Theoretical analysis

Terzaghi’s model is developed for small-strain deformation where a linear relationship is held for stress and strain, and the conductivity is constant.

As mentioned in verification with Morris’s analytical solution, Gibson’s equation can be rearranged as follows. Sill, 1985; Cargill, 1985; McVay et al., 1986).

For small-strain consolidation (Gibson et al., 1981):

t

It can be seen from the comparison between Equation 2.25 and 2.26, the small-strain model ignores the effect of self-consolidation that is the second term in Equation 2.25. When the effective stress is in linear relationship with void ratio (which is common in piecewise linear approximation), so that  becomes zero, or the unit weight of the sediment equals that of water, the large-strain consolidation theory will degenerate into small-strain theory ( in which gis not constant). Otherwise, a difference is always present, and may accumulate or amplify without correct handling when considering shallow compaction.

Xie and Leo, and Morris compared small-strain (Terzaghi) and large-strain (Gibson) compaction models for a column of sediments with variable volume compressibility. Their results are provided in Figure 2.20 and Figure 2.21 for completeness.

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Figure 2.20 The surface settlements calculated using large- and small-strain models vs. time (PTIB) (Xie and Leo, 2004). Note that _q m_vlq shown on the figure is positively related to the volume

compressibility of the sediments under consideration

Figure 2.21 Large and small strain surface settlement vs. time (PTIB) (Morris, 2002). Note that )

(r_s r_w l

N   , and l is thickness of soil layer in material coordinates. N is positively related to the volume compressibility of the sediment in this case.

According to the results of their analysis, the bigger the volumetric compressibility is, the bigger the difference between small-strain and large-strain result is, for both forced and self-consolidation. Hence, it is not acceptable to apply small-strain theory directly into shallow compaction - large strain.

If the unit weight of soil is equal to that of water, the whole system will not go through consolidation under gravity. This is taken as the reference case, with which all other cases with nonzero consolidation are compared. In order to study the proportion of

self-41

consolidation, a sensitivity comparison is carried out. Relative errors compared with zero consolidation (absolute value of the ratio between settlement of zero minus nonzero self-consolidation and settlement of zero self self-consolidation) are shown in Figure 2.23 - Figure 2.25.

Consolidation schematic diagram follows Figure 2.2, no overlying water and no surcharge, PTIB, self-consolidation to steady state. 90% clay content material is adopted for analysis (Ma and Couples, 2008), as shown in Figure 2.22. The unit weight of soil is chosen to cover a wide range of possible sediments. Other model parameters are shown in Table 2.8.

Figure 2.22 Properties and fitting function of 90% clay content, upper - conductivity-void ratio and fitting function of 90% clay content, lower - void ratio-effective stress and fitting function of 90%

clay content (Ma and Couples, 2008) y = 7E-06x²- 3E-06x + 2E-06

0.00E+00 5.00E-06 1.00E-05 1.50E-05 2.00E-05 2.50E-05 3.00E-05

0 0.5 1 1.5 2 2.5

conductivity (m/day)

void ratio

permeability-void ratio 多项式 (permeability-void …

y = -0.52ln(x) + 5.6247

0 0.5 1 1.5 2 2.5

0 10000 20000 30000 40000 50000 60000 70000

void ratio

effective stress(KPa) void ratio-effective stress 对数 (void ratio-effective stress)

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Parameter Value Units

unit weights of soil variation / _s [1.2,1.6,2.0,2.4,2.8,3.2]×9.8 kN/m³ initial effective stresses [200,600,1000,1400,1800,2200] kPa conductivity - void ratio/ke k710^6e²310^6e210^6 m/d(k ) void ratio - effective stress /e' e5.62470.52ln(') _kPa(_')

initial thickness variation [0.1,1,10,20,40,60,80] m

Table 2.8 Parameters utilized in self-consolidation evaluation

(1) Influence of initial thickness:

The influence of self-consolidation increases with the increase of initial thickness and unit weight of soil, and the increase rate gradually slows down (can be seen from curve’s slope change and variation of spaces between adjacent curves). The influence decreases with increase of effective stress, and the decrease rate gradually decreases (can be seen from variation of spaces between adjacent curves) as shown in Figure 2.23.

