Soluciones diplomáticas

The consolidation characteristics of a soil can be measured in the laboratory using the one-dimensional consolidation test, shown schematically in Figure 4.3.

(a) (b)

∆σ

Sand

Sand Clay H= 2H^dr

Rock (Impervious)

∆σ

Sand Clay H= H^dr

FIGURE 4.2 (a) Two- and (b) one-way drainage conditions.

Clay Porous Stone

Porous Stone

Constant Load

Rigid Ring Loading Plate

FIGURE 4.3 One-dimensional consolidation test apparatus.

Initial Settlement

log Time Primary

Consolidation Settlements

Secondary Compression

Deformation

FIGURE 4.4 Deformation versus time curve (semilog).

A cylindrical specimen of soil measuring 75 mm in diameter and approximately 15 mm in thickness is enclosed in a metal ring and subjected to staged static loads.

Each load stage lasts 24 hours, during which changes in thickness are recorded.

The load is doubled with each stage up to the required maximum (e.g., 100, 200, 400, 800 kPa). At the end of the final loading stage, the loads are removed and the specimen is allowed to swell. Figure 4.4 shows an example of the settlement versus time curve obtained from one loading stage. Three types of deformations are noted in the figure: initial compression, primary consolidation settlement, and secondary compression or creep. Initial compression is caused by the soil’s elastic response to applied loads. Primary consolidation settlement is caused by dissipation of the excess pore pressure generated by load application. Secondary compression is caused by the time-dependent deformation behavior of soil particles, which is not related to excess pore pressure dissipation. Primary consolidation settlement is our focus in this chapter.

Enclosing the soil specimen in a circular metal ring is done to suppress lateral strains. The specimen is sandwiched between two porous stones and kept sub-merged during all loading stages; thus, the specimen is allowed to drain from top and bottom. This is a two-way drainage condition in which the thickness of the specimen is 2H_dr. The void ratio versus logarithm vertical effective-stress relation-ship (e–logσ^v) is obtained from the changes in thickness at the end of each load stage of a one-dimensional consolidation test. An example of ane–logσ_v^ curve is shown in Figure 4.5.

Now define a preconsolidation pressure σ^_c as the maximum past pressure to which a clay layer has been subjected throughout time. A normally consolidated (NC) clay is defined as a clay that has a present (in situ) vertical effective stressσ^₀ equal to its preconsolidation pressureσ^c. An overconsolidated (OC) clay is defined as a clay that has a present vertical effective stress of less than its preconsolida-tion pressure. Finally, define an overconsolidapreconsolida-tion ratio (OCR) as the ratio of the preconsolidation pressure to the present vertical effective stress (OCR= σ^c/σ^₀).

To understand the physical meaning of OC clay, preconsolidation pressure, and OCR, imagine a 20-m-thick clay layer that was consolidated with a constant pres-sure of 1000 kPa caused by a glacier during the ice age. The glacier melted away and the pressure that had been exerted was gone totally. Thus, the preconsolida-tion pressure is 1000 kPa, the maximum past pressure exerted. Assuming that the groundwater table level is now at the top surface of the clay layer and that the sat-urated unit weight of the clay is 19.81 kN/m³, the present vertical effective stress in the middle of the clay layer is σ^₀ = (γsat− γw)H /2= (19.81−9.81)20/2 = 100 kPa. The present vertical effective stress, 100 kPa, is less than the precon-solidation pressure of 1000 kPa. Therefore, the clay is overconsolidated. The overconsolidation ratio of this clay isσ^_c/σ^₀= 1000 kPa/100 kPa = 10.

The preconsolidation pressure is a soil parameter that can be obtained from its e–log σ^_v curve deduced from the results of a one-dimensional consolidation test.

e₀

σ′0 σ′c log σ′v σ′0 σ′c log σ′v

C_c

C_s 1 1

(a) (b)

Loading

Unloading

e e

FIGURE 4.5 Void ratio versus vertical effective stress (semilog): (a) consolidation test results; (b) idealization.

The preconsolidation pressure is located near the point where thee–log σ^_v curve changes in slope, as shown in Figure 4.5a (Casagrande, 1936). Other consolidation parameters, such as the compression index(C_c) and swelling index (C_s), are also obtained from the e–log σ^v curve. The compression index is the slope of the loading portion, in the e-logσ^v curve, and the swelling index is the slope of the unloading portion, as indicated in Figure 4.5b.

The coefficient of consolidation,cv, is an essential parameter for consolidation rate calculations [Terzaghi’s one-dimensional consolidation equation (4.1)]. The coefficient of consolidation can be obtained from the changes in thickness recorded against time during one load stage of a one-dimensional consolidation test. This is done using the root-of-time method or the log-time method. The square-root-of-time method (Taylor, 1948) uses a plot of deformation versus the square root of time. Figure 4.6a shows a typical plot of deformation versus the square-root-of-time at a given applied load. The square-root-of-square-root-of-time method involves drawing a line AB through the early straight-line segment of the curve. Another line, AC, is drawn in such a way that OC= 1.15OB. The x-coordinate of point D, which is the intersection of AC with the curve, gives√

t90(the square root of time corresponding toU= 90%). One can use (4.6) to show that for a 90% degree of consolidation, the dimensionless time factorTvis 0.848. Then (4.4) yieldscv= 0.848H_dr²/t90. The value oft₉₀ can be obtained from Figure 4.6a as explained above. H_dr is equal to the thickness of the soil specimen in the one-dimensional consolidation test if it is allowed to drain from one side only (one-way drainage). If the specimen is allowed to drain from both sides (top and bottom),H_dr is equal to half the thickness of the soil specimen.

The log-time method (Casagrande and Fadum, 1940) uses a plot of deformation versus the logarithm of time. Figure 4.6b shows a typical plot of deformation versus the logarithm of time at a given applied load. The log-time method involves extending the straight-line segments of the primary consolidation and the secondary compression. The extensions will intercept at point A. They-coordinate of point

t₉₀

FIGURE 4.6 Graphical procedures for determining the coefficient of consolidation:

(a) square-root-of-time method; (b) log-time method.

A isd₁₀₀, which is the deformation corresponding to 100% consolidation. On the initial portion of the deformation versus log-time plot, select two points, B and C, at timest1 andt2, respectively, such thatt2= 4t1. The vertical difference between points B and C is equal to δ. Draw a horizontal line DE at a vertical distance BD equal to δ. DE intercepts with the y-axis at d0, which is the deformation corresponding to 0% consolidation. Now calculated50 as the average of d0 and d100. Draw a horizontal line atd50that will intercept with the curve at point F. The x-coordinate of point F is t50, which is the time corresponding to 50% consolidation.

Equation (4.6) can be used to calculate the dimensionless time factorTv(= 0.197), which corresponds toU= 50%. Then (4.4) can be used to calculate the coefficient of consolidation:cv = 0.197H_dr²/t50.

4.3 CALCULATION OF THE ULTIMATE CONSOLIDATION