What information does the shape of the deflection bowl give?

The shape of the deflection bowl allows detailed structural analysis of the pavement. Basically, the outer deflections define the stiffness of the subgrade while the bowl shape close to the loading plate allows analysis of the stiffness of the near surface layers. A broad bowl with little curvature, indicates that the upper layers of the pavement are stiff in relation to the subgrade. A bowl with the same maximum deflection but high curvature around the loading plate, indicates that the upper layers are weak in relation to the subgrade. With the critical layer identified in this manner, existing or potential distress mechanisms can be identified and therefore the most fitting treatment may be designed.

What analysis software is available?

There is now a large selection of software for determining the stresses, strains and deflections within a layered elastic system. A back-analysis procedure is therefore generally adopted to find moduli from an observed deflection bowl. The basic procedure comprises iterations, adjusting layer moduli until the computed deflections match the measured deflections. When the multi-layered elastic model is established, a forward analysis is carried out to determine strains for a modelled rehabilitation treatment such as overlay. Some packages eg EFROMD2 & CIRCLY are supplied as separate programs while others such as ELMOD combine both back and forward analysis into a single program. Most of the forward analysis programs (including CIRCLY, BISAR and MODULUS) are based on multi-layer elastic theory with numerical integration or finite element analysis (FLEA), while a few (eg ELMOD ) are based on the Odemark-Bousinesq transformed section approach. There are however many users of the latter form of software because of its rapid processing time.


EFROMD2 (Elastic properties from Deflections) was developed by the Australian Road Research Board. It uses CIRCLY iteratively to provide elastic layer moduli corresponding to a given deflection bowl. Field data from either the FWD or Instrumented Benkelman Beam may be used, and the program will apply one or two loading circles accordingly. The program also corrects for secondary effects if the beam support points are affected by the deflection bowl. When an appropriate model of the existing pavement has been established, then CIRCLY is used again in the forward analysis to evaluate rehabilitation options. For materials where the modulus is strongly dependent on stress level, sublayering is recommended to improve the accuracy of modelling. Seed moduli are required for EFROMD2, and maximum/minimum credible moduli can be specified. CIRCLY uses numerical integration and is one of the few programs which will accommodate materials with anisotropic moduli. It is very versatile and can include complex loading patterns.


MODULUS, provided by the Texas Transportation Institute, uses a deflection bowl fit to a library of bowl shapes with corresponding layer stiffnesses. This greatly increases the speed over iterative numerical integration methods. This was recently selected as the back-analysis program of choice by the Strategic Highway Research Progam (SHRP), and it can therefore be expected that MODULUS will gain increasing support in the United States. It allows only isotropic moduli to be considered.


ELMOD (Evaluation of Layer Moduli and Overlay Design) is supplied by Dynatest . It carries out back and forward analysis within the one program, originally using the Odemark-Boussineq transformed section approach. The program has recently been upgraded to include the capacity for deflection basin fit, and it can also provide results based on numerical integration methods. A facility is incorporated to find the appropriate adjustment factors so that Odemark-Bousinesq solutions will fit more closely with numerical integration methods if required. The upgrade also allows modulus limits to be applied. Unlike most other software, it has the capacity to analyse non-linear subgrade moduli as stress dependent (rather than depth dependent from sublayering), and has been widely used in Europe, Asia and North America. ELMOD will analyse only isotropic materials.

What is the merit of using non-linear moduli?

Only a few of the available packages provide for analysis of non-linear moduli. Ullidtz (1998) considers this feature to be of particular importance:

"Many subgrade materials are highly non-linear, and if this is neglected very large errors may result in evaluation of the moduli of the pavement materials. It should be noted that in a non-linear material the modulus increases with distance from the load, both in the vertical and in the horizontal direction. If one of the linear elastic programs is used to calculate the pavement response then the vertical increase in modulus may be approximated by subdividing the layer into a number of layers with increasing modulus, or by introducing a stiff layer at some depth. But this will not imitate the horizontal increase in modulus, and the deflection profiles derived will be quite different from those found on a non-linear material."

Do the theoretical analyses relate to real strains in the pavement?