(2) Influence of unit weight of soil:

Influence of self-consolidation increases with the increase of unit weight of soil (soil density).

The influence increases with increase of initial thickness, and the increase rate gradually slows down. The influence decreases with increase of effective stress, and the decrease rate gradually slows down as shown in Figure 2.24.

(3) Influence of initial effective stress:

It can be seen from Figure 2.25, the influence of self-consolidation decreases with the increase of effective stress. The influence increases with increase of initial thickness/ unit weight of soil, and the increase rate gradually slows down.

Results analyses show that influence of self-consolidation increases with the increase of soil density and initial thickness, while decreases with the increase of effective stress, resulting in the maximum error of 10%.

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Figure 2.23 Influence of initial thickness on relative error, (relative error is absolute value of the ratio between settlement of zero minus nonzero self-consolidation and settlement of zero self consolidation.

‘rs’ is unit weight of soil, equals _s, h is initial sediment thickness) 1E-05

0.0001 0.001 0.01 0.1 1

0 10 20 30 40 50 60 70 80 90

relative error

initial thickness(m)

rs=1.2 effective stress=200KPa rs=1.6 effective stress=200KPa rs=2.0 effective stress=200KPa rs=2.4 effective stress=200KPa rs=2.8 effective stress=200KPa rs=3.2 effective stress=200KPa

1E-05 0.0001 0.001 0.01 0.1 1

0 10 20 30 40 50 60 70 80 90

relative error

initial thickness(m) rs=1.2 effective stress=200KPa rs=1.2 effective stress=600KPa rs=1.2 effective stress=1000KPa rs=1.2 effective stress=1400KPa rs=1.2 effective stress=1800KPa rs=1.2 effective stress=2200KPa

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Figure 2.24 Influence of unit weight of soil on relative error, (relative error is absolute value of the ratio between settlement of zero minus nonzero self-consolidation and settlement of zero self

consolidation, ‘h’ is initial sediment thickness) 1E-05

0.0001 0.001 0.01 0.1 1

1 1.5 2 2.5 3 3.5

relative error

soil density(g/cm³) h=0.1 effective stress=200KPa h=1 effective stress=200KPa h=10 effective stress=200KPa h=20 effective stress=200KPa h=40 effective stress=200KPa h=60 effective stress=200KPa

1E-05 0.0001 0.001 0.01 0.1 1

1 1.5 2 2.5 3 3.5

relative error

soil density(g/cm³) h=10 effective stress=200KPa

h=10 effective stress=600KPa h=10 effective stress=1000KPa h=10 effective stress=1400KPa h=10 effective stress=1800KPa h=10 effective stress=2200KPa

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Figure 2.25 Influence of initial effective stress on relative error, (relative error is absolute value of the ratio between settlement of zero minus nonzero self-consolidation and settlement of zero self

consolidation, ‘rs’ equals _s, ‘h’ is initial sediment thickness) 1E-05

0.0001 0.001 0.01 0.1 1

0 500 1000 1500 2000 2500

relative error

initial effective stress(KPa) rs=1.2 h=10 rs=1.6 h=10 rs=2 h=10 rs=2.4 h=10 rs=2.8 h=10 rs=3.2 h=10

0.000001 0.00001 0.0001 0.001 0.01 0.1 1

0 500 1000 1500 2000 2500

relative error

initial effective stress(KPa) rs=1.2 h=0.1 rs=1.2 h=1 rs=1.2 h=10 rs=1.2 h=20 rs=1.2 h=40 rs=1.2 h=60 rs=1.2 h=80

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