A COMPUTATIONAL APPROACH TO THE DYNAMIC STABILITY ANALYSISOF PILE STRUCTURES BY FINITE ELEMENT METHOD.SUDIPTA CHAKRABORTY B.E(Cal),M.Tech(IIT),M.Engg(IHE,Delft),F.I.E(I),C.EManager(Infrastructure & Civic Facilities),Haldia Dock Complex,Kolkata Port TrustAbstractThe Finite Element Approach to the Dynamic Stability Analysis of Pile Structures subjected toperiodic loads considering the soil modulus to be varying linearly has been discussed. TheMathiew Hill type eigen value equation have been developed for obtaining the stability andinstability regions for different ranges of static and dynamic load factors..Key words: Eigen value equation of Mathiew Hill type , the stability and instability regions ,fstatic and dynamic load factors.Introduction :The stability and instability of structural elements in Offshore Structuresviz. pile are of great practical importance. Piles are often subjected to periodic axialand lateral forces. These forces result into parametric vibrations, because of largeamplitudes of oscillation.The studies on stability of structures subjected to pulsatingperiodic loads are well documented by Bolotin (5). The study with axial loads werecarried out first by Beliaev (4) and later by Mettler (11) .. For simply supportedboundary conditions there are well-known regions of stability and instability forlateral motion, the general governing equation for which being of Mathiew – Hill type(5). In cases of typical structures with arbitrary support conditions, either integralequations or the Galerkin’s method was used to reduce the governing equations of theproblem to a single Mathiew-Hill equation. Finite element method was used byBrown et. al. (6) for study of dynamic stability of a uniform bar with variousboundary conditions and was investigated by Ahuja and Duffield (2) by modifiedGalerkin Method. The behaviour of piles subjected to lateral loads was analysed inFinite Element Method by Chandrasekharan (8). A discrete element type of numericalapproach was employed by Burney and Jaeger (7) to study the parametric instabilityof a uniform column. The most recent publications on stability behaviour of structuralelements are provided by Abbas and Thomas (1).1. Analysis :The equation for the free vibration of axially loaded discretised system(9) in which rotaryand 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 governingequation of a pile (8) under lateral load is given asd2⎞dx22.................... .......( )2 ⎛⎜⎜

⎝EIddx2y ⎟⎟⎠=− ESy Where, EI, Es and y are the flexural rigidity, soil modulus and lateral deflectionrespectively at any point x along thelength of the pile. The analytical solution of theequation for y in case of a pile with flexural rigidity and soil modulus constant with depthis available which can lead to generate design data like Moment and Shear but innature the soil modulus and flexural rigidity may vary with depth (8). Moreover, the Esmay 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 casewhen variation of Es is linear of the form (C1 + C2 x), is also difficult and one has toresort 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 thedisturbing frequency, the static & time dependent components of load can berepresented as a fraction of the fundamental static buckling load P* viz. P = αP* + βP*

Cos Ωt with α & β as percentage of static and buckling load P*,the governing equationtransforms 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 asystem of second order differential equation with periodic co-efficient of Mathiew-Hilltype. The boundaries between stable and unstable regions are catered by periodsolutions 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] ≡ [St] ≡ [S].and the boundaries of the regions of dynamic instability can bedetermined (6) from the equation : ⎡ ⎢⎣[

Ke ] − ( α± 12

β )P [*S ] − Ω42[M ] ⎤⎥⎦⎧⎨⎩q ⎫⎬⎭=0 .......... .......( )4 This is resulting in two sets of Eigenvalues bounding the regions of instability as the twoconditions are combined under plus and minus sign. For finding out the zones ofdynamic stability, the disturbing frequency Ω is taken as, Ω=(Ω/ω1ms :

(i) Free [ [ Ke ] − Vibration λ

2 [ M ] ] q = 0 .......... ....( )5 = 0, λ = ω1

[ [

Ke ] − α P [* S ] − λ 2 [ M ] ] q = 0 .......... ....( )6 [ [ Ke

] − P [* S ] ] q = 0 .......... ....( )7 ) ω1,where ω1= thefundamental natural frequency as may be obtained from solution of equation (5).The above equation (4) represents cases of solution to a number of related problewith 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 fundamentalnatural 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 into a number of finite elements, (element shown inFig.1)each element has two nodes i & j. Three degrees of freedom i.e. axial and lateraldisplacement u, v and rotation θ = dv/dx are considered for each nodal point. Thegeneralised forces corresponding to these degrees of freedom are the axial & lateral forceP,Y and the moment M. The nodal displacement vector for the Finite Element Model usingDisplacement function for the element in Fig.1 is :qeumed to be generalised polynomials of the mostα-sthe element displacement vector for an element of lengthqe = [ xi yi θi xj yj θj ]T and thecorresponding elemental force vector is given byFe = [ Pi Yi Mi Pj Yj Mj ]T.

