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Federal Highway Administration
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Federal Highway Administration Research and Technology
Coordinating, Developing, and Delivering Highway Transportation Innovations
REPORT 
This report is an archived publication and may contain dated technical, contact, and link information 

Publication Number: FHWAHRT15036 Date: December 2015 
Publication Number: FHWAHRT15036 Date: December 2015 
The best fit method proposed previously for the backcalculation of LTPP rigid pavement deflection data was adopted in this project.^{(3)} All concretesurfaced LTPP sections were analyzed using the best fit procedure, which involves the backcalculation of a twolayered (PCC and base) slabongrade.^{(28,58,59)} The procedure was recommended after the completion of the case studies evaluation for the following reasons:
The best fit approach may be considered as one that determines the stiffness of a fictitious effective slab which deforms in a manner identical to the actual pavement under the applied FWD loads. Therefore, the procedure attempts to match the deflection of the simulated effective slab with the measured deflection from the FWD testing. The simulation uses the same level of load as the FWD test and positions the load in the slab interior comparable to the J1 point location in the LTPP database.
E_{PCC} = Elastic modulus of the PCC layer.
E_{base} = Elastic modulus of the base layer.
h_{PCC} = Thickness of the PCC layer.
H_{base} = Thickness of the base layer.
H_{eff} = Effective thickness of the pavement.
Figure 81. Illustration. Transformation of design structure to effective structure used by the neural networks to compute mechanistic responses.
In the figure, the effective thickness parameter derived from section transformation in step 2 changes with the interface bond condition (i.e., h_{eff)} is smaller for the unbond condition. Additionally, the JPCP faulting model uses an unbonded interface for the calculation of corner deflections, while CRCP is always modeled as an unbonded section.
The optimization procedure solves for the specific combination of the coefficient of subgrade reaction and the radius of relative stiffness, which result in slab deflections closest to the measured deflections at each sensor. The procedure allows for the use of weighting factors for the error at each sensor, and the convergence criterion is the minimization of the sum of the mean squared errors from each sensor, with the error weighted by a predefined factor for each sensor. Therefore, the solution is determined by the minimization of the error function expressed as follows:
Figure 82. Equation. Error function definition for best fit procedure.
Where:
F = Error function dependent on the elastic modulus and coefficient of subgrade reaction.
E = Elastic modulus.
k = Coefficient of subgrade reaction.
n = Number of increments.
α_{i} = Weighting factor.
w(r_{i}) = Calculated deflection.
W_{i} = Measured deflection.
The deflection at any point at a radial distance measured from the center of a circular load plate distributing uniform pressure may be expressed as a function of r as follows:
Figure 83. Equation. Calculated deflection.
Where:
r = Radial distance.
p = Pressure.
f = Function.
l = Relative stiffness.
Where f(r) is a function of two dimensionless parameters, the radius of the load plate normalized to the radius of relative stiffness, and the radial distance of the sensor normalized to the radius of relative stiffness. The pressure p is the uniform pressure under the applied load P, calculated as P/πa^{2}, where a is the loaded area under the loading plate.
The radius of relative stiffness and the subgrade kvalue define the flexural rigidity of the effective slab. Most commonly, the radius of relative stiffness is expressed as a function of the effective plate thickness, h, elastic modulus of the effective plate, E, Poisson’s ratio, μ, and the kvalue as follows:
Figure 84. Equation. Radius of relative stiffness.
Where D = Stiffness of the PCC slab.
The error function in figure 82 can be rewritten using figure 83 as follows:
Figure 85. Equation. Error function in alternate form.
The best fit procedure satisfies the two conditions shown in figure 86 and figure 87 as follows:
Figure 86. Equation. Partial derivative of F with respect to k.
Figure 87. Equation. Partial derivative of F with respect to l.
The substitution of the error function in figure 85 in the conditions represented by figure 86 and figure 87 yields the equations to solve for k and l, as shown in figure 88 and figure 89.
Figure 88. Equation. kvalue.
Figure 89. Equation. Radius of relative stiffness.
The computational effort involved in solving the equation in figure 89 forms the major component of the best fit procedure. The k and l values determined from the equations in figure 88 and figure 89 are substituted into the equation in figure 84 to determine the modulus of the plate or the effective slab.
The modulus backcalculated for the effective slab using the best fit procedure is subdivided between the slab and the base, which are assumed to provide a composite section in the calculated deflections. The effective contribution of the base layer in the overall stiffness of the composite section depends on two factors: the relative stiffness of the materials itself and the bond condition at the slab base interface.
