Given information:
The thickness of the clay layer (H) is 6 m.
The time (t) is 12 months.
The final consolidation settlement (Sp) is 220 mm.
The coefficient of consolidation of the clay (cv) is 3 m2/year.
Calculation:
Consider the unit weight of water (γw) is 9.81 kN/m3.
For the time (t) of 3 months:
Calculate the maximum drainage path (Hdr) for two way drainage using the relation.
Hdr=H2
Substitute 6 m for H.
Hdr=62=3 m
Calculate the time factor (Tv) using the relation.
Tv=(cvtHdr2) (1)
Substitute 3 m2/year for cv, 3 months for t, and 3 m for Hdr in Equation (1).
Tv=3 m2/year×3 months×1 year12 months(3 m)2=0.759=0.083
Refer Figure 9.33 “U−Tv relationships for ramp loading and instantaneous loading” in the Text Book.
Take the value of average degree of consolidation (U) as 21% for the value time factor (Tv) of 0.083.
Calculate the final consolidation settlement (Sp(f)) with 25% of the applied load using the relation.
Sp(f)=Percentage of load applied×Sp (2)
Substitute 25% for percentage of load applied and 220 mm for Sp in Equation (2).
Sp(f)=25100×220=55 mm
Calculate the consolidation settlement for 3 months using the relation.
Consolidation settlement=U×Sp(f) (3)
Substitute 21% for U and 55 mm for Sp(f) in Equation (3).
Consolidation settlement=21100×55=11.6 mm
For the time (t) of 6 months:
Calculate the time factor (Tv) as shown below.
Substitute 3.0 m2/year for cv, 6 months for t, and 3 m for Hdr in Equation (1).
Tv=3 m2/year×6 months×1 year12 months(3 m)2=1.59=0.167
Refer Figure 9.33 “U−Tv relationships for ramp loading and instantaneous loading” in the Text Book.
Take the value of average degree of consolidation (U) as 31% for the value time factor (Tv) of 0.167.
Calculate the final consolidation settlement (Sp(f)) with 50% of the applied load as shown below.
Substitute 50% for percentage of load applied and 220 mm for Sp in Equation (2).
Sp(f)=50100×220=110 mm
Calculate the consolidation settlement for 6 months using the relation.
Substitute 31% for U and 110 mm for Sp in Equation (3).
Consolidation settlement=31100×110=34.10 mm
For the time (t) of 12 months:
Calculate the time factor (Tv) as shown below.
Substitute 3.0 m2/year for cv, 12 months for t, and 3 m for Hdr in Equation (1).
Tv=3 m2/year×12 months×1 year12 months(3 m)2=39=0.333
Refer Figure 9.33 “U−Tv relationships for ramp loading and instantaneous loading” in the Text Book.
Take the value of average degree of consolidation (U) as 43% for the value time factor (Tv) of 0.333.
Calculate the final consolidation settlement (Sp(f)) with 50% of the applied load as shown below.
Substitute 100% for percentage of load applied and 220 mm for Sp in Equation (2).
Sp(f)=100100×220=220 mm
Calculate the consolidation settlement for 12 months using the relation.
Substitute 43% for U and 220 mm for Sp in Equation (3).
Consolidation settlement=43100×220=94.6 mm
For the time (t) of 18 months with t0 of 12 months.
Calculate the time factor Tv using the relation.
Tv=cv(t−t0)Hdr2 (4)
Substitute 3.0 m2/year for cv, 18 months for t,12 months for t0,and 3 m for Hdr in Equation (4).
Tv=3 m2/year×(12−6) months×1 year12 months(3 m)2=1.59=0.167
Refer Table 9.5“ U−Tv Values for Non-Uniform Initial Pore Pressure Distributions” in the Text Book.
Take the value of Ut−t0 as 0.3377 for the value Tv of 0.167 for half-sine.
Calculate the average degree of consolidation (Ut) using the relation.
Ut=U0+(1−U0)Ut−t0
Substitute 43% for U0 and 0.3377 for Ut−t0.
Ut=43100+(1−43100)×0.3377=0.43+0.1925=0.6225×100%=62.25 %
Calculate the consolidation settlement for 18 months using the relation.
Substitute 62.25% for U and 220 mm for Sp in Equation (3).
Consolidation settlement=62.25100×220=136.95 mm
For the time (t) of 36 months with t0 of 12 months.
Calculate the time factor (Tv) as shown below.
Substitute 3.0 m2/year for cv, 36 months for t, 12 months for t0, and 3 m for Hdr in Equation (4).
Tv=3 m2/year×(36−12) months×1 year12 months(3 m)2=69=0.667
Refer Table 9.5 “ U−Tv Values for Non-Uniform Initial Pore Pressure Distributions” in the Text Book.
Take the value of Ut−t0 as 0.7725 for the value Tv of 0.600 for half-sine.
Take the value of Ut−t0 as 0.8222 for the value Tv of 0.700 for half-sine.
Calculate the value of Ut−t0 for the value Tv of 0.667 by interpolation as shown below.
0.700−0.6670.700−0.600=0.8222−Ut−t00.8222−0.77250.33×0.0497=0.8222−Ut−t00.0164=0.8222−Ut−t0Ut−t0=0.806
Calculate the average degree of consolidation (Ut) using the relation.
Ut=U0+(1−U0)Ut−t0
Substitute 43% for U0 and 0.806 for Ut−t0.
Ut=43100+(1−43100)0.806=0.43+0.4594=0.8894×100%=88.9 %
Calculate the consolidation settlement for 36 months using the relation.
Substitute 88.9% for U and 220 mm for Sp in Equation (3).
Consolidation settlement=88.9100×220=195.58 mm
For the time (t) of 60 months with t0 of 12 months.
Calculate the time factor (Tv) as shown below.
Substitute 3.0 m2/year for cv, 60 months for t, 12 months for t0, and 3 m for Hdr in Equation (4).
Tv=3 m2/year×(60−12) months×1 year12 months(3 m)2=129.0=1.33
Refer Table 9.5 “ U−Tv Values for Non-Uniform Initial Pore Pressure Distributions” in the Text Book.
Take the value of Ut−t0 as 0.9152 for the value Tv of 1 for half-sine.
Take the value of Ut−t0 as 0.9753 for the value Tv of 1.5 for half-sine.
Calculate the value of Ut−t0 for the value Tv of 1.33 by interpolation as shown below.
1.5−1.331.5−1=0.9753−Ut−t00.9753−0.91520.34×0.0601=0.9753−Ut−t00.0204=0.9753−Ut−t0Ut−t0=0.955
Calculate the average degree of consolidation (Ut) using the relation.
Ut=U0+(1−U0)Ut−t0
Substitute 43% for U0 and 0.955for Ut−t0.
Ut=43100+(1−43100)0.955=0.43+0.5444=0.9744×100%=97.4 %
Calculate the consolidation settlement for 60 months using the relation.
Substitute 97.4% for U and 220 mm for Sp in Equation (3).
Consolidation settlement=97.4100×220=214.3 mm
Show the values of time and consolidation settlement as in Table 1.
Time (months) | Consolidation settlement (mm) |
3 | 11.60 |
6 | 34.10 |
12 | 94.60 |
18 | 136.95 |
36 | 195.58 |
60 | 214.30 |
Table 1
Refer to Table 1.
Sketch the plot between consolidation settlement and time as in Figure 1.