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Publication Number: FHWA-HRT-05-152
Date: February 2006
Guidelines for Review and Evaluation of Backcalculation Results
Chapter 2. Development of Forwardcalculation Methodology
Closed-form solutions for determining select layered elastic properties of pavement systems have been used extensively in the past.
In 1884, Boussinesq developed a set of closed-form equations for a semi-infinite, linear elastic half-space (a semi-infinite layer), including the modulus of elasticity of the median, based on a point load. Subsequently, it has been shown that the apparent or composite subgrade modulus derived from any FWD sensor at offset "r" can be calculated from the equation in: (3)
Figure 1. Equation. Composite subgrade modulus at an offset.
The suggested constant of 0.84 assumes that Poisson's ratio is 0.4 (from the calculation 1-m2). If dr is a reasonably large distance from the edge of the loading plate, the load can be assumed to be a point load, so the plate pressure distribution does not matter. Furthermore, small changes in Poisson's ratio have only a minimal impact on this equation.
Subsequent developments have allowed the use of the shape of the deflection basin to estimate various layered elastic (or slab-on-dense-liquid) moduli from FWD deflection readings.
One method to ascertain the approximate subgrade stiffness, or elastic modulus, directly under an imposed surface load and in the upper portion of the subgrade is the Hogg model. The Hogg model is based on a hypothetical two-layer system consisting of a relatively thin plate on an elastic foundation. The method in effect simplifies the typical multilayered elastic system with an equivalent two-layer stiff layer on elastic foundation model.
Depending on the choice of values along the deflection basin used to calculate subgrade stiffness, there can be a tendency to either over- or underestimate the subgrade modulus. The Hogg model uses the deflection at the center of the load and one of the offset deflections. The offset distance where the deflection is approximately one-half of that under the center of the load plate was shown to be effective at removing estimation bias. Variations in pavement thickness and the ratio of pavement stiffness to subgrade stiffness are taken into account, since the distance to where the deflection is one-half of the deflection under the load plate is controlled by these factors.
The underlying model development for a finite subgrade was first published in 1944 by Hogg.(4) The implementation of the model used in these guidelines was subsequently published in 1983 by Wiseman and Greenstein.(5)
The equations used are as follows:
Figure 2. Equation. Hogg subgrade modulus.
Figure 3. Equation. Offset distance where deflection is half of center deflection.
Figure 4. Equation. Characteristic length of deflection basin.
Figure 5. Equation. Theoretical point load stiffness/pavement stiffness ratio.
The implementation of the Hogg model described by Wiseman and Greenstein included three cases. Case I is for an infinite elastic foundation, while cases II and III are for a finite elastic layer with an effective thickness that is assumed to be approximately 10 times the characteristic length, l, of the deflection basin. These two finite thickness cases are for subgrades with Poisson's ratios of 0.4 and 0.5, respectively. The various constants used for the three versions of the Hogg model are shown in Table 1. Use of Case II is recommended to obtain realistic design values, and it has been used extensively to calculate subgrade moduli for purposes of pavement evaluation through forward calculation.
Case II of the Hogg model has been used extensively over the past 15 years or more, and it has been found to be reasonably stable on a wide variety of pavement types and locations, tending to have a high correlation with back calculated subgrade moduli but with significantly lower (and therefore more conservative) results than the corresponding back calculated values. This difference is generally due to the presence of apparent or actual subgrade nonlinearity (effectively, stress softening) as well as the calculation of a finite depth of subgrade = 10 x l (as defined by Case II) to an effective hard bottom layer of either deeper lying subgrade material or actual bedrock.
In addition, less variation is indicated between FWD test points along the same test section when the Hogg model is used in forward calculation when compared to virtually any Backcalculation approach.
Both as a screening tool and to derive relatively realistic, in situ subgrade stiffnesses, the Hogg model is effective and very easy to use compared to other methods.
A viable method to determine the apparent surface course stiffness of the uppermost bound layer(s), under an imposed surface load is called the AREA method.
This approach was first introduced in the National Cooperative Highway Research Program (NCHRP) Study 20-50(09), LTPP Data Analysis: Feasibility of Using FWD Deflection Data to Characterize Pavement Construction Quality.(6) More recently, the equations originally suggested have been updated and calibrated for both asphalt concrete (AC) and portland cement concrete (PCC) pavement surfaces.
For both pavement types, the radius of curvature method is based on the AREA concept (a deflection basin curvature index) and the overall composite modulus of the entire pavement structure, Eo, as defined in Figure 6.
Figure 6. Equation. Composite modulus under FWD load plate.
