Chapter 28: Consolidation -2 - Class Notes, Civil engineering Notes | EduRev

: Chapter 28: Consolidation -2 - Class Notes, Civil engineering Notes | EduRev

 Page 1


NPTEL- Advanced Geotechnical Engineering 
 
Dept. of Civil Engg. Indian Institute of Technology, Kanpur                                                                           1 
Module 5 
Lecture 28 
Consolidation-2 
Topics 
1.1.3 Relations of  
  
 and  
 
 for Other Forms of Initial Excess Pore Water 
Pressure Distribution  
1.1.4 Numerical Solution for One-Dimensional Consolidation 
? Consolidation in a layered soil  
1.1.5 Degree of Consolidation under Time-Dependent Loading 
 
1.1.3 Relations of  
  
 and  
 
 for Other Forms of Initial Excess Pore Water 
Pressure Distribution  
Using the basic equation for excess pore water pressure and with proper boundary conditions, relations for 
 
  
 and  
 
 for various other types of initial excess pore water pressure distribution can be obtained. Figure 
5.10 and 5.11 present some of these cases. 
 
 
Figure 5.10  Some forms of initial excess pore water pressure distribution 
Page 2


NPTEL- Advanced Geotechnical Engineering 
 
Dept. of Civil Engg. Indian Institute of Technology, Kanpur                                                                           1 
Module 5 
Lecture 28 
Consolidation-2 
Topics 
1.1.3 Relations of  
  
 and  
 
 for Other Forms of Initial Excess Pore Water 
Pressure Distribution  
1.1.4 Numerical Solution for One-Dimensional Consolidation 
? Consolidation in a layered soil  
1.1.5 Degree of Consolidation under Time-Dependent Loading 
 
1.1.3 Relations of  
  
 and  
 
 for Other Forms of Initial Excess Pore Water 
Pressure Distribution  
Using the basic equation for excess pore water pressure and with proper boundary conditions, relations for 
 
  
 and  
 
 for various other types of initial excess pore water pressure distribution can be obtained. Figure 
5.10 and 5.11 present some of these cases. 
 
 
Figure 5.10  Some forms of initial excess pore water pressure distribution 
NPTEL- Advanced Geotechnical Engineering 
 
Dept. of Civil Engg. Indian Institute of Technology, Kanpur                                                                           2 
 
 
 
 
Example 1 Consider the case of initial excess hydrostatic pore water that is constant with depth, i.e., 
 
 
  
 
 (Figure 5.12). For  
 
    , determine the degree of consolidation at a depth H/3 measured from 
the top of the layer. 
                        
Figure 5.12 
Solution From equation (32), for constant pore water pressure increase, 
 
 
 
    
 
 
 
   
 
 
 
       
 
 
 
 
   
   
      
Figure 5.11 Variation of  
  
       
 
 for initial excess pore water pressure diagrams shown in Figure 5. 10 
Page 3


NPTEL- Advanced Geotechnical Engineering 
 
Dept. of Civil Engg. Indian Institute of Technology, Kanpur                                                                           1 
Module 5 
Lecture 28 
Consolidation-2 
Topics 
1.1.3 Relations of  
  
 and  
 
 for Other Forms of Initial Excess Pore Water 
Pressure Distribution  
1.1.4 Numerical Solution for One-Dimensional Consolidation 
? Consolidation in a layered soil  
1.1.5 Degree of Consolidation under Time-Dependent Loading 
 
1.1.3 Relations of  
  
 and  
 
 for Other Forms of Initial Excess Pore Water 
Pressure Distribution  
Using the basic equation for excess pore water pressure and with proper boundary conditions, relations for 
 
  
 and  
 
 for various other types of initial excess pore water pressure distribution can be obtained. Figure 
5.10 and 5.11 present some of these cases. 
 
 
Figure 5.10  Some forms of initial excess pore water pressure distribution 
NPTEL- Advanced Geotechnical Engineering 
 
Dept. of Civil Engg. Indian Institute of Technology, Kanpur                                                                           2 
 
 
 
 
Example 1 Consider the case of initial excess hydrostatic pore water that is constant with depth, i.e., 
 
 
  
 
 (Figure 5.12). For  
 
    , determine the degree of consolidation at a depth H/3 measured from 
the top of the layer. 
                        
Figure 5.12 
Solution From equation (32), for constant pore water pressure increase, 
 
 
 
    
 
 
 
   
 
 
 
       
 
 
 
 
   
   
      
Figure 5.11 Variation of  
  
       
 
 for initial excess pore water pressure diagrams shown in Figure 5. 10 
NPTEL- Advanced Geotechnical Engineering 
 
Dept. of Civil Engg. Indian Institute of Technology, Kanpur                                                                           3 
                                     . We can now make a table to calculate  
 
  
1.                   
2.   
 
 0.3 0.3 0.3  
3.    0 1 2  
4.                   
5.                     
6.      1.273 0.4244 0.2546  
7.        
 
 
 
  0.4770 0.00128     
8.        
 
     0.5 1.0 0.5  
9.    
          
