# Resilient modulus of subgrade soils: Comparison of two constitutive equations

Tags: stress model, universal model, Georgia Department of Transportation, Washington, D.C., International Conference, deformation analysis, Department of Civil Engineering, constitutive models, stress, stress-strain, material, pavement analysis, Highway Research Record, Texas A&M University, College Station, subgrade, Transportation Facilities Group, Asphalt Pavements, Base Course Materials, Resilient Modulus, Constitutive Equations, physical properties, cohesive soils, J. Uzan, moduli, National Research Council, stress condition, stress conditions, TRANSPORTATION RESEARCH RECORD, J. Characterization, Transportation Engineer Journal, M. W. Witczak, Granular Materials, TRB, Granular Material
Content: Transportation Research RECORD 1462
79
Resilient Modulus of Subgrade Soils: Comparison of Two Constitutive Equations
ER = recoverable axial strain in the direction of principal stress cri, and 80
TRANSPORTATION RESEARCH RECORD 1462
(6) As mentioned by Uzan and Scullion (J 3), Equation 4 can be used as a universal model for all types of soils. For a constant modulus or linear elastic material, both k2 and k3 are set to zero (14). Also the bulk stress model (Equation 2) can be obtained by setting k3 to zero. The semi-log model (Equation 6) can be obtained by setting k2 to zero.
FIGURE 1 Illustration of principal stresses acting on a soil element.
servations. This model includes the influence of deviator stress on resilient modulus. (4) where ud is deviator stress (u1 - u 3) as defined in Figure 1, and ki, k1, k3 are material and physical property parameters. In 1987 Lade and Nelson (7) presented a constitutive model that shows that modulus of granular materials is a function of the first stress invariant (bulk stress) and the second invariant of the stress deviator tensor. This development provides support for the validity of Equation 4 instead of Equation 2. Brown and Pappin's (8) nonlinear stress-strain relationship also agrees with Equation 4 instead of Equation 2.
Cohesive Soils
Sneddon (9) conducted resilient modulus tests for sand and finegrained soils. Results indicated that the resilient modulus of sandy soils is a function of the applied deviator stress and the confining stress. However the resilient modulus of fine-grained soils is mainly a function of the applied deviator stress, when single confining stress level is considered. In general the resilient modulus of these soils decreases with increasing deviator stress. In 1976 Thompson and Robnett (J 0) introduced an arithmetic model (Equation 5) to describe the resilient properties of fine-grained soils. This model was successfully used in the ILLI-PAVE (J 1,12) computer program.
MR = k1 + k3 (k1 - (jd)
MR= k1 + k4 (ud - k1)
(5)
where MR resilient modulus of the fine-grained soil, ud deviator stress (u1 - u 3), and ki, k1, k3, k4 = material and physical property parameters. Another successful model often used to describe the behavior of cohesive soils is the semi-log model (Equation 6). This model has the advantage of having fewer material constants than the arithmetic model.
LABORATORY TESTS Subgrade soil samples collected from different locations in Georgia were classified and separated into cohesive and granular categories according to the AASHTO soil classification. Each soil was subjected to laboratory tests such as sieve analysis, Atterberg limits, percent swell and shrinkage, optimum moisture, maximum dry unit weight, and California Bearing Ratio. Resilient modulus tests were carried out according to the AASHTO T 274-82 (1986) test procedure. The tested samples had a diameter of 73 mm (2.875 in.) and a height of 142.2 mm (5.6 in.) and were statically compacted in three layers. A sample is placed in a triaxial device and subjected to the repetitive loads and stresses expected in a pavement system. These tests were performed on four replicate samples of each soil. One of these samples was compacted to 95 percent compaction; the other three were compacted to 100 percent. The ratio of sample dry unit weight to maximum dry unit weight of soil was taken as the measure of compaction. Low compacted samples had a moisture content of approximately 3 percent above the optimum moisture content. Two of the other three samples had moisture contents approximately 1.5 percent below and 1.5 percent above the optimum moisture. The moisture content of the fourth sample was kept close to optimum. Practical difficulties prevented obtaining the exact intended compactions and moisture contents. The results of the tests were recorded along with the other soil properties. STUDY DATA Data used in this study were obtained from a data base created from the aforementioned laboratory test results. To avoid inaccuracies, data were selected according to the following criteria, based on research findings and observations: 1. All soils, fine and coarse grained, that have decreasing resilient modulus with increasing deviator stress at least at lower deviator stresses (6-8,15,16). 2. All soils, fine and coarse grained, that have increasing resilient modulus with increasing confining stress (1,2). 3. All soils, fine and coarse grained, that have decreasing resilient modulus with increasing moisture content in the vicinity of optimum moisture (16) when other soil properties are kept constant. Fourteen cohesive and 15 granular data sets that satisfied the criteria were used for the study. Some of these soils did not have resilient modulus test results for low-compacted, high-moisture samples. Therefore the resilient modulus test results of low-compacted, high-moisture samples were not included in this study.