All of the mechanistic design methods in general use, assume the loading is static, the materials are in uniform, continuous, homogenous, layers and have simple stress-strain relationships. However, the calculated strains apply to a continuum, whereas pavements are comprised of a series of discrete particles which will experience much lower strains within individual particles and much higher strains at particle contact points. In other words "correct" analysis methods can only provide only an average of the combination of strains which occurs in practice. To put the difference between currently used mechanistic analysis programs in perspective, the implication of material variability inherent in pavement engineering is that a 1 metre shift along the road for any given FWD test point, is likely to produce greater variation in moduli results than that related to choice of any of the recognised software packages.

How important is sub-layering or use of non-linear moduli?

When back-calculating layer stiffnesses, the deflection bowl is initially analysed in conjunction with assumed or measured layer thicknesses to determine moduli, stresses and strains in each layer. Because most of the measured deflection is dominated by the nature of the subgrade, it is important that its stiffness is accurately modelled, otherwise back-analysis to provide the upper layer moduli can produce disproportionately large errors. The subgrade can therefore best be characterised by a non-linear elastic model, taking into account the stress dependency of that layer. Some packages provide for approximate analysis of non-linear subgrades by generating additional sub-layers with gradational elastic properties. AUSTROADS (1994) suggest that in this case (eg when using EFROMD2) the subgrade should be modelled as 4 sub-layers with thicknesses from top to bottom of 250, 350, 500 mm, and infinite thickness. The ELMOD package requires only one subgrade layer because it uses the deflections to calculate C and n in the non-linear subgrade modulus relationship:

E = C (sz/s')n

where: C and n are constants, sz is the vertical stress and s' is a reference stress. The reference stress is introduced to make the equation correct with respect to dimensions. E and C then both take dimensions of stress. This approach allows quick and accurate modelling with the additional benefit that the subgrade soil type may be broadly identified. The exponent n is a measure of the non-linearity of the subgrade modulus. If n is zero the material is linear elastic (eg hard granular materials). Soft cohesive soils may be markedly non-linear with n being between -0.3 and -1.

What is the difference between analyses which assume isotropic or anisotropic materials?

Anisotropic pavement materials (with a vertical to horizontal modular ratio, Ev/Eh of 2) are suggested for design by AUSTROADS (1992, Table 6.4). However, few analysis methods other than CIRCLY, allow for anisotropy. Also, there is substantial worldwide experience founded on analyses which have assumed isotropic conditions. To allow valid comparison of results from those software programs which use isotropic moduli, and CIRCLY when a degree of anisotropy of 2 is used, it is necessary to determine the applicable modulus constant (Ki-a) in the relationship:

Ev,n=1 = Ki-a * Ev,n=2

where Ev,n is the vertical modulus with modular ratio of n.

Logically it would be expected that the equivalent isotropic modulus (Ev,n=1) for a material with modular ratio n=Ev/Eh=2 must be somewhere between the extremes,


0.5 < Ki-a < 1

The comparison between pavement structures which are anisotropic and their isotropic equivalents cannot be determined directly, however the equations can be solved iteratively to provide the theoretical relationships. The constant Ki-a is found to be independent of stress but is very slightly dependent on the depth below the surface, Poisson's ratio, and the loaded area. The relevant data for highway situations are shown in the following figure.


For subgrade material (at depth of say 0.3 to 0.5 m or more, and Poisson's ratio of 0.45) a value of 0.67 for Ki-a provides a practical equivalent, ie a subgrade with anisotropic modulus (Ev,n=2 = 100 MPa) could be modelled as an material with 67 MPa isotropic modulus. Or use approxmiated formual Ki-a=(0.67-u/10)Z-0.06 to calculate Ki-a, where u is the Poisson's ratio and Z is the depth below pavement surface in metres.

For basecourse material (say 100 to 150 mm thick with Poisson's ratio of 0.35), Ki-a will be about 0.75, ie a typical M/4 modulus of about Ev,n=2 = 500 MPa is equivalent to a material with isotropic modulus of 375 MPa.

