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Publication Number: FHWA-HRT-12-068
Date: December 2012

 

Curl and Warp Analysis of The LTPP SPS-2 Site in Arizona

APPENDIX F. PSG ESTIMATION

Analyses were performed to segregate and quantify the portion of roughness caused by slab curl and warp within a profile and the portion associated with other sources of roughness such as built-in defects and surface distress.

The level of curl and warp present within each profile was estimated using slab-by-slab analysis of local profile segments. The automated procedure described in appendix D isolated the profile of each slab by seeking the locations where negative spikes appeared repeatedly in the same locations. The procedure described in this chapter quantifies the level of curl and warp on each slab using PSG, which is the gross strain gradient required to deform a slab into the shape that appears within its measured profile from a flat baseline.

Using PSG to quantify curl and warp represents an approach similar to the calculation of equivalent temperature gradient from measured slab profiles.(19, 20) PSG is used rather than equivalent temperature gradient because this study sought to relate curl and warp to IRI rather than relate content within the profile to the environment in which the pavement functions.

Other examples of this analytical strategy for processing slab profiles have been documented by Sixbey and Vandenbossche.(21, 22) Those studies applied curve fitting to profiles of individual slabs using appropriately selected polynomials to represent idealized slabs with a strain gradient. The specific approach applied in this study uses an idealized profile for a slab with curling and warping proposed by Westergaard in the classical literature.(23) With only minor exceptions, the step-by-step procedure used in this study follows the analysis methods proposed by Rasmussen and applied by Chang.(2, 24) Portions of this appendix duplicate the descriptions by Chang, with additional specific information pertinent to this study.(2, 24)

WESTERGAARD

The curve fitting procedure for measured slab profiles used an idealized shape proposed by Westergaard.(23) The idealized profile is based on the assumption of a linear temperature and moisture gradient through the depth of the slab, unrestrained slab ends, and an infinite slab along the undeformed axis. Westergaard provided the solution for an infinitely long pavement of finite width. This study applies the solution to an infinitely wide slab of finite length.

The idealized profile relates the slab elevation (z) to position along the slab as shown in figure 72.

The variable z equals the product of three terms. The first term is negative z subscript zero. The second term is a fraction. The numerator is 2 times the cosine of lambda times the hyperbolic cosine of lambda. The denominator is the sine of the quantity 2 times lambda minus the hyperbolic sine of the quantity 2 times lambda. The third term is the sum of two sub-terms. The first sub-term is the product of three items. Item 1 is the cosine of a quotient, where the quotient has x as the numerator and the product of l and the square root of 2 as the denominator. Item 2 is the hyperbolic cosine of a quotient, where the quotient has x as the numerator and the product of l and the square root of 2 as the denominator. Item 3 is the hyperbolic tangent of lambda minus the tangent of lambda. The second sub-term is the product of three items. Item 1 is the sine of a quotient, where the quotient has x as the numerator and the product of l and the square root of 2 as the denominator. Item 2 is the hyperbolic sine of a quotient, where the quotient has x as the numerator and the product of l  and the square root of 2 as the denominator. Item 3 is the hyperbolic tangent of lambda plus the tangent of lambda. The equation includes additional expressions for z subscript zero, lambda, and radius of relative stiffness. The value of z subscript zero is equal to l squared times the opposite of one plus mu, times the sum of the temperature gradient and the moisture gradient. The value of lambda is equal to a fraction with b as the numerator and l  times the square root of 8 as the denominator. l is equal to a fraction taken to the fourth root, where the numerator is the product of h to the third power and E. The denominator is the product of 12, k, and 1 minus the square of mu.

Figure 72. Equation. Relationship of Slab Elevation to Position.

 

Where:

x is the horizontal coordinate along the slab profile, referenced to the slab center (L).
z0 is the uplift at the slab ends (L).
l is the radius of relative stiffness (L).
b is the slab width (used as slab length here) (L).
E is slab elastic modulus (F/L2).
μ is Poisson’s ratio (–).
k is the modulus of subgrade support (F/L2/L).

Figure 72 is a specialized version of the Westergaard solution that includes both moisture gradient (Δεsh/h) and temperature gradient (αΔT/h).

The four independent pavement properties b, E, μ, and k are parametric inputs needed to estimate l. The curve fitting procedure applied in this study assumes a fixed value for the five aforementioned parameters along a pavement section and seeks the optimal value for total strain gradient on each slab segment as shown in figure 73.

PSG is the sum of the moisture gradient (delta epsilon subscript sh over h) and the temperature gradient (alpha delta T over h).

Figure 73. Equation. PSG

 

PSG is given the prefix “pseudo” because it is estimated empirically.

SLAB-BY-SLAB ANALYSIS

The PSG associated with each slab profile was estimated using the following procedure.

Step 1: Crop

Crop the profile of the slab to exclude the negative spikes at the joints. Appendix E describes the procedure for determining the location of the joints. Set the profile starting point to the end of the spike group found at the leading joint plus a small offset. Set the profile ending point to the start of the spike group found at the trailing joint minus a small offset. This study applied an offset of 1 inch.

Step 2: Shift

Shift the longitudinal scale to place the slab center at a value of 0. Assume that the center of the slab is at the midpoint between the two average spike position values, derived as described in appendix E. Figure 74 shows an isolated slab profile after offset of the longitudinal scale.