Santha
data analysis
Granular Soils
The measured resilient moduli values corresponding to confining stresses of 6.89 and 34.45 kPa (1 and 5 psi) and deviator stresses of 13.78, 34.45, 51.68, 68.9, and 103.35 kPa (2, 5, 7.5, 10, and 15 psi) were used to obtain a close representation of stress conditions of the subgrade. These stress conditions and the measured resilient moduli were used to develop relationships for each soil sample in which the resilient modulus is a function of both the bulk stress and the deviator stress. The form of this relationship is given in Equation 4, the universal model, which was transformed to linear form as shown in Equation 7 to carry out Linear regressions:
Log(MR) = Log(k1P,,) + k2 Log[~] + k3 Log[~:]
(7)
Linear regressions were performed for each set of data and ki. k2, and k3 were found for each soil sample. These developed relationships and the stress conditions of soil samples were then used to back-calculate the resilient moduli for each stress condition of each soil sample. These resilient moduli values were referred to as the predicted resilient moduli. Atmospheric pressure used in these analyses was 101.3 kPa (14.7 psi). The same data were used to develop relationships for each soil sample in which the resilient modulus is a function of bulk stress. The form of this relationship is given in Equation 2, the bulk stress model. Equation 8 gives the linear form of the bulk stress model used for regression analysis.
81 (8) The predicted resilient moduli obtained from the bulk stress model gave a poor correlation to actual resilient moduli (Figure 2). Figure 3 shows a close correlation between predicted resilient moduli obtained from the universal model and the actual resilient moduli. Symbols A, B, C, and Din the figures refer to one, two, three, and four overlapping data points, respectively. The statistics given in Table 1 provide evidence of better predictability in describing behavior of granular soils using the universal model than using the bulk stress model. Fifty percent of the predicted resilient moduli obtained using the bulk stress model were within ::±:: 16.6 percent of the actual values, whereas the predicted moduli were within ::±:: 5 percent using the universal model. The universal model also provided better predictions than Equation 2 for the 25th, 75th, and 90th percentiles (Table 1). The universal model is therefore more suitable to represent the relationship between resilient modulus and stress levels of granular soils used in this study. The coefficients ki. k2, and k3 obtained from least-square regressions of the universal model for each sample are listed in Table 2 with moisture content (MC), percent saturation (SATU), and compaction (COMP) of the sample. The values of these coefficients were used in further analyses. Table 2 also gives the optimum moisture content (MOIST) of each soil and the ratio of MC and MOIST (MCR). A multiple regression analysis approach was used to obtain the relationships among k parameters (Dependent variables) and other soil properties such as percent passing #40 sieve (S40), percent passing
I I I JlCIUrlJ.. I t>R I I 22000 + I 20000 + I 18000 + I 16000 + I 14000 + I 12000 + I 10000 + I 8000 + I 6000 + I 4000 + I 2000 + I 0+ I I I 0
IEXDD R:R 'IHE PI.Or CF JlCIUrlJ.. IB vs. PWICIED IB: A= 1 CBS, B = 2 CBS, ETI:. SYM3JL USED lN LIN:: CF EIJNtl1Y IS *
AA A
A
A
B AA
A AA~
A NlAf3 N>B
AB BA. A Be.AA. A
BB CA BA.
BA. AB BA.
A APN>D
A. N!F!I!AA.
A AA CAA.
A A A AA.AA B BA
AA AA
AA A
~K//~
+ I
+
I +
I +
i +
I +
I +
I +
I +
I +
I +
I +
I
I
I
2000 4000 6000 8000 10000 12000 14000 16000 10000 20000 22000
PREDICIID IB
FIGURE 2 Actual resilient moduli versus predicted resilient moduli obtained from bulk stress model.
82
TRANSPORTATION RESEARCH RECORD 1462
I I AC1lFlL I IB I I 22000 + I 20000 + I 18000 + I 1600) + I 14000 + I 12000 + I 10000 + I 8000 + I 6000 + I 4000 + I 2000 + I 0+ I I I 0
IE:E:D KR 'IBE PIDI' CF JCilN., IB vs. HIDICIED IB: A 1 CBS, B = 2 CBS, EIC. SXM3:L C6ED IN Ll1'E CF EQJA1TIY IS *
AM A
A+
A
I
AA
+
AM A
I
A
+
BA
I
+
I
I
+
I +
I
+
I
+
I
+
I
+
I +
I
I
I
2000 4000 600) 8000 10000 12000 14000 16000 18000 20000 22000
PPEDICIED IB
FIGURE 3 Actual resilient moduli versus predicted resilient moduli obtained from universal model.
