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Since laboratory or field experiments are generally expensive and time consuming,
- a. Use the Osman et al. (2008) method [Eqs. (6.15) through (6.18)].
- b. Use the Gurtug and Sridharan (2004) method [Eqs. (6.13) and (6.14)].
- c. Use the Matteo et al. (2009) method [Eqs. (6.19) and (6.20)].
- d. Plot the calculated wopt against the experimental wopt, and the calculated γd(max) with the experimental γd(max). Draw a 45° line of equality on each plot.
- e. Comment on the predictive capabilities of various methods. What can you say about the inherent nature of empirical models?
(a)
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Find the optimum moisture content and maximum dry unit weight using Osman et al. (2008) method.
Explanation of Solution
Calculation:
Determine the plasticity index PI using the relation.
PI=LL−PL
Here, LL is the liquid limit for soil 1 and PL is the plastic limit for soil 1.
Substitute 17 % for LL and 16 % for PL.
PI=17−16=1%
Similarly, calculate the PI for remaining soils.
Determine the optimum moisture content for soil 1 using the relation.
wopt=(1.99−0.165lnE)(PI)
Here, E is the compaction energy for soil 1.
Substitute 2,700 kN-m/m3 for E and 1 % for PI.
wopt=(1.99−0.165ln2,700)(1)=0.69 %
Similarly, calculate the optimum moisture content for remaining soils.
Determine the value of L using the relation.
L=14.34+1.195lnE
Substitute 2,700 kN-m/m3 for E.
L=14.34+1.195ln2,700=23.78
Determine the value of M using the relation.
M=−0.19+0.073lnE
Substitute 2,700 kN-m/m3 for E.
M=−0.19+0.073ln2,700=0.387
Determine the maximum dry unit weight of the soil 1 using the relation.
γd(max)=L−Mwopt
Substitute 23.78 for L, 0.387 for M, and 0.69 % for wopt.
γd(max)=23.78−0.387×0.69=23.51 kN/m3
Similarly, calculate the maximum dry unit weight for remaining soils.
Summarize the calculated values as in Table 1.
Soil |
LL (%) |
PL (%) |
PI (%) |
E (kN-m/m3) |
wopt(%) (Exp) |
γd(max) (kN/m3) (Exp) |
wopt(%) | L | M |
γd(max)(kN/m3) |
1 | 17 | 16 | 1 | 2,700 | 8 | 20.72 | 0.69 | 23.782 | 0.387 | 23.51 |
600 | 10 | 19.62 | 0.93 | 21.984 | 0.277 | 21.73 | ||||
354 | 10 | 19.29 | 1.02 | 21.354 | 0.238 | 21.11 | ||||
2 | 68 | 21 | 47 | 2,700 | 20 | 16.00 | 32.26 | 23.782 | 0.387 | 11.30 |
600 | 28 | 13.80 | 43.92 | 21.984 | 0.277 | 9.82 | ||||
354 | 31 | 13.02 | 48.01 | 21.354 | 0.238 | 9.91 | ||||
3 | 56 | 14 | 42 | 2,700 | 15 | 18.25 | 28.82 | 23.782 | 0.387 | 12.63 |
1,300 | 16 | 17.50 | 33.89 | 22.908 | 0.333 | 11.61 | ||||
600 | 17 | 16.50 | 39.24 | 21.984 | 0.277 | 11.12 | ||||
275 | 19 | 15.75 | 44.65 | 21.052 | 0.220 | 11.23 | ||||
4 | 66 | 27 | 39 | 600 | 21 | 15.89 | 36.45 | 21.984 | 0.277 | 11.89 |
5 | 25 | 21 | 4 | 600 | 18 | 16.18 | 3.740 | 21.984 | 0.277 | 20.95 |
6 | 35 | 22 | 13 | 600 | 17 | 16.87 | 12.15 | 21.984 | 0.277 | 18.62 |
7 | 23 | 18 | 5 | 600 | 12 | 18.63 | 4.67 | 21.984 | 0.277 | 20.69 |
8 | 29 | 19 | 10 | 600 | 15 | 17.65 | 9.35 | 21.984 | 0.277 | 19.39 |
(b)
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Find the optimum moisture content and maximum dry unit weight using Gurtug and Sridharan (2004) method.
Explanation of Solution
Calculation:
Determine the optimum moisture content for soil 1 using the relation.
wopt=(1.95−0.38(logE))(PL)
Substitute 2,700 kN-m/m3 for E and 16 % for PL.
wopt=(1.95−0.38(log2,700))16=10.34 %
Determine the maximum dry unit weight of the soil 1 using the relation.
γd(max)=22.68e−0.0183wopt
Substitute 10.34 % for wopt.
γd(max)=22.68e−0.0183×10.34=18.77 kN/m3
Similarly, calculate the maximum dry unit weight for remaining soils.
