CAPITAL HUMANO
A) El capital humano innato
5. Elaboración de un indicador global de capital humano
Interactive Concept Learning and Self-Assessment
Access Chapter 3, Section 3.6 on CD to learn about the liquid, plastic, and shrinkage limits and to view digital movies on index tests. Take Quiz 3.6 to test your understanding.
3.6.1 Casagrande Cup Method
The liquid limit is determined from an apparatus (Fig. 3.7) that consists of a semispherical brass cup that is repeatedly dropped onto a hard rubber base from a height of 10 mm by a cam-operated mechanism. The apparatus was developed by A. Casagrande (1932) and the procedure for the test is called the Casagrande cup method.
A dry powder of the soil is mixed with distilled water into a paste and placed in the cup to a thickness of about 12.5 mm. The soil surface is smoothed and a groove is cut into the soil using a standard grooving tool. The crank operating the cam is turned at a rate of 2 revolutions per second and the number of blows required to close the groove over a length of 12.5 mm is counted and recorded. A specimen of soil within the closed portion is extracted for determination of the water content. The liquid limit is defined as the water content at which the groove cut into the soil will close over a distance of 12.5 mm following 25 blows. This is difficult to achieve in a single test. Four
Soil
Cam
Hard rubber base
11 mm Groove
2 mm
FIGURE 3.7 Cup apparatus for the determination of liquid limit. (Photo courtesy of Geotest.)
or more tests at different water contents are usually required for terminal blows (number of blows to close the groove over a distance of 12.5 mm) ranging from 10 to 40. The results are presented in a plot of water content (ordinate, arithmetic scale) versus terminal blows (abscissa, logarithm scale) as shown in Fig. 3.8.
The best-fit straight line to the data points, usually called the flow line, is drawn. We will call this line the liquid state line to distinguish it from flow lines used in describing the flow of water through soils. The liquid limit is read from the graph as the water content on the liquid state line corresponding to 25 blows.
The cup method of determining the liquid limit has many shortcomings. Two of these are:
1. The tendency of soils of low plasticity to slide and to liquefy with shock in the cup rather than to flow plastically.
2. Sensitivity to operator and to small differences in apparatus.
3.6.2 Plastic Limit Test
The plastic limit is determined by rolling a small clay sample into threads and finding the water content at which threads approximately 3 mm in diameter will just start to crumble. Two or more determina-tions are made and the average water content is reported as the plastic limit.
3.6.3 Fall Cone Method to Determine Liquid and Plastic Limits
A fall cone test, popular in Europe and Asia, appears to offer a more accurate (less prone to operator’s errors) method of determining both the liquid and plastic limits. In the fall cone test (Fig. 3.9), a cone with an apex angle of 30 and total mass of 80 grams is suspended above, but just in contact with, the soil sample. The cone is permitted to fall freely for a period of 5 seconds. The water content corresponding to a cone penetration of 20 mm defines the liquid limit.
The sample preparation is similar to the cup method except that the sample container in the fall cone test has a different shape and size (Fig. 3.9).
Four or more tests at different water contents are also required because of the difficulty of achieving the liquid limit from a single test. The results are plotted as water content (ordinate, logarithmic scale) versus penetration (abscissa, logarithm scale) and the best-fit straight line (liquid state line) linking the data points is drawn (Fig. 3.10). The liquid limit is read from the plot as the water content on the liquid state line corresponding to a penetration of 20 mm.
The plastic limit is found as follows.
Best-fit straight line called the liquid state line
10 20 25 30 40 50 60708090100
Number of blows – logarithmic scale 35
40 45 50 55 60
Water content (%)
LL = 46.2%
FIGURE 3. 8 Typical liquid limit results from the Casagrande cup method.
3.6 DETERMINATION OF THE LIQUID, PLASTIC, AND SHRINKAGE LIMITS 51
Project the best fit-straight line backward to intersect the water content axis at a depth of penetration of 1 mm. The water content at this depth of penetration (1 mm) is c. The plastic limit is given as (Feng, 2000)
PL¼ cð2Þm (3.24)
where m is the slope (taken as positive) of the best-fit straight line. If you use a spreadsheet program, you can obtain c and m from a power trend line function that gives the best fit equation.
3.6.4 Shrinkage Limit
The shrinkage limit is determined as follows. A mass of wet soil, m1, is placed in a porcelain dish 44.5 mm in diameter and 12.5 mm high and then oven-dried. The volume of oven-dried soil is determined by using mercury to occupy the vacant spaces caused by shrinkage. The mass of the mercury is determined and the volume decrease caused by shrinkage can be calculated from the known
30° Cone
40 mm 55 mm
35 mm
FIGURE 3. 9 Fall cone apparatus.
