This document summarizes an analytical study on how the natural frequency of a single pile foundation varies with soil-pile interaction effects. It presents the theoretical formulation for calculating the depth of fixity and natural frequency of a pile based on the soil modulus and pile diameter. A finite difference method and MATLAB code were used to model the soil-pile system and conduct a parametric study. The results show that the natural frequency non-linearly decreases with lower soil modulus and smaller pile diameter. An equation was developed through regression analysis to predict the natural frequency based on the soil and pile properties. Understanding how natural frequency varies with soil-pile interaction is important for designing foundations subjected to dynamic loads.

1. Proceedings of the International Conference on Emerging Trends in Engineering and Management (ICETEM14)
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ANALYTICAL STUDY ON SOIL-PILE INTERACTION
EFFECT IN THE VARIATION OF NATURAL
FREQUENCY OF A SINGLE PILE
Kavitha P. E.1
, Dr. Beena K. S.2
, Dr. Narayanan K. P.3
1
( Research Scholar, Cochin University of Science and Technology, Kalamassery, India)
2
(Professor, School of Engineering, Cochin University of Science and Technology, Kalamassery, India)
3
(Associate Professor, Dept. of Ship Technology, Cochin University of Science and Technology, Kalamassery, India )
ABSTRACT
Natural frequency is a fundamental property which determines the behaviour of the structure under dynamic
loading. It depends on the stiffness and mass of the structure. The stiffness of a structure with pile foundation varies with
the properties of soil. This paper is an analytical study on the variation of natural frequency of a pile embedded in
different soil conditions. The influence of soil modulus and pile diameter on the depth of fixity and the natural frequency
of the pile is studied. The governing equation representing the soil pile system is solved using a finite difference
procedure suggested by Reese et al. The procedure being highly iterative, a MATLAB-7 code is developed and a
parametric study has been conducted.
Keywords: Depth of Fixity, Natural Frequency, Soil Modulus, Soil-Pile Interaction, Surface Fitting.
1. INTRODUCTION
Pile supported structures are essential in coastal areas and at locations where the soil supporting the
superstructure is very loose and weak. Pile supported structures are subjected to significant amount of lateral forces due
to wind, wave, current, operating machinery and also the accidental forces. Structural designs are primarily based on
limiting deflection criteria. The deflection may be amplified or de-amplified based on the dynamic characteristics of the
soil-pile system. The non-linear behaviour of soil causes a variation in the dynamic characteristics of the soil-pile system.
2. LITERATURE REVIEW
Pile response to dynamic loading is dependent on a number of factors such as the material properties of soil and
pile, which in turn causes wave reflection, wave refraction, radiation damping and also the development of gap at the
soil-pile interface during the dynamic loading. To develop a comprehensive solution to the problem is quite complex.
However in the last few decades significant research has been done to understand the fundamental properties of pile
foundation under dynamic loading. Novak et al. (1974) have developed semi-analytical elastic continuum approach to
determining the stiffness and damping constants of single piles by assuming linear soil behaviour. Kuhlemeyer et al.
(1979) have attempted 2D finite element method to study the dynamic soil-pile interaction. Sawant et al. (1999) have
developed 3D finite element method to analyze the seismic response of piles using simple nonlinear models. Mostafa et
al. (2002) have developed models by accounting nonlinear soil behaviour and gapping. Gazetas et al.(2005) developed
simplified models to study the kinematic bending behaviour of piles based on Beam–on–Dynamic–Winkler–Foundation
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2. Proceedings of the International Conference on Emerging Trends in Engineering and Management (ICETEM14)
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(BDWF) models with linear behaviour of soil. The use of these simplified models is restricted to the situation where
linear soil behaviour prevails, and not appreciable to use where the soil nonlinearity governs the pile response. The
present study is also a simplified approach to study the change in the fundamental behaviour of soil-pile system due to
the change in the properties of soil and pile.
3. THEORETICAL FORMULATION
The natural frequency of a structure depends on the stiffness and mass of the structure. The stiffness of the
structure embedded in soil varies with the depth of fixity of the pile. The depth of fixity of the pile is calculated using the
sub-grade modulus method of Reese et al. [1]
The variation of un-drained shear strength along the depth of soil can be obtained from the soil investigation
report. A co-relation between the un-drained shear strength and soil modulus is obtained based on IS 2911 (Part 1/ Sec
2): 2010, Annex C, Analysis of laterally loaded pile [2]. The recommended values of soil modulus (Es) for cohesive soil
are adopted as per the equation Es , where D is the diameter of the circular pile and k1 is the modulus of sub-
grade reaction. Soil modulus values for a pile with diameter 1m are tabulated in Table1.
Table 1: Recommended Values of soil modulus [2]
Unconfined Compression Strength,
qu kN/m2
Modulus of Subgrade Reaction,
k1 X 103
N/m3
Soil Modulus,
Es N/m2
25 4.5 900
50 9.0 1800
100 18.0 3600
200 36.0 7200
400 72.0 14400
The finite difference method of analysis of the pile is used to calculate the deflection and bending moment
variation along the length of the pile and the depth of fixity of the pile [Reese et al.]. The soil modulus Es is assumed to
be linearly increasing with depth. The equilibrium equation representing a laterally loaded pile is,
(1)
The boundary conditions to solve the above equation are as follows. At the bottom of the pile, bending moment
and displacement are considered to be zero. At the top of the pile, the bending moment is the moment due to the point
load applied at a height above the ground level and the shear force is the applied load at the top of the pile. The soil-pile
idealization is represented in Fig.1.
Fig.1: Idealization of soil-pile system
Solving the equation no. (1), the variation of bending moment along the length of the pile was plotted. The finite
difference code generated with equation no. (1) is validated with the studies conducted by Reese et al. [1]. Fig.1 shows
the general nature of bending moment variation along the length of pile. The length measured from the ground level to

