Module 1 (Lecture 3) GEOTECHNICAL PROPERTIES OF SOIL AND OF REINFORCED SOIL

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1 Module 1 (Lecture 3) GEOTECHNICAL PROPERTIES OF SOIL AND OF REINFORCED SOIL Topics 1.1 CAPILLARY RISE IN SOIL 1.2 CONSOLIDATIONS-GENERAL 1.3 CONSOLIDATION SETTLEMENT CALCULATION 1.4 TIME RATE OF CONSOLIDATION CAPILLARY RISE IN SOIL When a capillary tube is placed in water, the water level in the tube rises (figure 1.15a). The rise is caused by the surface tension effect. According to figure 1.15a, the pressure at any point A in the capillary tube (with respect to the atmospheric pressure) can be expressed as uu = γγ ww zz (for zz = 0 to h cc ) And uu = 0 (for z h c )

2 Figure 1.15 Capillary rise In a given soil mass, the interconnected void spaces can behave like a number of capillary tubes with varying diameters. The surface tension force may cause in the soil to rise above the water table, as shown in figure 1.15b. The height the capillary rise will depend on the diameter of the capillary tubes. The capillary rise will decrease with the increase of the tube diameter. Because the capillary tube in soil has variable diameters, the height of capillary rise will be nonuniformly. The pore water pressure at any point in the zone of capillary rise in soil cause approximated as uu = SSγγ ww zz [1.52] Where SS = degree fo saturation of soil [equation (7)] zz = distance measured above the water table CONSOLIDATION-GENERAL In the field, when the stress on a saturated clay layer is increased-for exam by the construction of a foundation-the pore water pressure in the clay increase. Because the hydraulic conductivity of clays is very small, sometime be required for the excess pore water pressure to dissipate and the stress increase to be transferred to the soil skeleton gradually. According to figure 1.16 if a surcharge at the ground surface over a very large area, the increase of total structure σσ, at any depth of the clay layer will be equal to pp, or σσ = pp

3 Figure 1.16 Principles of consolidation However, at time tt = 0 (that is, immediately after the stress application), the excess pore water pressure at any depth, uu, will equal pp, or uu = h 1 γγ ww = Δpp (at time tt = 0) Hence the increase of effective stress at time tt = 0 will be Δσσ = Δσσ Δuu = 0 Theoretically, at time tt =, when all the excess pore water pressure in the clay layer has dissipated as a result of drainage into the sand layers, Δuu = 0 at time tt = ) Then the increase of effective stress in the clay layer is Δσσ = Δσσ Δuu = Δpp 0Δpp This gradual increase in the effective stress in the claylayer will cause settlement over a period of time and is referred to as consolidation. Laboratory tests on undisturbed saturated clay specimens can be conducted (ASTM Test Designation D-2435) to determine the consolidation settlement caused by various incremental loadings. The test specimens are usually 2.5 in. (63.5 mm) in diameter and 1 in. (25.4 mm) in height. Specimens are placed inside a ring, with one porous stone at the top and one at the bottom of the specimen (figure 1.17a). Load on the specimen is then applied so that the total vertical stress is equal to pp. Settlement readings for the specimen are taken for 24 hours. After that, the load on the specimen is doubled and settlement readings are taken. At all times during the test the specimen is kept under water. This procedure is continued until the desired limit of stress on the clay specimen is reached.

