Ch. 5 - Strength and Stiffness of Sand Page 1 Ch 5 Strength and Stiffness of Sands Reading Assignment Ch. 5 Lecture Notes Sections 5.1-5.7 (Salgado) Other Materials Homework Assignment Problems 5-9, 5-12, 5-23
Ch. 5 - Strength and Stiffness of Sand Page 2 Stress-Strain Behavior, Volume Change and Shear of Sands What factors contribute to the strength of sand? Soil Fabric Density (void ratio) Confinement Sources of shear resistance (at the particle (i.e., fabric level) Friction between soil particles Particle rearrangement Interlocking between particles Contractive behavior Dilative behavior
Ch. 5 - Strength and Stiffness of Sand Page 3 Stress-Strain Behavior of Sands (cont.) (D R = relative density) (e max - e)/(e max - e min ) Effects of confinement Peak strength Residual strength Note that the sand's post-yield behavior changes as a function of the confining stress. The sample with the lowest ' 1 / ' 3 ratio has the most reduction in residual strength when compared with the peak strength. Samples with higher ratios have less reduction. At large strain, the residual strength is reached and the soil has reached the critical state.
Ch. 5 - Strength and Stiffness of Sand Page 4 Stress-Strain Behavior of Sands (cont.)
Ch. 5 - Strength and Stiffness of Sand Page 5 Stress-Strain Behavior of Sands (cont.) Note that at large axial strain the ratio of ' 1 / ' 3 remains unchanged. Because ' 3 remains constant during an axial compression test, this means that ' 1 is not changing with axial strain. When this point is reached, this is called the critical state. Note that the volumetric strain (y axis) remains unchanged with axial strain when the critical state is reached. This means that the sample is neither contracting or dilating, but straining at a constant void ratio. Note that the dilatancy angle is reducing with axial strain and is near zero at the critical state.
Ch. 5 - Strength and Stiffness of Sand Page 6 Stress-Strain Behavior of Sands (cont.) (loose sand) (dense sand)
Ch. 5 - Strength and Stiffness of Sand Page 7 Critical State Friction Angle - Contractive Sand How does the critical state phi angle differ from the peak friction angle we have previously developed?
Ch. 5 - Strength and Stiffness of Sand Page 8 Critical State Friction Angle - Dilative Sand FLOW Number N ' 1 / ' 3 = (1+ sin ) / (1 - sin ) = tan 2 (45 + /2)
Ch. 5 - Strength and Stiffness of Sand Page 9 Critical State Friction Angle What friction angle do we use for design? Do we use the peak friction angle p or critical state friction angel c? (see discussion on p. 203 of Salgado)
Ch. 5 - Strength and Stiffness of Sand Page 10 Correlation for Drained Shear Strength of Sands The shear strength of sand has a component due to interparticle friction and particle rearrangement (i.e., critical-state shear strength) and another due to dilatancy or contraction during shear. de Josselin de Jong (1976) showed that this can be expressed mathematically for plane-strain conditions as: N = MN c where N is the flow number, M is the dilatancy number and N c is the critical-state flow number. These are related by: N = (1 + sin / (1 - sin f f for c = 0 (from Lecture 4a) M = (1 + sin / (1 - sin N c = (1 + sin / (1 - sin Where is the friction angle, is the dilatancy angle and c is the critical state friction angle. sin = - (d V /(d 1 - kd 3 )) see Eq. 4-19 in Salgado where k = 1 for plane-strain conditions and 2 for triaxial conditions. For k =1 (plane-strain conditions), then sin = - d V /(d 1 - kd 3 ) Bolton (1986) examined a large number of triaxial compression and plane-strain compression tests and concluded that, for both types of loading, the following relationship held: - (d V / d 1 ) p = 0.3I R where the p subscript indicates that quantity in parenthesis should be calculated at the peak strength.
Ch. 5 - Strength and Stiffness of Sand Page 11 Correlation for Drained Shear Strength of Sands (cont.) Bolton defined the relative dilatancy index for the peak strength as: I R = I D [Q - ln (100 'mp /p A )] - R Q where I D = D R /100 = relative density (%) divided by 100, Q and R Q = fitted parameters that depend on the intrinsic characteristics of the sand, p A is the reference stress (100 kpa = 0.1 Mpa 1 tsf = 2000 psf, and 'mp = mean effictive stress at the peak shear strength. 'mp = ( ' 1p + ' 2 + ' 3 )/3 For triaxial compression test during shear phase, ' 2 = ' 3 = ' c where ' c is the confining or consolidation stress applied on the outer cell Bolton found that the following equation describes the peak friction angle very well for triaxial and plane-strain conditions. p = c + A I R (Eq. 5-16) Salgado where A = 3 for triaxial conditions and A = 5 for plane-strain conditions
Ch. 5 - Strength and Stiffness of Sand Page 12 Plane Strain vs. Triaxial Strain Conditions Triaxial Strain Plane Strain (See Eq. 5-16 to relate p and c) Valid only for a confining stress of 1 atm p = peak friction angle C = critical state friction angle
Ch. 5 - Strength and Stiffness of Sand Page 13 Estimation of the peak friction angle from critical state friction angle Iteration to estimate peak friction angle from stress state and void ratio Practical application If we know the critical state friction angle of a soil, the horizontal earth pressure coefficient Ko, and the relative density of the deposits, we can estimate the peak friction angle. This is valuable for design because most often, the peak friction angle is used to define the strength (i.e., resistance) of the soil in foundation calculations. The mean effective stress (in situ) was used to calculate the average consolidation stress for the sample because the soil is anisotrophically consolidated in situ. Mean effective stress at the end of consolidation phase for Ko condition. This is kept constant during the shear phase of the test This is the peak mean effective principle stress
Ch. 5 - Strength and Stiffness of Sand Page 14 Estimation of the peak friction angle from critical state friction angle Note that in the above example, the peak friction angle calculate from the above equation, is not consistent with the assumed value of 40 degrees. Therefore, the assumption needs to be revised. Thus, the mean stress of 30.6 is somewhat inconsistent with the calculated peak friction angle of 39.1 degrees. Hence, another iteration is required. This is done by adjusting the assumed peak friction angle to a new estimate of 39.1 degrees and recalculating the mean stress and resulting friction angle until convergence is reached. In practice, friction angles are usually reported to the rounded nearest whole number, so once the iteration converges to a stable whole number value, then iteration can cease. For more information on the iterative process, see Example 5-2 in Salgado.
Ch. 5 - Strength and Stiffness of Sand Page 15 Undrained Shear Tests in Sands Note that there is no change in volume or void ratio during undrained shear. Thus, the sample responds to shear by increasing or decreasing the pore pressures, which in turn changes the effective stress during shear, as shown by changes in p' The implications of pore pore pressures generated during shearing are further discussed in Ch. 6. Undrained tests will be more fully explained in Ch. 6.
Ch. 5 - Strength and Stiffness of Sand Page 16 Blank