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Publication Number:  FHWA-HRT-15-036    Date:  December 2015
Publication Number: FHWA-HRT-15-036
Date: December 2015

 

Long-Term Pavement Performance Program Determination of In-Place Elastic Layer Modulus: Backcalculation Methodology and Procedures

Appendix A: Best Fit Method—Calculating PCC Elastic Layer Moduli

The best fit method proposed previously for the backcalculation of LTPP rigid pavement deflection data was adopted in this project.(3) All concrete-surfaced LTPP sections were analyzed using the best fit procedure, which involves the backcalculation of a two-layered (PCC and base) slab-on-grade.(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.

Click on graphic to see description.

EPCC = Elastic modulus of the PCC layer.

Ebase = Elastic modulus of the base layer.

hPCC = Thickness of the PCC layer.

Hbase = Thickness of the base layer.

Heff = 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., heff) 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 pre-defined 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. F open parenthesis E, k closed parenthesis equals the summation from i equals 0 under n of alpha subscript i times open parenthesis w open parenthesis r subscript i closed parenthesis minus W subscript i closed parenthesis squared.

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.

αi = Weighting factor.

w(ri) = Calculated deflection.

Wi = 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. w open parenthesis r closed parenthesis equals the ratio of p divided by k times f times open parenthesis r, l closed parenthesis.

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/πa2, where a is the loaded area under the loading plate.

The radius of relative stiffness and the subgrade k-value 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 k-value as follows:

Figure 84. Equation. Radius of relative stiffness. l equals the ratio of open parenthesis D divided by k closed parenthesis raised to the power of 1 divided by 4 which equals open parenthesis E times cubed h all divided by and the quantity of 12 times open parenthesis 1 minus mu squared closed parenthesis times k closed parenthesis raised to the power of 1 divided by 4.

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. F open parenthesis E, k closed parenthesis equals F open parenthesis l, k closed parenthesis equals the summation of sensors from i equals 0 to n of alpha subscript i times open bracket p divided by k times f subscript i times open parenthesis l closed parenthesis minus W subscript i closed bracket squared.

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. The partial derivative of F divided by the partial derivative of k equals 0.

Figure 86. Equation. Partial derivative of F with respect to k.

Figure 87. Equation. Partial derivative of F with respect to l. The partial derivative of F divided by the partial derivative of l equals 0.

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. k-value. k equals p times the ratio of the summation of i equals 0 to n of the quantity of alpha subscript i times open bracket f subscript i times open parenthesis l subscript k closed parenthesis closed bracket squared divided by the summation of i equals 0 to n of the quantity of alpha subscript i times W subscript i times f subscript i times open parethesis l subscript k closed parenthesis.

Figure 88. Equation. k-value.

Figure 89. Equation. Radius of relative stiffness. The ratio of the summation of i equals 0 to n of the quantity alpha subscript i times f subscript i times open parenthesis l subscript k closed parenthesis times f prime subscript i times open parethesis l subscript k closed parenthesis all divided by the summation of i equals 0 to n of the quantity alpha subscript i times open bracket f subscript i times open parenthesis l subscript k closed parenthesis closed bracket squared equals the ratio of the summation of i equals 0 to n of the quantity alpha subscript i times W subscript i times f prime subscript i times open parenthesis l subscript k closed parenthesis all divided by the summation of i equals 0 to n of the quantity alpha subscript i times W subscript i times open bracket f subscript i times open parenthesis l subscript k closed parenthesis closed bracket.

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.

Estimation of Base Layer Modulus in Best Fit Procedure

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. Beta asterisk equals E subscript Base divided by E subscript PCC.

Figure 90. Equation. Modular ratio.

Where:

EBase = Base modulus.

EPCC = 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.