Figure 6 has been extensively used over the past three to four decades. An excellent introduction to this approach is presented by Ullidtz in Pavement Analysis. (3)
Figure 6 is the most commonly used version of this equation. In theory, it is based on an evenly distributed and uniform FWD test load and a Poisson's ratio of 0.5. Generally, Poisson's ratio is less than 0.5 (between 0.15 and 0.20 for PCC layers and between 0.3 and 0.5 for most other pavement materials). On the other hand, the distribution of the load under the FWD plate will not be exactly uniform, but rather somewhat nonuniform because of the rigidity of the loading plate. These two offsetting factors have resulted in the widely used and straightforward 1.5-times composite modulus formula, which was therefore chosen for the development of the forward calculation spreadsheets.
AREA, as used for rigid pavements in this study and reported by the American Association of State Highway and Transportation Officials (AASHTO) in the 1993 AASHTO Guide for Design of Pavement Structures, (7) has been historically calculated as:
Figure 7. Equation. 914-millimeter (mm) (36-inch) AREA equation for rigid pavements.
When calculating AREA36, the diameter of the loading plate must be between 300 mm (11.8 in) and 305 mm (12 in). An AREA36 calculation of 36 is achieved if all 4 deflection readings, at the 0-, 305-, 610-, and 914-mm (0-, 12-, 24-, and 36-inch) offsets, are identical, which is tantamount to an infinitely stiff upper layer.
While the equation in Figure 7 is well suited for rigid pavements with a large radius of curvature, flexible pavements generally have a much smaller radius of curvature (i.e., a steeper deflection basin). Accordingly, for AC pavements a new version of the AREA concept based on FWD sensors placed at 0-, 203-, and 305-mm (0-, 8-, and 12-inch) offsets was derived:
Figure 8. Equation. 305-mm (12-inch) AREA equation for flexible pavements.
An AREA12 calculation of 12 is achieved if all three deflection readings, at the 0-, 203-, and 305-mm (0-, 8-, and 12-inch) offsets, are identical, which is tantamount to an infinitely stiff upper layer.
For AC and PCC pavement types, respectively, a series of calculations were made to see what the relevant AREA terms become if all layers in a multilayered elastic system have identical stiffnesses or moduli (and Poisson's ratios). This can be carried out using, for example, the CHEVRON, CHEVLAY2, ELSYM5, or BISAR multilayered elastic programs (CHEVLAY2 was used in this case). It turns out that, no matter which modulus value is selected, as long as all of the layers are assigned the same identical modulus of elasticity and there is continuity between layers (generally assumed in Backcalculation as well), the AREA36 term is always equal to 11.04 for rigid pavements (assuming no bedrock) and AREA12 is always equal to 6.85 if bedrock is assumed for flexible pavements. (Note: The AREA12 calculation for identical moduli with no bedrock = 6.91-nearly identical.) The reason that bedrock was assumed for AC and not PCC pavements is that FWD deflection readings generally reflect the presence of an apparent underlying stiff layer for flexible pavements but not for rigid pavements. (Using either approach, however, the resulting calculations for upper layer pavement stiffness will be nearly the same, whether or not bedrock is assumed.)
The minimum AREA values of 11.04 and 6.85 for the 914-mm (36-inch) and 305-mm (12-inch) areas, respectively, are important in the following equations because they can now be used to ascertain whether the upper layer has a significantly higher stiffness than the underlying layer(s), and to what extent this increase affects the stiffness of the upper, bound pavement layers. For example, if the AREA36 term is much larger than 11.04, then the concrete layer is appreciably stiffer than the underlying (unbound) layer(s). The value 11.04 is therefore used in Figure 9, below, while Figure 10 can be thought of as a radius of curvature stiffness index, based on the stiffness of the bound upper layer(s) compared to the composite stiffness of the underlying unbound layers.
The calculation of Eo was previously explained in connection with the presentation of Figure 6. To reiterate, Eo is a composite, effective stiffness of all the layers under the FWD loading plate. If these two terms are combined such that the boundary conditions are correct and the logic of the two AREA concepts, for PCC and AC pavements respectively, are adhered to, the following equations result:
Figure 9. Equation. AREA factor for rigid pavements.
Figure 10. Equation. AREA factor for flexible pavements.
Figure 11. Equation. Stiffness or modulus of the upper PCC layer.
Figure 12. Equation. Stiffness or modulus of the upper AC layer.
