(i) The document discusses a computational approach using finite element method to analyze the dynamic stability of pile structures subjected to periodic loads. (ii) It develops the governing Mathieu-Hill type eigenvalue equation to determine stability and instability regions for different ranges of static and dynamic load factors. (iii) Key steps involve discretizing the pile into finite elements, developing element stiffness, mass and stability matrices, and assembling them to solve the eigenvalue problem and analyze dynamic stability conditions for the pile structure.

1. A COMPUTATIONAL APPROACH TO THE DYNAMIC STABILITY ANALYSIS
OF PILE STRUCTURES BY FINITE ELEMENT METHOD.
SUDIPTA CHAKRABORTY B.E(Cal),M.Tech(IIT),M.Engg(IHE,Delft),F.I.E(I),C.E
Manager(Infrastructure & Civic Facilities),Haldia Dock Complex,Kolkata Port Trust
Abstract
The Finite Element Approach to the Dynamic Stability Analysis of Pile Structures subjected to
periodic loads considering the soil modulus to be varying linearly has been discussed. The
Mathiew Hill type eigen value equation have been developed for obtaining the stability and
instability regions for different ranges of static and dynamic load factors..
Key words: Eigen value equation of Mathiew Hill type , the stability and instability regions ,f
static and dynamic load factors.
Introduction :The stability and instability of structural elements in Offshore Structures
viz. pile are of great practical importance. Piles are often subjected to periodic axial
and lateral forces. These forces result into parametric vibrations, because of large
amplitudes of oscillation.The studies on stability of structures subjected to pulsating
periodic loads are well documented by Bolotin (5). The study with axial loads were
carried out first by Beliaev (4) and later by Mettler (11) .. For simply supported
boundary conditions there are well-known regions of stability and instability for
lateral motion, the general governing equation for which being of Mathiew – Hill type
(5). In cases of typical structures with arbitrary support conditions, either integral
equations or the Galerkin’s method was used to reduce the governing equations of the
problem to a single Mathiew-Hill equation. Finite element method was used by
Brown et. al. (6) for study of dynamic stability of a uniform bar with various
boundary conditions and was investigated by Ahuja and Duffield (2) by modified
Galerkin Method. The behaviour of piles subjected to lateral loads was analysed in
Finite Element Method by Chandrasekharan (8). A discrete element type of numerical
approach was employed by Burney and Jaeger (7) to study the parametric instability
of a uniform column. The most recent publications on stability behaviour of structural
elements are provided by Abbas and Thomas (1).
1. Analysis :
The equation for the free vibration of axially loaded discretised system(9) in which rotary
and longitudinal inertia are neglected is :[M] {q˚ ˚
}+[Ke]{q} – [S]{q} = 0 ………(1),
in which {q} = generalised co-ordinate, [M] = Mass matrix, [Ke] = elastic stiffness matrix,
and [S] = Stability matrix , which is a function of the axial load.The general governing
equation of a pile (8) under lateral load is given as
)2.......(....................2
2
2
2
yE
dx
yd
EI
dx
d
S−=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
Where, EI, Es and y are the flexural rigidity, soil modulus and lateral deflection
respectively at any point x along thelength of the pile. The analytical solution of the
equation for y in case of a pile with flexural rigidity and soil modulus constant with depth
is available which can lead to generate design data like Moment and Shear but in
nature the soil modulus and flexural rigidity may vary with depth (8). Moreover, the Es
may also depend on the deflection y of the pile, the soil behaviour, making Es non-
linear, the analytical solution for which is highly cumbersome. Even with a single case
when variation of Es is linear of the form (C1 + C2 x), is also difficult and one has to
resort to numerical approaches like finite difference or finite element method.
Considering a system subjected to periodic force P(t) = Po+Pt Cos Ω t, where Ω is the
disturbing frequency, the static & time dependent components of load can be
represented as a fraction of the fundamental static buckling load P* viz. P = αP* + βP*

2. Cos Ωt with α & β as percentage of static and buckling load P*,the governing equation
transforms to the form
[M]{q˚˚
}+( [Ke] – αP*[Ss] – βP*CosΩt[St] ){q} = 0 …….. …………….(3)
The matrices [Ss] and [St] reflect the influences of Po & Pt. The equation represents a
system of second order differential equation with periodic co-efficient of Mathiew-Hill
type. The boundaries between stable and unstable regions are catered by period
solutions of period T and 2T where T=2π/Ω.
