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Publication Number: FHWARD98085 Date: November 2003 
Like the Delta basin shape factors, the "Ratio" basin shape factors date back to the early days of deflection testing, and are quite simple in concept. Simply put, a "Ratio" factor is the ratio between the center deflection (defl_{0}) and the deflection at some distance away from the load plate (defl_{x}). Therefore, the general equation is as follows:
Ratio_{x} = defl_{0} / defl_{x}
Where:
x = Offset from the center of the load plate, in.defl_{0} = Deflection measured at an offset of 0 in., µm
defl_{x} = Deflection measured at and offset of x in., µm
Following industry practice at the expense of using mixed units, the sensor offsets used in the Ratio factors presented here are in US customary units. Correction factors for Ratio_{8}, Ratio_{12}, Ratio_{18}, Ratio_{24}, Ratio_{48} and Ratio_{60} are presented here.
The Ratio_{x} factor is primarily dependent on the HMA thickness and stiffness, however as "x" increases, so does the influence of the base and subgrade layers. The influence of the lower layers is dependent on many things, such the ratio of "x" to the thickness of the HMA layer and the stiffness of the HMA layer relative to the stiffness of the lower layers, so care should be exercised in the use of the Ratio factors to characterize properties of the HMA layer.
As discussed earlier in this document, because a factor is dependent on the stiffness of an HMA layer, it necessarily follows that the factor is also temperature dependent. The Ratio factors are no exception. The dependence of the Ratio_{12} on temperature is illustrated in Figures 2527. Similar figures for the other Ratio factors are available here.
Figure 25. Sample Deflection Basins Measured at the Same Point
Figure 26. Ratio_{12} Factors Calculated from Basins in Figure 25
Figure 27. Ratio_{12} Factors versus Temperature at a Single Test Location
The LTPP SMP data was used to develop regression equations relating the Ratio basin shape factors to pavement characteristics including the thickness of the HMA layer, the temperature at the middepth of the HMA layer, the 9 kip loadnormalized deflection at the 36 in. offset (which provides a substitute for subgrade stiffness) and the latitude (which is used as a surrogate for binder stiffness).
Ratio_{8}
Log(Ratio_{8}) = 0.183 + 0.0118*Log(ac)*Log(defl_{36}) + 0.00980*T + 0.0696*Log(theta)  0.133*Log(ac)*0.00416*T*Log(defl_{36})
Ratio_{12}
Log(Ratio_{12}) = 0.200  0.117*Log(ac)*Log(defl_{36}) + 0.126*Log(theta)*Log(defl_{36} + 0.00861*T  0.00183*T*Log(theta)*Log(defl_{36})
Ratio_{18}
Log(Ratio_{18}) = 0.952  0.450*Log(ac)  0.169*Log(defl_{36}) + 0.327*Log(theta) + 0.00212*T*Log(ac)
Ratio_{24}
Log(Ratio_{24}) = 1.16  0.587*Log(ac)  0.210*Log(defl_{36}) + 0.481*Log(theta) + 0.00257*T*Log(ac)
Ratio_{36}
Log(Ratio_{36}) = 0.0912  0.367*Log(ac)*Log(defl_{36}) + 0.489*Log(defl_{36} + 0.691*Log(theta) + 0.00298*T*Log(ac)
Ratio_{60}
Log(Ratio_{60}) = 0.0726  0.367*Log(ac)*Log(defl_{36}) + 0.334*Log(defl_{36}) + 0.872*Log(theta) + 0.00246*T*Log(ac)
Where:
ac = Total thickness of the HMA, mmtheta = Latitude of the pavement section
defl_{36} = Deflection (loadnormalized to 40.5 kN (9 kip)) at 915 mm (36 in.) from the center of the load plate, µm
T = Temperature at middepth of the HMA, °C
All of the Ratio factors increase as the log of the thickness of the HMA increases, and increase as the temperature of the HMA increases. Latitude is used here as a surrogate for binder stiffness, as generally stiffer binders are used in northern areas, and softer binders in southern areas. The coefficients for the latitude terms are positive (except for Ratio_{12} which includes a minor, secondorder latitude effect) indicating that the Ratio factor increases as the latitude increases, which is consistent with the relationship between the Ratio factors and stiffness.
Use of the Ratio basin factors generally will require that the calculated Ratio values be adjusted for temperature. Temperature adjustment factors can be calculated by using the respective Ratio equation to calculate an Ratio factor for the pavement tested using the middepth HMA temperature at the time of the test (T_{m}). The Ratio is then calculated again using the middepth HMA temperature, or reference temperature (T_{r}) that it is to be adjusted to. The Basin Adjustment Factor for Ratio_{x}, or BAFRatio_{x}, is the Ratio_{x} at the reference temperature divided by the Ratio_{x} at the measured temperature. This procedure is illustrated by the following equation:
BAFRatio_{x} = Ratio_{x, Tr} / Ratio_{x, Tm}
Where:
BAFRatio_{x} = Basin Adjustment Factor for Ratio_{x}Delta_{x, Tr} = Ratio_{x} factor calculated for the reference temperature
Ratio_{x, Tm} = Ratio_{x} factor calculated for the reference temperature
Source code is included for implementing the Ratio_{8}, Ratio_{12}, Ratio_{18}, Ratio_{24}, Ratio_{36} and Ratio_{60} is included, as is sample data for checking code.
Source code for implementing the basin adjustment factors for Ratio_{8}, Ratio_{12}, Ratio_{18}, Ratio_{24}, Ratio_{36} and Ratio_{60} is also included.
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