#60 sieve (S60), percentage of clay (CLY), percentage of silt (SLT), percent swell (SW), percent shrinkage (SH), maximum dry unit weight (DEN), optimum moisture content (MOIST), California Bearing Ratio (CBR), sample moisture content (MC), sample compaction (COMP), and percent saturation (SATU). Table 3 gives the index properties of each soil used for the study. Three separate regression analyses were done for the three coefficients k1, k2, and k3· To select the best subset of these variables and their interaction terms, The "PROC STEPWISE" procedure in SAS (17) was used with maximum R-square improvement technique (MAXR). This regression procedure does not settle on a single model, but tries to find the best one-variable model, the best two-variable model, and so forth. The stepwise procedure with the MAXR method begins by finding the one-variable model producing the highest R2· Then another variable, the one that yields the greatest increase in R2, is added. Once the two-variable model is obtained, each variable in the
model is compared to each variable not in the model. For each comparison, MAXR determines whether removing one variable and replacing it with the other variable increases R2· After comparing all possible switches, MAXR makes the switch that produces the largest increase in R2· By doing this MAXR finds the best twovariable model. Another variable is then added to the model and the comparing and switching process is repeated to find the best threevariable model, and so forth. The number of parameters that correspond to the lowest coefficient of performance value (Cp) was selected to describe the data (J 8). In each model developed, the multicolinearity effect of independent variables was evaluated and interaction terms were selected to minimize the multicolinearity effect. The difference between the forward or backward stepwise technique and the MAXR technique is that all switches are evaluated before any switch is made in the MAXR method. In the forward or
TABLE 1 Difference in Actual Resilient Moduli and Predicted Resilient Moduli as a Percentage of Actual Resilient Moduli for Granular Soil Data
Percentile
25
Using Equation 2
9.0
(Bulk stress model)
Using Equation 4
3.0
(Universal model)
50 16.6 5.0
75 27.6 7.3
90 35.4 10.4
Santha
83
TABLE 2 Physical Properties of Granular Soil Samples and Their Least-Square regression results
ID
MOIST MC
MCR
COMP SATU kl
1-I 1-II 1-III 2-I 2-II 2-III 3-I 3-II 3-III 4-I 4-II 4-III 5-I 5-II 5-III 6-I 6-II 6-III 7-I 7-II 7-III 8-I 8-II 8-III 9-I 9-II 9-III 10-I 10-II 10-III 11-I 11-II 11-III 12-I 12-II 12-III 13-I 13-II 13-III 14-I 14-II 14-III 15-I 15-II 15-III
12
11.2 0.93
1. 00
80.3
392
12
12.5 1. 04
1. 00
91. 0
349
12
9.7
0.81
1. 00
70.1
543
18
17.0 0.94
1. 01
78.7
401
18
19.5 1. 08
0.98
82.6
326
18
14.9 0.83
1. 02
69.8
715
16
16.0 1. 00
1. 00
91. 0
451
16
17. 4 1. 09
1. 00
99.4
413
16
14.8 0.93
1. 00
83.8
642
25
24.5 0.98
1. 01
82.2
528
25
25.9 1. 04
1. 01
87.0
403
25
22.9 0.92
1. 01
77.1
703
16
15.2 0.95
1. 01
78.3
356
16
17.0 1. 06
1. 01
86.8
335
16
13.8 0.86
1. 01
70.9
547
17
15.7 0.92
1. 02
80.8
573
17
17.2 1. 01
1. 02
88.5
423
17
14.0 0.82
1. 02
72 .4
832
19
19.3 1. 02
1. 00
81. 5
214
19
20.6 1. 08
1. 00
87.2
173
19
17 .1 0.90
1. 03
73.3
299
16
15.7 0.98
1. 01
80.0
241
16
17.0 1. 06
1. 01
86.7
211
16
14.0 0.88
1. 01
71. 3
284
11
13.4 1.22
0.99
69.3
280
11
14.3 1. 30
0.99
78.8
252
11
11.1 1. 01
1. 00
61. 6
324
16
15.7 0.98
1. 01
71. 5
430
16
16.6 1. 04
1. 01
76.8
338
16
13. 4 0.84
1. 01
62.2
534
14
13 .1 0.94
1. 01
70.2
458
14
14.4 1. 03
1. 01
78.7
384
14
11. 6 0.83
1. 02
63.0
573
14
13.6 0.97
1. 01
78.3
668
14
15.2 1. 09
1. 00
87.2
494
14
12.1 0.86
1. 01