If the static and time dependent component of loads are applied in the time manner, then
[ ]{ } )5....(..........0][][ 2
=− qMKe λ
[ ]{ } )6....(..........0][][*] 2
=−− qMSPKe λα[
[ ]{ } )7....(..........0][*][ =− qSPKe
determined (6) from the equation :
values bounding the regions of instability as the two
ms :
(i) Free Vibration = 0, λ = ω1/2 the natural frequency,
(ii λ= Ω/2
(iii) Static Stability with α = 1, β = 0 and Ω = 0
(iv Dynamic stability when all terms are present.
The problem then remains with generation of [Ke], [S] and [M] for the pile. The fundamental
nto a number of finite elements, (element shown in
umed to be generalised polynomials of the most
α-s
the element displacement vector for an element of length
{qe} ……… (9)
[Ss] ≡ [St] ≡ [S].and the boundaries of the regions of dynamic instability can be
This is resulting in two sets of Eigen
( ) )4.......(..........0][
4
][*
2
1][
2
=
⎭
⎬
⎫
⎩
⎨
⎧
⎥
⎦
⎤
⎢
⎣
⎡ Ω
−±− qMSPKe βα
conditions are combined under plus and minus sign. For finding out the zones of
dynamic stability, the disturbing frequency Ω is taken as, Ω=(Ω/ω1) ω1 ,where ω1 = the
fundamental natural frequency as may be obtained from solution of equation (5).
The above equation (4) represents cases of solution to a number of related proble
with α = 0, β
) Vibration with static axial load: β = 0,
)
natural frequency and the critical static buckling load are to be solved from equations (5) and
(7). The regions of dynamic stability can then be solved from the equation (4).
Element Stiffness & Mass Matrices.
Assuming that the pile is discretized i
Fig.1)each element has two nodes i & j. Three degrees of freedom i.e. axial and lateral
displacement u, v and rotation θ = dv/dx are considered for each nodal point. The
generalised forces corresponding to these degrees of freedom are the axial & lateral force
P,Y and the moment M. The nodal displacement vector for the Finite Element Model using
Displacement function for the element in Fig.1 is :{qe} = [ xi yi θi xj yj θj ]T
and the
corresponding elemental force vector is given by
{Fe} = [ Pi Yi Mi Pj Yj Mj ]T
.
The displacement functions are ass
common form v(x) = α1 + α2 x + α3 x2
+ α4 x3
or, {v(x)} = [p(x)]{ α}………………(8)
The no. of terms in the polynomial determines the shape of displacement model where
determine the amplitude. The generalised displacement models for any element are as
follows: u = α1 + α2 x; v = α3 + α4 x +α5 x2
+ α6 x3
& θ = dv/dx = α4 + 2α5 x + 3α6 x2
.
Substituting the nodal co-ordinates
“l”, {q} can be written as
{q} = [A] {α} or, {α} = [A]-1

3. 3
EI
DΔ
L
⎥
⎦
⎢
⎣
−
L
AE
L
AE2 ⎥
⎥
⎤
⎢
⎢
⎡ −
= L
AE
L
AE
K U][
u1
u2
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎥
⎥
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
−
−−
−−
15
2
10
1
5
6
3010
1
15
2
105105
[
L
L
LL
LL
⎤⎡− 1616
=] PK 3U
V1 ⎤⎡ SSSS
θ1
V2
⎥
⎥
⎥
⎦
⎥
⎢
⎢
⎢
⎢
⎣
=
44
3433
24232221
14131211
4
][
S
SS
SSSS
K U
= [N(x)]{q} ……………..(10),
l expansions for u
nd v, the strain energy expression becomes
he strain energy U of an elemental length l of a pile subjected to an axial load & lateral load
From the first term of U, the stiffness matrix from U1 only, for bending only is [K]U1 as given in
2
2(a)StiffnessMatrix (for bending) 2 (b) StiffnessMatrix (for axial load)
2 (c) StiffnessMatrix (Beam Column Action) 2 (d) StiffnessMatrix(All Action)
Figure. 2. Stiffness Matrices
lly and axially the expression for
is given by,
A {u2
+ v2
}dx………………………………..(13)
Therefore,from (8), {v(x)} = [p(x)] [A]-1
{qe}
where matrix [N(x)] is the element shape function. Assuming polynomia
a
T
Fig. 2(a).The stiffness matrix from 2nd term U2 for axial deformation only will be [K]U2 as
given in Fig. 2(b).For axial load only i.e. by considering the beam column action the stiffness
matrix due to U3 will be [K]U3 as in Fig. 2(c).Using equation (8) and equation (9), equation
(12) can be simplified and stiffness matrix can be evaluated as [K]U4 as in Fig. 2(d).