The issue does not arise with cemented materials or asphalt for which AUSTROADS indicates isotropic moduli should be used. There is little information presented in the AUSTROADS Guide on sensitivity of analyses to anisotropy. It remains as one factor in the stiffness expression which is determined by the back-analyses and cannot be deduced explicitly. In the anisotropic model it is still necessary to assume 3 other variables (Poisson's ratio and layer thickness as well as modular ratio), in order to determine in-situ vertical modulus. Adding the capability for variable anisotropy has been considered for a future ELMOD upgrade, but is not receiving high priority. Ullidtz (pers comm) comments:

"Including anisotropy would introduce one more unknown parameter, and a parameter that is very difficult to measure, but it would be uncertain whether this would bring you closer to or further away from the actual stresses and strains in the pavement."

What is the correlation between FWD subgrade modulus and CBR?

The CBR test imposes high strain, plastic deformation, in marked contrast to the loading applied to the subgrade to determine resilient modulus which imposes low strain, elastic conditions. There is hence little reason to expect good correlations between CBR and resilient modulus and any values inferred from a mean value relationship could be in error by a factor of 2 or more (AUSTROADS, 1992).

It is important to note that the E-CBR relationship for the subgrade used by AUSTROADS, is given by:

Ev = 10 CBR, Eh = 5 CBR

(because modular anisotropy of 2 is adopted).

It follows from the earlier discussion on anisotropy, that the equivalent isotropic modulus of the subgrade, implied by AUSTROADS is:

E isotropic = 6.7 CBR

This assumes that the subgrade is at a depth of about 300 mm and has a Poisson's Ratio of 0.45 although there is very little sensitivity to these parameters.

The above relationship is clearly more conservative than relationships adopted by other organisations for estimating the subgrade modulus from CBR. The compensating consequence of this difference is that the AUSTROADS subgrade strain criterion (derived by back analysis of subgrade CBR design curves) is somewhat less conservative than strain criteria recognised by others.

In practice, the moduli of most subgrades are not linear (ie do not obey Hooke's Law) but are dependent on the applied stress level. Accordingly, the subgrade modulus is usually much stiffer beneath a thick pavement than under a thin one. CBR will increase only slightly as pavement depth increases. Modulus-CBR relationships are therefore very doubtful and CBR itself need not be used during normal mechanistic design of pavements. CBR is used as a component of the Adjusted Structural Number (SNP), and for any one region or soil type, consistent modulus-CBR relationships should be adopted.

Where the pavement is very thick back-analysis normally identifies the region of greatest vertical strain for assigning the "subgrade" modulus. For example a 2 m thick layered granular embankment on a soft subgrade may have the greatest strains at about 500 mm depth and the pavement structural life would be unrelated to the strains beneath the embankment in the true subgrade.

The CBR's inferred from the FWD are in situ values which may of course be considerably higher than laboratory soaked values. Seasonal correction is discussed in a later section.

How important is accurate layer thickness information?

It is usually important to know the thickness of any structural AC layer, if the adopted method of analysis calculates tensile strains at the base of that layer. It is less important if the method uses only curvature functions. If thicknesses of the granular layers are not known, then sensitivity anlyses may be carried for a series of possible thicknesses to find out what differences in overlay requirements are indicated, and also decide on layer thicknesses that result in moduli consistent with the values typically achieved in subbase and basecourse materials. Comparisons with moduli found in the layers of other pavements in the same area are also used to arrive at likely layer thicknesses. Preferably, some test pits information or as-builts are desirable, but much fewer than are needed if FWD bowls have not been measured.

In some instances, test pit information from old roads may not fit closely with the back-analysed model. Reasons for this are that the test pit may relate only to an isolated section of a road of variable construction. Also intrusion of one layer into another may make the cause a shift in the effective boundaries between layers - especially where a granular subbase meets a cohesive subgrade. Also there are practical limitations in modelling thin layers close to the FWD loading plate (which has a diameter of 300 mm). Any layers thinner than about 75 mm need to be modelled as combined with the underlying layer in the back-analysis. Alternatively the modulus of a thin layer (eg 30 mm AC surfacing) can be assigned from typical values, then the underlying layer modulus can be calculated separately. The back analysed moduli for any bound layer should be regarded as providing relative stiffnesses rather than absolute values and appropriate judgement with primary dependence on the visual survey is important, especialy when the top layer is cement stabilised or thin AC.