To estimate the relative stiffness of the slab and the base materials, an additional parameter is introduced to quantify the modular ratio of the slab and base. The modular ratio, β, is defined as follows (where the asterisk represents an adjustment to the modular ratio):
Figure 90. Equation. Modular ratio.
Where:
E_{Base} = Base modulus.
E_{PCC} = PCC modulus.
The β* values used in the analyses were based on the LTPP material codes for the base layer and were borrowed from Khazanovich et al. (see table 17).^{(3)} Some material codes were assigned β* values under the current study and have been identified in the table. The selected values are based on β* values previously assigned for similar materials. In addition, for PCC layers overlaid using an unbonded PCC overlay, assuming the layer has undergone a fair level of damage, the stiffness was considered to be lower than a concrete slab in good condition ( β* = 1) but higher than an AC layer ( β* = 10). Therefore, a β* value of 5 was assigned for PCC layers that have an unbonded overlay according to the TST_L05B layer structure. Also, a material code of 1000 was assigned within the best fit software code to identify PCC layers with unbonded overlay.
LTPP Code  β* = 1/β  Base Type 

1  10  HMA, dense graded 
2  15  HMA, open graded 
3  50  Sand asphalt 
4  1  JPCP 
5  1  JRCP 
6  1  CRCP 
7  1  PCC (prestressed) 
8  1  PCC (fiberreinforced) 
9  20  Plant mix (emulsified asphalt) material, cold laid 
10  20  Plant mix (cutback asphalt) material, cold laid 
13  10  Recycled asphalt concrete (AC), hot laid, central plant mix 
14  15  Recycled AC, coldlaid, central plant mix 
15  15  Recycled AC, coldlaid, mixedinplace 
16  15  Recycled AC, heater scarification/recompaction 
17  100  Recycled JPCP 
18  100  Recycled JRCP 
19  100  Recycled CRCP 
78*  10  Densegraded AC interlayer 
80*  20  Opengraded AC interlayer 
181  100  Finegrained soils: limetreated soil 
182  50  Finegrained soils: cementtreated soil 
183  100  Bituminoustreated SS 
292  150  Crushed rock 
302  200  Gravel, uncrushed 
303  150  Crushed stone 
304  175  Crushed gravel 
305  175  Crushed slag 
306  250  Sand 
307  400  Soilaggregate mixture (predominantly finegrained) 
308  250  Soilaggregate mixture (predominantly coarsegrained) 
309*  250  Finegrained soils 
310*  250  Other (specify, if possible) 
319  15  HMA 
320  50  Sand asphalt 
321  50  Asphalttreated mixture 
322  10  Densegraded, hotlaid, central plant mix AC 
323  15  Densegraded, coldlaid, central plant mix AC 
324  15  Densegraded, coldlaid, mixedinplace HMA 
325  15  Opengraded, hotlaid, central plant mix HMA 
326  15  Opengraded, coldlaid, central plant mix HMA 
327  15  Opengraded, coldlaid, mixedinplace HMA 
328  10  Recycled HMA, plant mix, hot laid 
329  15  Recycled HMA, plant mix, cold laid 
330  15  Recycled HMA, mixedinplace 
331  5  Cement aggregate mixture 
332  4  Econocrete 
333  50  Cementtreated soil 
334  2  Lean concrete 
335  100  Recycled PCC 
337*  150  Limerock, caliche 
338  100  Limetreated soil 
339  10  Soil cement 
340  100  Pozzolanicaggregate mixture 
341  25  Cracked and seated PCC layer 
351  100  Treatment: lime, all classes of quick lime and hydrated lime 
352  150  Treatment: limeflyash 
353  150  Treatment: lime and cement flyash 
354  50  Treated: PCC 
355  100  Treatment: bitumen (includes all classes of bitumen and asphalt treatments) 
700  15  HMA 
730  1  PCC 
999  10000  No base (fictitious base) 
1000*^{#}  5  Existing PCC underneath unbonded overlay assumed damaged with lower modulus 
*Represents material codes for which the modular ratios were established under this study.
^{#}Represents a material code assigned in the software program for all PCC layers that have an unbonded overlay.
The best fit procedure considers two distinct cases for the bond condition at the interface between the upper (PCC) and lower (base) slabs: unbonded slabbase interface and fully bonded slabbase interface. Intermediate levels of interface bond are not analyzed.
In the case of unbonded PCCbase condition, flexural stiffness of the effective slab can be presented as follows:
Figure 91. Equation. Flexural stiffness.
Where:
D_{e} = Flexural stiffness of the effective slab.
D_{PCC} = Flexural stiffness of the PCC.
D_{Base} = Flexural stiffness of the base.