4 Figure 1.17 (a) Schematic diagram of consolidation test arrangement; (b) ee log pp curve for a soft clay from East St. Louis, Illinois Figure 1.17 continued Based on the laboratory tests, a graph can be plotted showing the variation of the void ratio ee at the end of consolidation against the corresponding vertical stress pp

5 (semilogarithmic graph: ee on the arithmetic scale and pp on the log scale). The nature of variation of ee against log pp for a clay specimen is shown in figure 1.17b. After the desired consolidation pressure has been reached, the specimen can be gradually unloaded, which will result in the swelling of the specimen. Figure 1.17b also shows the variation of the void ratio during the unloading period. From the ee log pp curve shown in figure 1.17b, three parameters necessary for calculating settlement in the field can be determined. 1. The preconsolidation pressure, pp cc, is the maximum past effective overburden pressure to which the soil specimen has been subjected. It can be determined by using a simple graphical procedure as proposed by Casegrande (1936). This procedure for determining the preconsolidation pressure, with reference to figure 1.17b, involves five steps: a. Determine the point O on the ee log pp curve that has the sharpest curvature (that is, the smallest radius of curvature). b. Draw a horizontal line OA. c. Draw a line OB that is tangent to the ee log pp curve at O. d. Draw a line OC that bisects the angle AOB, e. Produce the straight-line portion of the ee log pp curve backward to intersect OC. This is point D. the pressure that corresponds to point pp is the preconsolidation pressure, pp cc. Natural soil deposits can be normally consolidated or overconsolidated (or preconsolidated). If the present effective overburden pressure pp = pp aa is equal to the preconsolidated pressure pp cc the soil is normally consolidated. However, if pp oo < pp cc, the sol is overconsolidated. Preconsolidation pressure (pp cc ) has been correlated with the index parameters by several investigators. Stas and Kulhawy (1984) suggested that pp cc σσ aa = 10 ( LLLL) [1.53a] Where σσ aa = atmospheric stress in derived unit LLLL = liquidity index The liquidity index of a soil is defined as LLLL = ww PPPP LLLL_PPPP [1.53b] Where ww = iiii ssssssss moisture content

6 LLLL = liquid limit PPPP = plastic limit Nagaraj and Murthy (1985) provided an empirical relation to calculate pp cc, which is as follows: kn/m 2 log pp cc = eeoo ee LL log pp oo kn/m 2 [1.54] Where ee oo = iiii ssssssss void ratio pp oo = iiiiiiiiiiii effective overburden pressure ee LL = void ratio of the soil at liquid limit ee LL = LLLL(%) 100 GG ss [1.55] The U. S. Department of the Navy (1982) also provided generalized relationships between pp cc, LLLL and the sensitivity of clayey soils (SS tt ). This relationship was also recommended by Kulhawy and Mayne (1990). The definition of sensitivity is given in section. Figure 1.18 shows the relationship. Figure 1.18 Variation of pp cc with LI (after U. S. Department of the Navy, 1982)

7 2. The compression index, CC cc, is the slope of straight-line portion (latter part of the loading curve), or CC cc = ee 1 ee 2 log pp 2 log pp 1 = ee 1 ee 2 log pp 2 pp 1 [1.56] where ee 1 and ee 2 are the void ratios at the end of consolidation under stresses pp 1 and pp 2, respectively The compression index, as determined from the laboratory ee log pp curve, will be somewhat different from that encountered in the field. The primary reason is that the soil remolds to some degree during the field exploration. The nature of variation of the ee log pp curve in the field for normally consolidated clay is shown in figure It is generally referred to as the virgin compression curve. The virgin curve approximately intersects the laboratory curve at a void ratio of 0.42ee oo (Terzaghi and Peck, 1967). Note that ee oo is the void ratio of the clay in the field. Knowing the values of ee oo and pp cc you can easily construct the virgin curve and calculate the compression index of the virgin curve by using equation (56). Figure 1.19 Construction of virgin compression curve for normally consolidated clay

