INFLUENCE OF FOUNDATION NONLINEARITY ON

OFFSHORE TOWERS RESPONSE

By Mohamed H. EI Naggar1 and Milos Novak/ Fellow, ASCE

ABSTRACT: The response of a fixed, offshore tower is greatly affected by the nonlinear behavior of thesupporting piles. An analysis of the foundation piles response to transient dynamic l~ading that accounts forsoil nonlinearity, discontinuity conditions at the pile-soil interface, and group effect IS used to assem~le thestiffness matrix of the foundation piles. A foundation stiffness matrix is added to the global structural stiffnessmatrix at the pertinent degrees of freedom (DOF). A model for the evaluatio~ of wa~e forces on .the towermembers that takes into account the spatial incoherence effect on wave forces IS used in the analySIS. Hydro­dynamic damping is evaluated, and the effect of the tower foundation behavior on it is assessed. Furthermore,the tower response to random wave forces is conducted, and the effect of foundation parameters on the towerresponse is examined. It was found that both the hydrodynamic damping and tower response to wave forces aregreatly influenced by the foundation nonlinearity and by pile-soil-pile interaction.

INTRODUCTION

Offshore structures are subjected to severe, environmentalloads from time to time. The effect of such severe conditionsbe studied to avoid unnecessarily high safety factors. Struc­tural members of an offshore tower behave linearly within acertain deformation range beyond which they behave nonlin­early.

Bea (1991) performed a series of "static, push over" anal­yses to determine at what loading and displacements a~d

where the major nonlinear developments might be expected in

an offshore platform. It was found that the first nonlinear ac­tion would develop in the foundation piles. The first nine non­linear events were all concentrated in the foundation piles.

The accuracy of the prediction of the tower response toenvironmental loads depends on several factors. These factorsinclude the parameters used in the analysis, the modeling ofthe structure and foundation, as well as environmental loads,and the method used to solve the governing equations. How­ever, large uncertainties are associated with these factors andgood approximations are accepted.

In the present study, the superstructure is idealized as aspace frame, masses are lumped at the structural nodes, andthe structural damping is modeled by equivalent viscousdamping. Power spectra of the structural response to waveforces are evaluated using the modal superposition method.The response evaluation is carried out in an iterative procedureto account for the nonlinear behavior of the supporting pilesas well as the hydrodynamic damping, which depends on thestructural displacements.

each end, and each node has six DOF, three translations, andthree rotations. The rotational DOFs are condensed except forthose at the structure-foundation interface because they are im­portant as a significant part of the overall flexibility, and damp­ing of the foundation is associated with these DOFs. Thestatic-condensation technique is used to attain the globalstiffness matrix pertaining to the DOF of interest. To accountfor the inertia forces of the tower, a lumped mass matrix isused. The lumped mass matrix is formed by lumping themasses of the individual members at the nodes, and their ro­tatory inertias are ignored. The mass matrix obtained is diag­onal and pertains to the translational DOE

The structural damping incorporated in the analysis is thematerial damping and is hysteretic in nature. Structural damp­ing is expressed as equivalent viscous damping, and the damp­ing matrix of the offshore tower C, is obtained as

(1)

where K, =condensed stiffness matrix of the offshore tower;~ = material damping ratio; and w = frequency.

PILE-FOUNDATION STIFFNESS MATRIX

A computationally efficient model for the analysis of thevertical response of single piles and pile groups is developedby El Naggar and Novak (l994a), and a similar model for thelateral response is developed by El Naggar and Novak(1995a). The preceding nonlinear soil models account for non­linear soil behavior, gapping, and slippage between the pile

y

Mean WattrLevel

Super - StructureSpace Frame

Elements

Sfructure - Foundationlnferlace

Mud LineX

Sub - Slructure

Ptone' of / Main Le;sSymmetry

--f-- X,

zFIG.1. Offshore Tower Ide.llzatlon

STRUCTURE GLOBAL MATRICES

A space-frame idealization is used to model the offshoretower as shown in Fig. I. The global structural stiffness matrixis constructed from stiffness matrices of the individual ele­ments, following the standard displacement method. In theglobal stiffness matrix, only the translational degrees of free­dom (DOF) are retained. Members of the offshore tower aremodeled by space-frame elements that have two nodes, one at

'Asst. Prof., Facu. of Engrg. Sci., The Univ. of Western Ontario, Lon­don, Ontario, Canada, N6A 5B9.

2Prof. (Deceased), Facu. of Engrg. Sci., The Univ. of Western Ontario,London, Ontario, Canada, N6A 5B9.

Note. Discussion open until February I, 1997. To extend the closingdate one month, a written request must be filed with the ASCE Managerof Journals. The manuscript for this paper was submitted for review andpossible publication on May 31, 1994. This paper is part of the ]ou17UI1of Geotechnical Engineering, Vol. 122, No.9, September, 1996.©ASCE, ISSN 0733-9410/96/0009-0717-0724/$4.00 + $.50 per page.Paper No. 8571.