This figure shows the profile over one slab. The vertical scale shows measured profile from -0.3 to 0.1 inches. The horizontal scale shows longitudinal distance from -100 to 100 inches. The longitudinal scale is shifted so that the center of the slab appears at 0 inches. As such, the plot covers a range from -90 to 90 inches. The profile includes many asperities over the length of the slab. However, the profile is at a local peak of about -0.1 inches near the left end. The height reduces as longitudinal position progresses from left to right and reaches a minimum of about -0.27 inches at a longitudinal position of -30 inches. The height then increases as longitudinal position progresses further from left to right and reaches a peak of about 0.02 inches at the right end of the slab.

Figure 74. Graph. Measured Slab Profile for Section 0215.

 

Step 3: Detrend

Detrend the isolated profile. Apply a least-squares linear fit to the profile segment and subtract the best-fit line point by point. The resulting signal will have a zero mean. Figure 75 shows the detrended profile.

This figure shows the profile over one slab. The vertical scale shows detrended profile with a range from -0.1 to 0.15 inches. The horizontal scale shows longitudinal distance from -100 to 100 inches. The longitudinal scale is shifted so that the center of the slab appears at 0 inches. As such, the plot covers a range from  90 to 90 inches. The plot shows the same profile that was shown in figure 72 after the linear trend for that slab was subtracted and the resulting profile was shifted vertically to remove the mean. Like the profile in figure 74, this profile is concave up. The profile is highest at both ends, with an approximate height of 0.125 inches. The profile is lowest near the center, and the lowest point is about -0.8 inches at a longitudinal position of -30 inches.

Figure 75. Graph. Detrended Slab Profile for Section 0215.

 

Step 4: Estimate Pavement Properties

Estimate the pavement properties needed to apply the Westergaard equation for the slab under examination. The LTPP database includes measurements of elastic modulus (E), Poisson’s ratio (μ), and slab thickness (h) for each section. For this study, these values were held constant for all slabs within a given section. Values for modulus of subgrade support (k) were not available, and a value of 200 psi/inch was used for the entire experiment. Table 32 lists the values of E, μ, and h for each section, as well as the radius of relative stiffness (l).

Table 32. Pavement Properties.

Section Elastic Modulus (ksi) Poisson’s Ratio (–) Slab Thickness (inches) Radius of Relative Stiffness (inches)
0213 5,100 0.15 7.9 32.2
0214 4,100 0.13 8.3 31.6
0215 4,850 0.15 11.0 40.7
0216 4,300 0.19 11.2 40.2
0217 5,050 0.16 8.1 32.7
0218 4,050 0.14 8.3 31.5
0219 4,900 0.14 10.8 40.2
0220 4,150 0.15 11.2 39.7
0221 4,750 0.15 8.1 32.2
0222 4,075 0.135 8.6 32.4
0223 5,000 0.14 11.1 41.3
0224 5,150 0.15 10.6 40.2
0262 5,150 0.09 8.1 32.7
0263 5,300 0.12 8.2 33.3
0264 5,250 0.13 11.5 42.9
0265 4,300 0.16 10.8 39.0
0266 4,900 0.15 12.3 44.4
0267 5,350 0.13 11.3 42.5
0268 4,850 0.13 8.5 33.5

 

The joint-finding procedure provided the length of each slab (b). To obtain slab length, calculate the longitudinal distance between the average spike positions at the leading and trailing joint. Each slab requires a distinct estimate.

Step 5: Curve Fit

Perform a non-linear curve fit of the Westergaard model to the measured slab profile. This provides an estimate of the PSG required to optimize the linear fit. In this study, curve fitting was performed using functions MRQMIN and MRQCOF provided in Numerical Recipes.(25)

Figure 76 shows a sample curve fit for a slab on section 0215. For this slab, the routine used a slab length of 15.1 ft and radius of relative stiffness of 40.7 inches. For this slab, the routine returned a PSG value of 70.57 × 10-6 in units of strain per inch or 70.57 με/inch.

This figure shows the profile over one slab and a curve fit to the profile. The vertical scale shows slab profile from -0.1 to 0.15 inches. The horizontal scale shows longitudinal distance from -100 to 100 inches. The longitudinal scale is shifted so that the center of the slab appears at 0 inches. As such, the plot covers a range from  90 to 90 inches. The plot shows the same profile that was shown in figure 75. The plot also shows a curve fit to the profile, which is symmetric about the center of the slab and is a smooth function relative to the measured profile. The fitted trace reaches a minimum value of -0.06 inches at the center of the slab, where it has zero slope. The trace is highest at the ends, with values of about 0.12 inches. In addition, the slope of the trace roughly matches the slope of the measured profile at both ends. Over much of the range shown, the fitted and measured traces agree to within 0.02 inches, but the measured trace is up to 0.04 inches below the fitted trace in the area from -40 to  20 inches along the horizontal scale.

Figure 76. Graph. Curve Fit for Section 0215.

 

SUMMARY INDEX

Summary PSG values for a section were compiled using a weighted average of the absolute slab-by-slab PSG values. A weighting factor was applied to the PSG values in proportion to their length, relative to the overall section length. For slabs that appeared at the ends of a section, a weighting factor was applied to the PSG values in proportion to the length that appeared within the section, relative to the overall section length.

 


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