INTRODUCTION
Propagation characteristics of vibrations generated by various vibration sources depend on the type of the generated waves which can be assessed by measuring particle motions. Even human activities such as pile driving, traffic or trains passing, may cause seismic waves propagating in the superficial soil layers, typically within a few tens of meters from the ground surface. The vibrations may be within an intolerable limit for adjacent structures and sensitive equipment therefore wave barrier are used to mitigate vibration energy. Installing a wave barrier near the vibration source to alleviate adverse effects of vibrations is known as active isolation. Whereas, passive isolation is distant from the source surrounding or in the immediate vicinity of the structure to be protected.
With reference to the literature on groundborne vibrations, Woods
(1967, 1968) conducted a series of field tests to
scrutinize the screening performance of different governing parameters of trenches
both in active and passive isolation systems. Woods (1974)
defined Amplitudereduction ratio (Arr) and deduced that a minimum trench depth
of 0.6 times the Rayleigh wave length is required to achieve a 75% reduction
in ground displacement amplitude. Using Finite Element Method (FEM) in the frequency
domain and under the assumption of a plane strain condition, Lysmer
and Waas (1972), Haupt (1977) and Segol
et al. (1978) assessed the vibration screening isolation.
Beskos et al. (1985, 1986a,
b, 1990, 1991)
employed Boundary Element Method (BEM) to investigate open and infilled trenches
as well as pile wave barriers. In an extensive parametric study, AlHussaini
and Ahmad (1996, 2000) concentrated on simplified
design methodologies for vibration screening of machine foundations by trenches
using a threedimensional boundary element algorithm.
Kattis et al. (1999a, b)
developed an advancedfrequency domain BEM code to study the screening efficiency
of open, infilled trenches and pile barriers. They reported that trenches are
more proficient than pile barriers, except for vibrations with large wavelength,
where deep trenches are impractical. Hollow piles also are observed to be more
efficient than concrete piles and circular crosssection piles have a similar
behavior to those of square crosssection. Shrivastava and
Kameswara Rao (2002) examined the efficiency of open and filled trenches
for screening Rayleigh waves due to impulse loads in a 3D FE model considering
the effects of the geometric parameters of trench barrier. Soil, in his research,
was idealized as linear, isotropic continuum. Adam and Estorff
(2005) studied the effectiveness of open and infilled trenches in reducing
the sixstorey building vibrations due to passing trains using a twodimensional
FE analysis. An 80% reduction in the building vibrations and internal forces
was reported. El Naggar and Chehab(2005) explored the
efficiency of soft and stiff barriers in screening pulseinduced waves for shallow
foundations resting on an elastic halfspace. The efficiency of different types
of wave barriers in vibration isolation for shockproducing equipment was assessed
and results were presented in Arr. Celebi (2006) presented
two mathematical models and numerical techniques for solving problems associated
with the wave propagation in a track and an underlying soil owing to passing
trains in the frequency domain. They utilized BEM to investigate the threedimensional
dynamic response of the free field nearby railway lines induced by the moving
loads acting on the surface of a homogeneous soil deposit. In addition, Celebi
et al. (2006) conducted comprehensive numerical investigation to
indicate the influence of wave barriers on the complex dynamic stiffness coefficients
of the surface supported foundations under dynamic loads.
Tsai (2007) conducted numerical research using 3D BEM in frequency domain
to scrutinize the screening effectiveness of circular piles in a row for a massless
square foundation subject to harmonic vertical loading. They reported that screening
effectiveness of steel pipe piles is generally better than that of solid piles
and that a concrete hollow pile barrier can be ineffective due to its stiffness.
From the above review, researches mainly focused on vibration reduction of open and infilled trenches induced by shallow foundations in which the Rayleigh waves play a significant role in transmission of ground vibrations. There has been no literature relating to the isolation of deep foundations by open trenches. Thus, this study performs an extensive parametric study on open trenches to reduce the ground vibrations using a 3D FEA.
PROPAGATION AND ATTENUATION OF GROUND VIBRATIONS
Theoretical background of vibration attenuation: Vibration energy decays
during propagating through the ground and the amplitude of the vibrations decreases
with increasing distance from the source. This is due to two components; geometric
(radiation) damping and material damping. The general equation modeling propagation
of ground vibration from point a (a location at distance r_{a} from
the source) to point b (a location at distance r_{b} from the source)
is given by Eq. 1:
where, γ depends upon the type of propagation mechanism and α is
a material damping coefficient (Amick and Gendreau, 2000).
Geometric damping occurs due to a decrease in energy density with distance from
the source. This coefficient can be analytically determined by assessing the
type of the propagating wave, source type and location as shown in Table
1. Geometric damping occurs even in a perfectly elastic media. The ground
is not perfectly elastic and the vibration energy is reduced due to the friction
and cohesion between soil particles. This attenuation is affected by the soil
type and frequency of vibration.
Distribution of body waves from a deep foundation on a homogeneous, isotropic
and elastic half space: Emanated waves from deep pile foundations in the
ground are elastic waves in the form of shear wave compression waves and surface
waves (Fig. 1). Vertically polarized shear waves are generated
by soilshaft contact which propagate radially from the shaft on a cylindrical
surface; meanwhile, shear and compression waves propagate in all directions
from the toe, on a spherical wave front especially at the pile toe and Rayleigh
waves propagate radially on a cylindrical wave front along the surface. In an
elastic half space, both body waves and Rayleigh waves decrease in amplitude
with increasing distance from the pile foundation due to geometrical damping.
Theoretically, ground vibrations in the far field attenuate are inversely proportional
to the square of the area of the wave front or according to r^{n} where,
r is the distance and n is the geometrical attenuation coefficient which is
equal to 0.5 for surface waves propagating on a cylindrical wave front and equal
to 1 for body waves propagating on a spherical wave front in the interior of
the half space; for body waves propagating along the surface, n is equal to
2 (Wolf, 1994).
PROBLEM DEFINITION AND ASSUMPTIONS
The study examines the effects of geometrical and material parameters such
as depth and location of trench, on performance of an open trench in active
isolation system of a deep foundation. A rigid circular footing of radius 9
m (Bf), with 17 piles of diameter 80 cm and length L resting on a soil layer
of a limited thickness underlain by a hard stratum at a depth of H and length
Lh is subjected to a harmonic compressive surface load P sin(ωt) (Fig.
2). An annular open trench of depth D and width W is located at distance
of R from the edge of foundation (Table 2).
Table 2: 
Assumed geometrical properties of the trench and the piles 


