A sensitivity of the proposed model with respect to changes in the hypoplastic parameters was performed with the parameters of Jurong sand and the machine parameters of a Keller S 340 vibrator. The compaction process was simulated for the following initial conditions σz = −128 kPa, K0 = 0.5 and e0 = 0.863, which corresponds to the stress state in z = −11 m depth and the initial density as obtained from pre-CPT results (cf. Sec. 5.3).
Variation of the limit void ratios ei0, ec0 and ed0: On the basis of emax and emin, which were increased by 10% and decreased by 10%, respectively, the limit void ratios
ed0, new = 0.9 · emin = 0.621 ec0, new = 1.1 · emax = 1.133 ei0, new = 1.15 · ec0,new = 1.303
were newly determined. In Figure D.1 the temporal change of the void ratio at three distances to the vibrator axis (r1 = 0.50 m, r2 = 1.00 m and r3 = 2.00 m) is depicted.
The influence of this variation on the compaction and its range is negligible. If the densest state is reached, as here in the case of r1 and r2, e is obviously smaller, here by 10%. If the relative density ID is taken for the judgement in both cases ID = 1.00 is obtained, if not e is practically identical (r3).
Variation of the critical friction angle ϕc: The results of the vibro compaction sim-ulations with ϕc = ϕc,0± 10% are depicted in Figure D.2. For the distance r3 the reduction of ϕc by 10% leads to an increase in the relative density ∆ID = 11.8%
(∆e = −0.04), the increase of ϕc by 10% yields ∆ID = −11.8% (∆e = 0.04)after 40 s. The difference in time needed for reaching the densest state in the case of r1
and r2 is negligible.
Variation of the compressibility parameters hs and n: Figure 5.18 shows the ef-fect of the change of the granular hardness hs and the exponent n by ± 10% on the calculation results of oedometric compression tests. Since compressibility depends more on n than on hs, only the former parameter is varied in the vibro compaction simulations. The results are depicted in Figure D.3.
The variation of n has a strong influence on the simulation results. For the distance r3 the reduction of n yields ∆ID = −59% (∆e = −0.20). In the case of r2 the
5.4. Sensitivity Analysis 97
0.70 0.75 0.80 0.85 0.90
10 100 1000
e
p’ in kPa h
s=80 MPa
n=0.56 n=0.51 n=0.46
0.70 0.75 0.80 0.85 0.90
10 100 1000
e
p’ in kPa n=0.51
h
s=88 MPa h
s=80 MPa h
s=72 MPa
Figure 5.18: Influence of hs and n on the calculated oedometric compression lines.
densest possible state is reached within the first few seconds for n = 0.56 while for n = 0.46 it is not reached at all at the end of the vibration time.
Variation of the exponent α: The exponent α controls the dilatancy behaviour and the influence of density on the peak friction angle. The simulation results for α = 0.1 ± 10% are shown in Figure D.4. The results are almost identical, independent from the distance to the vibrator axis.
Variation of the exponent β: The exponent β controls the influence of density on the stiffness of the granulate material. Higher values of β result in higher stiffnesses for
the same density and stress state. The influence of the variation of β (Fig. D.5) is somewhat larger than in the case of α. For r3 ∆ID(β = 0.9) = 11.8% (∆e = −0.04) and ∆ID(β = 1.19) = −14.8% (∆e = 0.05) after 40 s. The times for reaching the densest state in the case of r1 and r2 are similar.
5.5 Conclusions
A novel method for estimating the grid spacing and improving control and the effectiveness of Deep Vibratory Compaction of cohesionless soils was worked out. For a given set of machine parameters (frequency, mass, diameter, length of the vibrator and static moment of the eccentric rotating mass), the method allows the evaluation of density changes at different distances to the vibrator, taking into account the initial state (stress and density) and the granulometric properties of the soil. For analytical purposes the dynamic, three-dimensional boundary value problem has been idealized as the one-three-dimensional dynamic expansion of a cylindrical cavity subjected to a sinusoidal pressure. The amplitude of the cavity pressure has been estimated using a lumped mass-spring-damper model, which simulates the soil-vibrator interaction. The soil behaviour is modelled by a hypoplastic constitutive equation with intergranular strain. Solutions to the one-dimensional dynamic cavity expansion problem show a rate of change of density which is higher than the values observed in the field during DVC. This is mainly due to the fact that the evolution of vertical stresses cannot be predicted realistically by the proposed one-dimensional model.
For this reason, the density changes are recalculated by means of quasi-static truly triaxial element test simulations, in which the two principal stresses are equal to the radial and hoop stresses from the one-dimensional cavity expansion solutions, and the third one is assumed to be a function of the initial geostatic overburden pressure and the cavity pressure amplitude.
The novel method was tested for a well documented deep vibratory compaction mea-sure carried out by Keller Grundbau in Singapore. Based on the available laboratory data (grain size distribution, void ratios in loosest and densest state, one-dimensional compression tests) the hypoplastic parameters could be estimated for the numerical sim-ulation of the compaction process. The resulting grid spacing of approximately 2.5 to 3.0 m is quantitatively close to the one carried out in situ of 3.5 m. It is noted here that this result was a class A prediction, as the actual grid spacing was kept secret by Keller until the end of this study. Considering the the simplifications and assumptions entailed in the proposed method, this is a very encouraging result.
The results of a sensitivity analysis are presented in Section 5.4. It was investigated