69.7
918
12
11. 5 0.96
0.99
66.6
354
12
12.7 1. 06
0.99
74.4
334
12
9.8
0.82
0.99
57.3
446
16
14.7 0.92
1. 01
70.5
440
16
16.3 1. 02
1. 01
78.1
346
16
13.4 0.84
1. 01
64.1
507
20
19.8 0.99
1. 01
76.4
183
20
21. 4 1. 07
1. 01
82.4
130
20
18.2 0.91
1. 01
70.3
201
k2 0.291 0.316 0.268 0.239 0.328 0.175 0.301 0.316 0.199 0.304 0.318 0.292 0.285 0.293 0.203 0.201 0.250 0.145 0.404 0.412 0.319 0.379 0.441 0.295 0.328 0.349 0.267 0.457 0.479 0.368 0.401 0.444 0.345 0.398 0.469 0.326 0.484 0.498 0.436 0.429 0.454 0.397 0.400 0.430 0.342
k3 -0.487 -0.531 -0.402 -0.484 -0.627 -0.330 -0.501 -0.574 -0.403 -0.364 -0.385 -0.259 -0.304 -0.369 -0.213 -0.272 -0.317 -0.152 -0.343 -0.403 -0.351 -0.319 -0.340 -0.292 -0.336 -0.322 -0.301 -0.340 -0.373 -0.298 -0.353 -0.385 -0.294 -0.302 -0.363 -0.159 -0.403 -0.459 -0.367 -0.382 -0.446 -0.330 -0.450 -0.451 -0.437
R2 0.97 0.89 0.98 0.98 0.93 0.94 0.97 0.94 0.97 0.94 0.84 0.93 0.93 0.90 0.75 0.82 0.83 0.68 0.91 0.98 0.95 0.95 0.92 0.91 0.90 0.94 0.91 0.95 0.90 0.91 0.96 0.89 0.94 0.94 0.94 0.93 0.95 0.90 0.94 0.95 0.90 0.96 0.89 0.90 0.93
backward stepwise method, the worst variable may be removed without considering what adding the best remaining variable might accomplish. The MAXR method could require much more computer time than the stepwise method. Cohesive Soils The measured resilient moduli values corresponding to the confining stress of 20.67 kPa (3 psi) and the deviator stresses of 13.78,
27.56, 55.12, and 68.9 kPa (2, 4, 8, and 10 psi) were used to obtain a close representation of stress conditions of subgrade. The stress conditions and the measured resilient moduli were used. to develop relationships in which the resilient modulus is a function of deviator stress. The form of this relationship is given in Equation 6, the deviator stress model, which was transformed into the following linear form to carry out linear regressions: (9)
84
TRANSPORTATION RESEARCH RECORD 1462
TABLE 3 Index Properties of Granular Soils
ID
SLT
CLY
CBR
SW
SH
S40
DEN
1
8
22
2.5
4
4
72
121
2
9
16
4.4
13
2
50
108
3
8
13
8.7
19
2
38
113
4
19
52
8.1
7
4
96
92
5
13
23
2.5
15
7
69
108
6
9
32
4.7
7
10
75
108
7
15
23
4.3
19
2
82
103
8
12
17
9.8
24
1
68
110
9
8
15
2.8
20
4
68
114
10
28
27
4.7
18
2
97
106
11
21
26
8.1
17
2
99
111
12
12
22
2.5
4
2
71
112
13
13
19
13 .5
13
3
69
114
14
19
24
13.0
12
4
98
107
15
3
16
32.4
1
0
50
123
Linear regressions were performed for each set of data and k1 and k3 were found for each cohesive soil sample. Atmospheric pressure of 101.3 kPa (14.7 psi) was also used. The coefficients ki and k3 that were obtained from least-square regressions of the deviator stress model for each sample are listed in Table 4 with sample MC, SATU, and COMP of the samples. The multiple regression analysis approach, described in the discussion of granular soils, was used to obtain relationships for ki and k3· In these regressions, the independent variable CBR, which was used in granular soil regressions, was replaced by liquid limit (LL) and plasticity index (Pl). Table 5 gives the index properties of each soil used for the study.
RESULTS
Granular Soils
Table 6 presents the descriptive statistics of the k values. Mean and median of each k value were close. Mean, maximum, and minimum values of k1 were 421, 918, and 130, respectively. These results show that the k values of granular soils are subject to a great degree of variability. Equations 10-12 represent the regression equations found for ki, k2, and k3, respectively, from the multiple regression procedure. Complete regression results, including Analysis of Variance tables, are given in the Georgia Department of Transportation Special Report 91001 (19).