Fig. 2: 
Problem definition: active isolation by open trench in deep
foundation 
Table 3: 
Soil properties 

Nearly saturated
clayey soil layer with uniform dynamic elastic modulus, Poisson’s ratio,
density and material damping is selected (Table 3). Material damping considered 0.05. The hard stratum was assumed to be very rigid
compared to the soil layer.
GEOMETRICAL MODEL
Taking advantage of axissymmetry in plan, only 1/4 of the actual model was required to be built resulting in significant reduction in computation time and efforts. Model dimensions were selected optimally large enough to prevent the surface wave reflection from boundaries. The depth of the model to prevent any base wave reflection is computed through trial and error as three times of the pile length and shall not be less than 30 m.
FINITE ELEMENT MODELING STRATEGIES
DruckerPrager yield criterion, for plastic deformations and yielding was applied to simulate the soil behavior which is defined by the cohesion value (C), the angle of internal friction (φ) and the dilatancy angle (ψ) (Drucke and Prager, 1952). The angle of dilatancy was assumed to be equal to the angle of internal friction (associated flow rule) and no strain hardening was assumed thus progressive yielding was neglected. Solid 45 element used to model the soil and the foundation block (Fig. 3). The element is defined by eight nodes having three degrees of freedom at each node: translations in the node x, y and z directions. The element has plasticity, creep, swelling, stress stiffening, large deflection and large strain capabilities (ANSYS Manual). To simulate behavior of the soil and the pile foundation such as sliding or any probable separation at the soilstructure interface more realistically, three dimensional pointtosurface contact elements have been employed.
Meshing and boundary condition: Precision is achieved by using smaller
element size in the vicinity of the foundation and the trench increasing in
size gradually with distance from the outer edge of the trench. Boundary conditions
are assigned by restraining the displacement in the x and y directions: the
y displacements in x direction together with x displacement along y axis are
restrained. The outer edge of model is also restrained in the x and y directions.
The hard stratum underlying the soil layer is assumed to be a rigid boundary
therefore the degrees of freedom in the three directions were restrained.