When all the four cases are considered, i.e. all the four terms of U1, U2, U3, U4 are involved
the stiffness matrix KU1, KU2, KU3, KU4 are super imposed which yields final stiffness
matrix [K]e as given in Fig. 3(a).
v1 θ1 v1 θ2
Where a = 12 D, b = 6LD, c = 4L2
D, d = 2L D Where
The expression for kinetic energy for a pile loaded latera
strain energy
l l
T = ½ ∫μ{u2
+ v2
}dx = ½ ∫ ρ
0 0
dx
dx
du
EAdx
dx
vd
EIU
ll
00
2
2
2
1
2
1
∫∫ ⎟
⎠
⎞
⎜
⎝
⎛
+⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
22
)11...(..........
2
1
2
1 2
00
dxvEdx
dx
dv
P S∫∫ +⎟
⎠
⎞
⎜
⎝
⎛
−
2 ll
.UUUU +++= 4321
)12(....................
0
dxuv
⎥
⎥
⎦⎣
⎟
⎠⎝
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−
−
=
c
ba
baba
K U1
][
2
1
2
1 22
4
12
1
L
EE
vEU
l
SS
S∫
⎤
⎢
⎢
⎡
⎟
⎞
⎜
⎜
⎛ −
+=
⎢ − dbc
(for bending only) (for axial load only)

4. ⎟
⎞
⎜
⎛
+⎟
⎞
⎜
⎛
−=
x
u
x
uu 1
⎠⎝⎠⎝ ll
21
420
M
ALρ
=
3
DΔ
al and transverse
(14)
………………………………… (15)
For bending vibration only :
6)
(10) as
Fig. 3(a) Element Stiffness Matrix
2 2
in [K]U4
iven in.fig 3(b).
al and μ , the mass per unit length = A x ρ
m
The solution to the problem follows the well known displacement approach which
L
Where μ = mass per unit length of the pile, u and v are the axi
displacement. Using expressions for u & v, T = ½ [q]T
[M]{q}
For axial vibration only : l
T = ½ ∫μu2
dx …………………….
0
The displacement model for axial displacement is taken as
l
T = ½ ∫μv2
dx…………………….(1
0
The displacement model for lateral displacement is given by
v = N1v1 + N2θ1+ N3v2 +N4θ2 …………………………………………… (17)
Where Ni i =1,4 are the standard shape functions as derived from equation
EI
Where a = 12 D, b = 6LD, c = 4L D, d = 2L D & S11 – 44 as
So, from the expression of T. Mass Matrix [ M ] can be determined as g
Fig.3(b) Mass Matrix
Where , A is area of c/s. ρ is the density of materi
&
2. Analysis of the whole proble
consists of the following main steps :
• Formulation of overall stiffness and mass matrices by assembling the elemental
matrices.
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+−=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+−=
⎟
⎠
⎜
⎝
2
32
4
3
3
2
2
3
2
32
2
21
11
1
2
1
3
11
2
131
xx
N
xx
N
xx
xN
N
[ ]
⎟
⎞
⎜
⎛
+−=
32
23
1
xx P
e
S
L
c
SbS
L
a
L
AE
S
L
dSbSc
SbS
L
aSS
L
a
LL
K
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
+−
+−−+−
−
+++−−+−
++++−+++−
=
44
3433
242322
14131211
15
2
00000
10
1
5
6
0000
00000
3010
1
0
15
2
00
10
1
5
6
0
10
1
5
6
0
0000
[ ]
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣ −−−
−
−
−
=
MLLMMLLM
LMMLMM
MM
MLLMMLLM
LMMLMM
M
22
22
42203130
22156013540
001400070
31304220
13540221560
AEAE⎡ −
L
b
⎥
⎤
⎢
⎡ MM 007000140

5. [ ] [ ]e
qun.(6).
f boundary conditions also yield solution for the nodal
ich of course also leads
ations of the assembly may be written as [ K ] { δ } = {F}
nown displacement conditions are introduced in the equation and the equations are
and the
3.
es of static load factor, α , which will give
.
Notation :A Area of Gross section. E Modulus of Elasticity.
[ K ] Stiffness Matrix l Elemental length
]
P * Static
ates
t Variable time T Kinetic Energy
v
te y Lateral Co-ordinate
F
Ω requency μ Mass per unit length.