For the unbonded conditions, PCC and base moduli are defined as follows:
Figure 92. Equation. Slab modulus from effective modulus for unbonded condition.
Where:
E_{PCC} =Elastic modulus of the PCC slab.
h_{PCC} = PCC thickness.
h_{Base} = Base thickness.
E_{e} = Effective modulus from best fit backcalculation.
Figure 93. Equation. Base modulus from effective modulus for unbonded condition.
For the case of the bonded base condition, the flexural stiffness of the effective slabs is derived using the parallel axis theorem. The PCC and base moduli are defined as follows, where x is the depth of the parallel axis from the surface as defined in figure 96 :
Figure 94. Equation. Sab modulus from effective modulus for bonded condition.
Figure 95. Equation. Base modulus for bonded condition.
Figure 96. Equation. Depth of the parallel axis from the surface.
The project team developed procedures to identify layers in the pavement structure that could be treated as the base layer in the best fit analysis. Table 18 lists the layers below the PCC slab that were included in the base layer for different layer structures. The selection of the base layer was based on the following general rules:
Number of Layers Between PCC Surface Layer and Subgrade 
Layer Types in the Layer Structure Included in TST_L05B Table 
Number of layers beneath the PCC layer included in the effective base layer 

1  PCC, GB, SS  1 
1  PCC, GS, SS  1 
1  PCC, TB, SS  1 
2  PCC, AC, PC, SS  2 
2  PCC, AC, TB, SS  2 
2  PCC, GB, GS, SS  1 
2  PCC, GB, TS, SS  1 
2  PCC, TB, GB, SS  1 
2  PCC, TB, GS, SS  1 
2  PCC, TB, TS, SS  1 
3  PCC, AC, AC, GB, SS  2 
3  PCC, AC, AC, PC, SS  3 
3  PCC, AC, PC, GB, SS  2 
3  PCC, AC, PC, TB, SS  3 
3  PCC, AC, TB, GS, SS  2 
3  PCC, AC, TB, TS, SS  2 
3  PCC, GB, GB, GS, SS  2 
3  PCC, GB, GS, GS, SS  1 
3  PCC, GB, GS, TS, SS  1 
3  PCC, TB, AC, GS, SS  2 
3  PCC, TB, AC, TS, SS  2 
3  PCC, TB, GB, GS, SS  1 
3  PCC, TB, GB, TS, SS  1 
3  PCC, TB, GS, GS, SS  1 
3  PCC, TB, GS, TS, SS  1 
3  PCC, TB, TS, GS, SS  2 
3  PCC, TB, TS, TS, SS  1 
4  PCC, AC, AC, PC, GB, SS  3 
4  PCC, AC, PC, GB, GS, SS  2 
4  PCC, AC, PC, GB, TS, SS  2 
4  PCC, AC, PC, TB, GS, SS  3 
4  PCC, GB, GS, GS, GS, SS  1 
4  PCC, TB, GB, GB, GS, SS  1 
4  PCC, TB, GB, GS, GS, SS  1 
4  PCC, TB, GB, GS, TS, SS  1 
4  PCC, TB, GS, GS, GS, SS  1 
6  PCC, AC, AC, AC, PC, GB, GS, SS  4 
6  PCC, AC, AC, AC, PC, GB, GS, SS  4 
7  PCC, AC, AC, AC, AC, PC, TB, TS, SS  6 
GS = granular subbase.
TB = Treated base.
In the case of more than one layer between the PCC and subgrade, all base layers were combined into one layer. Using parallel axis theorem and defining the equivalent base thickness, h_{eq}, as the sum of the base layers thicknesses, it was possible to calculate the equivalent base modular ratio, β_{eq}. β_{eq}_{ }calculations for a structure with up to five layers combined into the base layer is further described.
If h_{i}_{ }is the thickness of the i^{th} base layers and β_{i} is the modular ratio of the i^{th} layer (from table 17), then for the case of five base layers, the equivalent base modular ratio can be calculated as follows:
Figure 97. Equation. Equivalent beta when multiple layers are combined into the base.
Where:
Expri = A model parameter defined in figure 99 and figure 100.
The variables in figure 97 are defined as shown in figure 98 through figure 103.
Figure 98. Equation. Equivalent thickness.
Figure 99. Equation. Expr1.
Figure 100. Equation. Expr2.
Figure 101. Equation. Depth of parallel axis from the surface when multiple layers are combined into the base.
Figure 102. Equation. Expr3.
Figure 103. Equation. Expr4.
Note that in these equations, the β values are the inverse of the β* listed in table 17. The β_{eq}value calculated in figure 97 provides the combined effective multilayer β value used in figure 89 to calculate the modulus of the effective base layer.