Cyclic Shear Tests: Dynamic Properties 131
Fig. 5.23 Variation of damping ratio with shear strain for f = 0.5–4.0 Hz and σʹc = 50–150 kPa
Dynamic properties of saturated cohesionless soil using on-sample
Cyclic Shear Tests: Dynamic Properties 132 variation and accumulation of local strains, when attached at different sections of the specimen.
Fig. 5.24 Strain time histories from external and on-sample LVDTs at peak axial strains (a) 0.03% (b) 0.2% (c) 0.66%
Fig. 5.25 represents the stress-strain response of the soil specimens based on both the global and local strain measurements. As obvious, the peak global strain, measured by the external LVDT, is found to be constant for each of the tests. However, for each particular test configuration, the measured local strains are found to be significantly varying throughout the test, and are noticeably different than the applied global strain. Hence, it is understood that along the length of the specimen, distribution of axial strain takes place due to elemental response of particles, which result in such deviation between the local and global strains. Thus, evaluation of the soil stiffness, based on the presumption that global strain is uniform all along the soil sample, will yield conservative results. Hence, it would be judicious enough to utilize the measured local strains for the estimation of the shear stiffness of the soil sample, as the stiffness is also bound to vary along the length of the sample.
0 10 20 30 40
-0.04 -0.02 0.00 0.02 0.04
0 10 20 30 40
-0.2 -0.1 0.0 0.1 0.2 0.3 0.4
0 10 20 30 40
-1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 (c) (b) Global Local
Time (s) Time (s)
Axial strain (%)
Time (s)
(a)
0 10 20 30 40
-0.04 -0.02 0.00 0.02 0.04
0 10 20 30 40
-0.2 -0.1 0.0 0.1 0.2 0.3 0.4
0 10 20 30 40
-1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 (c) (b) Global Local
Time (s) Time (s)
Axial strain (%)
Time (s)
(a)
0 10 20 30 40
-0.04 -0.02 0.00 0.02 0.04
0 10 20 30 40
-0.2 -0.1 0.0 0.1 0.2 0.3 0.4
0 10 20 30 40
-1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 (c) (b) Global Local
Time (s) Time (s)
Axial strain (%)
Time (s)
(a)
Cyclic Shear Tests: Dynamic Properties 133
Fig. 5.25 Stress-strain plots based on external and on-sample LVDTs at Dr = 90%, σʹc = 100 kPa and f = 1 Hz
Fig. 5.26 represents the importance of the measurement of local strains to estimate the shear modulus. It can be observed that the on-sample LVDTs are able to measure low shear strains induced in the specimen, and thus, the shear modulus evaluated at these local strains can be considered as maximum shear modulus (Gmax). The shear strain (γ) shown in Fig. 5.26 was evaluated from the axial strain (ε) using Eqn. (5.3) (ASTM D3999), and the Poisson’s ratio was considered to be 0.5 for saturated undrained specimens (Rollins et al., 1998).
(1 )
(5.3)
Fig. 5.26 also depicts that the variations of shear modulus evaluated from the external LVDT. It can be observed that beyond cyclic shear strain amplitude of 0.1%, the shear modulus obtained from the measurements of on-sample and external LVDT are in extremely close agreement, thus indicating that the on-sample transducers are also capable enough in manifesting the measurement of the shear modulus at higher strain ranges as well. Table 5.4
-0.04 -0.02 0.00 0.02 0.04 -40
-20 0 20 40
-0.06 -0.03 0.00 0.03 0.06 -45
-30 -15 0 15 30 45
-0.2 -0.1 0.0 0.1 0.2 -60
-40 -20 0 20 40 60
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 -80
-40 0 40 80 120
-0.50 -0.25 0.00 0.25 0.50 -120
-80 -40 0 40 80 120
-0.75 -0.50 -0.25 0.00 0.25 -80
-40 0 40 80 120
-0.8 -0.4 0.0 0.4 0.8 1.2 1.6 -80
-40 0 40 80 120 160
-3.0 -1.5 0.0 1.5 3.0 4.5 6.0 7.5 -120
-60 0 60 120 180 240
Global Local
= 0.03% = 0.05% = 0.1%
= 0.2% = 0.3%
Deviatoric stress (kPa)
= 0.4%
= 0.66%
Axial strain (%)
= 1.33%
Cyclic Shear Tests: Dynamic Properties 134 lists the values of Gmax as well as the secant shear modulus (G, computed from 1st cycle of the test) for different test conditions based on both local and global strains. Researchers have performed low-strain tests on fine sands and proposed standard correlations for the estimation of Gmax (Chung et al., 1984; Hardin and Drnevich 1972; Kokusho 1980). It is observed from Table 5.4 that the Gmax obtained from the present experimental investigation are in appreciable agreement with those obtained using the above-mentioned standard correlations.