Summarize the calculated values as in Table 2.
Soil |
PL (%) |
E (kN-m/m3) |
wopt(%) (Exp) |
γd(max) (kN/m3) (Exp) |
wopt(%) |
γd(max)(kN/m3) |
1 | 16 | 2,700 | 8 | 20.72 | 10.34 | 18.77 |
600 | 10 | 19.62 | 14.31 | 17.46 | ||
354 | 10 | 19.29 | 15.70 | 17.02 | ||
2 | 21 | 2,700 | 20 | 16.00 | 13.57 | 17.69 |
600 | 28 | 13.80 | 18.78 | 16.08 | ||
354 | 31 | 13.02 | 20.61 | 15.55 | ||
3 | 14 | 2,700 | 15 | 18.25 | 9.05 | 19.22 |
1,300 | 16 | 17.50 | 10.73 | 18.64 | ||
600 | 17 | 16.50 | 12.52 | 18.04 | ||
275 | 19 | 15.75 | 14.32 | 17.45 | ||
4 | 27 | 600 | 21 | 15.89 | 24.15 | 14.58 |
5 | 21 | 600 | 18 | 16.18 | 18.78 | 16.08 |
6 | 22 | 600 | 17 | 16.87 | 19.67 | 15.82 |
7 | 18 | 600 | 12 | 18.63 | 16.10 | 16.89 |
8 | 19 | 600 | 15 | 17.65 | 16.99 | 16.62 |
(c)
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Find the optimum moisture content and maximum dry unit weight using Matteo et al. method.
Explanation of Solution
Calculation:
Determine the optimum moisture content for soil 1 using the relation.
wopt=−0.86(LL)+3.04(LLGs)+2.2
Substitute 17 % LL and 2.67 for Gs.
wopt=−0.86(17)+3.04(172.67)+2.2=6.94 %
Determine the maximum dry unit weight of the soil 1 using the relation.
γd(max)=40.316(w−0.295opt)(PI0.032)−2.4
Substitute 6.94 % for wopt and 1 % for PI.
γd(max)=40.316(6.94−0.295)(10.032)−2.4=20.37 kN/m3
Similarly, calculate the maximum dry unit weight for remaining soils.
Summarize the calculated values as in Table 3.
Soil |
Specific gravity (Gs) |
PI (%) |
wopt(%) (Exp) |
γd(max) (kN/m3) (Exp) |
wopt(%) |
γd(max)(kN/m3) |
1 | 2.67 | 1 | 8 | 20.72 | 6.94 | 20.37 |
10 | 19.62 | 6.94 | 20.37 | |||
10 | 19.29 | 6.94 | 20.37 | |||
2 | 2.73 | 47 | 20 | 16.00 | 19.44 | 16.60 |
28 | 13.80 | 19.44 | 16.60 | |||
31 | 13.02 | 19.44 | 16.60 | |||
3 | 2.68 | 42 | 15 | 18.25 | 17.56 | 17.11 |
16 | 17.50 | 17.56 | 17.11 | |||
17 | 16.50 | 17.56 | 17.11 | |||
19 | 15.75 | 17.56 | 17.11 | |||
4 | 2.68 | 39 | 21 | 15.89 | 20.31 | 16.25 |
5 | 2.67 | 4 | 18 | 16.18 | 9.16 | 19.52 |
6 | 2.71 | 13 | 17 | 16.87 | 11.36 | 18.97 |
7 | 2.69 | 5 | 12 | 18.63 | 8.41 | 20.25 |
8 | 2.72 | 10 | 15 | 17.65 | 9.67 | 19.82 |
(d)
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Plot the graph between the calculated wopt values versus experimental wopt values, and the calculated γd(max) values versus experimental γd(max) values. Draw the 45° line of equality on the each plot.
Explanation of Solution
Refer Table 1, 2, and 3.
Draw the graph between the calculated wopt values versus experimental wopt values as in Figure 1.
Draw the graph between calculated γd(max) values versus experimental γd(max) values as in Figure 2.
(e)
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Comment on the predictive capabilities of various methods and comment about the inherent nature of empirical models.
Explanation of Solution
Prediction of optimum moisture content:
Refer Figure (1), several data points are closely packed around the 45° line for both the method of Osman et al. (2008) and Gurtug and Sridharan (2004) models signifying reasonable agreement between the calculated and experimental values. For the remaining soils, a poor agreement is observed. The one point from the Matteo method in the model plot showing the close agreement near the equality line and the other two points are slightly away from the equality line.
Prediction of maximum dry unit weight:
Most data points for all models show good covenant between the calculated and the experimental values.
The empirical models are often limited to the materials, test methods and environmental conditions under which the experiments were conducted and the developed models. For new materials and conditions, the predicted values may not be reliable.
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Chapter 6 Solutions
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