Table 17. Modular ratio to estimate the relative stiffness between PCC slab and base in the best fit procedure (for both bonded and unbonded conditions).
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 (fiber-reinforced)
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, cold-laid, central plant mix
15 15 Recycled AC, cold-laid, mixed-in-place
16 15 Recycled AC, heater scarification/recompaction
17 100 Recycled JPCP
18 100 Recycled JRCP
19 100 Recycled CRCP
78* 10 Dense-graded AC interlayer
80* 20 Open-graded AC interlayer
181 100 Fine-grained soils: lime-treated soil
182 50 Fine-grained soils: cement-treated soil
183 100 Bituminous-treated 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 Soil-aggregate mixture (predominantly fine-grained)
308 250 Soil-aggregate mixture (predominantly coarse-grained)
309* 250 Fine-grained soils
310* 250 Other (specify, if possible)
319 15 HMA
320 50 Sand asphalt
321 50 Asphalt-treated mixture
322 10 Dense-graded, hot-laid, central plant mix AC
323 15 Dense-graded, cold-laid, central plant mix AC
324 15 Dense-graded, cold-laid, mixed-in-place HMA
325 15 Open-graded, hot-laid, central plant mix HMA
326 15 Open-graded, cold-laid, central plant mix HMA
327 15 Open-graded, cold-laid, mixed-in-place HMA
328 10 Recycled HMA, plant mix, hot laid
329 15 Recycled HMA, plant mix, cold laid
330 15 Recycled HMA, mixed-in-place
331 5 Cement aggregate mixture
332 4 Econocrete
333 50 Cement-treated soil
334 2 Lean concrete
335 100 Recycled PCC
337* 150 Limerock, caliche
338 100 Lime-treated soil
339 10 Soil cement
340 100 Pozzolanic-aggregate mixture
341 25 Cracked and seated PCC layer
351 100 Treatment: lime, all classes of quick lime and hydrated lime
352 150 Treatment: lime-flyash
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 slab-base interface and fully bonded slab-base interface. Intermediate levels of interface bond are not analyzed.

In the case of unbonded PCC-base condition, flexural stiffness of the effective slab can be presented as follows:

Figure 91. Equation. Flexural stiffness. D subscript e equals D subscript PCC plus D subscript Base.

Figure 91. Equation. Flexural stiffness.

Where:

De = Flexural stiffness of the effective slab.

DPCC = Flexural stiffness of the PCC.

DBase = 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. E subscript PCC equals the ratio of h cubed subscript PCC divided by the quantity of h cubed subscript PCC plus beta times h cubed subscript Base all times E subscript e.

Figure 92. Equation. Slab modulus from effective modulus for unbonded condition.

Where:

EPCC =Elastic modulus of the PCC slab.

hPCC = PCC thickness.

hBase = Base thickness.

Ee = Effective modulus from best fit backcalculation.

Figure 93. Equation. Base modulus from effective modulus for unbonded condition. E subscript Base equals the ratio of the quantity beta times h cubed subscript PCC divided by the quantity of h cubed subscript PCC plus beta times h cubed subscript Base all times E subscript e.

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. Slab modulus from effective modulus for bonded condition. E subscript PCC equals the ratio of h cubed subscript PCC divided by the quantity of h cubed subscript PCC plus the quantity of beta times h cubed subscript Base plus the quantity of 12 times h subscript PCC times open parenthesis x minus 0.5 times h subscript PCC closed parenthesis squared plus the quantity of 12 times beta times h subscript Base times the quantity of open parenthesis h subscript PCC minus x minus 0.5 times h subscript Base closed parenthesis squared all times E subscript e.

Figure 94. Equation. Sab modulus from effective modulus for bonded condition.

Figure 95. Equation. Base modulus for bonded condition. E subscript Base equals beta times E subscript PCC.

Figure 95. Equation. Base modulus for bonded condition.

Figure 96. Equation. Depth of the parallel axis from the surface. x equals the ratio between the quantity of 0.5 times h squared subscript PCC plus the quantity of beta times h subscript Base times the quantity of open parenthesis h subscript PCC plus 0.5 times h subscript Base closed parenthesis divided by the quantity of h subscript PCC plus the quantity of beta times h subscript Base.

Figure 96. Equation. Depth of the parallel axis from the surface.

Defining the Base Layer in the Pavement Structure

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:

Table 18. Effect of layer structure on base layer—number of layers underneath the PCC slab included in the effective base layer.
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, heq, as the sum of the base layers thicknesses, it was possible to calculate the equivalent base modular ratio, βeq. βeq calculations for a structure with up to five layers combined into the base layer is further described.

If hi is the thickness of the ith base layers and βi is the modular ratio of the ith 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. Beta subscript eq equals the ratio between Expr1 plus 12 times Expr2 divided by h cubed subscript eq all times beta subscript 1.

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. h subscript eq equals the summation from i equals 1 to 5 of h subscript i.

Figure 98. Equation. Equivalent thickness.

Figure 99. Equation. Expr1. Expr1 equals h cubed subscript 1 plus beta subscript 21 times h cubed subscript 2 plus beta subscript 31 times h cubed subscript 3 plus beta subscript 41 times h cubed subscript 4 plus beta subscript 51 times h cubed subscript 5.

