During installation process of stone columns the probe is continuously vibrated imparting energy into the surrounding soil (Fig. 6.3). There have been several research efforts that focused how the amplitude of ground velocity attenuates during man-made ground vibrations. Richart et al. (1970) modeled this attenuation by considering geometrical damping and material damping; Wiss (1981) introduced relationships based on (i) pseudo-attenuation, and (ii) pseudo-attenuation with scaled-distance; Woods and Jedele (1985) analysed above three approaches based on field observations, and found the model proposed by Richart et al. (1970) to generally agree with the observations, while pseudo-attenuation with scaled-distance being comparably more appropriate for the case histories they studied, which consisted mostly vibrations due to impact loads such as dynamic compaction, ball dropping, and pile driving. The main shortcoming in the models proposed by Wiss (1981) is that they does not specifically account for amount of energy dissipated within the soil medium, which is necessary to predict excess pore pressure generation. It should be noted that in all three models, the vibration amplitude was considered only at the ground surface.
Due to the fact that the only model, which deals with material damping, is that proposed by Richart et al. (1970), it has been selected for determining the dissipated energy within the surrounding soil during vibro stone column installation process. In this study, the energy source in the probe is considered as an in-depth point-source (Fig.6.3),
and the energy propagates spherically outward. It is further assumed that there is no energy loss within a spherical region of radius r0, where r0 is the radius of the probe.
Assuming the material damping presented in their analysis applies to subsurface vibrations as well, an equation can be developed for the dissipated energy.
Let the energy imparted by the probe into the soil per unit time be W0. Energy delivered at the probe-soil interface attenuates due to: (i) geometrical damping, and (ii) material damping. Generally, attenuation relationship for vibration amplitude due to material damping for ground vibration is of the form A=A1exp[-α(r-r1)], where, A1 is amplitude at distance r1 from the source, A is amplitude at distance r from the source, and α is the coefficient of attenuation. α depends on several factors such as source characteristics, frequency of vibration, wave velocity, soil profile, stress and strain field within the surrounding soil, soil type, degree of saturation, changes in excess pore pressure and soil density during vibration, etc. Ground vibration studies indicate that, typically, α ranges from 0.02~0.26 m-1 (Richart et al. 1970, Dowding 1996). Woods and Jedele (1985) classify the soil based on the value of α (Table 6.1).
Table 6.1 Classification of Earth Materials by Attenuation Coefficient Class Attenuation Coefficient
α (m-1) at 50 Hz Description of Material I 0.10 – 0.33
Soft soils: loose soils, dry or partially saturated peat and muck, mud, loose beach sand, recently ploughed ground, etc.
II 0.033 – 0.10 Competent soils: most sands, sandy clays, silty clays, gravel, silts, and weathered rock.
III 0.0033 – 0.033 Hard soils: dense compacted sand, dry consolidated clay, consolidated glacial till, and some exposed rock.
IV <0.0033 Hard, competent rock: bedrock, and freshly exposed hard rock.
Source: Woods and Jedele (1985).
Attenuation coefficient values reported in Table 1 are at 50 Hz. Values at different frequencies can be obtained by (Woods and Jedele 1985):
( )
2 1 f f2 1
α =α
(6.1)
where, α1 is known value of a at frequency f1, and α2 is unknown value of α at frequency f2.
It is further considered that energy density is proportional to square of amplitude, which is supported by field observations as well (Wiss 1981). Considering material damping and radial (geometric) damping, the energy density at a radius r is given by:
( )
0
2exp 2 0
4
W W r
r α r
π ⎡ ⎤
= ⎣− − ⎦
(6.2)
where, W = energy per unit time passing through a unit area of the spherical surface at radius r. Energy loss per unit time per unit volume of soil at distance r is,
( )
0 2 2
w W 2 Exp r r
r 0
α α
π ⎡ ⎤
= ⎣− − ⎦
(6.3)
In the soil around the vibratory probe, as excess pore pressure develops due to vibration, the soil becomes weak. Since the amplitude of vibration of the probe is limited (FHWA 2001), the energy imparted to the surrounding soil would decrease resulting in a reduced efficiency. When the pore pressures dissipate, and the soil is sufficiently densified, the energy transfer rate would increase. In this study, this phenomenon has been taken into account considering the energy transfer rate to decay with increasing excess pore pressure:
( ) ( )
0 2 2 0 .
2 u av
w W Exp r r Exp r
r
α α β
π ⎡ ⎤ ⎡ ⎤
= ⎣− − ⎦ ⎣− ⎦
(6.4)
where, Wo = η0P0, P0 = power rating of the vibratory probe, η0 = probe efficiency, (ru)av = the average excess pore pressure ratio within the soil surrounding the probe up to an effective radial distance re, which is assumed to be the same as the center to center spacing between stone columns in this study, and β = a constant.
VibroProbe
P-Waves
2r0
r
Energy Wave Front Probe
Fig.6.3 Radiation of Vibratory Energy - Schematic
Based on the experimental data presented in Ch.5 and theoretical considerations, excess pore water pressure generated due to cyclic loading has been related to frictional energy loss in the soil by Thevanayagam et al. (2003a,b) as:
0.5 10 100 , 0.05
7 , 0.05
c c
u
L L
c c
L L
E E
r Log
E E
E E
E E
⎛ ⎞
= ⎜ ⎟ ≥
⎝ ⎠
<
(6.5)
where, ru = excess pore pressure ratio (u/σ0’), σ0’ = initial mean effective confining pressure, Ec = cumulative energy loss per unit volume of soil, and EL = energy per unit volume required to cause liquefaction.