Cyclic Shear Tests: Dynamic Properties 136
Cyclic Shear Tests: Dynamic Properties 137
Fig. 5.28 (a-d) Typical test results obtained from stress-controlled cyclic triaxial test of SBS
Fig. 5.29 Variation of shear modulus of SBS with shear strain from stress-controlled loading
0 10 20 30 40
-0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2
0 10 20 30 40
-40 -20 0 20 40
0 10 20 30 40
0.0 0.2 0.4 0.6 0.8 1.0
-0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 -40
-20 0 20
40 (d)
(c)
(b)
Axial strain (%)
Number of cycles (a)
Deviator stress (kPa)
Number of cycles
Excess PWP ratio
Number of cycles
Deviator stress (kPa)
Axial strain (%)
0.01 0.1 1 10
0 10 20 30 40 50 60
'c = 100kPa Dr = 60%
f = 1 Hz
Shear modulus (MPa)
Shear strain (%)
CSR 0.1 0.2 0.3 0.4
Cyclic Shear Tests: Dynamic Properties 138 Fig. 5.30 depicts the variations in damping ratio with shear strain at different CSR ranging from 0.1-0.4. Damping ratio, D and D#, was evaluated based on the both symmetrical and asymmetrical approach, respectively. It is seen that, for CSR = 0.1, D and D# are almost same, due to the formation of symmetrical hysteresis loop at all 40 loading cycles. At CSR = 0.2, it was been seen that the D and D# up to γ = 0.2% are nearly same, and beyond γ = 0.2%, D exceeds from D#. This condition arises, when loop is not symmetrical because the symmetrical approach estimates lesser stored strain energy by considering the area of triangle in first quadrant, as shown in Fig. 5.31 and Fig. 5.32, for N = 1 and 23, respectively. Damping ratio calculated by symmetrical approach is overestimated because the area of stored strain energy (calculated by area of triangle) in the first quadrant is less at N = 1 than the third quadrant.
However, in strain-controlled approach, D was observed to be lesser than D# because the area of stored strain energy (calculated by area of triangle) in the first quadrant was higher at N = 1 than the third quadrant, and thus D was underestimated. Thus, based on the both strain- controlled and stress-controlled, it can be stated that the asymmetrical approach can evaluate the precise damping ratio because of the accurate estimation of stored strain energy during the loading cycle. The degradation in damping ratio beyond γ = 1.0% has been observed, which is similar to the strain-controlled cyclic loading.
Fig. 5.33 presents the variations in γ with N at different CSR. It is seen that, at CSR = 0.1, the amount of γ till 40 cycles are substantially less i.e. less than or equal to 0.02%, whereas, γ was observed nearly ~ 1.0%, 1.75% and 2.4% for CSR values 0.2, 0.3 and 0.4, respectively.
Fig. 5.34 presents the variations in ru with N at different CSR. At CSR = 0.1, the magnitude of ru is nearly 0.1 at the end of 40 cycles, which is attributed to the development of very low shear strain (almost constant γ = 0.02%). However, at CSR ≥ 0.2, significant variation in ru was observed during cyclic loading, depicted in Fig. 5.34. The excess PWP ratio (ru) reached 1, at 4th, 7th and 28th number of loading cycles when γ was nearly ~ 1.0%, 1.75% and 2.4% for CSR
Cyclic Shear Tests: Dynamic Properties 139 values 0.2, 0.3 and 0.4, respectively. Fig. 5.35 presents the development of ru along with γ accumulation. It shows that ru increases with the increase in γ and, higher γ poses higher vaue of ru, for a constant Dr and σ′c. The accumulation of γ and development of ru depends on the soil type, stress applied or tests conditions.