3. Proceedings of the International Conference on Emerging Trends in Engineering and Management (ICETEM14)
30 – 31, December 2014, Ernakulam, India
228
the point of maximum bending moment is considered to be the depth of fixity ( ) of the pile [3]. When the soil pile
system is considered for the dynamic analysis, the length of pile is considered to be the sum of the height (h) of the pile
above the ground level and the depth of fixity ( ) below the ground level.
The equation to calculate the circular natural frequency (w) of the pile, considering the mass (m), Young’s
modulus (E), moment of inertia (I) and length (l) of the pile is, [4]
(2)
The natural frequency,
(3)
Concrete circular piles are considered for this study with modulus of concrete N/m2
and density of
concrete kg/m3
. The cross sectional area , the mass , and .
The natural frequency of a soil-pile system with the pile head in level with the ground (h=0) is calculated with as,
(4)
Hence, the natural frequency of the soil-pile system varies with diameter of the pile ( ) and the depth of fixity
( ) as in equation no.(5)
(5)
4. RESULTS AND DISCUSSION
A parametric study has been conducted with the soil modulus ranging from 500 N/m2
to 40000 N/m2
and the
diameter of the pile is varied from 0.5 m to 2.0 m. The range of soil modulus has been considered as per IS2911 (part-1/
sec-2)-2010, as listed in Table 1. A MATLAB-7 coding has been developed to solve equation no.(1) and to predict the
variation of bending moment along the length of the pile. This can be used to predict the variation of the depth of fixity
of the system. The variation of depth of fixity with the soil modulus and the diameter of the pile is plotted and studied
using surface fitting and multiple regression analysis.
Fig.2: Variation of Depth of fixity with soil modulus and diameter of pile
From the Fig.2, it is clear that as the soil modulus decreases the variation of depth of fixity with the change in
the diameter of the pile becomes nonlinear. The variation of depth of fixity is predicted using multivariate regression
analysis with R2
=0.94 as,

4. Proceedings of the International Conference on Emerging Trends in Engineering and Management (ICETEM14)
30 – 31, December 2014, Ernakulam, India
229
(6)
Where, D is the diameter of the pile in metre and Es is the soil modulus in N/m2
.
The parametric study is now extended to calculate the natural frequency of the soil-pile system using equation
(5).The expected non-linear variation of natural frequency of the pile with variation in Es and D is plotted using surface
fitting in Fig.3 using MATLAB-7. It is extended to multiple regression analysis and an equation is predicted, with R2
=
0.84 to represent the natural frequency of the pile.
(7)
Fig.3: Variation of natural frequency with soil modulus and diameter of pile
5. CONCLUSION
Natural frequency of a soil-pile system becomes significant when it is subjected to dynamic loading. The
resonance of a structure occurs when the applied load is having a frequency nearer to the natural frequency of the
structure. Hence the natural frequency of the structure has to be kept well away from the predominant frequencies of the
external dynamic loading.
Coastal and offshore structures such as berthing structures, oil rig and wind mills are often subjected to
significant amount of lateral dynamic forces. Accurate assessment of the natural frequency of such a structure is
necessary to predict its behaviour under dynamic loading.
The equation no.(7) clearly shows the significance of soil modulus and the diameter of the pile in predicting the
natural frequency of a single pile. Hence for a particular soil condition, by changing the diameter of the pile within the
range of the parameters considered, the natural frequency of a soil-pile system can changed, using the equation no. (7).
If the predominant frequency of the dynamic loading happens to be same as the natural frequency of the soil-
pile system, there is a possibility of changing the natural frequency of the system by adjusting the diameter of the pile as
suggested by the equation no. (7). A detailed study on soil-pile interaction analysis is necessary for the satisfactory
design and performance of pile foundations especially for low rise structures and for structures embedded in soft soil.
REFERENCES
[1] Lymon C. Reese, William M. Isenhower and Shin Tower Wang, Analysis and Design of Shallow and Deep
foundations, John Wiley and Sons Inc., 2006
[2] IS 2911 (Part 1/ Section 2) – 2010, Design and Construction of Pile Foundations -Code of Practice, Part 1-
Concrete Piles, Section 2- Bored Cast In-situ Concrete Piles.
[3] Ayothiraman R and Boominathan A. , Depth of fixity of Piles in Clay Under Dynamic Lateral Load, Geotechnical
and Geological Engineering, 2013, 31, 447-461.
[4] William T. Thomson, Theory of Vibrations with Applications, Pearson Education, Eighth Impression, 2008.
[5] Anuj Chandiwala, “Fem Modeling for Piled Raft Foundation in Sand”, International Journal of Civil Engineering
& Technology (IJCIET), Volume 4, Issue 6, 2013, pp. 239 - 251, ISSN Print: 0976 – 6308, ISSN Online:
0976 – 6316.