are also required, as previously discussed in connection with the determination of the AREA term. The notes shown in cells W-6 through AA-6 indicate which of these are required, at which positions, and the required units.
  8. The constant in cell AO-5 is needed only if you wish to run a calculation for stresses, strains, and deflections after the layer moduli are obtained. This allows for the use of a stiff or hard layer at depth in such calculations, together with the thickness of the upper subgrade and the modulus of this hard layer. In each example provided, the hard layer is assumed to have an effective modulus of three times the subgrade modulus. Change this factor if you have evidence that there is bedrock near the surface and this factor should be higher than three.
  9. In cells AH-8 through AH-xxx (as far down in the worksheet as the deflection data are entered), enter the thickness of the bound surface course in the specified units. This can either be the same for the entire worksheet or different for every station, as desired. Please recall that the bound surface course thickness is the sum of all the bound layers in the pavement, not only the AC or PCC surfacing. An exception to this rule occurs if you intend to apply the formulas presented in the equations shown in Figure 11, Figure 12, Figure 15, and Figure 16, in which case you should enter only the thickness of the PCC slab.
  10. If the surface course thickness is the same for the entire worksheet, it is possible to enter a constant value in cell AI-5 and create a formula for the entire column AH from cell AH-8 to the bottom of the data, to copy in this constant value. Otherwise, cell AI-5 is not used. (Note: It is highly recommended that the minimum surface course thickness used be 3 inches or 75 mm.)
  11. For the flexible spreadsheets only, cells AV-8 through AV-xxx (as far down as the deflection data are entered) are available to employ the Dorman and Metcalf relationship discussed above, provided the base and subgrade consist of unbound materials. Enter the thickness of the unbound base material in the specified units. This can either be the same for the entire worksheet or different for every station, as desired. Please note that only one intermediate layer is allowed, so this must be the sum of all improved intermediate layers. These intermediate base course layers should be in the 2-24 inch or 50-600 mm range.
  12. If the base course thickness is the same for the entire worksheet, it is possible to enter this constant value in cell AZ-5 and create a formula for the entire row AV, from cell AV-8 to the bottom of the file, to copy in this constant value. Otherwise, cell AZ-5 is not used. (Note: It is recommended that the use of the Dorman and Metcalf relationship is limited to granular-type bases, not improved subgrade materials, which should form part of the subgrade, not the base.)
  13. The formulae and columns that are not shaded gray will need to be copied from the first few lines of data after all input data has been entered into the gray cells.
  14. The statistical results displayed in the examples provided at the top of the worksheet are for the entire file, down to around 1000 lines of FWD load-deflection data. Of course, any portion of the data may be manipulated, as desired, for example, by eliminating spurious input data (e.g., nondecreasing deflections) or dividing the file into uniform subsections, with a separate worksheet for each, etc.

What To Do With the Results of Back- and Forwardcalculation

After either back- or forward calculation has been carried out to the satisfaction of the analyst, all values (or averages and coefficients of variance thereof) should be checked for reasonableness before using these data for pavement design purposes. Table 3 provides a broad range of modulus values for various pavement materials that may be considered reasonable.

When using the broad ranges shown in Table 3, common sense should be exercised as well. For example, the range for asphalt-bound surface courses covers a very broad temperature range, with the higher parts of the range shown covering colder pavement temperatures (down to freezing) and the lower part covering temperatures as high as 45° C (~ 115° F).

If the analyst feels that either back- or forward calculation resulted in any modulus values outside of the ranges shown in Table 3, or any other appropriate ranges, these should be rejected as either unrealistic or unreasonable. If both methods produced values within the ranges shown in Table 3, the pairs of values may be compared (or corresponded) and designated as recommended in Table 4.

As can be seen in Table 4 and with the variability of in situ materials in mind, it is felt that an acceptable correspondence between back- and forward calculated moduli can be considered to be within a factor of 1.5 (times or divided by) the screening value calculated through forward calculation. (Again, this is as long as both values are still within the reasonable ranges shown in Table 3.) In such a case, either value may be selected for pavement design purposes.