8 The value of CC cc can vary widely depending on the soil. Skempton (1944) has given am empirical correlation for the compression index in which CC cc = 0.009(LLLL 10) [1.57] Where LLLL = liquid limit Besides Skempton, other investigators have proposed correlations for the compression index. Some of these correlations are summarized in table The swelling index, CC ss, is the slope of the unloading portion of the ee log pp curve. In figure 1.17b, it can be defined as CC ss = ee 3 ee 4 log pp 4 pp 3 [1.58] In most cases the value of the swelling index (CC ss ) is 1 to 1 of the compression index. 4 5 Flowing are some representative values of CC ss /CC cc for natural soil deposits. The swelling index is also referred to as the recompression index. Description of soil CC ss /CC cc Boston Blue clay Chicago clay New Orleans clay St. Lawrence clay Table 14 Correlations for Compression Index Reference Correlation Azzouz, Krizek, and Corotis (1976) CC cc = 0.01 ww nn (Chicago clay) CC cc = ee oo (Chicago clay) CC cc = ww nn (organic soils, peat) CC cc = (LLLL 9) (Brazillian clay)

9 Rendon-Herrero (1980) CC cc = 0.141GG 12 ss 1 + ee 2.38 oo GG ss Nagaraj and Murthy (1985) Wroth and Wood (1978) CC cc = LLLL 100 GG ss CC cc = 0.5GG ss PPPP 100 Leroueil, Tavenas, and LeBihan (1983) Note: GG ss = specific gravity of soil solids LLLL = liquid limit PPPP = plasticity index SS tt = sensitivity ww nn = natural moisture content The swelling index determination is important in the estimation of consolidation settlement of overconsolidated clays. In the field, depending on the pressure increase, an overconsolidated clay will follow an e-log pp path aaaaaa, as shown in figure Note that point aa with coordinates of pp oo and ee oo corresponds to the field condition before any pressure increase. Point bb corresponds to the preconsolidation pressure (pp cc ) of the clay. Line aaaa is approximately parallel to the laboratory unloading cure cccc (Schmertmann, 1953). Hence, if you know ee oo, pp oo, pp cc, CC cc, and CC ss, you can easily construct the field consolidation curve.

10 Figure 1.20 Construction of field consolidation curve for over consolidated clay Nagaraj and Murthy (1985) expressed the swelling index as CC ss = LLLL 100 GG ss [1.59] It is essential to point out that any of the empirical correlations for CC cc and CC ss given in the section are only approximate. It may be valid for a given soil for which the relationship was developed but may not hold good for other soils. As an example, figure 1.21 shows the plots of CC cc and CC ss with liquid limit for soils from Richmond, Virginia (Martin et al., 1985). For these soils,

11 Figure 1.21 Variation of CC cc and CC ss with liquid limit for soils from Richmond, Virginia (after Martin et al., 1995) CC cc = (LLLL 43.4) [1.60] And CC ss = (LLLL ) [1.61] The CC ss / CC cc ratio is about 1 25 ; whereas, the typical range is about1 5 to CONSOLIDATION SETTLEMENT CALCULATION The one-dimensional consolidation settlement (caused by an additional load) of a clay layer (figure 1.22a) having a thickness HH cc may be calculated as

12 Δee SS = HH 1+ee cc [1.62] oo Figure 1.22 One-dimensional settlement calculation: (b) is for equation (64); (c) is for equations (66 and 68) Where SS = settlement Δee = total change of void ratio caused by the additional load application ee oo = the void ratio of the clay before the application of load Note that Δee 1+ee oo = εε vv = vertical strain For normally consolidated clay, the field ee log pp curve will be like the one shown in figure 1.22b. If pp oo = initial average effective overburden pressure on the clay layer and Δpp = average pressure increase on the clay layer caused by the added load, the change of void ratio caused by the load increase is