Fig. 5.30 Variation of damping ratio of SBS with shear strain from stress-controlled loading
Fig. 5.31 Hysteresis loop at N = 1 for different CSR
0.01 0.1 1 10
0 10 20 30
40 '
c = 100kPa Dr = 60%
f = 1 Hz
Damping ratio (%)
Shear strain (%) CSR= 0.1: D, D#
CSR= 0.2: D, D# CSR= 0.3: D, D# CSR= 0.4: D, D#
-1.2 -0.8 -0.4 0.0 0.4
-45 -30 -15 0 15 30 45
Shear stress (kPa)
Shear strain (%) CSR
0.1 0.2 0.3 0.4
'c = 100 kPa Dr = 60%
f = 1 Hz N = 1
Cyclic Shear Tests: Dynamic Properties 140
Fig. 5.32 Hysteresis loop at N = 23 for different CSR
Fig. 5.33 Variation of shear strain with number of cycles for different CSR
-6 -4 -2 0 2 4 6 8 10
-20 -10 0 10
20 CSR 0.1 0.2 0.3 0.4
Shear stress (kPa)
'c = 100 kPa Dr = 60%
f = 1 Hz N = 23 Shear strain (%)
0 10 20 30 40
0 1 2 3 4 5
'c = 100kPa Dr = 60%
f = 1 Hz CSR
0.1 0.2 0.3 0.4
Shear strain (%)
Number of cycles
Cyclic Shear Tests: Dynamic Properties 141
Fig. 5.34 Variation of excess PWP ratio with number of cycles for different CSR
Fig. 5.35 Variation of excess PWP ratio with shear strain for different CSR Effect of loading frequency on stress-controlled dynamic shear properties
The effects of loading frequency on the variations of shear modulus and damping ratio have
0 10 20 30 40
0.0 0.2 0.4 0.6 0.8 1.0
Excess PWP ratio(
r u
)
Number of cycles
CSR 0.1 0.2 0.3 0.4
'c = 100kPa Dr = 60%
f = 1 Hz
0.01 0.1 1 10
0.0 0.2 0.4 0.6 0.8 1.0
'c = 100kPa Dr = 60%
f = 1 Hz
Excess PWP ratio (
r u
)
Shear strain (%) CSR
0.1 0.2 0.3 0.4
Cyclic Shear Tests: Dynamic Properties 142 also been studied using stress-controlled cyclic loading. Fig. 5.36-Fig. 5.38 depicts the results at different loading frequency ranging from 0.1 Hz to 4 Hz for CSR = 0.2 at Dr = 60% and σʹc
= 100 kPa. It is seen from Fig. 5.36, that the shear modulus and damping ratio are significantly affected by loading frequency. It has also been seen that the shear modulus increases up to the frequency of 1 Hz and decreases afterward, whereas damping ratio follow decreasing trend. In contrary to this, Dash and Sitharam (2016) reported that the shear modulus decreases and damping ratio increases with increase in frequency from 0.1 Hz to 0.5 Hz because of the increase in excess pore water pressure from 0.1 Hz to 0.5 Hz at first loading cycle. To observe the effect of excess pore water pressure ratio and shear strain developed on shear modulus and damping ratio at first loading cycle for different loading frequencies, plots are presented in Fig.
5.37.
Fig. 5.36 Variations of shear modulus and damping ratio with loading frequency for N = 1 It can be seen from Fig. 5.37, that the excess pore water pressure ratio and shear strain, for f = 0.1 Hz, are significantly higher than the other frequency resulting lower shear modulus and higher damping ratio. The shear modulus, at f = 1 Hz, is higher and corresponding excess pore water pressure ratio and shear strain are lower from other loading frequency. Fig. 5.38
0 1 2 3 4 5
0 5 10 15 20
G D
Frequency, f (Hz)
Shear modulus, G (MPa)
0 5 10 15 20 25 30
Dr= 60%
'c= 100 kPa CSR = 0.2 N = 1
Damping ratio, D (%)
Cyclic Shear Tests: Dynamic Properties 143 describes that the rate of generation of excess pore water pressure affected significantly by frequency of cyclic loading. The loading frequency of 1 Hz shows lesser rate of generation of excess pore water pressure compared to other frequency.
Fig. 5.37 Variations of ru and shear strain with loading frequency for N = 1
Fig. 5.38 Variations of ru with number of loading cycles at different loading frequencies