Both Figure 11 and Figure 12 have been calibrated using a large number of CHEVLAY2 runs, and they work very well for typical pavement materials and modular ratios when the underlying materials are unbound. It should be noted that this approach is not totally rigorous but rather is empirical in nature. The approach can therefore be used effectively to approximate the stiffness of the upper (bound) layer(s) in a pavement cross section, for QC, comparative, or routine testing and analysis purposes.
The advantage of using the equations in Figure 9 through Figure 12, or similar equations developed elsewhere, is that forward calculation techniques, together with commonly used deflection-based quantities (such as AREA), can be employed. Only the composite modulus or stiffness of the pavement system, the AREA, and the pavement thickness normalized to the diameter of the loading plate, are needed to calculate the relative stiffness of the bound upper layer(s) of pavement.
Forwardcalculation techniques, as shown in the two previous sections concerning the subgrade and bound surface courses, can in turn be used to estimate the modulus of any intermediate layer through the use of modular ratios. For example, the modulus relationship developed by Dorman and Metcalf between two adjacent layers of materials can be used if the base and subgrade layers are unbound.(2) The Dorman and Metcalf method computes the base modulus, as shown by the equations in Figure 13 (U.S. Customary units) and Figure 14 (SI units).
Figure 13. Equation. Modulus (psi) of the unbound base using the Dorman and Metcalf relationship.
Figure 14. Equation. Modulus (MPa) of the unbound base using the Dorman and Metcalf relationship.
Another technique, sometimes used for rigid pavement sections with bound base courses, is to relate the apparent modulus of the PCC layer, Epcc,app., to a calculated modulus of the PCC layer (E1) and the base course layer (E2) expressed as a ratio between these two layers. This calculation is a function of the thickness of each layer and whether these layers are bonded or unbonded (i.e., whether there is slip between the two uppermost layers, under load).
As previously shown, only the PCC surface course and the subgrade moduli are forward calculated, essentially ignoring the effect of the base layer. Therefore, the forward calculated EPCC actually reflects the effect of both the upper PCC layer and the underlying base layer. In other words, EPCC is really an apparent modulus of the PCC layer, and needs to be divided into two parts, especially when a bound base layer is involved: the actual modulus of the PCC and the calculated modulus of the base. In these cases, EPCC is called Epcc,app., which is the apparent modulus of the PCC layer alone when it is significantly influenced by the base.
The method to divide the calculated Epcc,app.-value into two parts is adopted from Khazanovich, et al.(8) The upper PCC surface layer and the base layer may be bonded or unbonded, as appropriate, and are assumed to act as plates. Thus, no through-the-thickness compression is assumed. The details of this method are given below for an unbonded and a bonded condition between the PCC slab and the base, respectively.
For the unbonded case, the PCC slab modulus is computed from Figure 15.
Figure 15. Equation. PCC slab modulus-100 percent unbonded case.
For the bonded case, the PCC slab modulus is computed from Figure 16.
Figure 16. Equation. PCC slab modulus-100 percent bonded case.
Figure 17. Equation. Layer thickness relationship-both cases.
Figure 18. Equation. Modular ratio ß-both cases.
The procedures presented above require the modular ratio as an input parameter. This ratio should be assigned based on engineering judgment. It is assumed that if the ratio is assigned within reasonable limits, the PCC modulus (= E1) results are insensitive to the chosen ratio. Table 2 presents a set of recommended modular ratios (ß) of the PCC (E1) and base (E2) moduli for each type of base layer. It should be noted that ßfrom Figure 18 is defined as a ratio of base to PCC moduli. This was done to make it stable for the case of a weak base (i.e., when ßapproaches 0). Therefore, if the modular ratios from Table 2 are used, these should be inverted before applying them in the procedures described above.
Given the values for ßand for the actual plate thicknesses, h1 and h2, the equations in Figure 15 and Figure 16 may used with the forward calculated Epcc,app.-value to yield E1 and E2 for the two upper layers. Alternatively, any other modular ratio may be used, as appropriate, depending on the actual materials present in any given project.
In summary, it should be emphasized that forward calculated modulus data are not intended to replace Backcalculation or any other form of modulus of elasticity measurements. Forwardcalculation, like all other methods of determining in situ stiffnesses or moduli, merely provide the analyst with approximations or estimates of such values. The only question is: how realistic are such estimates for pavement evaluation or design purposes?
Accordingly, forward calculation is designed for routine FWD-based project use, and for screening purposes for data derived using Backcalculation techniques. It is ultimately intended to ascertain whether back calculated modulus values-which are also estimates-are reasonable, since two distinctly different methods of deriving stiffnesses or moduli from the same FWD load-deflection data should not produce vastly dissimilar results.
In the following section, the use of the provided forward calculation spreadsheets is presented.