The document discusses a computational approach to analyzing the dynamic stability of pile structures subjected to periodic loads using finite element methods. It develops a Mathieu-Hill type eigenvalue equation to determine stability and instability regions for piles under varying static and dynamic load factors. The approach discretizes piles into finite elements and develops element stiffness, mass, and stability matrices. It then derives a governing equation of Mathieu-Hill type to analyze the pile's dynamic stability for different load conditions.

3. 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 PortTrust
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
d
2
⎞
dx
2
2
..........
.......... .......( )2 ⎛
⎜
⎜

4. ⎝
EI
d
dx
2
y ⎟
⎟
⎠
=
− E
S
y 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*

7. Cos Ωt with α & β as percentage of static and buckling load P*,the governing equation
transforms to the form
[M]{q˚˚}+( [Ke] – αP*[S
s
] – βP*CosΩt[S
t
] ){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
[Ss] ≡ [S
t
] ≡ [S].and the boundaries of the regions of dynamic instability can be
determined (6) from the equation :
⎡ ⎢
⎣
[
Ke ] − ( α
± 1
2
β )
P [*
S ] − Ω
4
2
[
M ] ⎤
⎥
⎦
⎧
⎨
⎩
q ⎫
⎬
⎭
=
0 .......... .......( )4 This is resulting in two sets of Eigen
values bounding the regions of instability as the two
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/ω
1
ms :
(i) Free [ [ Ke ] − Vibration λ

8. 2 [ M ] ]{ q }
= 0 .......... ....( )5 = 0, λ = ω
1
[ [
Ke ] − α P [* S ] − λ 2 [ M ] ]{ q }
= 0 .......... ....( )6 [ [ Ke
] − P [* S ] ]{ q }
= 0 ..............( )7 ) ω
1
,where ω
1
= 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
/2 the natural frequency,
(ii ) Vibration with static axial load: β = 0,
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
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
nto a number of finite elements, (element shown in
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 :{q
e
umed to be generalised polynomials of the most
α-s
the element displacement vector for an element of length
{q
e
} = [ xi yi θi xj yj θj ]T and the
corresponding elemental force vector is given by
{F
e
} = [ Pi Yi Mi Pj Yj Mj ]T.

9. The displacement functions are ass
common form v(x) = α
1
x2 + α
4
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 = α
1
+ α
2
x + α
3
+ α
2
x; v = α
3
+ α
4
x +α
5
x2 + α
6
x3
& θ = dv/dx = α
4
+ 2α
5
x + 3α
6
x2 .
Substituting the nodal co-ordinates
“l”, {q} can be written as
{q} = [A] {α} or, {α} = [A]-1
} ……… (9)

12. Therefore,from (8), {v(x)} = [p(x)] [A]-1{q
e
}
= [N(x)]{q} ……………..(10),
where matrix [N(x)] is the element shape function. Assuming polynomia
l expansions for u
a
nd v, the strain energy expression becomes
U
=
1 2
2
2 l
2
l 2
..........
...( 11 ) T
he strain energy U of an elemental length l of a pile subjected to an axial load & lateral load
=
U 1
+ U 2 + U 3 + U 4 . From the first term of U, the stiffness matrix from U
1
Fig. 2(a).The stiffness matrix from 2nd term U2 for axial deformation only will be [K]
U2
..........
.......... ( 12 ) 0
l
EI ∫ ⎛
⎜
0
d
2
v ⎞
dx
⎟
dx
+
1
2
l ⎜
⎝
2
⎟

13. ⎠
EA ∫ 0 ⎛
⎜
⎝
du
dx
⎞
⎟
⎠
dx
−
1
2
P
∫ 0
⎛
⎜
⎝
dv
dx
⎞
⎟
⎠
dx
+ 1
∫ 0 E S
v 2
dx 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)
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

14. 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).
U
4
l
=
∫
⎡
⎢
⎢
⎣
1
2
E S
1 v 2
+ 1
2
⎛ ⎜
⎜
⎝
E
S
2
−
E S 1 L
⎞
⎟
⎟
⎠ uv
2 ⎤
⎥
⎥
⎦ dx v
1
θ
1
v
1

15. θ
2
⎡
⎢
a
b c
−
a − b b d (for bending only)
⎥
(for axial load only)
Where a = 12 D, b = 6LD, c = 4L2D, 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
u
1
K
U
1
=
⎢
−
u
2
c
⎥
⎤
⎥
⎥
⎢
⎣
⎦
⎡ −
=
AE
AE
[
] [
K
] U
2
⎢
⎢

16. ⎢
⎣
−
L
AE
AE
L
⎢
a
b ⎥
L
L
⎤
⎥
⎥
⎥
⎦
DΔ
EI
L
3
⎡ − 6
1 6 1 ⎤ K
] = P 3
⎢
⎥
V
1
⎡ S
11
S 12 S 13 S 14 ⎤ θ1
[
K
] U
4
=
V2
⎢
⎢
⎢
⎢
⎣
S
21

17. S 22 S 23 S 24 ⎥
⎥
⎥
S
44
⎦
⎢
⎣
10 [
⎢
⎢
⎢
5
L
−
10 2
L
5 − L 1
15
10
L U
−
6
30 − 1
⎢
⎢
⎢
⎥
⎥
⎥
⎥
5
L
−
10
2
L
S
33
S 34 ⎥
15
⎦
⎥
⎥
/