13 Δee = CC cc log pp oo +Δpp pp oo [1.63] Now, combining equations (62 and 63) yields SS = CC cchh cc 1+ee oo log pp oo +Δpp pp oo [1.64] For overconsolidated clay, the field ee log pp curve will be like the one show figure 1.22c. In this case, depending on the value of Δpp, two conditions may at. First, if pp oo + Δpp < pp cc, Δee = CC ss log pp oo +Δpp pp oo [1.65] Combining equations (62 and 65) gives SS = HH cccc ss 1+ee oo log pp oo +Δpp pp oo [1.66] Second, if pp oo < pp cc < pp oo + pp, ee = ee 1 + ee 2 = CC zz log pp cc pp oo + CC cc log pp oo +Δpp pp oo [1.67] Now, combining equations (62 and 67) yields SS = CC sshh cc 1+ee oo log pp cc pp oo + CC cchh cc 1+ee oo log pp oo +Δpp pp cc [1.68] TIME RATE OF CONSOLIDATION In section we showed that consolidation is the result gradual dissipation of the excess pore water pressure from a clay layer. Pore water pressure dissipation, in turn, increases the effective stress, which induces settlement. Hence, to estimate the degree of consolidation of a clay layer at some time t after the load application, you need to know the rate of dissipation of the excess pore water pressure. Figure 1.23 shows a clay layer of thickness HH cc that has highly permeable sand layers at its top and bottom. Here, the excess pore pressure at any point at any time t after the load application is uu = ( h)γγ ww. For a vertical drainage condition (that is, in the direction of z only) from the clay layer, Terzaghi derived the following differential equation:

14 Figure 1.23 (a) Derivation of equation (71); (b) nature of variation of uu with time ( uu) = CC vv 2 ( uu) 2 [1.69] Where CC vv = coefficient of consolidation CC vv = kk kk = ee mm vv γγ ww pp (1+eeaaaa ) γγ ww [1.70] Where kk = hydraulic conductivity of the clay ee = total change of void ratio caused by a stress increase of p ee aaaa = average void ratio during consolidation mm vv = volume coefficient of compressibility = ee/[ pp(1 + ee aaaa )] Equation (69) can be solved to obtain uu as a function of time t with the following boundary conditions:

15 1. Because highly permeable sand layers are located at zz = 0 and zz = HH cc, the excess pore water pressure developed in the clay at those points will be immediately dissipated. Hence uu = 0 at zz = 0 uu = 0 at zz = HH cc = 2HH Where HH = Length of maximum drainage path (due to two-way drainage condition-that is, at the top and bottom of the clay) 2. At time tt = 0, uu = uu oo = initial excess pore water pressure after the load application With the preceding boundary conditions, equation (69) yields uu = mm= 2( uu oo ) ssssss MMMM MM HH ee MM2 TT vv 0 [1.71] Where MM = [(2mm + 1)ππ]/2 mm = an integer = 1, 2, TT vv = nondimensional time factor = (CC vv tt)/hh 2 [1.72] Determining the field value of CC vv is difficult. Figure 1.24 provides a first-order determination of CC vv using the liquid limit (u. A. Department of the Navy, 1971). The value of uu for various depths (that is, zz = 0 to zz = 2HH) at time given time t (thus TT vv ) can be calculated from equation (71). The nature of this variation of uu is shown in figure 1.23b.

16 Figure 1.24 Range of CC vv (after U. S. Department of the Navy, 1971) The average degree of consolidation of the clay layer can be defined as UU = Where SS tt SS mmmmmm [1.73] UU = average degree of consolidation SS tt = settlement of a clay layer at time tt after the load application SS mmmmmm = maximum consolidation settlement that the clay will undergo under given loading If the initial pore water pressure ( uu oo ) distribution is constant with depth as shown in figure 1.25a, the average of consolidation can also be expressed as

17 Figure 1.25 Drainage condition for consolidation: (a) two-way drainage; (b) oneway drainage UU = SS tt = 2HH ( uu oo )dddd 2HH 0 0 ( uu)dddd 2HH SS mmmmmm ( uu oo )dddd 0 [1.74] Or UU = ( uu oo )2HH 2HH 0 ( uu)dddd ( uu oo )2HH = 1 2HH 0 ( uu)dddd 2HH( uu oo ) [1.75] Now, combining equations (71 and 75) we obtain SS tt mm= UU = = 1 2 SS mmmmmm MM 2 ee MM2 TT vv mm=0 [1.76] The variation of UU with TT vv can be calculated from equation (76) and is plotted in figure Note that equation (76) and thus figure 1.26 are also valid when an impermeable layer is located at the bottom of the clay layer (figure 1.25b). In that case, excess pore water pressure dissipation can take place in one direction only.