Log(k1) = 3.479 0.07 *MC+ 0.24 * MCR + 3.681
* COMP + 0.011 * SLT + 0.006 * CLY - 0.025
* SW - 0.039 * DEN + 0.004 * (SW 2/CLY)
+ 0.003 * (DEN2/S40) R2 = 0.94
(10)
k2 = 6.044 0.053 *MOIST - 2.076 * COMP + 0.0053
* SATU - 0.0056 * CLY + 0.0088 * SW 0.0069 * SH
- 0.027 *DEN+ 0.012 * CBR + 0.003 * {(SW2/CLY)}
- 0.31 * (SW + SH)ICLY R2 = 0.96
(11)
k3 = 3.752 - 0.068 *MC+ 0.309 * MCR - 0.006 * SLT + 0.0053 * CLY+ 0.026 * SH 0.033 * DEN
- 0.0009 * (SW 2/CLY) + 0.00004 * (SATU 2/SH)
0.0026 * (CBR * SH) R2 = 0.87
(12)
The success of multiple regression analysis to obtain relationships for ki, k2, and k3 is shown in Figures 4-6. Actual material and physical properties of soils (given in Tables 2 and 3) were used to calculate ki, k2, and k3 from Equations 10-12, respectively. These k values were labeled as predicted k values. The plots of predicted k values against actual k values with the line of equality clearly indicate the good fit of the regressions. The coefficients of determination for each regression (also given in the figures) were 0.94, 0.96, and 0.87 for ki, k2, and k3, respectively. As a measure of model evaluation, predicted k values and the stress levels were used to back-calculate the resilient moduli values using the universal model. These resilient moduli values were labeled as predicted resilient moduli values. These predicted resilient moduli values versus actual laboratory-observed resilient moduli values and the line of equality are shown in Figure 7. This plot shows a close correlation between actual and predicted resilient moduli values. The descriptive statistics for the actual and predicted resilient moduli values show that 50 percent of the predicted resilient moduli values were within ±8 percent of actual values. They also show that 25 percent, 75 percent, and 90 percent of the predicted values were within ±4 percent, ± 15 percent, and ±21 percent of the actual values, respectively. Thus these results show that for granular soils the k parameters of Equation 4 have a direct relationship to material and physical properties of each soil.
Cohesive Soils Table 7 gives the descriptive statistics of the k values. Mean and median of each k value were very close. Mean, maximum, minimum, and standard deviation of k1 were 645, 1263, 188, and 252, respectively. These results show that the k values of cohesive soils are also
TABLE 4 Physical Properties of Cohesive Soil Samples and Their Least-Square Regression Results
ID 1-I 1-II 1-III 2-I 2-II 2-III 3-I 3-II 3-III 4-I 4-II 4-III 5-I 5-II 5-III 6-I 6-II 6-III 7-I 7-II 7-III 8-I 8-II 8-III 9-I 9-II 9-III 10-I 10-II 10-III 11-I 11-II 11-III 12-I 12-II 12-III 13-I 13-II 13-III 14-I 14-II 14-III
MOIST 20 20 20 20 20 20 20 20 20 19 19 19 20 20 20 17 17 17 19 19 19 18 18 18 15 15 15 16 16 16 21 21 21 16 16 16 18 18 18 22 22 22
MC 19.2 21. 7 18.4 20.2 20.8 17.5 19.9 21.1 18.1 18.1 19.9 16.4 19.2 20.6 17.7 16.6 18.2 14.6 18.4 19.1 16.6 17.7 18.6 15.9 14.2 15.5 12.2 15.6 16.3 13.6 20.2 21.5 18.5 15.5 17.1 13.6 19.1 20.5 17.7 19.9 21. 7 18.5
MCR 0.96 1.09 0.92 1.01 1.04 0.88 1.00 1. 06 0.91 0.95 1.05 0.86 0.96 1.03 0.89 0.98 1.07 0.86 0.97 1.01 0.87 0.98 1.03 0.88 0.95 1.03 0.81 0.98 1. 02 0.85 0.96 1.02 0.88 0.97 1.07 0.85 1. 06 1.14 0.98 0.90 0.99 0.84
COMP 1.01 1.00 1. 01 1.00 1.01 1. 01 1.00 1.00 1. 00 1. 00 1.00 1. 00 1. 00 1.00 1.00 1.00 1.00 1.01 1.01 1. 01 1.01 1.01 1. 01 1. 01 1. 00 1.00 1. 01 1. 01 1. 01 1.01 1.00 1. 00 1. 01 1. 01 . 1. 01 1.01 0.99 0.99 0.99 1. 02 1.02 1.02
SATU 89.3 98.7 84.3 93.0 97.7 82.7 86.3 92.1 79.0 88.7 96.8 80.8 89.5 96.1 82.6 85.0 93.1 75.1 84.8 89.2 76.6 85.0 90.7 77.0 78.1 85.6 68.0 83.3 89.0 73.9 77.1 90.3 77.8 83.1 91.3 73.7 83.0 89.1 76.8 82.5 89.5 76.7
kl 382 287 574 276 188 450 657 431 745 608 423 774 641 442 657 777 473 913 651 549 943 460 299 599 650 474 823 917 685 1169 916 748 1263 541 310 808 967 734 1181 560 442 6~1
k3 -0.466 -0.478 -0.322 -0. 511 -0.598 -0.368 -0.188 -0.261 -0.128 -0.264 -0.272 -0.251 -0.219 -0.312 -0.134 -0.169 -0.235 -0.079 -0.273 -0.260 -0.136 -0.323 -0.424 -0.177 -0.243 -0.366 -0.072 -0.204 -0.211 -0.074 -0.184 -0.216 -0.090 -0.414 -0.501 -0.274 -0.109 -0.176 -0.068 -0.221 -0.262 -0.206
R2 0.94 0.97 0.89 0.98 0.98 0.96 0.80 0.80 0.51 0.95 0.97 0.88 0.99 0.90 0.78 0. 71 0.74 0.70 0.94 0.90 0.98 0.90 0.97 0.78 0.93 0.97 0.97 0.98 0.90 0.97 0.95 0.84 0.99 0.89 0.98 0.86 0.89 0.82 0.98 0.91 0.93 0.95
TABLE 5 Index Properties of Cohesive Soils
ID
SLT
CLY
LL
PI
SW
1
14
52
40.3
20.9
3
2
11
29
35
10.8
5
3
14
55
38.9
19.2
3
4
10
36
36
17. 5
6
5
43
39
40.5
17 .8
13
6
6
32
46.5
30.4
10
7
11
39
43
18
7
8
14
31
40
13 .1
16
9
10
27
33
11. 3
17
10
12
32
49
32
9
11
10
51
59
18
3
12
11
31
30
12
7
13
7
40
34
14
2
14
10
43
39
14
3
SH
540
DEN
9
79
105
7
67
107
11
90
106
8
84
108
8
89
106
12
88
109
5
73
104
7
72
107
4
64
112
10
95
111
14
83
103
4
50
110
5
87
111
1
95
101
·
TABLE 6 Mean, Median, Standard Deviation, Maximum, and Minimum of k Values of Granular Soils
Samples
Mean
Median Std.Dev.