20. al Where μ = mass per unit length of the pile, u and v are the axi
and transverse
displacement. Using expressions for u & v, T = ½ [q]T[M]{q}
For axial vibration only : l
T = ½ ∫μu2 dx …………………….
(14)
0
The displacement model for axial displacement is taken as
u =
u 1
⎛
⎜
⎝ 1
− x
l
⎞
⎟
⎠ + u
2 ⎛
⎜
⎝ x
l ⎞
⎟
⎠ …………………………………(15)
For bending vibration only :
l
T = ½ ∫μv2 dx…………………….(1
6)
0
The displacement model for lateral displacement is given by
v = N
1
v
1
+ N
2
θ
1
+ N
3
v
2
+N
4

21. θ
2
…………………………………………… (17)
Where N
i
i =1,4 are the standard shape functions as derived from equation
(10) as
N
1
=
⎛
⎜
⎜
⎝
1
− 3
1
x
2
2 + 2 13 x 3 ⎞
⎟
⎟
⎠
⎡ AE 0
DΔ
EI
0 −
AE 0 0 ⎤
P
L
[ ]
e
N
x
x 1
1 2
⎢
S
Fig. 3(a) Element Stiffness Matrix
Where a = 12 D, b = 6LD, c = 4L 2 D, d = 2L 2 D & S11 – 44 as
in [K]U4
So, from the expression of T. Mass Matrix [ M ] can be determined as g
iven in.fig 3(b).
[ ]

22. ⎥
Fig.3(b) Mass Matrix
Where , A is area of c/s. ρ is the density of materi
al and μ , the mass per unit length = A x ρ
&
420
2. Analysis of the whole proble
m
The solution to the problem follows the well known displacement approach which
consists of the following main steps :
• Formulation of overall stiffness and mass matrices by assembling the elemental
matrices.
⎥
N
2
=
⎛
⎜
⎜
x − 2
x
2
+ 3 ⎞
⎢
⎢
⎢
⎝
1
x 1 2
⎟
⎟
⎠
L
⎢
0
a
−
5
6
L
+ S
b
+ 0
L
1

23. 3
10
+ S K
=
11
12 0 c
2
15
S L 0
− a + 5
6
L
+ S
b + 1
+ S 14 ⎥
⎥
0
⎣
44
⎥
⎦
⎥
13 ⎢
⎢
⎢
b 1
S d 10
L
⎥
N
3
=
⎛
⎜
⎜
⎝
3
x
2
1
2
− 2
x 3
⎞
⎟
⎟

24. ⎠
⎢
0
0 −
+ 22
0 −
AE
− − 10
+ 23 0 + 30 + S
24 1
3
⎢
⎢
⎢
0
0 0 L
0 0 4
=
⎛
⎜
⎜
⎝
− 2
+ 3 ⎞
⎟
⎠
⎢
0
0 0 0 a
−
5
6
L
+ S
33
0 − ⎟
c
b −
− 2
15
10
L
1
+ + S 34 ⎥
⎥
⎥

25. ⎥
⎥
⎥
⎥
⎥
⎡ 140
M 0 0 70 M M
=
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
LM
L M 0 0 ⎤
0
156 M
22 LM 0 54 M −
13 LM 0
22 LM
4 L 2
M 0 13 LM −
3 L 2 M 70
M
0 0 140 M 0 0 0
54 M
13 LM 0 156 M −
22 LM 0
−
13 − 3 2
0 − 22 LM 4 L 2 M ⎥
⎥
⎥
⎥
⎥
⎥
⎦ M
=
ρ
AL

28. • Solution for the fundamental natural frequency from equn. (3) & critical static buckling
load from e
qun.(6).
• Solution for the dynamic stability regions from equation (10).
The application o
f boundary conditions also yield solution for the nodal
displacements from the generalised equilibrium equation wh
ich of course also leads
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
[ K
]
=
∑ e
E
1=
[ K ]
e
The equilibrium equ
ations of the assembly may be written as [ K ] { δ } = {F}
K
nown displacement conditions are introduced in the equation and the equations are
solved for unknown nodal displacements (8). Commonly the symmetry
and the
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/ω
1
.
3.
Conclusion : The characteristic non-dimensionalised regions in (β, Ω/ω
1
) parameter
space can be extrapolated for different valu
es of static load factor, α , which will give
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

29. .
Notation :A Area of Gross section.E Modulus of Elasticity.
[ K ] Stiffness Matrix l Elemental length
[ M ]
Mass Matrix N Shape Function
P Axial Periodic load P * Fundamental
Static
Buckling load
{q} Generalised Co-ordin ates
[ S ] Stability Matrix
t Variable time T Kinetic Energy
U Strain Energy
u Axial displacementofnode v
Lateral displacementofnode
x Axial co-ordina
te y Lateral Co-ordinate
α Static load factor β Dynamic load factor
ρ Density
ω
1
F
undamental Natural Frequency
Ω Disturbing F
requency μ Mass per unitlength.
P
o
, P t
Time independentamplitudes ofload
S
11
= 156B + 72C
S
12
= L (22B + 14C) S 23
S
24
= =
L L
2
(13B ( - 3B + – 14C)
3C)
S
13
= 54B + 54C S
33
= 156B + 240C
S
14
= )
S
22
= L L 2
( (4B -13B + –12C) 3C )
S

30. 34
= S
44
= L L 2
( (4B - 22B + 5C)
– 30C
Where, B = E
S1
. L/420 C = (E
S2
– E
S1
). L/840

33. y,v Es
1
x,u
i
l
ui,vi, θi
j
u
j
, v
j
, θ
j
Es
2
x,u
Figure 1 : Typical Pile Element
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Journal of sound and vibration, Vol. 39, No.2, PP 159 – 174, 1975.
3. Beilu, E.A. and Dzhauelidze, G.- Survey of work on thedynamic stability of elastic systems, PMM, Vol. 16 PP635 – 648,
1952.
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–
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12,
1940.
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No .6, PP 534 – 551, 1964.
θ
P(t)