Fig. 3: 
Eightnoded element (solid 45) 


Fig. 4: 
The geometry of finite element model. (a) 3D model, (b) plane
and (c) elevation 
The geometry of FEM, meshing method and boundary conditions are shown in Fig.
4.
Damping model: The damping matrix [C] used in the harmonic, damped modal
and transient analyses as well as substructure generation is defined as:
Where:
[C] 
= 
Structure damping matrix 
[M] 
= 
Structure mass matrix 
[K] 
= 
Structure stiffness matrix 
α_{r} and β_{r} 
= 
Rayleigh damping 
β_{c} 
= 
Materialindependent damping multiplier 
β_{J} 
= 
Materialdependent damping multiplier 
N_{Mat} 
= 
Number of materials with DAMP or DMPR input 
[K_{J}] 
= 
Portion of structure stiffness matrix based on material j 
[C_{K}] 
= 
Element damping matrix 
[C_{s}] 
= 
Frequencydependent damping matrix 
In addition, the simplified form of the damping matrix [C] is calculated by
multiplying the following constants to the mass matrix [M] and stiffness matrix
[K]:
And the Rayleigh damping is materialdependent damping (βs) calculated
by Eq. 4:
In which ωi donates the natural circular frequency of mode i.
In many practical soil related problems, alpha damping is ignored, β_{s}
= 0, since it can lead to undesirable results, in case of introducing a large
mass into the FE model. Assuming β_{s} = 0.05, α_{r}
and β_{r} could be determined as following:
MODEL VERIFICATIONS
There is no literature relating to the isolation of deep foundations by trenches
in clayey soil and earlier researchers mostly have concentrate on vibration
reduction of trenches by shallow foundations in sandy soil, for the credibility
of the FE model. Firstly, a zero length pile foundation with materials similar
to Woods’s field study (19681969) and Ahmad’s
et al. (1996) BEM researches were utilized to validate the result
of current study. Figure 5 demonstrates a good agreement between
the current FEM model and the published data; however, the variation of amplitude
reduction ratio along the radial distance from the source is not identical.
This maybe accounted for the local inhomogeneity of the site test and errors
in the estimation of shear module of the soil layer.
Finally, the accuracy of the obtained results from the deep foundations is
validated by the assessment of the body wave propagation of deep foundations
by Eq. 1. Pile foundation can be classified as a point on
source generating body waves and the travel distance can be estimated as a horizontal
distance from the source. Figure 6 also exhibits a reasonable
agreement between the current FEM model and the Eq. 1. The
variation of displacements is not identical; this may be due to the approximations
used in deriving the equation.

Fig. 5: 
Comparative study for active isolation by open trench 

Fig. 6: 
Comparative study of obtained results with theoretical formulation 
DISCUSSION ON THE RESULTS OF FINITE ELEMENT ANALYSIS
Amplitude reduction ratio concept: Woods (1968)
studied the problem of screening of elastic waves by trenches to evaluate the
trench effectiveness and established Amplitude reduction ratio (Arr) which is:
The axissymmetric nature of the problem, amplitude reduction along all radial
lines should be identical. Therefore, the average value of Arr is computed along
a radial line (Ahmad et al., 1996):
Where:
r_{b} 
= 
The radial distance between the trench and the outer edge
of the barrier 
n 
= 
Represents the number of points along the radial distances 
For the sake of generalization, the geometric parameters and the results are
presented in a dimensionless form. The curves of Arr plotted against the depth
of trench and trench location normalized by length of pile and Rayleigh wavelength,
respectively, is shown in Table 4.
The ground displacement amplitudes are negligible after a distance of 10 λ_{r}
from the trench compared to those immediately after the trench thus the crucial
zone for screening lays within a distance of 10 λ_{r} from the
trench. Therefore, in all computations, the average Arr was calculated over
an area extending to a distance of 10 λ_{r} after the trench (Ahmad
and AlHussaini, 1991).
Due to similar behaviour of different soil types on trench efficiency, in the following only diagrams related to soil type II are presents.
Effect of geometrical parameters: Due to similar behavior of different soil types on trench efficiency, in the following only diagrams related to soil type II are presents.
Effect of trench depth: Figure 7 demonstrates the
effect of trench depth on vibration isolation of an open rectangular trench.
The horizontal axis shows normalized depth and the vertical axis indicates Arr.
In these diagrams, Normalized trench location is considered a constant parameter.
By and large, increasing the depth of the trench improves the performance as
deeper trenches reduce (or even eliminate) a larger amount of vibrations in
the path of the wave train causing less displacements beyond the trench.
For shallow piles (L = 5 m), Arr varies with a constant slope whereas the asymptotic
value of Arr can be observed in deeper pilefoundations (L ≥ 10 m) where
the value of Arr decreases dramatically with increasing the pile length. These
two regions are separated at d = 0.5. The efficiency of shallow trenches (d<0.5)
of deep pile foundation is function of trench location and varies significantly
with trench location. Exceeding the depth of a trench more than this limit (d>0.5)
has merely slight effects on the screening performance owning to the fact that
a large amount of vibration energy is transmitted away from surface energy and
increasing the depth is often of no use or its effect is incredibly small. In
case of pilefoundation with (L≥10 m) and trench depth more than 0.5 (L≥10
m), Arr is mostly independent of trench location, which may lead to an exceptionally
enormous vibration reduction; therefore feasibility studies are recommended
in terms of economy of construction a trench barrier since other vibration reductions
methods may be in scale of economy.
Table 4: 
Normalized geometrical parameters of trench with their normalized
values 



