Consolidation theory on the basis of Terzaghi's consolidation theory

Time rate of consolidation On the basis of Terzaghi's consolidation theory

1. 1 Time rate of consolidation ( Terzaghi’s consolidation theory)

2. Time-rate of consolidation 2 The time required to reduce the excess pore water pressure is primarily a function of the distance the water must travel to exit the stressed area. Karl Terzaghi proposed a theory in 1925 that predicted the time-rate of consolidation. This theory is still in use today. In essence, Terzaghi derived the basic differential equation that described the change of the excess pore water pressure, u, as a function of time and space.

3. Time-rate of consolidation 3

4. Time-rate of consolidation 4 Terzaghi’s differential equation can be solved with following boundary condition The solution is

5. Time-rate of consolidation 5 The solution is T v is a nondimensional number. Tv provides a useful expression to estimate the settlement in the field from the results of a lab consolidation test.

6. Graphical solution of differential equation 6

7. Analytical solution of 1D-consolidation Eq. 7 Data E = 1,000 kPa = 0.0 k = 0.001 m/day  w = 10 kN /m 3 H = 1.0 m p o = 1 kPa E oed = 1,000 kPa c v = 0.1 m 2 /day T v = c v .t /H 2 c v = k.E oed /  w E oed = (1- )E/[(1+ )(1-2 )] H Free drainage Impermeable boundary p o

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10. Degree of consolidation 10 The degree of consolidation is

11. Degree of consolidation 11

12. Average degree of consolidation 12

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14. Average degree of consolidation 14 Fro U = 0 to 60% Fro U = 0 to 60% For T v ≤ 0.217 (U ≤ 52.6%) For T v > 0.217 (U > 52.6%)

15. Average degree of consolidation 15 Relationship between Tv & average degree of consolidation for a uniform and triangular distribution of initial excess porewater pressure

16. Logarithm-of-time Method 16

17. Square-root-of-time Method 17

18. Consolidation Test - data Specimen data At the beginning of test Diameter of specimen, D = 2.50 in. Initial height of specimen, H = 0.780 in. Mass of specimen ring + specimen = 208.48 g Mass of specimen ring = 100.50 g Initial moisture content mass of wet soil + can = 79.97 g mass of dry soil + can = 66.82 g mass of can = 35.60 g specific gravity = 2.72 At the end of test mass of entire wet specimen + can = 234.54 g mass of entire dry specimen plus can = 203.11 g mass of can = 127.20 g 18

19. Consolidation Test - data 19 Time-versus-deformation data date time deformation dial reading (in.) 06/08/96 9:15 AM 0.0000 9:15.1 0.0067 9:15.25 0.0069 9:15.5 0.0071 9:16 0.0077 9:17 0.0084 9:19 0.0095 9:23 0.0107 9:30 0.0120 9:45 0.0132 10:15 0.0144 11:15 0.0152 1:15 PM 0.0158 5:15 0.0160 06/09/96 8:15 AM 0.0162   11:15 0.0162 Loading increment from 0 to 500 lb/ft 2 )

20. Calculations specimen parameters 20 Specimen parameters ( at the beginning of test) 1. Initial wet unit weight 2. Initial moisture content

21. 21 Estimated from initial wet mass and initial moisture content 3. Initial degree of saturation Calculations specimen parameters

22. 22 Specimen parameters ( at the end of test) 1. Final moisture content 2. Initial void ratio Determined at the end of test Calculations Initial void ratio

23. 23 Height of solid in specimen Note: 2.54 cm = 1 in. Change in void ratio, D e To convert in 2 into cm 2 To convert in into cm Void ratio for 500 lb/ft 2 loading Calculations change in void ratio

24. 24 Time (min.) Deformation dial reading (in.) < = log scale 0.1 1 10 100 1000 10,000 0.0040 0.0060 0.0080 0.0100 0.0120 0.0140 0.0160 0.0180 0% consolidation, d = 0.0058 D d D d t 1 Mean of d and d 50 50% consolidation, d 50 = 0.0108 The deformation at t 2 should be greater than ¼, but less than ½ of total deformation for the load increment. time-deformation curve from data points t 50 = 8.2 min 100% consolidation, d 100 = 0.0158 t 2 = 4t 1 Pressure increment from 0 to 500 lb/ft 2 Calculations Time vs deformation – time for 50% consolidation

25. 25 Coefficient of consolidation, c v where t 50 = time required for 50% consolidation H = half the thickness of test specimen at 50% consolidation For 500 lb/ft 2 loading: t 50 = 8.2 min where H o = initial height of specimen at beginning of test d 50 = deformation dial reading at 50% consolidation Calculations Coefficient of consolidation, c v