A soft subgrade can also limit the modulus that can be achieved in the subbase, causing an apparent shift. In very thick pavments (eg as an extreme example consider a 3 metre thick granular embankment on a soft subgrade) the FWD analysis will consider the "subgrade" to be the granular material as the true subgrade is too deep to have any significant impact on the deflection bowl. This is the correct way to model the pavement as the greatest strains will occur in the granular layer. It is important to check that the back-analysed model does not give unrealistically conservative or unconservative results by adhering to strictly to any one test pit profile.

How are unbound granular materials modelled?

A complication in pavements with unbound granular surfacing, is the non-linearity of the basecourse modulus. Brown and Pell (1967) suggested the use of the now widely adopted relationship:

E = K1 ?K2

where: ? is the sum of the principal stresses at maximum deviatoric stress and K1, K2 are material parameters.

To express the relationship of modulus of unbound granular materials to their degree of compaction and stress state. Typical values for K1 and K2 are given by Sweere (1990), and some of these (closely complying with TNZ M/4 grading and crushing resistance) are given in the following figure:


These show that a non-linear elastic model would be preferable. However, for the widely used linear elastic models, Sweere recommends as a first approximation that thick granular basecourses be divided into sub-layers, to minimise the effects of stress dependency of the back calculated moduli. At some future time, a rigorous a finite element method that fully characterises this range of values is likely to be adopted by practitioners, but no such procedure is in general use. Meanwhile the assumptions will need to be kept in mind, while using the widely recognised packages currently available, as the latter still do provide practical working models for analysis and design.

Considering the principal stresses under an ESA loading, at the top and bottom of a 125 mm layer of unbound basecourse, Sweere's data suggest a range of moduli mainly between about 200 and 300 MPa. These values are isotropic and relate to freshly compacted laboratory samples. Substantially higher values are typically obtained on good quality basecourses that have experienced either with repetitive loading in the laboratory (Jameson, 1991) or sustained trafficking in the field.

It is important to appreciate that the modulus of any unbound layer is not simply a function of the component material, but is also dependent to a large degree on the stiffness of the underlying material. In a multi layer system, Heukelom and Foster (1960) found using linear elastic analyses, that the ratio of the E modulus of an unbound base layer Ei to that of the underlying soil Ei+1 was limited to Ei / Ei+1 < 2.5. Their rationale was that an unbound material cannot be properly compacted on a soft subgrade. Alternatively, it may be appreciated that if a stiff dense layer is placed on a yielding foundation, then tensile strains will develop and the upper layer will de-compact. Heukelom and Foster supported this practical explanation theoretically, showing that tensile horizontal stresses would develop at the bottom of layer i if the Ei / Ei+1 ratio exceeded 2.4. Under repeated loading these stresses would lead to de-compaction of the overlying unbound layer until its stiffness reduced to a limiting value at which tensile stresses would not occur.

Subsequently the Shell Pavement Design Manual (1978) used the concept of modular ratio limitations in successive unbound layers in the relationship:

Ei / Ei+1 = 0.2 hi 0.45 and 2 < Ei / Ei+1 < 4

where hi is the height of the overlying layer in mm.
More recently, Brown and Pappin (1985) found using more rigorous non-linear finite element analyses, that the above limitations were too restrictive and reported:

1.5 < Ei / Ei+1 < 7.5

AUSTROADS (1992), requires sub-layering of granular materials placed directly on the subgrade with constraints that the sub-layer thickness must be approximately in the range of 50-150 mm and that the ratio of moduli of adjacent sublayers does not exceed 2.

The above relationship are intended for forward design. However back-analysed moduli should be checked using the above criteria to check that a reasonable pavement model has been obtained when carrying out sensitivity analyses with respect to varying layer thicknesses. Clearly, only unbound layer moduli are restricted in this manner as the moduli of bound materials are influenced much less by the stiffnesses of underlying layers.