Consider as well that the subgrade modulus back calculated through the Hogg model as presented herein is usually lower than that obtained through classical Backcalculation. This is mainly because the Hogg model only calculates the effective subgrade modulus under the load plate and to a finite depth, as indicated in the forward calculation spreadsheet output. Backcalculation, in most cases, assumes that the subgrade extends to an infinite (or even a fixed finite) depth, and is the same at all deflection basin offset distances (sensors #2 through #7 or more), including under the load plate (sensor #1).

As a result, in the LTPP database, for example, the forward calculated subgrade modulus was around half of the back calculated subgrade modulus, on average. Therefore, if one uses the traditional AASHTO design formula, where the design subgrade modulus is assumed to be one-third that of the back calculated modulus, then the forward calculated subgrade moduli should not be divided by three, or the design will be far too conservative.

It is not recommended that some of the modulus values from Backcalculation and some of the values from forward calculation be used for the design of a single uniform section, but rather the entire set of either back calculated or forward calculated values should be used, depending on the reasonableness of the values obtained.

Microsoft® Excel spreadsheets have been prepared containing all formulae used in phase I of this study. All forward calculation input quantities are totally transparent to those who wish to use the methodology, whether for screening or in rehabilitation design. To this end, four spreadsheets are available-two for asphalt-bound surfaces (using SI and U.S. Customary units) and two for cement-bound surfaces (SI and U.S. Customary). These spreadsheets can be obtained by contacting LTPP Customer Support Services: by phone at 202-493-3035 or by e-mail at ltppinfo@fhwa.dot.gov.

Table 3. Reasonable ranges for various pavement layers from the LTPP database.


LTPP Code Base Materials Min. Range (MPa) Max. Range (MPa) Min. Range (psi) Max. Range (psi)
321 Asphalt-treated mixture, not permeable asphalt-treated base (PATB) 700 25,000 101,500 3,625,000
302 Gravel, uncrushed 50 750 7,250 108,750
303 Crushed stone 100 1,500 14,500 217,500
304 Crushed gravel 75 1,000 10,875 145,000
306 Sand 40 500 5,800 72,500
307 Soil-aggregate mixture (predominantly fine-grained) 50 700 7,250 101,500
308 Soil-aggregate mixture (predominantly coarse-grained) 60 800 8,700 116,000
309 Fine grained soil or base 35 450  5,100  65,000
319 Hot-mixed AC 700 25,000 101,500 3,625,000
320 Sand asphalt 700 25,000 101,500 3,625,000
323 Dense-graded, cold-laid, central plant mix AC 700 25,000 101,500 3,625,000
325 Open-graded, hot-laid, central plant mix AC (PATB) 350 3,500 50,750 507,500
331 Cement aggregate mixture 2,000 20,000 290,000 2,900,000
332 Econocrete 3,500 35,000 507,500 5,075,000
334 Lean concrete 4,500 45,000 652,500 6,525,000
339 Soil cement 1,000 7,000 145,000 1,015,000
327 Open-graded, cold-laid, in-place mix AC 200 3,000 29,000 435,000
337 Limerock; caliche 150 1,500 21,750 217,500
350 Other-treated base 400 8,000 58,000 1,160,000
  Bound surface courses:        
  Concrete surface (uncracked) 10,000 70,000 1,450,000 10,150,000
  AC surface (>0° C-<45°C, not alligatored) 700 25,000 101,500 3,625,000
  Unbound subgrades:        
  Any unbound type 15 650 2,175 94,250

Table 4. Flagging codes used to screen the back calculated LTPP database.


Description of the Correspondence Between the Forward calculated and the Backcalculated Modulus Values Correspondence Codes Ratio Between the Forward calculated and Backcalculated Modulus Values
Acceptable 0 2/3 < Ratio = 1.5
Marginal 1 1/2 < Ratio = 2 (& not code 0)
Questionable 2 1/3 < Ratio = 3 (& not codes 0 or 1)
Unacceptable 3 Ratio = 1/3 or Ratio > 3


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