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where a" =standard deviation of the relative velocity betweenthe structure and water particles at the position and directionof DOF i. The total mass matrix M = sum of the structuraldiagonal mass matrix and added mass matrix, which is definedas

(5)

(7)

(8)

N.

(CH);i =2: 1I2pCD dtv'Sha"rtL tk-l

Mii + Cli + Ku = P(t)

The total damping matrix is given by

C = C, + Cf + CH (6)

in which the hydrodynamic damping matrix, CH , is a diagonalmatrix with elements

coordinate of member k, respectively; and N. = number ofmembers attached to joint n at which DOF i is located. Valuer t = function of direction cosines of member k; and p = water­mass density. Drag and inertia coefficients CD and CM are as­sumed to be constant with the respective values 1.4 and 2.

Using the linearized form of the drag term and transferringboth the added mass and the hydrodynamic damping to theleft side of the equations of motion, (3) is rewritten as

and soil. The Winkler soil model is adopted, and soil displace­ments are assumed to be in either the vertical or horizontaldirection, depending on the direction of the pile-shaft model.Formulations of the models are in the time domain to allowthe nonlinearity to be accounted for.

The proposed analytical models may be applied to analyzethe response of the entire pile group, accounting directly forthe nonlinearity and interaction among all piles at the sametime. An alternative approach is to solve just two piles at atime, with the results to be superimposed to assemble theglobal flexibility matrix of the entire group. The second ap­proach may be very advantageous if it is implemented in con­junction with pile-soil-pile interaction factors, along with sin­gle-pile solutions. To approximately account for groupnonlinearity in the analysis, the equivalent, linear, single-pileparameters and interaction factors have to be established de­pending on the P/Pu ratio, where P = load at the pile head;and Pu = pile ultimate capacity. To this end, EI Naggar andNovak (1994b, 1995b) proposed an approach to establish thesingle-piles-impedance functions and interaction factors foraxial and lateral vibrations, respectively. For a basic range ofsoil and pile parameters, they established a set of charts forsingle-pile impedances and interaction factors.

The equivalent, linear, single-pile impedances along withthe equivalent linear-interaction factors are used herein to as­semble the global flexibility matrix of the entire group ac­counting, approximately, for the nonlinearity of the foundationbehavior.

The foundation stiffness and damping matrices are to be addedto the tower stiffness and damping matrices at the pertinentDOE

Pile-Group Flexibility Matrix

The complex foundation stiffness matrix Kj is obtained asthe inverse of the complex foundation flexibility matrix Ff . Fora group of n piles, the foundation flexibility matrix is a sym­metric 6n X 6n matrix. The construction of the foundationflexibility matrix is described in detail in El Naggar and Novak(1995c).

The foundation stiffness matrix, K f , is then calculated asthe real part of complex stiffness matrix Kj. The foundationdamping matrix, Cf , is computed as the equivalent viscousdamping as

Cf = Im(K*)/w (2)

In modal analysis approach, the structural response is ob­tained using the superposition technique, which is valid forlinear systems only. However, for real offshore towers, theresponse is dominated by the first mode only, and the errordue to the application of the modal analysis is small. The anal­ysis starts with the solution of the free-vibration problem tocompute the natural frequencies, mode shapes, and modaldamping ratios. For the case of nonproportional damping, asit is herein, classical normal modes do not uncouple the damp­ing matrix. Foster (1970) used the nonclassical, complex, ei­genvalued modal superposition to uncouple the equations ofmotion. In this study, the modal superposition is adopted, andthe complex eigenvalue approach is used to solve the equa­tions of free vibration. Foundation stiffness and damping ma­trices are evaluated at the first natural frequency, which is ob­tained by iteration starting with an undamped system andclassical analysis.

GOVERNING EQUATIONS OF MOTION

(3)

FREE-VIBRATION ANALYSIS

Setting force vector P(t) to zero, the free-vibration equationsof motion are

RESPONSE TO WAVE FORCES

where <Pi} = modal coordinate associated with DOF i in modej; and T1J(t) = generalized coordinate of mode j. From the the-

The actual coordinates can be related to the generalized co­ordinates using the mode shapes that have been evaluated fromthe free-vibration analysis by

(9)

(10)

N

u;(t) =2: <Pu T1J(t)j-I

Mil + Cli + Ku =0

This homogeneous equation is solved for free-vibrationmodes, natural frequencies, and modal damping ratios usingthe complex eigenvalue analysis (Novak and EI Hifnawy1983). The free-vibration analysis is detailed in El Naggar andNovak (1995c).

