2.3.1 Root t method [11]
Introduction 2.3.1.1
The method proposed based on the equation for the equal vertical strain condition [4]. In Eq. (2.3) have Ur = f[Tr, F(n)] then the author can find Tr = f[Ur, F(n)].
(n) ln(1 U ) 8
r r
T F
(2.21) T90 is the time factor at 90% average degree of consolidation so the value of T90 can calculated by F(n), Ur.
Berry (1969) commented that all the curve show linear portions between about 20%
- 60% average degree of consolidation. Thus a straight line is drawn through the experimental volume change –t0.5 results between about 20% to 60% consolidation, and a second line is then constructed having an abscissa 1.17 time that of the first [11].
The radial (horizontal) coefficient of consolidation is determined in this case:
2 90
90 e r
c T d
t
(2.22) The procedure for determine cr
2.3.1.2
The steps for determining the radial (horizontal) coefficient are the same as described in section.2.2.1.2
Evaluation of the method 2.3.1.3
Advantages
- This method is easy to practice for all engineers.
16
- Identify a straight line within Ur = 20% to Ur = 60% on the curve.
- This method does not need to find value 0 and 100. Disadvantages
- Straight line depends on the subjective evaluation of data processing engineers
2.3.2 Matching log (de2/t) and Ur method [12]
Introduction 2.3.2.1
The method proposed based on the equation for the equal vertical strain condition [4]. This approach solves the equation of Eq. (2.3) to find the dependence of Tr on Ur & F(n) then replaces Tr = f[Ur, F(n)] into Eq.(2.4).
Thus, the radial (horizontal) coefficient of consolidation (cr) is determined
2 8
( ) ln(1 U )
e r
r
d c
t F n
(2.23) The procedure for determine cr
2.3.2.2
- Step 1: Plot log (de2/t) versus Ur curve considering the δ - t data on Figure 2.7.
- Step 2: Identify a zone where the experimental curve is parallel to the theoretical curves
- Step 3: Using graphical or Eq. (2.23) can be determined cr.
Figure 2.7. Theoretical log(de2/t) versus Ur curves [12]
17 Evaluation of the method
2.3.2.3 Advantages
- cr can determine easily by the graph.
Disadvantages
- Matching between theoretical and experimental curve does not always occur - The variable Ur needs to determine exact values for 0 &100.
2.3.3 Inflection point method [13]
Introduction 2.3.3.1
The method was developed based on [13] and [4]. Eq. (2.3) can show the relationship Ur = f[log (Tr)].
According to the mathematical definition, the value of Ur maximum when the derivative d(Ur)/dlog Tr the maximum.
Figure 2.8. (a) Theretical Ur - log Tr curve and (b) d(Ur)/dlog Tr plot [13]
In this case, The degree of consolidation at the inflection point also the same for all the curves at Ur = Ur,inf = 63,21% with maximum derivative .
Thus, the value of Ur = Ur,inf = 63,21% can be calculated by Eq. (2.3)
18
r,inf
(n)ln(1 0.631) (n)
8 8
F F
T
(2.24) The radial (horizontal) coefficient of consolidation is determined in this case:
2
inf
( ) 8
e r
d c F n
t
(2.25) The procedure for determine cr
2.3.3.2
- Step 1: Plot (Ur - log t) to t then finding time at max value (Ur - log t). This is the value tinf.
- Step 2: cr can be determined cr by Eq.(2.25) Evaluation of the method
2.3.3.3 Advantages
- 0, 100 does not need to be identified.
- In this method, the author finds tinf value.
Disadvantages
- There is no method yet to find tinf from Experimental data.
- The accuracy of results depends on the time distance between measurement results.
2.3.4 Non-graphical method [14]
Introduction 2.3.4.1
The method proposed based on the equation for the equal vertical strain condition [4].
The laboratory time (t) – compression r is described by the equation.
100 0
r Ur
(2.26) Ur is change in Eq. (2.26) from [4].
100 ,0 2 0
1 exp 8 ( )
r
r r
e
C t F n d
(2.27)
Combine Eq of (2.27) and constant values of de and 100, 0. F(n) can be found by matching between theoretical and experimental curve. Solve r – t curve can find cr.
19 The procedure for determine cr 2.3.4.2
Finding cr with100, 0, F(n), de and r – t curve by A source code or program on Eq.
(2.27).
Evaluation of the method 2.3.4.3
Advantages
- Use a coding program to resolve results independent of the implementer.
Therefore, the value has high accuracy.
- Results processing time is fast.
Disadvantages
- Matching between theoretical and experimental curve does not always occur - The value depends on the data range.
- Eq of (2.27) variable Ur needs to determine the exact values for 0 and 100. 2.3.5 Log - log method [15]
Introduction 2.3.5.1
The value of o can be calculated by selecting two time – settlement in the range Ur < 20% at experimental data(1, t1) & (2, t2).
2 2
1 1
o o
t t
(2.28) Two straight lines depicted in Figure 2.9. The radial (horizontal) coefficient of consolidation is determined in this case:
2 ,66
66
(T )r e
r
c d
t
(2.29) The procedure for determine cr
2.3.5.2
- Step 1: Calculate the initial compression (0) using Eq. (2.28) from the time – compression data by choosing two points in the early stages of consolidation. The value of 0 can be calculated by selecting two time – settlement data within Ur < 20%.
20
- Step 2: Plot the time t – corrected settlement ( – 0) in a log – log plot.
- Step 3: Identify the initial linear portion and draw line.
- Step 4: Identify the linear secondary compression portion by drawing a line and extending it to intersect the initial straight line. The time at the point of intersection (t66) corresponds to a degree of consolidation of 66%.