Fig. 7: 
Arr vs. normalized depth of trench for soil type II with pile
length of: (a) 5 m, (b) 10 m, (c) 15 m and (d) 20 m 
The Arr value for different normalized trench depths for short pile foundations
(L = 5 m) varies between 0.8 and 0.1, for 10 mlong pile foundations Arr ranges
between 0.3 and 0 and for longer piles L≥15 m it is between 0.15 and 0. Short
pile foundations are found to be more critical since practically all of vibratory
wave propagates in a zone close to the ground surface (beneath the ground surface,
with some distance from the soil surface). Owning to fact that surface waves
decay much more slowly with distance than the body waves therefore higher ground
displacements are predicted and special care should be taken. Conversely, the
body waves are typically generated by deeper pilefoundations. This vibration
energy may reach the ground surface and convert into surface Rayleigh waves
at far distance from the vibration source and will be damped due to geometrical
damping and material damping. For this reason Arr is minute. Consequently installing
open trench barriers may not be the best solution.
Effect of trench location: Figure 8 demonstrates the
effect of trench location on vibration isolation of a deep foundation. The horizontal
axis shows normalized trench locations and the vertical axis indicates Arr.
In these diagrams, Normalized depth considered a constant parameter. As the
trench location increases, Arr trend is upwards reaching a peak value and then
decreases.
The peak value and Arr curvature are more notable in shallow trenches. The
Arr curvature reduces and changes to a straight line with increasing trench
depth. The Mentioned peak value for short pile (L = 5 m) is located at:
Whereas, for longer pile foundation (L≥10 m) this value is measure at:
Location of a deep trenches to reduce vibration of deep pilefoundations, (L≥10
m), is found to be insignificant parameter in both shallow and deep trenches.
In these cases, similar behavior is observed and the average values of Arr are
similar. This may be justified by the theory of distribution of body waves in
a homogeneous, isotropic, elastic half space soil. Vibrations induced by deep
pilefoundation are mainly transmitted away from body waves passing through
the soil medium. A portion of vibration energy reaches the ground surface converting
to surface waves and part of it will be reflected. The mentioned vibration energy
is not capable of passing through a shallow trench located in vicinity of a
deep pilefoundation as the FE results indicate, better trench performance is
obtained by comparison. Increasing the distance between the vibration source
and trench provides adequate distance to convert body waves to Rwaves which
decay much more slowly with distance than the body waves.

Fig. 8: 
Arr vs normalized trench location for soil type II with pile
length of: (a) 5 m, (b) 10 m, (c) 15 m and (d) 20 m 