Figure 99. Equation. Expr1.

Figure 100. Equation. Expr2. Expr2 equals h subscript 1 times open parenthesis 0.5 times h subscript 1 minus x closed parenthesis squared plus h subscript 2 times beta subscript 21 times open parenthesis h subscript 1 plus 0.5 times h subscript 2 minus x closed parenthesis squared plus h subscript 3 times beta subscript 31 times open parenthesis h subscript 1 plus h subscript 2 plus 0.5 times h subscript 3 minus x closed parenthesis squared plus h subscript 4 times beta subscript 41 times open parenthesis h subscript 1 plus h subscript 2 plus h subscript 3 plus 0.5 times h subscript 4 minus x closed parenthesis squared plus h subscript 5 times beta subscript 51 times the quantity of open parenthesis h subscript 1 plus h subscript 2 plus h subscript 3 plus h subscript 4 plus 0.5 times h subscript 5 minus x closed parenthesis squared.

Figure 100. Equation. Expr2. Expr2 equals h subscript 1 times open parenthesis 0.5 times h subscript 1 minus x closed parenthesis squared plus h subscript 2 times beta subscript 21 times open parenthesis h subscript 1 plus 0.5 times h subscript 2 minus x closed parenthesis squared plus h subscript 3 times beta subscript 31 times open parenthesis h subscript 1 plus h subscript 2 plus 0.5 times h subscript 3 minus x closed parenthesis squared plus h subscript 4 times beta subscript 41 times open parenthesis h subscript 1 plus h subscript 2 plus h subscript 3 plus 0.5 times h subscript 4 minus x closed parenthesis squared plus h subscript 5 times beta subscript 51 times the quantity of open parenthesis h subscript 1 plus h subscript 2 plus h subscript 3 plus h subscript 4 plus 0.5 times h subscript 5 minus x closed parenthesis squared.

Figure 100. Equation. Expr2.

Figure 101. Equation. Depth of parallel axis from the surface when multiple layers are combined into the base. x equals Expr3 divided by Expr4.

Figure 101. Equation. Depth of parallel axis from the surface when multiple layers are combined into the base.

Figure 102. Equation. Expr3. Expr3 equals 0.5 times h squared subscript 1 plus the quantity of beta subscript 21 times h subscript 2 times the quantity of open parenthesis h subscript 1 plus 0.5 times h subscript 2 closed parenthesis plus the quantity of beta subscript 31 times h subscript 3 times the quantity of open parenthesis h subscript 1 plus h subscript 2 plus 0.5 times h subscript 3 closed parenthesis plus the quantity of beta subscript 41 times h subscript 4 times the quantity of open parenthesis h subscript 1 plus h subscript 2 plus h subscript 3 plus 0.5 times h subscript 4 closed parenthesis plus beta subscript 51 times the quantity of open parenthesis h subscript 1 plus h subscript 2 plus h subscript 3 plus h subscript 4 plus 0.5 times h subscript 5 closed parenthesis.

Figure 102. Equation. Expr3. Expr3 equals 0.5 times h squared subscript 1 plus the quantity of beta subscript 21 times h subscript 2 times the quantity of open parenthesis h subscript 1 plus 0.5 times h subscript 2 closed parenthesis plus the quantity of beta subscript 31 times h subscript 3 times the quantity of open parenthesis h subscript 1 plus h subscript 2 plus 0.5 times h subscript 3 closed parenthesis plus the quantity of beta subscript 41 times h subscript 4 times the quantity of open parenthesis h subscript 1 plus h subscript 2 plus h subscript 3 plus 0.5 times h subscript 4 closed parenthesis plus beta subscript 51 times the quantity of open parenthesis h subscript 1 plus h subscript 2 plus h subscript 3 plus h subscript 4 plus 0.5 times h subscript 5 closed parenthesis.

Figure 102. Equation. Expr3.

Figure 103. Equation. Expr4. Expr4 equals the quantity of h subscript 1 plus beta subscript 21 times h subscript 2 plus beta subscript 31 times h subscript 3 plus beta subscript 41 times h subscript 4 plus beta 51 times h subscript 5.

Figure 103. Equation. Expr4.

Note that in these equations, the β values are the inverse of the β* listed in table 17. The βeqvalue calculated in figure 97 provides the combined effective multilayer β value used in figure 89 to calculate the modulus of the effective base layer.

 

 

 

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