18 Figure 1.26 Plot of time factor against average degree of consolidation ( uu oo = constant) The length of the maximum drainage path then is equal to HH = HH cc. The variation of TT vv with UU shown in figure 1.26 can also be approximated by TT vv = ππ 4 UU% (for UU = 0 60%) [1.77] And TT vv = log(100 UU%) (for UU > 60%) [1.78] Sivaram and Swamee (1977) have also developed an empirical relationship between TT vv and UU that is valid for U varying from 0 to 100%. It is of the form TT vv = ππ 4 UU % UU % [1.79] In some cases, initial excess pore water pressure may not be constant with depth as shown in figure Following are a few cases of those and the solutions for the average degree of consolidation.

19 Trapezoidal Variation Figure 1.27 shows a trapezoidal variation of initial excess pore water pressure with two-way drainage. For this case the variation of TT vv with UU will be the same as shown in figure Figure 1.27 Trapezoidal initial excess pore water pressure distribution Sinusoidal Variation This variation is shown in figures 1.28a and 1.28b. For the initial excess pore water pressure variation shown in figure 1.28a, z Figure 1.28 Sinusoidal initial excess pore water pressure distribution

20 uu = uu oo ssssss ππππ 2HH [1.80] Similarly, for the case shown in figure 1.28b, uu = uu oo cccccc ππππ 4HH [1.81] The variations of TT vv with UU for these two cases are shown in figure 1.29 Figure 1.29 Variation of UU with TT vv sinusoidal variation of initial excess pore water pressure distribution Triangular Variation Figures 1.30 and 1.31 show several types of initial pore water pressure variation and the variation of TT vv with the average degree of consolidation.

21 z Figure 1.30 Variation of UU with TT vv triangular initial excess pore water pressure distribution Figure 1.31 triangular initial excess pore water pressure distribution-variation of UU with TT vv

22 Example 9 A laboratory consolidation test on normally consolidated clay showed the following Load, pp(kn/m 2 ) Void ratio at the end of consolidation, e The specimen tested was 25.4 mm in thickness and drained on both sides. The time required for the specimen to reach 50% consolidation was 4.5 min. A similar clay layer in the field, 2.8 m thick and drained on both sides, is subjected to similar average pressure increase (that is, pp oo = 140 kn/m 2 and p o + p = 212kN/m 2 ). Determine the a. Expected maximum consolidation settlement in the field b. Length of time required for the total settlement in the field to reach 40 mm (assume uniform initial excess pore water pressure increase with depth) Solution Part a For normally consolidated clay [equation 56] CC cc = ee 1 ee 2 = llllll pp 2 pp llllll From equation (64) SS = CC cchh cc 1+ee oo llllll pp oo + pp Part b = = (0.333)(2.8) pp oo llllll 212 = m = 87.5 mm 140 From equation (73) the average degree of consolidation is UU = SS tt = 40 (100) = 45.7% SS mmmmmm 87.5 The coefficient of consolidation, CC vv, can be calculated from the laboratory test. From equation (72) TT vv = CC vvtt HH 2

23 For 50% consolidation (figure 1.26), TT vv = 0.197, tt = 4.5 min, and HH = HH cc /2 = 12.7 mm, so CC vv = TT 50 HH 2 tt = (0.197)(12.7)2 4.5 = mm 2 /min Again, for field consolidation, UU = 45.7%. From equation (77) TT vv = ππ 4 UU% = ππ = But TT vv = CC vvtt HH 2 Or tt = TT vvhh 2 CC vv = = 45,523 min = 31.6 days