Maximum
kl
45
421
401
173
918
k2
45
0.34
0.33
0.089
0.50
k3
45
-0.37
-0.36
0.095
-0.15
Minimum 130 0.15 -0.63
I I ACru\L I kl I I 1000 + I I I I 800 + I I I I 600 + I I I I 400 + I I I I 200 + I I I I 0+ I I 0
I.EX»D KR 'IHE PlOI' CF ~ kl vs. P.FIDICIED kl: A= 1 CBS, B = 2 CBS, EIC. SYM3'.Jl, USED IN I.JIB CF EQ:JiLTIY IS * R-SJJl'FE = 0. 94
A A
AA
A
* AA.
AA A
A
A
AA B
A
~
AA B A * A BA
A
A
200
400
600
800
I=REDICIED kl
FIGURE4 Actual k1 versus predicted k1 for granular soils.
+ I I I I + I I I I + I I I I + I I I I + I I I I + I I 1000
subject to a great degree of variability. Equations 13 and 14 represent the regression equations found for k1 and k3, respectively, from the multiple regression procedure. Complete regression results, including analysis of variance table are presented elsewhere (19). Results of an evaluation of the success of multiple regression analysis to obtain the relationships between k1 and k3 are shown in Figures 8 and 9. Actual material and physical properties of soils (given in Tables 4 and 5) were used to calculate k1 and k3 from Equations 13 and 14, respectively. These k values were labeled as predicted k values. The plots of predicted k values against actual k values show the effectiveness of the regressions. The coefficients of determination for each regression (also given in the figures) are 0.95 and 0.88 for k1 and k3, respectively.
Log(k1)
19.813 - 0.045 *MOIST- 0.131 *MC - 9.171 *COMP+ 0037 * SLT + 0.015 *LL - 0.016 *PI
- 0.021 *SW - 0.052 *DEN+ 0.00001
* (S40 * SATU) R2 0.95
(13)
k3 = 10.274 0.097 *MOIST 1.06 * MCR - 3.471 *COMP+ 0.0088 * S40 0.0087 *PI+ 0.014 *SH
- 0.046 *DEN R2 0.88
(14)
As a measure of model evaluation, predicted k values and the stress levels of soil samples were used to back-calculate the resilient
I I I JCIUll.L I k2 I 0.55 + I O.:D + I 0.45 + ! 0.40 + I 0.35 + I 0.)) + I 0.25 + I 0.20 + I 0.15 + I 0.10 + I 0.05 + I 0.00 + I I 0.00
I
IfilID KR 'IRE PlOl' CF l\CilN. k2 vs. Pm:>ICIID k2: A= 1 CBS, B = 2 CBS, EIC.
I
SiMil, !.EID IN I...Il'E CF EQJ>J.JTf IS *
I
I
R-~=0.96
/ fI
AA.%
+
~A'A
I
,.B)lMA A
~
*'B A
+
B
I
+
I
+
I
+
I
+
I
+
I
+
I
+
I
I
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.5-0 0.55
PFIDICIID k2
FIGURE 5 Actual k2 versus predicted k2 for granular soils.
I
I
IfilID KR 'IRE PlOl' CF JCIUllL k3 vs. Pm:>ICIID K.3: A = 1 CBS, B = 2 CBS, EIC.