100
50 40 30
20
101
Penetration (mm) – log scale
10 20 30 4050 100
Water content (%) – log scale
80-gram cone Best-fit straight line
LL = 48%
c
Slope = m
FIGURE 3.10 Typical fall cone test results.
density of mercury. The shrinkage limit is calculated from
SL¼ m1 m2
m2
V1 V2
m2
gw g
100 (3.25)
where m1is the mass of the wet soil, m2is the mass of the oven-dried soil, V1is the volume of wet soil, V2is the volume of the oven-dried soil, and g is the acceleration due to gravity (9.8 m/s2). The range of water content from the plastic to the shrinkage limits is called the shrinkage index (SI),
SI¼ PL SL ð3:26Þ
The essential points are:
1. Fine-grained soils can exist in one of four states: solid, semisolid, plastic, and liquid.
2. Water is the agent that is responsible for changing the states of soils.
3. A soil gets weaker if its water content increases.
4. Three limits are defined based on the water content that causes a change of state. These are the liquid limit—the water content that caused the soil to change from a liquid to a plastic state;
the plastic limit—the water content that caused the soil to change from a plastic to a semisolid; and the shrinkage limit—the water content that caused the soil to change from a semisolid to a solid state. All these limiting water contents are found from laboratory tests.
5. The plasticity index defines the range of water content for which the soil behaves like a plastic material.
6. The liquidity index gives a measure of strength.
7. The soil strength is lowest at the liquid state and highest at the solid state.
EXAMPLE 3.6 Interpreting Casagrande’s Cup Data
A liquid limit test conducted on a soil sample in the cup device gave the following results:
Number of blows 10 19 23 27 40
Water content (%) 60.0 45.2 39.8 36.5 25.2
Two determinations for the plastic limit gave water contents of 20.3% and 20.8%. Determine (a) the liquid limit and plastic limit, (b) the plasticity index, (c) the liquidity index if the natural water content is 27.4%, and (d) the void ratio at the liquid limit, if Gs¼ 2:7. If the soil were to be loaded to failure, would you expect a brittle failure?
Strategy To get the liquid limit, you must make a semi-logarithm plot of water content versus number of blows. Use the data to make your plot, then extract the liquid limit (water content on the liquid state line corresponding to 25 blows). Two determinations of the plastic limit were made and the differences in the results are small. So, use the average value of water content as the plastic limit.
Solution 3.6
Step 1: Plot the data.
See Fig. E3.6.
3.6 DETERMINATION OF THE LIQUID, PLASTIC, AND SHRINKAGE LIMITS 53
LL = 38%
25
10 20 25 30 40 50 60708090100
30 35 40 45 50 55 60
Water content (%)
Number of blows – logarithmic scale Best-fit straight line
FIGURE E3.6 Plot of the liquid state line for the liquid limit by the Casagrande cup method.
Step 2: Extract the liquid limit.
The water content on the liquid state line corresponding to a terminal blow of 25 gives the liquid limit.
LL¼ 38%
Step 3: Calculate plastic limit.
The plastic limit is
PL¼20:3 þ 20:8
2 ¼ 20:6%
Step 4: Calculate PI
PI¼ LL PL ¼ 38 20:6 ¼ 17:4%
Step 5: Calculate LI
LI¼ðw PLÞ
PI ¼27:4 20:6 17:4 ¼ 0:39 Step 6: Calculate the void ratio.
Assume the soil is saturated at the liquid limit. For a saturated soil, e¼ wGs. Thus, eLL¼ LLGs¼ 0:38 2:7 ¼ 1:03
Brittle failure is not expected as the soil is in a plastic stateð0 < LI < 1Þ. &
EXAMPLE 3.7 Interpreting Fall Cone Data
The results of a fall cone test are shown in the table below.
Parameter 80 gram cone
Penetration (mm) 5.5 7.8 14.8 22 32
Water content (%) 39.0 44.8 52.5 60.3 67
Determine (a) the liquid limit, (b) the plastic limit, (c) the plasticity index, and (d) the liquidity index if the natural water content is 46%.
Strategy Adopt the same strategy as in Example 3.6. Make a plot of water content versus penetration, both at logarithm scale. Use the data to make your plot, then extract the liquid limit (water content on the liquid state line corresponding to 20 mm). Find the water content difference between the two liquid state lines at any fixed penetration. Use this value to determine the plastic limit or use Eq. (3.24).
Solution 3.7
Step 1: Plot the data.
See Fig. E3.7.
Y = 23.6 X0.3 LL = 60%
23.6
80 gram cone
101 100
10
Penetration (mm) – log scale
100
Water content (%) – log scale
FIGURE E3.7 Plot of fall cone results.
Step 2: Extract the liquid limit.
LL¼ 60%
Step 3: Determine the plastic limit.
The best-fit straight line for the 80-gram cone is Y¼ 23:6X0:3where Y is water content and X is penetration. Therefore, C¼ 23:6 and m ¼ 0:3. From Eq. (3.25):
PL¼ Cð2Þm¼ 23:6ð2Þ0:3
¼ 29%
Step 4: Determine PI.
PI¼ LL PL ¼ 60 29 ¼ 31%
Step 5: Determine LI.
LI¼w PL
PI ¼46 29
31 ¼ 0:55 &
What’s next. . .We now know how to obtain some basic soil dataparticle size and indicesfrom quick, simple tests. The question that arises is: What do we do with these data? Engineers would like to use the data to get a first impression on the use and possible performance of the soil for a particular purpose such as a construction material for an embankment. This is currently achieved by classification systems. Next, we will study two of these systems.