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Research Article

Three Dimensional Analysis of Active Isolation of Deep Foundations by Open Rectangular Trenches

M. Jesmani, M.R. Shafie and R.S. Vileh
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Present study examines effects of geometrical parameters of annular open trenches, such as depth and location, on active isolation of deep foundations in different clayey soil. Three dimensional Finite Element Analyses (FEA) with ANSYS software are employed to carry out a parametric study. The point to surface contact element for the nonlinearity state of the soil-pile interface as well as nonlinear Drucker-Prager soil behavior under full transient analysis has been utilized to attain more realistic results. In this research, the critical trench location for pile foundations is obtained based on the FE results. Also, a minimum trench depth of 0.5 times the pile length is the optimal depth to attain an ideal vibration reduction and construction of deeper trench is uneconomically. Lastly based on results, the contour plots for Arr against trench depths and locations are presented for practical applications purposes. Simplified mathematical formulations based on the contour plots are derived to evaluate efficiency of a trench in active isolation system of deep foundation.

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M. Jesmani, M.R. Shafie and R.S. Vileh, 2009. Three Dimensional Analysis of Active Isolation of Deep Foundations by Open Rectangular Trenches. Journal of Applied Sciences, 9: 2544-2555.

DOI: 10.3923/jas.2009.2544.2555



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 ground-borne 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 Amplitude-reduction 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 in-filled trenches as well as pile wave barriers. In an extensive parametric study, Al-Hussaini and Ahmad (1996, 2000) concentrated on simplified design methodologies for vibration screening of machine foundations by trenches using a three-dimensional boundary element algorithm.

Kattis et al. (1999a, b) developed an advanced-frequency domain BEM code to study the screening efficiency of open, in-filled 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 cross-section piles have a similar behavior to those of square cross-section. 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 in-filled trenches in reducing the six-storey building vibrations due to passing trains using a two-dimensional 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 pulse-induced waves for shallow foundations resting on an elastic half-space. The efficiency of different types of wave barriers in vibration isolation for shock-producing 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 three-dimensional 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 in-filled 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 3-D FEA.


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 ra from the source) to point b (a location at distance rb from the source) is given by Eq. 1:


Table 1: Theoretical geometric attenuation coefficients Amick and Gendreau (2000)

Fig. 1: Wave propagation induced by deep foundation (Attewell and Farmer, 1973)

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 soil-shaft 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).


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.


Taking advantage of axis-symmetry 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.


Drucker-Prager 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 soil-structure interface more realistically, three dimensional point-to-surface 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: Eight-noded 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:


[C] = Structure damping matrix
[M] = Structure mass matrix
[K] = Structure stiffness matrix
αr and βr = Rayleigh damping
βc = Material-independent damping multiplier
βJ = Material-dependent damping multiplier
NMat = Number of materials with DAMP or DMPR input
[KJ] = Portion of structure stiffness matrix based on material j
[CK] = Element damping matrix
[Cs] = Frequency-dependent 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 material-dependent 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, βs = 0, since it can lead to undesirable results, in case of introducing a large mass into the FE model. Assuming βs = 0.05, αr and βr could be determined as following:



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 (1968-1969) 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


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 axis-symmetric 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):


rb = 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 λr from the trench compared to those immediately after the trench thus the crucial zone for screening lays within a distance of 10 λr from the trench. Therefore, in all computations, the average Arr was calculated over an area extending to a distance of 10 λr after the trench (Ahmad and Al-Hussaini, 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 pile-foundations (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 pile-foundation 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 m-long 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 pile-foundations. 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 pile-foundations, (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 pile-foundation 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 pile-foundation 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 R-waves 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 Bfr, (b) r = 1.0 Bfr, (c) r = 1.5 Bfr and (d) r = 2.0 Bfr

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 deep-foundation 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 pile-foundations (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 Arr-depth-location: Based on FE results, the contour plots for Arr against non-normalized trench depths and locations are presented due to practical applications (Fig. 10-12). 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 Arr-depth-location 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 Arr-depth-location 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 Arr-depth-location 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.


Three-dimensional 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
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