I
SlMil. USED IN I...Il'E CF EttFJJlY IS *
JCIUll.L I
k3 I
R-S'J,:VIFE = 0. 87
--0. 7 +
+
I
I
I I --0.6 +
I
A
I
+
I
A
I
I I --0.5 +
I
A
I
A
+
I
I
I
A
I
I --0.4 +
A
I
+
I
I
I
I
I
I
--0.3 +
+
I
I
I
I
I
I
--0.2 +
+
I
I
I
A
A
I
I
I
--0.1 +
+
I
I
I
I
I
I
0.0 +
+
0.0
--0.1
--0.2
--0.3
--0.4
--0.5
--0.6
--0. 7
Pf®ICIID k3
FIGURE 6 Actual k3 versus predicted k3 for granular soils.
88
TRANSPORTATION RESEARCH RECORD 1462
I
I.film FIB 'IEE Prill CF ~ M<. vs. Pm)JCIED M<.: A = 1 CBS, B = 2 CBS, EIC.
I
S'YM3JL USED JN LnE CF EQ.:N.JTI IS *
~I
M\ I
25000 +
+
I
I
225/JJ +
+
I 20000 +
A
I
+
I 175/JJ +
A
A
I
A AA A
+
I
I
15000 +
+
I
I
12500 +
+
I
I
10000 +
+
I
I
75/JJ +
+
I
I
5000 +
+
I
I
25/JJ +
+
I
I
0+
+
I
I
I
I
I
I
0 25/JJ 5000 7500 10000 12500 15/JJO 17500 20000 225/JJ 25000
PFEDICIED M<.
FIGURE 7 Actual resilient moduli versus predicted resilient moduli for granular soils.
moduli values using the deviator stress model. These values, named as predicted resilient moduli values, and the line of equality are shown in Figure 10. This plot shows a close correlation between actual and predicted resilient moduli values. The descriptive statistics for actual and predicted resilient moduli values show that 50 percent of the predicted resilient moduli values were within ±7 percent of actual values. They also show that 25 percent, 75 percent, and 90 percent of the predicted values were within ± 3 percent, ± 11 percent, and ± 18 percent of the actual values, respectively. Thus these results show that the k parameters of Equation 6 have a direct relation to the material and physical properties of each soil. CONCLUSIONS AND RECOMMENDATIONS Conclusions 1. Equation 4 (the universal model) is capable of describing the behavior of granular soils better than Equation 2 (the bulk stress model).
2. Both granular and cohesive soils have a wide spread in their k parameters. These k parameters depend on the material and physical properties of soil. Recommendations 1. When a complete data base of resilient modulus test results and material and physical properties of soils is available, it is possible to develop a regression model to predict resilient modulus. 2. The universal model is recommended for use with cohesive soils if the model development data have more than one confining stress level. 3. It is important to omit any incorrect data from the data base. ACKNOWLEDGMENTS The data used for this study were obtained from Georgia Department of Transportation (DOT) Research Project 8801, sponsored by FHWA. The author would like to express his sincere appreciation
TABLE 7 Mean, Median, Standard Deviation, Maximum, and Minimum k Values of Cohesive Soils
Samples
Mean
Median Std.Dev.
Maximum
kl
42
645
645
252
1263
k)
42
-0.26
-0.24
0.13
-0.07
Minimum 188 -0.60
I I I ACrrnLI kl I 1400 + I I I 1200 + I I I 1000 + I I I 800 + I I I 6CO + I I I 400 + I I I 200 + I I I 0+ 0
I.EGID FCR '.IEE PIOI' CF 1CIU\I.i kl vs. :EFEDIC'.IID kl: A= 1 CBS, B = 2 CBS, EIC. SYMn., lEED IN LINE CF m,wJ'IY IS *
R-~=0.95 A AA A AA AA A
.i
+
I
I
I
A+
A
I
I
I +
I
I
I
+
I
I
I
+
I
AB AAA
A
+
I
PA
I
I
+
I
I
I
+
200
400
800
1000
1200
1400
PPEDICIED kl
FIGURES Actual k1 versus predicted k1 for cohesive soils.
I
I I
1EGID FCR 'IHE PIOI' CF AC1lN.. k3 vs. PPEDICIED k3: A= 1 CBS, B = 2 CBS, EfC. SYM3JL UID IN Ll1'E CF EQNnY IS *
ACrrnL I
k3 I -0.60 + I -0.55 + I -0.50 + I -0.45 + I -0.40 + I -0.35 + I -0.30 + I -0.25 + I -0.20 + I --0.15 + I --0.10 + IA --0.05 + I 0.00 + I I
R-\$Jll\RE = 0. 88
A A AA A A
A
A
A
A
AA
A
A
A
m.
A
+
I
+
I
A
+
I
+
I
+
I
+
I +
I
+
I
+
I
+
I
+
I
+
I
+
I
I
0.00
:EFEDICIID k3
FIGURE 9 Actual k3 versus predicted k3 for cohesive soils.
