Module 4 Lecture 20 Pore water pressure and shear strength - 4 Topics 1.2.6 Curvature of the Failure Envelope Effect of angularity of soil particles Effect of rate of loading during the test 1.2.7 Shear strength of Granular Soils under Plane Strain Condition 1.2.6 Curvature of the Failure Envelope Since Mohr s envelope is actually curved, a higher effective normal stress will yield lower values of. This fact is demonstrated in Figure 4.10, which is a plot of the results of direct shear tests on standard Ottawa Sand. For loose sand, the value of decreases from about to less than when the normal stress is increased form. Similarly, for dense sand (initial void ratio approximately decreases from about to about due to a sixteen-fold increase of. Figure 4.10 Variation of peak friction angle,, with effective normal stress for direct shear tests on standard Ottawa sand. (Redrawn after D. W. Taylor, Fundamentals of Soil Mechanics, Wiley, New York, 1948). Dept. of Civil Engg. Indian Institute of Technology, Kanpur 1
For high values of confining pressure (greater than about ), Mohr s failure envelope sharply deviates from the assumption given by equation (3). This is shown in Figure 4.11. Skempton (1960, 1961) introduced the concept of angle of intrinsic friction for a formal relation between shear strength and effective normal stress. Based on Figure 4.11, the shear strength can be defined as Figure 4.11 Failure envelope at high confining pressure (7) Where is the angle of intrinsic friction. For quartz, skempton (1961) gave the values of. Effect of angularity of soil particles Other factors remaining constant, a soil possessing angular soil particles will show a higher friction angle than one with rounded grains because the angular soil particles will have a greater degree of interlocking and, thus, cause a higher value of. Effect of rate of loading during the test The value of tan in triaxial compression tests is not greatly affected by the rate of loading. For sand, Whitman and Healy (1963) compared tests conducted in 5 and in 5 ms and found that tan decreases at the most by about 10%. 1.2.7 Shear strength of Granular Soils under Plane Strain Condition Dept. of Civil Engg. Indian Institute of Technology, Kanpur 2
The results obtained from triaxial tests are widely used for the design of structures. However, under structures such as continuous wall footings the soils are actually subjected to a plane strain type of loading, i.e., the strain in the direction of the intermediate principal stress is equal to zero. Several investigators have attempted to evaluate the effect of plane strain type of loading (Figure 4.12) on the angle of friction of granular soils. Figure 4.12 Plane strain conditions Figure 4.13 shows the results of the initial tangent modulus for various confining pressures. For gives values of the initial tangent modulus for plane strain loading shows a higher value than that for triaxial loading, although in both cases E increases exponentially with the confining pressure. The variation of Poisson s ratio v with the confining pressure for plane strain and triaxial loading condition is shown in Figure 4.14. The values of v were calculated by measuring the change of the volume of specimens and the corresponding axial strains during loading. The derivation of the equation used for finding v can be explained with the aid of Figure 4.15. Assuming compressive strain to be positive, for the stresses shown in Figure 4.15 Dept. of Civil Engg. Indian Institute of Technology, Kanpur 3
Figure 4.13 Strength of Antioch sand under drained condition. (Redrawn after K.L. Lee, Comparison of Plane Strain and Triaxial Tests on Sand, J. Soil Mech. Found. Div., ASCE, vol. 96, no. SM3, 1970) Figure 4.14 Initial tangent modulus from drained tests on Antioch sand. (Redrawn after K.L. Lee, Comparison of Plane Strain and Triaxial Tests on Sand, J. Soil Mech. Found. Div., ASCE, vol. 96, no. SM3, 1970) Dept. of Civil Engg. Indian Institute of Technology, Kanpur 4
Figure 4.15 Poisson s ratio from drained tests on Antioch sand. (Redrawn after K.L. Lee, Comparison of Plane Strain and Triaxial Tests on Sand, J. Soil Mech. Found. Div., ASCE, vol. 96, no. SM3, 1970) (12) (13) (14) Where The volume of the specimen before load application is equal to after the load application is equal to. Thus, and the volume of the specimen Where is change in volume. Neglecting the higher order terms such as, equation (15) gives (15) Dept. of Civil Engg. Indian Institute of Technology, Kanpur 5
(16) Where v is the change in volume per unit volume of the specimen. For triaxial tests,, and they are expansions (negative sigh). So,. Substituting this into equation (16) we get or (17) With plane strain loading conditions,. Hence, from equation (16),, or (18) Figure 4.15 shows that for a given value of higher than that obtained from triaxial loading. the Poisson s ratio obtained from plane strain loading is Hence, based on the available information at this time, it can be concluded that exceeds the value of. The greatest difference is associated with dense sands at low confining pressures. The smaller differences are associated with loose sands at all confining pressures, or dense sand at high confining pressures. Although still disputed, based on the studies described in this section several suggestions have been made to use a value of for calculation of the bearing capacity of strip foundations. For rectangular foundations, the stress conditions on the soil cannot be approximated by either triaxial or plane strain loadings. Meyerhof (1963) suggested for this case that the friction angle to be used for calculation of the ultimate bearing capacity should be approximated as (19) Where is the length of foundation and the width of foundation. Figure 4.16 Dept. of Civil Engg. Indian Institute of Technology, Kanpur 6