Fig. 5.26 Comparative plot of shear modulus of SBS based on the external and on-sample LVDTs
Table 5.4 Estimated Gmax values for BS
Parameter σʹc (kPa) 50 100 150
Dr (%) 30 90 30 90 30 90
Shear modulus (G, MPa) Present study
G (based on External
LVDT, γ = 0.045%) 13 20 24 31 32 37
Gmax.(value based on On- sample LVDT)
60 70 71 91 102 120
Low-strain tests on fine sands
Chung et al. (1984) 50 67 69 94 84 114 Hardin and
Drnevich (1972) 59 79 83 112 102 137
Kokusho (1980) 61 90 86 127 105 156
1E-30 0.01 0.1 1 10
30 60 90 120 150
1E-30 0.01 0.1 1 10
30 60 90 120 150
Shear modulus, G (MPa)
Dr = 30%
N = 1 f = 1
Shear strain (%) Shear strain (%)
Dr = 90%
N = 1 f = 1 50 kPa 100 kPa 150 kPa External LVDT
50 kPa 100 kPa 150 kPa Onsample LVDT
Cyclic Shear Tests: Dynamic Properties 135 Fig. 5.27 illustrate the variation of modulus reduction curve, represented by G/Gmax, based on both global and local strains. The estimated Gmax based on the measurements from on-sample LVDTs was used to normalize G (i.e. the secant modulus obtained either from global strain or local strain measurements). Fig. 5.27 shows the variation of G/Gmax for BS, as obtained from the on-sample LVDT measurements, and the same has been compared with that obtained from external LVDT measurement from test specimens subjected to high cyclic strains (Kumar et al., 2017). Based on the agreeable match beyond a shear strain of 0.1%, it can be stated apart from obtaining the shear modulus at low strains (approximately 3×10-3 % to 0.1 %), the on- sample LVDTs can be effectively used even to obtain the same at high strain range (greater than 1.0%). Based on the application of on-sample transducers for cyclic triaxial testing, Gookin et al. (1996) have also reported the modulus reduction curve over a wide range of strain for Monterey No. 30 sand prepared (Dr = 89%), and the same has been illustrated in Fig. 5.27.
Thus, these findings highlight the importance of the use of on-sample LVDT for the measurement of local strains which can be used to obtain the dynamic response of cohesionless soil over a wide range of strain.
Fig. 5.27 Comparative modulus reduction of SBS based on external and on-sample LVDTs
1E-4 1E-3 0.01 0.1 1 10
0.0 0.2 0.4 0.6 0.8 1.0
1.2 Present study data (N = 1, f = 1Hz)
Dr = 30%
50 kPa 100 kPa 150 kPa Dr = 90%
50 kPa 100 kPa 150 kPa Upper bound Lower bound Average curve (based on local strains) Kumar et al. (2017) (based on global strains)
Gookin et al. (1996)
Seed and Idriss (1970) - Upper bound Seed and Idriss (1970) - Lower bound
G/G max
Shear strain (%)
Cyclic Shear Tests: Dynamic Properties 136