P t
S11 72C
S12 L2
( - 3B – 3C)
)
E
e KK 1=∑=
to solution for design data, like shear force and bending moments at nodal points.If n
denotes the number of nodes, then the total number of degrees of freedom for the
problem is equal to 3n. The expanded element stiffness matrices Ke are constructed
by inserting the stiffness co – efficients in the appropriate locations and filling the
remaining with zeros. If E is the number of elements then the overall stiffness matrix [
K ] is given by
• Solution for the fundamental natural frequency from equn. (3) & critical static buckling
load from e
• Solution for the dynamic stability regions from equation (10).
The application o
displacements from the generalised equilibrium equation wh
The equilibrium equ
K
solved for unknown nodal displacements (8). Commonly the symmetry
banded nature of the resulting equations are utilized for efficient computing.After
assemblage of stiffness and mass matrices, the eigen value problem in equation (10)
can be solved for the frequency ratio Ω/ω1.
Conclusion : The characteristic non-dimensionalised regions in (β, Ω/ω1) parameter
space can be extrapolated for different valu
rapid convergence characteristics of the boundary frequencies for the first few
instability regions (9).After obtaining the results for lower boundary and upper
boundary for instability regions the may be compared with Mathiew’s diagram (5).
4
[ M Mass Matrix N Shape Function
P Axial Periodic load Fundamental
Buckling load
{q} Generalised Co-ordin [ S ] Stability Matrix
U Strain Energy
u Axial displacement of node Lateral displacement of node
x Axial co-ordina
α Static load factor β Dynamic load factor
ρ Density
ω1 undamental Natural Frequency
Disturbing F
o
= 156B + S = L (13B + 14C)
, P Time independent amplitudes of load
23
= L (22B + 14C) S24 =
S13 = 54B + 54C S33 = 156B + 240C
S14 = L ( -13B –12C) S34 = L ( - 22B – 30C
2
)
2
S22 = L (4B + 3C S44 = L (4B + 5C)
Where, B = ES1. L/420 C = (ES2 – ES1). L/840

6. y,v Es1
x,u
θ
P(t)
Es2
l
x,u
ui,vi, θi
i
j
uj, vj, θj
Figure 1 : Typical Pile Element
REFERENCES:
1. Abbas, B.A.H. and Thomas, J – Dynamic stability of Timoshenko beams
resting on an elastic foundation.- Journal of sound and vibration, vol. – 60, N0. PP – 33 –44, 1978.
2. Ahuja, R. and Duffield, R.C.. – Parametric instability of variable cross – section beams resting on an elastic foundation.–
Journal of sound and vibration, Vol. 39, No.2, PP 159 – 174, 1975.
3. Beilu, E.A. and Dzhauelidze, G.- Survey of work on the dynamic stability of elastic systems, PMM, Vol. 16 PP635 – 648, 1952.
4. Beliaev N.M. – Stability of prismatic rods subjected to variable longitudinal force, Engineering Constructions and Structural
Mechanics, PP, 149 – 167, 1924.
5. Bolotin V.V.- The dynamic stability of elastic systems, Holden – Day Inc, 1964.
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1425, 1968.
7. Burney, S.Z. H and Jaeqer, L.G. –m A method of deter – mining the regions of instability of column by a numerical metos
approach, Journal of sound and vibration, Vol .15, No.1 PP- 75 – 91, 1971.
8. Chandrasekharan, V.S. – Finite Element Analysis of piles subjected to lateral loads – Short term courseon design of off shore
structures 3 – 15, July,1978, Civil Engineering Department, I.I.T. Bombay – Publications.
9. Dutta, P.K. and Chakraborty, S. – Parametric Instability of Tapered Beams by Finite Element Method – Journal of Mechanical
Engineering Science, London, Vol. –24, No. 4, Dec. 82, PP 205 –8.
10. Lubkin, S. and Stoker, J.J. – Stability of columns and strings under periodically varying forces. Quarterly of Applied
Mathematics, Vol. –1, PP 216 – 236, 1943.
11. Mettler, E. – Biegeschwingungen eins stabes unter pulsierenre axiallast, Mith . Forseh.- Anst. GHH Korzeren, Vol. 8, PP 1-12,
1940.
12. Pipes L.A. – Dynamic stability of a uniform straight column excited by pulsating load, Journal of the Franklin Institute, Vol . 277
No .6, PP 534 – 551, 1964.