The displacement functions are asscommon form v(x) = α1x2 + α4x3 or, v(x) = [p(x)] α………………(8)The no. of terms in the polynomial determines the shape of displacement model wheredetermine the amplitude. The generalised displacement models for any element are asfollows: u = α1+ α2x + α3+ α2x; v = α3+ α4x +α5x2 + α6x3

& θ = dv/dx = α4+ 2α5x + 3α6x2 .Substituting the nodal co-ordinates“l”, q can be written asq = [A] α or, α = [A]-1 ……… (9)

Therefore,from (8), v(x) = [p(x)] [A]-1qe= [N(x)]q ……………..(10),where matrix [N(x)] is the element shape function. Assuming polynomial expansions for uand v, the strain energy expression becomesU=1 222 l2

l 2.............( 11 ) The 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 U1Fig. 2(a).The stiffness matrix from 2nd term U2 for axial deformation only will be [K]U2

..........

.......... ( 12 ) 0l

EI ∫ ⎛⎜0

d2

v ⎞dx⎟dx+12l ⎜⎝2

⎟

⎠EA ∫ 0 ⎛⎜⎝dudx⎞⎟⎠dx−12P

∫ 0

⎛⎜⎝dvdx⎞⎟⎠dx+ 1

∫ 0 E Sv 2dx only, for bending only is [K]U1as given in22(a)StiffnessMatrix (for bending) 2 (b) StiffnessMatrix (for axial load)2 (c) StiffnessMatrix (Beam Column Action) 2 (d) StiffnessMatrix(All Action)Figure. 2. Stiffness Matriceslly and axially the expression foris given by,A u2 + v2dx………………………………..(13)asgiven in Fig. 2(b).For axial load only i.e. by considering the beam column action the stiffnessmatrix 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]U4as in Fig. 2(d).When all the four cases are considered, i.e. all the four terms of U1, U2, U3, U4 are involvedthe stiffness matrix KU1, KU2, KU3, KU4 are super imposed which yields final stiffnessmatrix [K]e as given in Fig. 3(a).U4l

=

∫⎡⎢⎢⎣12E S1 v 2+ 12 ⎛ ⎜⎜⎝ES2

−E S 1 L⎞⎟⎟ ⎠ uv

2 ⎤⎥⎥ ⎦ dx v

1θ1v1

θ2⎡⎢ab c−a − b b d (for bending only)

⎥(for axial load only)Where a = 12 D, b = 6LD, c = 4L2D, d = 2L D WhereThe expression for kinetic energy for a pile loaded laterastrain energyl lT = ½ ∫μu2 + v2dx = ½ ∫ ρ0 0u1KU1=⎢−u2c⎥⎤⎥⎥⎢⎣⎦ −⎡

=AEAE[] [K] U2⎢⎢

⎢⎣−LAEAEL⎢ab ⎥LL⎤⎥⎥⎥⎦DΔEIL3− ⎡ 6

1 6 1 ⎤ K] = P 3⎢⎥V1

⎡ S11

S 12 S 13 S 14 ⎤ θ1[K] U4

=V2⎢⎢⎢⎢⎣S21

S 22 S 23 S 24 ⎥⎥⎥S44

⎦⎢⎣10 [⎢⎢⎢5L−10 2L5 − L 11510L U−630 − 1⎢⎢⎢⎥⎥⎥⎥5L−102LS33

S 34 ⎥15⎦⎥⎥

/

al Where μ = mass per unit length of the pile, u and v are the axiand transversedisplacement. Using expressions for u & v, T = ½ [q]T[M]qFor axial vibration only : lT = ½ ∫μu2 dx …………………….(14)0The displacement model for axial displacement is taken asu =u 1⎛⎜ ⎝ 1

− xl⎞⎟ + ⎠ u

2 ⎛⎜ ⎝ x

l ⎞⎟ ⎠ ………………………………… (15)

For bending vibration only :lT = ½ ∫μv2 dx…………………….(16)0The displacement model for lateral displacement is given byv = N1v1+ N2θ1+ N3v2+N4

θ2…………………………………………… (17)Where Nii =1,4 are the standard shape functions as derived from equation(10) asN1

=⎛⎜⎜⎝1− 31x2

2 + 2 13 x 3 ⎞⎟⎟⎠ ⎡ AE 0

DΔEI0 −AE 0 0 ⎤PL

[ ]e

Nxx 11 2⎢SFig. 3(a) Element Stiffness MatrixWhere a = 12 D, b = 6LD, c = 4L 2 D, d = 2L 2 D & S11 – 44 asin [K]U4So, from the expression of T. Mass Matrix [ M ] can be determined as given in.fig 3(b).[ ]