 
 
 
     
  
 
  
0.3036 0.00005    
         
Using the value of 0.3041 calculated in step 9, the degree of consolidation at depth H/3 is 
 
     
                         
Note that in the above table we need not go beyond    , since the expression in step 9 is negligible for 
   . 
Example 2 Due to certain loading conditions, the excess pore water pressure in a clay layer (drained at top 
and bottom) increased in the manner shown in Figure 5.13. For a time factor  
 
    , calculate the average 
degree of consolidation. 
 
Figure 5.13 
Solution The excess pore water pressure diagram shown in Figure 5.13 can be expressed as the difference 
of   two diagrams, as shown in Figure 5.14b and c. the excess pore water pressure diagrams in Figure 5. 
14b shows a case where  
 
 varies linearly with depth. Figure 5.14c can be approximated as a sinusoidal 
variation. 
 
Page 4


NPTEL- Advanced Geotechnical Engineering 
 
Dept. of Civil Engg. Indian Institute of Technology, Kanpur                                                                           1 
Module 5 
Lecture 28 
Consolidation-2 
Topics 
1.1.3 Relations of  
  
 and  
 
 for Other Forms of Initial Excess Pore Water 
Pressure Distribution  
1.1.4 Numerical Solution for One-Dimensional Consolidation 
? Consolidation in a layered soil  
1.1.5 Degree of Consolidation under Time-Dependent Loading 
 
1.1.3 Relations of  
  
 and  
 
 for Other Forms of Initial Excess Pore Water 
Pressure Distribution  
Using the basic equation for excess pore water pressure and with proper boundary conditions, relations for 
 
  
 and  
 
 for various other types of initial excess pore water pressure distribution can be obtained. Figure 
5.10 and 5.11 present some of these cases. 
 
 
Figure 5.10  Some forms of initial excess pore water pressure distribution 
NPTEL- Advanced Geotechnical Engineering 
 
Dept. of Civil Engg. Indian Institute of Technology, Kanpur                                                                           2 
 
 
 
 
Example 1 Consider the case of initial excess hydrostatic pore water that is constant with depth, i.e., 
 
 
  
 
 (Figure 5.12). For  
 
    , determine the degree of consolidation at a depth H/3 measured from 
the top of the layer. 
                        
Figure 5.12 
Solution From equation (32), for constant pore water pressure increase, 
 
 
 
    
 
 
 
   
 
 
 
       
 
 
 
 
   
   
      
Figure 5.11 Variation of  
  
       
 
 for initial excess pore water pressure diagrams shown in Figure 5. 10 
NPTEL- Advanced Geotechnical Engineering 
 
Dept. of Civil Engg. Indian Institute of Technology, Kanpur                                                                           3 
                                     . We can now make a table to calculate  
 
  
1.                   
2.   
 
 0.3 0.3 0.3  
3.    0 1 2  
4.                   
5.                     
6.      1.273 0.4244 0.2546  
7.        
 
 
 
  0.4770 0.00128     
8.        
 
     0.5 1.0 0.5  
9.    
          
 
 
 
     
  
 
  
0.3036 0.00005    
         
Using the value of 0.3041 calculated in step 9, the degree of consolidation at depth H/3 is 
 
     
                         
Note that in the above table we need not go beyond    , since the expression in step 9 is negligible for 
   . 
Example 2 Due to certain loading conditions, the excess pore water pressure in a clay layer (drained at top 
and bottom) increased in the manner shown in Figure 5.13. For a time factor  
 
    , calculate the average 
degree of consolidation. 
 
Figure 5.13 
Solution The excess pore water pressure diagram shown in Figure 5.13 can be expressed as the difference 
of   two diagrams, as shown in Figure 5.14b and c. the excess pore water pressure diagrams in Figure 5. 
14b shows a case where  
 
 varies linearly with depth. Figure 5.14c can be approximated as a sinusoidal 
variation. 
 
NPTEL- Advanced Geotechnical Engineering 
 
Dept. of Civil Engg. Indian Institute of Technology, Kanpur                                                                           4 
 
Figure 5.14 
 
The area of the diagram in Figure 5.14b is 
 
 
    
 
 
                       
The area of the diagram in Figure 5.14c is 
 
 
        
  
  
         
  
  
        
  
 
      
  
  
 
 
  
 
   
 
    
    
 
            
  
 
    
   
  
The average degree of consolidation can now be calculated as follows: 
                                                                                                                       
                    
   
    
 
     
 
                
      
 
   
 
     
 
 
  
   
 
     
 
 
 
 
  
 
 
  
                                            
    
 
 
 
 
 
 
 
 
  
Form Figure 5. 6, for  
 
      
  
               
 
   
  
                 
 
. So 
 
  
 
                     
           
 
         
        
        
Example 3uniform surcharge of             
 
 is applied on the ground surface as shown in Figure 5. 
15a. 
(a) Determine the initial excess pore water pressure distribution in the clay layer. 
(b) Plot the distribution of the excess pore water pressure with depth in the clay layer at a time for which 
 
