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  NPTEL- Advanced Geotechnical Engineering Dept. of Civil Engg. Indian Institute of Technology, Kanpur 1   Module 6 Lecture 40 Evaluation of Soil Settlement - 6 Topics 1.5   STRESS-PATH METHOD OF SETTLEMENT CALCULATION 1.5.1 Definition of Stress Path  1.5.2 Stress and Strain Path for Consolidated Undrained Undrained Triaxial Tests  1.5.3 Calculation of Settlement from Stress Point 1.5   STRESS-PATH METHOD OF SETTLEMENT CALCULATION Lambe (1964) proposed a technique for calculation of settlement in clay which takes into account both the immediate and the primary consolidation settlements. This is called the  stress-path method. 1.5.1 Definition of Stress Path In order to understand what a stress path is, consider a normally consolidated clay specimen subjected to a consolidated drained triaxial test ( Figure 6.31a ). At any time during the test, the stress condition in the specimen can be represented  by a Mohr’s circle ( Figure 6.31b ). Note here that, in a drained test, total stress is equal to effective stress. So,  3 = ′ 3  (minor principal stress)    1 =  3 + ∆ = ′ 1  (major principal stress)    NPTEL- Advanced Geotechnical Engineering Dept. of Civil Engg. Indian Institute of Technology, Kanpur 2   At failure, the Mohr’s circle will touch a line that is the Mohr  -Coulomb failure envelope; this makes an angle ∅  with the normal stress axis ( ∅  is the soil friction angle). We now consider another concept; without drawing the M ohr’s circles, we may represent each one by a  point defined by the coordinates  ′ = ′ 1 + ′ 3 2  (59) And ′ = ′ 1 −′ 3 2  (60) This is shown in Figure 6.31b   for the smaller of the Mohr’s circles. If the points with  ′    ′  coordinates of all the Mohr’s circles are joined, this will result in the line  AB . This line is called a  stress path . The straight line joining the srcin and the point  B  will be defined here as the     line. The     line makes an angle   with the normal stress axis. Now, tan  =  = (  ′ 1   − ′ 3   )/2(  ′ 1   +  ′ 3   )/2  (61) Where  ′ 1    and  ′ 3    are the effective major and minor principal stresses at failure. Similarly, sin ∅ =  = (  ′ 1   − ′ 3   )/2(  ′ 1   +  ′ 3   )/2  (62) From equations (61 and 62), we obtain tan α = sin∅  (63) Figure 6. 31  Definition of stress path  NPTEL- Advanced Geotechnical Engineering Dept. of Civil Engg. Indian Institute of Technology, Kanpur 3  Again let us consider a case where a soil specimen is subjected to an oedometer (one-dimensional consolidation) type of loading ( Figure 6.32 ). For this case, we can write ′ 3 =   ′ 1  (64) Where    is the at-rest earth pressure coefficient and can be given by the expression (Jaky, 1944)   =1 − sin ∅  (65) For the Mohr’s circle shown in Figure 6. 32 , the coordinates of point  E   can be given by  ′ = ′ 1 −′ 3 2 =  ′ 1 (1 −  )2    ′ = ′ 1 + ′ 3 2 =  ′ 1 (1+   )2  Thus,  =  − 1  ′′  =  − 1  1 −  1+     (66) Where ,   is the angle that the line   (    line)  makes with the normal stress axis. For purposes of comparison, the    line  is also shown in Figure 6. 31b . In any particular problem, if a stress path is given in a  ′  . ′  plot, we should be able to determine the values of the major and minor principal stresses for any given point on the stress path. This is demonstrated in Figure 6. 33 , in which  ABC   is an effective stress path. Figure 6.32  Determination of the slope of    line  NPTEL- Advanced Geotechnical Engineering Dept. of Civil Engg. Indian Institute of Technology, Kanpur 4   1.5.2 Stress and Strain Path for Consolidated Undrained Triaxial Tests Consider a clay specimen consolidated under an isotropic stress  3 = ′ 3  in a triaxial test. When a deviator stress ∆  is applied on the specimen and drainage is not permitted there will be an increase in the pore water  pressure, ∆  ( Figure 6. 34a ). ∆ =     ∆  (67) Figure 6. 33  Determination of major and minor principal stresses for a point on a stress path Figure 6. 34  Stress path for consolidation undrained triaxial test
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