Fig. 9: 
Arr vs pile length for soil type II at: (a) r = 0.5 B_{f}/λ_{r},
(b) r = 1.0 B_{f}/λ_{r}, (c) r = 1.5 B_{f}/λ_{r}
and (d) r = 2.0 B_{f}/λ_{r} 
This may lead to an
increase in Arr values and a peak value in Arr diagram. For greater distances more than specified limit mentioned previously due to
significant damping, vibration energy decay and a downward trend in Arr diagram
is evident.
Pile length: Figure 9 shows the effect of pile length
in active isolation of deep foundations by rectangular trenches. The horizontal
axis shows length of pile and vertical axis indicates Arr. Normalized depth
was hold constant in proposed diagrams. Increasing pile length results in less
vertical displacements of deepfoundation therefore, larger piles transmit further
vibration energy transmits to deeper parts of the soil medium providing better
conditions for vibration isolation performance.
Two regions can be distinguished from the diagrams: A downward trend of Arr
together with an asymptotic region. These two regions are separated at a pile
length of 10 m. This border line may be considered as the depth where body waves
are typically produced and surface waves have merely a minute portion of vibration
energy. In all three different soil types, d≥0.5 are reported to be an identical
point in all diagrams highlighting that increasing trench depth does not noticeably
improve isolation performance of a trench barrier. A short pilefoundations
(L<5 m) located at:
causes higher ground vibrations by comparison and poor isolation performance
of open trench (Arr = 0.8) is achieved.
In cases of other pile lengths (L = 10, 15, 20 m), FE results predict lower ground vibrations, especially when the pile length is large and the material is dense. Based on FE results, Arr varies between 0.3 and 0.
Effect of soil properties: Soil type has only a slight effect on vibration reduction. Generally speaking, increasing soil density improves trench barrier performance. The effect of trench location will be reduced as the soil density increases.
Contour plots of Arrdepthlocation: Based on FE results, the contour
plots for Arr against nonnormalized trench depths and locations are presented
due to practical applications (Fig. 1012).
Selecting the desired Arr, the required trench location and depth can be achieved
from the contour plot. This may be useful when geometrical limitation, such
as trench location or depth, governs the design procedure to obtain moderated
vibrations energy to tolerable limit for sensitive structures and equipment.
Simplified mathematical formulation: The depth and location of an open
trench are two major parameters affecting the trench performance in very complicated
manners. The 3D surfaces of Arr (Fig. 13) against the depth
(d) and location (r) of a trench are plotted and attempts are made to formulate
the obtained FE results in to mathematical equations. Normalized trench location
(r) and trench depth (d) together with the effect of Rayleigh wavelength and
pile length are employed to provide a numerical formulation to evaluate Arr.

Fig. 10: 
Contour plots of Arrdepthlocation for soil type I, (a) L
= 5 m, (b) L = 10 m, (c) L = 15 m and (d) L = 20 m 

Fig. 11: 
Contour plots of Arrdepthlocation for soil type II, (a)
L = 5 m, (b) L = 10 m, (c) L = 15 m and (d) L = 20 m 
The vertical axis indicates Arr and horizontal axes are the normalized trench
depth and location.
As can be observed, for 5 m long pile foundation in all three soil types, linear
surfaces are seen and Eq. 9 can be devised to evaluate the
Arr.
For pile lengths equal or greater than 10 m, as mentioned before, the surface
are separated at d = 0.5, therefore, two different equation are proposed:

Fig. 12: 
Contour plots of Arrdepthlocation for soil type III. (a)
L = 5 m, (b) L = 10 m, (c) L = 15 m and (d) L = 20 m 

Fig. 13: 
Diagrams of Arr against normalized trench depth and location
for pile length of: (a) 5 m, (b) 10 m, (c) 15 and (d) 20 m 
The FE results agree reasonably well with those of obtained from the devised
equations.
CONCLUSION
Threedimensional FE analysis of a vibration isolation system of deep foundation
in active system has been conducted employing transient analysis. Since, there
no background in active isolation of deep foundation, this research performs
an extensive parametric study on open trenches attempting to fill the gap. A
deep foundation with 17 plies with four different lengths together a circular
trench of three different depths located at four locations are considered to
evaluate the performance of an open trench. More than 240 analysis were carried
out to obtain the results and the following conclusions may be drawn:
• 
A minimum trench depth of 0.5 times the pile length is the
optimal depth to attain an ideal reduction in ground displacement amplitude.
Construction of deeper trench is uneconomically practical 
• 
The critical trench location for short pile foundations (L = 5 m) is 

and for deep pile foundations (L≥10 m) is 
• 
The major portion of vibration energy transmits by Rayleigh
waves induced by short pile foundation therefore trench location and depth
should be carefully determined 
• 
Increasing soil density improves trench barrier performance 
• 
Amplitude reduction ratios for very deep piles are exponentially small 
• 
Efficiency of trench in very deep pile foundations majorly is independent
of trench depth and location 
• 
Proposed equations are to evaluate the efficiency of an open trench in
active isolation system of deep foundation 