In view of the above, correlation of the modulus of a granular layer with CBR is very poor. An approximation (based on observations of moduli determined on basecourses that have a known CBR of at least 80), Transit NZ suggests the use of the following relationship to estimate the CBR of an unbound granular basecourse material:

Ev (MPa) = 5 CBR for Ev/Eh = 2

The equivalent relationship for an isotropic basecourse is approximately:

Eisotropic (MPa) = 4 CBR for Ev/Eh = 1

Sweere (1990) presents data which are consistent with the above relationships (to within a factor of 2) provided the applied stresses (sum of principal stresses) are about 750 kPa. However, the constant of proportionality in the above equations decreases by a factor of 4 as the applied stresses reduce to 50 kPa. For sands (eg subbase materials) the constant of proportionality was found to be about 3 to 4 times higher than for gravels. Therefore by fortuitous cancellation, the above equations should apply (very approximately) for either basecourse close to the wheelload or sandy subbase at depth.

Moduli for granular materials are clearly very sensitive to test conditions requiring close replication of in-service density, grading, applied stresses and underlying support for meaningful measurement of modulus or correlation with CBR.

How are seasonal effects taken into account with the FWD?

The back analysis of a deflection bowl provides results for the specific moisture condition at the time of testing. Seasonal variations in moduli must therefore be considered prior to calculation of residual life and overlay requirements. Software packages vary in the way seasonal effects are incorporated. One option is to increase deflections by a multiplier in the range of 1.1 to 1.6 if measurements are not carried out during a wet period. Another approach is to assume an annual sinusoidal variation in moduli between a maximum and minimum value (usually, the subgrade modulus alone would be varied but the factor could be applied to all unbound layers, with similar end result).

In a long term study of deflection changes with season in Australia, Rallings & Chowdhury (1995) found a generally sinusoidal variation in peak deflection each year, and concluded that a seasonal adjustment factor of 1.1 would be appropriate for defection measurements made between mid-summer and the end of autumn. The data they obtained, include both 'wet' and 'dry' rainfall areas and there is clearly more seasonal fluctuation of deflection in the case of the dry areas. If the design condition for the subgrade is taken towards the wetter state rather than at the median condition, then an adjustment factor of about 1.3 would be indicated by the data.

Another similar study undertaken at Delft University (van de Pol et al, 1991) produced comparable sinusoidal seasonal fluctuations in subgrade moduli, from FWD measurements taken over a 2 year period, but no specific guidelines for assessing seasonal effects generally, were indicated.

A considerable degree of judgment will be required to assess seasonal adjustment factors for specific sites. However as a guide, the following table is suggested provisionally for temperate climates. This draws on the above references and is supported by studies in progress locally. The subgrade moisture condition at the time of testing should be assessed relative to expected ranges in that locality.


How is residual life determined?

Residual life, ie the number of ESAs that can be accommodated by a pavement before it is no longer serviceable, can be estimated by comparing the existing roughness with a terminal roughness condition, and using established relationships for allowable material strain versus number of load repetitions. The following figure shows a number of different strain criteria for unbound materials. Most methods are based on the AASHO Road Test and the criteria are applied to the subgrade only, although the Denmark relationship is used on all unbound layers and is only an implicit strain criterion (based on stress and modulus, Ullidtz, 1998). It is of note that the AUSTROADS relationship is not based on the AASHO Road Test and passes well above all others as the number of repetitions increases, ie it is significantly less conservative for high traffic loadings.


The procedure for determination of residual life from empirical data relating to the AASHO Road Test is clearly simplistic as it is based only on roughness progression, therefore prediction will be less reliable when other factors govern the pavement life.

Alternative residual life predictions, based on the AASHTO structural number approach are given by Paterson (1991). Where only roughness is available, the remaining life may be determined from:

Rit = 1.04 emt {RI0 + 263 (1+SNC) -5 NEt}


Rit = roughness at pavement age t (m/km international roughness index IRI)

RI0 = initial roughness

SNC = structural number modified for subgrade strength

NEt = cumulative ESA at age t (million ESA/lane)

t = pavement age since rehabilitation or construction (years)

m = environmental coefficient (0.023 for wet non-freeze climate)

The appropriateness of these two predictive methods for unbound granular pavements is the subject of ongoing local research (Transit NZ Research Project PR3-0171). Preliminary indications are that the AASHO method tends to give slightly optimistic but useful predictions for New Zealand unbound granular pavements, while the AASHTO structural number approach may produce excessively optimistic residual life predictions. Both residual life determinations appear to be good relative predictors for comparison or ranking of pavements of similar construction within a given area (eg in network surveys), but absolute life predictions should be regarded with caution until calibrated to local conditions.