As before, *K*1 was considered to be a function of

the frozen state. Also shown are the data for the

the degree of saturation and the dry density (eq 8).

thawed, undrained condition, which are assumed

In this case, the stress parameter, *f*(σ), was the

to be valid at a temperature of 0C and were in-

normalized bulk stress, *J*1. This form of the equa-

cluded as the warmest data point of the regression

tion was able to accommodate negative stress val-

analyses. Superimposed on the data are lines show-

ues that were generated in the layered elastic analy-

ing the moduli predicted by the three types of

sis portion of the predictive model.

regression equations. Where the predictive equa-

tion requires normalization to the total water con-

tent, that parameter was set to be the average value

for all the specimens tested. For the predictive

equation with volumetric unfrozen water, the dry

density was also set to be the average value for all

Appendices A and B give a tabulation of all the

the specimens tested.

laboratory test results of the frozen, thawed, and

Figure 4 shows that the frozen modulus does

never-frozen soil specimens. Appendix A contains

vary primarily as a function of the unfrozen water

the data from the current study, which includes

content. A minor amount of variation results from

the two Mn/ROAD subgrade samples 1206 and

1232, the class 6 special base, and the class 3

mens, as shown from the vertical spread in the

special subbase. Appendix B contains the data

data at any particular temperature. To illustrate

determined previously from dense graded stone,

this, Figure 5 shows the data from a few individ-

the substitute for Mn/DOT's class 5 special sub-

ual deviator stress levels plotted separately for the

base, and from Albany, New York, taxiway A sub-

subgrade samples.

base, the substitute for class 4 special subbase.

All three types of predictive equations appear

Data for the never-frozen 1206 subgrade speci-

to represent the data fairly well. The moduli re-

mens were acquired on a different testing ma-

sulting from the governing parameter, normalized

chine than all the other data. After testing of all

to the total water content, increases less rapidly

specimens was completed, we discovered that this

with decreasing temperature at temperatures just

second machine was out of calibration, such that

below freezing than do moduli from the other two

the moduli reported here are much higher than

equations. Unfortunately, the temperature varia-

they should be. However, data from the low density

tion of the environmental chamber of the testing

(CE 5) samples are close to moduli back-calcu-

machine was too great to allow acquisition of data

lated from falling weight deflectometer (FWD)

at temperatures close to the freezing point for the

tests on subgrade at the site during fall of 1991.*

Mn/ROAD materials. When the two substitute ma-

Table 6 summarizes the equations that resulted

terials were tested, a different chamber tempera-

from the regression analysis performed on the data,

ture controlling system was used, and it was pos-

with the frozen and unfrozen equations given in

sible to obtain data nearer to the freezing point. In

separate sections. The number "*n*" in Table 6 re-

these cases, shown in Figures 4d and 4e, the pre-

fers to the number of points evaluated in the analy-

dictive equations without normalization to total

sis. Each stress combination at a given moisture

water appear to pass nearer to the center of the

level or temperature results in one data point; thus,

range of data collected at temperatures warmer

the test of a single specimen results in many data

than 2.0C.

points. The table also lists the coefficients of de-

Figure 4 also shows that the predictions from

termination (*r*2) for these analyses.

the equations whose governing parameters are the

gravimetric and volumetric unfrozen water nor-

malized to a unit unfrozen water are not very dif-

Figures 4a through 4f illustrate the resilient

ferent. Predictions from the volumetric form rise

modulus data vs. temperature for each material in

less rapidly at temperatures just below freezing,

while at the colder temperatures, they are slightly

larger than the gravimetric form.

*D. Van Deusen, Mn/ROAD, pers. comm. 1992 .

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