⎥Fig.3(b) Mass MatrixWhere , A is area of c/s. ρ is the density of material and μ , the mass per unit length = A x ρ&4202. Analysis of the whole problemThe solution to the problem follows the well known displacement approach whichconsists of the following main steps :• Formulation of overall stiffness and mass matrices by assembling the elementalmatrices.⎥N2

=⎛⎜⎜x − 2x2

+ 3 ⎞⎢⎢⎢⎝1x 1 2⎟⎟⎠L⎢0a−56L+ Sb+ 0L1

310+ S K=1112 0 c215S L 0− a + 56L+ Sb + 1+ S 14 ⎥⎥0⎣44⎥⎦⎥13 ⎢⎢⎢b 1S d 10L⎥N3

=⎛⎜⎜⎝3x2

12

− 2x 3⎞⎟⎟

⎠⎢00 −+ 220 −AE− − 10+ 23 0 + 30 + S24 13⎢⎢⎢00 0 L0 0 4=⎛⎜⎜⎝− 2+ 3 ⎞⎟⎠⎢00 0 0 a−56L+ S33

0 − ⎟cb −− 21510L1+ + S 34 ⎥⎥⎥

⎥⎥⎥⎥⎥ ⎡ 140

M 0 0 70 M M=⎢⎢⎢⎢⎢⎢⎢⎢⎣LML M 0 0 ⎤0156 M22 LM 0 54 M −13 LM 022 LM4 L 2M 0 13 LM −3 L 2 M 70M0 0 140 M 0 0 054 M13 LM 0 156 M −22 LM 0−13 − 3 20 − 22 LM 4 L 2 M ⎥⎥⎥⎥⎥⎥ ⎦ M

=ρAL

• Solution for the fundamental natural frequency from equn. (3) & critical static bucklingload from equn.(6).• Solution for the dynamic stability regions from equation (10).The application of boundary conditions also yield solution for the nodaldisplacements from the generalised equilibrium equation which of course also leadsto solution for design data, like shear force and bending moments at nodal points.If ndenotes the number of nodes, then the total number of degrees of freedom for theproblem is equal to 3n. The expanded element stiffness matrices Ke are constructedby inserting the stiffness co – efficients in the appropriate locations and filling theremaining with zeros. If E is the number of elements then the overall stiffness matrix [K ] is given by

[ K]=∑ eE1=

[ K ]eThe equilibrium equations of the assembly may be written as [ K ] δ = FKnown displacement conditions are introduced in the equation and the equations aresolved for unknown nodal displacements (8). Commonly the symmetryand thebanded nature of the resulting equations are utilized for efficient computing.Afterassemblage of stiffness and mass matrices, the eigen value problem in equation (10)can be solved for the frequency ratio Ω/ω1.3.Conclusion : The characteristic non-dimensionalised regions in (β, Ω/ω1) parameterspace can be extrapolated for different values of static load factor, α , which will giverapid convergence characteristics of the boundary frequencies for the first fewinstability regions (9).After obtaining the results for lower boundary and upperboundary for instability regions the may be compared with Mathiew’s diagram (5).4

.Notation :A Area of Gross section. E Modulus of Elasticity.[ K ] Stiffness Matrix l Elemental length[ M ]Mass Matrix N Shape FunctionP Axial Periodic load P * FundamentalStaticBuckling loadq Generalised Co-ordin ates[ S ] Stability Matrixt Variable time T Kinetic EnergyU Strain Energyu Axial displacement of node vLateral displacement of nodex Axial co-ordinate y Lateral Co-ordinateα Static load factor β Dynamic load factorρ Densityω1Fundamental Natural FrequencyΩ Disturbing Frequency μ Mass per unit length.Po, P tTime independent amplitudes of loadS11= 156B + 72CS12= L (22B + 14C) S 23S24= =L L2(13B ( - 3B + – 14C)3C)S13= 54B + 54C S33= 156B + 240CS14= )S22= L L 2( (4B -13B + –12C) 3C )S

34= S44= L L 2( (4B - 22B + 5C)– 30CWhere, B = ES1. L/420 C = (ES2– ES1). L/840

y,v Es1x,uilui,vi, θijuj, vj, θjEs2x,u

Figure 1 : Typical Pile ElementREFERENCES:1. Abbas, B.A.H. and Thomas, J – Dynamic stability of Timoshenko beamsresting 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 StructuralMechanics, PP, 149 – 167, 1924.5. Bolotin V.V.- The dynamic stability of elastic systems, Holden – Day Inc, 1964.6. Brown, J.E, Hutt, J.M. and Salama, A.E. – Finite element solution to dynamic stability of bars, AIAA Journal, Vol. 6, PP 1423 –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 metosapproach, 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 shorestructures 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 MechanicalEngineering 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 AppliedMathematics, 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 . 277No .6, PP 534 – 551, 1964.θP(t)