90
TRANSPORTATION RESEARCH RECORD 1462
I I ACTI.FiL I M\ I 25{)00 + I 22:JX> + I 20000 + I 17500 + I 15{)00 + I 12500 + I 10000 + I 7500 + I 5{)00 + I 2500 + I 0+ I I I 0
:IEG1D FIB THE PlOI' CF ACTI.FiL M\ vs. WEDICIID M\; A 1 CBS' B = 2 CBSI Elt. SYMn, U3ED lli LINE CF EQ:.W..TIY IS *
+ I +
A
+
MA
I
1'BM
+
A
I
A
+
I
+
I
+
I
+
I
+
A
I
+
I +
I
I
I
2500 5{)00 75/JO 10000 125{)0 15000 175/JO 20000 22500 25{)00
PFEDICIID M\
FIGURE 10 Actual resilient moduli versus predicted resilient moduli for cohesive soils.
for this support. The author gratefully acknowledges the assistance and encouragement given by Wouter Gulden, Lamar Caylor, Steve Valdez, Gerald Bailey, and staff of the Office of Materials and Research of Georgia DOT. REFERENCES 1. Witczak, M. H., and J. Uzan. The Universal Airport Design System, Report I ofIV: Granular Material Characterization. Department of Civil Engineering, University of Maryland, College Park, 1988. 2. Barksdale, R. D. Repeated Load Testing Evaluation of Base Course materials. GHD Research Project 7002, Final Report. FHWA, U.S. Department of Transportation, 1972. 3. Hicks, R. G., and C. L. Monismith. Factors Influencing the Resilient Response of Granular Materials. In Highway Research Record 345, HRB, National Research Council, Washington, D.C., 1971. 4. Shook, J. F., F. N. Finn, M. W. Witczak, and C. L. Monismith. Thickness Design of Asphalt Pavement-The Asphalt Institute Method. Presented at 5th International Conference on the Structural Design of Asphalt Pavements, Delft, The Netherlands, 1982. 5. May, R. W., and M. W. Witczak. Effective Granular Modulus to Model Pavement Responses. In Transportation Research Record 810, TRB, National Research Council, Washington, D.C., 1981, pp. 1-9. 6. Uzan, J. Characterization of Granular Materials. In Transportation Research Record 1022, TRB, National Research Council, Washington, D.C., 1985,pp.52-59. 7. Lade, P. V., and R. B. Nelson. Modelling the Elastic Behavior of Granular Materials. International Journal for Numerical and analytical methods in Geomechanics, Vol. 11, No. 5, Sept.-Oct. 1987, pp. 521-542. 8. Brown, S. F., and J. W. Pappin. Analysis of Pavement with Granular Bases, Layered Pavement System. In Transportation Research Record 810, TRB, National Research Council, Washington, D.C., 1981, pp. 17-23.
9. Sneddon, R. V. Resilient Modulus Testing of 14 Nebraska Soils. Research Project RESI (0099). University of Nebraska, Lincoln, 1988. 10. Thompson, M. R., and Q. L. Robnett. Resilient Properties of Subgrade Soil. Transportation Engineer Journal, Jan. 1979, pp. 1-89. 11. Figueroa, J. L., and M. R. Thompson. Simplified structural analysis of Flexible Pavements for Secondary Roads Based on ILLI-PAVE. In Transportation Research Record 776, TRB, National Research Council, Washington, D.C., 1980, pp. 5-10. 12. ILLI-PAVE User's Manual. Transportation Facilities Group, Department of Civil Engineering, University of Illinois, Urbana-Champaign, May 1982. 13. Uzan, J., and T. Scullion. Verification of Backcalculation Procedures. Proc., 3rd International Conference on the Bearing Capacity ofRoads and Airfields, Trondheim, Norway, 1990. 14. Rodhe, G. T. The Mechanistic Analysis ofFWD Deflection Data on Sections with Changing Subgrade St?-ffness with Depth. Ph.D. dissertation. Texas A&M University, College Station, 1990. 15. Crockford, W.W., K. M. Chua, W. Yang, and S. K. Rhee. Response and Performance of Thick Granular Layers. Draft Interim Report, USAF Contract F08635-87-C-0039. Texas A&M University, College Station, May 1988. 16. Wilson, B. E., S. M. Sargand, G. A. Hazen, and R. Green, Multiaxial Testing of Subgrade. In Transportation Research Record 1278, TRB, National Research Council, Washington, D.C., 1990, pp. 91-95. 17. SAS User's Guide: Statistics. Version 5 Edition. 18. Neter, J., W. Wasserman, and M. H. Kutner. Applied Linear regression models, 2nd ed. Richard D. Irwin, Inc., Homewood, Ill., 1989. 19. Santha, B. Lanka. An Alternative Procedure to Estimate Resilient Mod- ulus of Subgrade Soils Using Their Materials and Physical Properties. Special Report 91001. Georgia Department of Transportation, 1991. The views expressed in this paper are those of the author, who is responsible for the facts and accuracy ofthe data analysis. The contents do not necessarily reflect the official views orpolicies ofGeorgia DOT. The paper does not constitute a standard, specification, or regulation. Publication of this paper sponsored by Committee on Soil and R.ock Properties.