- Step 5: cr can be determined by Eq. (2.29)
Figure 2.9. Log( - 0) versus log t plot [15]
Evaluation of the method 2.3.5.3
Advantages
- This method can determine 0 & 66.
- Methods Inheriting advantages of graphical method.
Disadvantages
- From Experimental data the value of 0 within Ur < 20% is not constant 2.3.6 Steepest tangent fitting method [16]
Introduction 2.3.6.1
The method Inflection point in Section 2.3.3 has disadvantages, Inflection point is difficult to determine exactly with experimental data. Vinod (2010) found a straight line through an Inflection point [16].
The equation of tangent through Inflection point on the semi-log graph (Figure 2.10) is determined:
= b - alog(t)
(2.30)
21
where a, b = constant and (t, ) value of experimental data.
One log cycle, the author chooses value (1, t1), (2, t2) on the condition (t1 = 10 time), (t2 = 100 time) & (1 - 2 = h). Substituting (1, t1), (2, t2) into Eq. (2.30)
1 = b – a & 2 = b – 2a Thus 1 - 2 = a = h
(2.31) Put a from Eq. (2.31).
= b - hlog(t)
(2.32) P(o, to) is the corrected initial experimental data (Figure 2.10). Put b with P(o, to) from Eq. (2.32).
= [0 + hlog(t0)]- alog(t) = 0 + hlog(t0 / t)
(2.33)
Figure 2.10. Steepest tangent fitting method for determination of cr
Similarly, a straight line through Inflection point on Ur-log Tr and d(Ur) /dlog Tr as shown Figure 2.8. Function for tangent on Ur - log Tr.
Ur = c - Slog(Tr)
(2.34) where c is constant and (Tr, Ur) value of predicted curve.
The value of S is defined by Section 2.3.3.
r,inf
0.847 63.2%
(log )
r r
S dU
d T U
(2.35) Similarly, The corrected initial P(0, t0, Tr,0, Ur,0) is given by Ur,0 = c - Slog(T0) Substituting b with P(0, t0, Tr,0, Ur,0)
Ur = [Ur,0 + Slog(Tr,0)] - Slog(Tr) = Ur,0 + Slog(Tr,0 / Tr)
(2.36) Then dial reading corresponding to x % consolidation
22 (Ur - Ur,0) / S = x / S = log(Tr,0 / Tr) = log(t0 / t)
(2.37) Eq. (2.37) and put log(to / t) from Eq. (2.33). Thus, function for steepest tangent is described as follows:
x = 0 + h x / S
(2.38) Author (Vinod, 2010) has determined the value of o as follows:
1 1
0 1 2
2 2
t / 1 t
t t
(2.39) The procedure for determine cr
2.3.6.2
- Step 1: Plot the dial reading against time on semi log graph as show in Figure 2.10.
- Step 2: Determine o in Eq. (2.39)
- Step 3: Draw a tangent PQ to the steepest part of the consolidation curve.
- Step 4: Find h, which is the slope of the tangent PQ.
- Step 5: Find x use Eq.(2.38)
- Step 6: cr is calculated using Eq. (2.35).
Evaluation of the method 2.3.6.3
Advantages
- This method finds the 0 values.
- Only conduct experiments to Ur = 60%.
- Overcoming method disadvantages Inflection point.
Disadvantages
- Experimental data the value of 0 is not constant.
2.3.7 Log t method [17]
Introduction 2.3.7.1
Put Ur = f(0, , 100) from Eq. (2.3)
23
,t
0
100 0
1 exp 8Tr F n
(2.40)
Replace t time in t1, t2 and 2t1 = t2
,1
1 0
1
100 0
1 exp 8Tr 1 exp F n A
(2.41)
,2
2 0
2
100 0
1 exp 8Tr 1 exp F n A
Substituting A1, A2 with Eq.(2.42)
1 1 0 100 1
100 0 100 0
exp A 1
(2.42)
2 2 0 100 2
100 0 100 0
exp A 1
Then
100 1
1
100 0
ln
A
(2.43)
100 2
2
100 0
ln
A
Substituting A1,/A2 with Eq.(2.43)
100 1
100 0
2
1 100 2
100 0
ln
2 ln
A A
(2.44)
In Eq. (2.44)
2
100 1 100 2 100 2
100 0 100 0 100 0
ln 2ln ln
(2.45)
2
100 1 100 2
100 0 100 0
Then
2
100 1 2 1
100 2
(2 )
o
(2.46)
24
Tr relationship fits approximately as a straight line within 0 < Ur < 20%.
The initial compression may be also determined graphically.
If the data points are selected such that t2 = 2t1, then Eq. (2.28) can show that
2 – 1 = 1 – 0
(2.47) The radial (horizontal) coefficient of consolidation is determined in this case:
2 50 50
(T )r e
r
c d
t
(2.48) The procedure for determine cr
2.3.7.2
- Step 1: Plot the time (t) – settlement () in a log t – plot.
- Step 2: Draw a tangent through the inflection point.
- Step 3: identify the asymptotic secondary compression portion, draw a line, and extend it to intersect the tangent line. The point of intersection corresponds to a degree of consolidation of 100 % (100).
- Step 4: The value of 0 can be obtained using Eq. (2.28) or graphically using Eq.(2.46).
- Step 5: cr is calculated using Eq. (2.48).
Evaluation of the method 2.3.7.3
Advantages
- The value 0 can determine in this method.
- Only conduct experiments to Ur = 60%.
- Overcoming method disadvantages Inflection point.
Disadvantages
- Experimental data the value of 0 is not constant.
2.3.8 Full – match method [10]
This method is identical for both CD & PD. The method is mentioned in the section (2.2.3)
The radial (horizontal) coefficient of consolidation is determined in this case:
25
2
8
n e r
c F d