 
    . 
Page 5


NPTEL- Advanced Geotechnical Engineering 
 
Dept. of Civil Engg. Indian Institute of Technology, Kanpur                                                                           1 
Module 5 
Lecture 28 
Consolidation-2 
Topics 
1.1.3 Relations of  
  
 and  
 
 for Other Forms of Initial Excess Pore Water 
Pressure Distribution  
1.1.4 Numerical Solution for One-Dimensional Consolidation 
? Consolidation in a layered soil  
1.1.5 Degree of Consolidation under Time-Dependent Loading 
 
1.1.3 Relations of  
  
 and  
 
 for Other Forms of Initial Excess Pore Water 
Pressure Distribution  
Using the basic equation for excess pore water pressure and with proper boundary conditions, relations for 
 
  
 and  
 
 for various other types of initial excess pore water pressure distribution can be obtained. Figure 
5.10 and 5.11 present some of these cases. 
 
 
Figure 5.10  Some forms of initial excess pore water pressure distribution 
NPTEL- Advanced Geotechnical Engineering 
 
Dept. of Civil Engg. Indian Institute of Technology, Kanpur                                                                           2 
 
 
 
 
Example 1 Consider the case of initial excess hydrostatic pore water that is constant with depth, i.e., 
 
 
  
 
 (Figure 5.12). For  
 
    , determine the degree of consolidation at a depth H/3 measured from 
the top of the layer. 
                        
Figure 5.12 
Solution From equation (32), for constant pore water pressure increase, 
 
 
 
    
 
 
 
   
 
 
 
       
 
 
 
 
   
   
      
Figure 5.11 Variation of  
  
       
 
 for initial excess pore water pressure diagrams shown in Figure 5. 10 
NPTEL- Advanced Geotechnical Engineering 
 
Dept. of Civil Engg. Indian Institute of Technology, Kanpur                                                                           3 
                                     . We can now make a table to calculate  
 
  
1.                   
2.   
 
 0.3 0.3 0.3  
3.    0 1 2  
4.                   
5.                     
6.      1.273 0.4244 0.2546  
7.        
 
 
 
  0.4770 0.00128     
8.        
 
     0.5 1.0 0.5  
9.    
          
 
 
 
     
  
 
  
0.3036 0.00005    
         
Using the value of 0.3041 calculated in step 9, the degree of consolidation at depth H/3 is 
 
     
                         
Note that in the above table we need not go beyond    , since the expression in step 9 is negligible for 
   . 
Example 2 Due to certain loading conditions, the excess pore water pressure in a clay layer (drained at top 
and bottom) increased in the manner shown in Figure 5.13. For a time factor  
 
    , calculate the average 
degree of consolidation. 
 
Figure 5.13 
Solution The excess pore water pressure diagram shown in Figure 5.13 can be expressed as the difference 
of   two diagrams, as shown in Figure 5.14b and c. the excess pore water pressure diagrams in Figure 5. 
14b shows a case where  
 
 varies linearly with depth. Figure 5.14c can be approximated as a sinusoidal 
variation. 
 
NPTEL- Advanced Geotechnical Engineering 
 
Dept. of Civil Engg. Indian Institute of Technology, Kanpur                                                                           4 
 
Figure 5.14 
 
The area of the diagram in Figure 5.14b is 
 
 
    
 
 
                       
The area of the diagram in Figure 5.14c is 
 
 
        
  
  
         
  
  
        
  
 
      
  
  
 
 
  
 
   
 
    
    
 
            
  
 
    
   
  
The average degree of consolidation can now be calculated as follows: 
                                                                                                                       
                    
   
    
 
     
 
                
      
 
   
 
     
 
 
  
   
 
     
 
 
 
 
  
 
 
  
                                            
    
 
 
 
 
 
 
 
 
  
Form Figure 5. 6, for  
 
      
  
               
 
   
  
                 
 
. So 
 
  
 
                     
           
 
         
        
        
Example 3uniform surcharge of             
 
 is applied on the ground surface as shown in Figure 5. 
15a. 
(a) Determine the initial excess pore water pressure distribution in the clay layer. 
(b) Plot the distribution of the excess pore water pressure with depth in the clay layer at a time for which 
 
 
    . 
NPTEL- Advanced Geotechnical Engineering 
 
Dept. of Civil Engg. Indian Institute of Technology, Kanpur                                                                           5 
 
      Figure 5.15 
 
 
Solution Part (a): the initial excess pore water pressure will be           
 
 and will be the same 
throughout the clay layer (Figure 5.15b; refer to Prob. 1 in chapter 4). 
Part (b): From equation (31),  
 
             
 
    
 
 . For  
 
                    
 
 can be 
obtained from the top half of Figure 5. 5 as shown in Figure 5.16a. 
          
 
    
 
    
 
    
   
 
 
0 0 0.63 740 
0.2 2 0.65 700 
0.4 4 0.71 580 
0.6 6 0.78 440 
0.8 8 0.89 220 
1.0 10 1 0 
 
Figure 5.16 
Figure 5.16b shows the variation of excess pore water pressure with depth. 
 
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