(4)

where db Lb and St = diameter, tributary length, and element

The equations of motion of an offshore tower subjected towave forces are written where K = sum of the stiffness ma­trices of structure K, and foundation K f ; Cb =sum of dampingmatrices of structure C, and foundation Cf ; and M, = structurallumped mass matrix. Values u, li, and ii =vectors of structuraldisplacements, velocities, and accelerations, respectively. Fi­nally, W(t) = vector of wave forces, the elements of which aredefined by

N

n [l L

• [ d2

W;(t) = 2: r t p1T .2. CMV(Sb t)t-I 0 4

1 ] d2

+ '2 pdtCDv'Sha"v(Sb t) dSt - p1T 4t

(CM - l)Ltii;(t)

718/ JOURNAL OF GEOTECHNICAL ENGINEERING / SEPTEMBER 1996

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FIG. 2. Fixed Offshore Tower Used In Analysis

(17)

Here, CT., = standard deviation of the response at the pile head;and "Ym = peak factor given by Davenport (1964)

"Ym =V21njT + O.5772/V2 InfT (18)

where T =duration of the storm; and f = /2'lT.The iteration procedure is terminated when convergence is

achieved.

complex eigenvalue analysis to establish the natural frequen­cies, mode shapes, and modal damping ratios. For free-vibra­tion analysis in still water, water-particle velocity v is assumedto be zero; then the drag term is proportional to the square ofthe structural velocity, and due to its quadratic nature can beneglected for small amplitudes. However, the hydrodynamicdamping is included in the structure-response analysis for cer­tain sea states.

The response analysis proceeds by specifying the sea stateand is performed in an iterative procedure. In the first cycle,structural velocities are assumed to be zero; then the relativevelocity is equal to the water-particle velocity, and hydrody­namic damping matrix CH can be evaluated. Also, the foun­dation stiffness and damping are computed assuming thatdisplacement amplitudes are small (linear conditions). A free­vibration analysis is carried out to determine the hydrody­namic modal damping ratios, ~hj' This is done by setting otherdamping matrices to zero. During each cycle, the system islinear and superposition holds; thus the total modal dampingratio, ~j, is the sum of ~sj, ~ and ~hj' Because of the iterativenature of the solution and to avoid excessive computationalcosts, the classical mode shapes are used in the evaluation ofthe response spectra. This is a reasonable approximation be­cause the difference between classical and nonclassical modeshapes is significant only for systems with high damping ra­tios, and for higher modes. After the response spectra are eval­uated, new values of the variances of relative velocities areevaluated, and another cycle of iteration is carried out. Onlyin the second cycle, the foundation stiffness and damping areadjusted to the level of displacements at pile heads to accountfor the nonlinear behavior of piles. These displacements arecalculated as mean peak values, i.e.

(II)

(13)

N

Fj(t) =2: <!>ljP,(t)...1

ory of random vibration, the spectral density of the responseat DOF i, S././(w) is given as

N N

S./.,(w) =2: 2: <!>A>irS'1i".(oo)}-1 ,..1

where S'1 ....(00) = cross spectrum of generalized coordinates'fli(t) and

l1'\r(t), which is given by

S'1I'1'(oo) =Ht(ioo)H,(ioo)SFjF,(oo) (12)

In (12), HJCiw) and H,(iw) = mechanical admittance func­tions of modesj and r respectively; and SF1F,,(W) = cross spec­trum of generalized forces Fj(t) and Fr(t). The generalizedforce in mode j, Fj(t), is given by

Stiffness, damping, and mass matrices are assembled. A so­lution of the free-vibration problem is carried out using the

where Sp/Pm(w) = cross spectrum of nodal loads P,(t) and Pmet).Mitwally and Novak (1989) developed a model for the eval­uation of wave forces on the tower members that takes intoaccount the spatial incoherence effect on wave forces. In thisstudy, they found that a stepwise approximation to the velocityfield and lumping the areas and volumes of the members atthe nodes have little effect for longer waves, where most ofthe energy lies. Also, they found that the cancelling effecttends to reduce the error. These approximations are adoptedhere. Consequently, Sp/Pm(w) is obtained as

Hence, the cross spectrum of the generalized forces isN N

SFjF,(oo) =2: 2: <!>ij<!>mrSp,pJoo);'1 m-I

Nn Ns

Sp,pJoo) =2: 2: rtr/LtL,{(w2a t a , + iooa'~t<Tflkwi 1-1

+ ~t~,<Tf,<TfJS"'''2(oo) + at~,<TfJiwS"'.2(oo)]*}

where

at = CM(p'lTd~)/4; ~t = 1I2pdt CD~

SOLUTION PROCEDURE

(14)

(IS)

(lOO,h)

EXAMPLE

A typical fixed offshore tower, shown in Fig. 2, is analyzedto illustrate the effect of soil-structure interaction on its modalproperties. The tower is 122.0 m high and stands in 104.5 mof water. It is supported by 10 piles with an outer diameter of1.45 m and penetration of 50 m. Six main piles extend throughthe main legs all the way to the top of the structure, and fourskirt piles are cut off and welded at the level of the first brac­ing. The tower is symmetric about both X- and Z-axes. Onlyone quarter of the structure is idealized, and two different setsof boundary conditions are applied: sway in X-direction (inthis case, X is the axis of symmetry and Z the axis of antisym­metry) and the torsional mode where both X and Z are axesof antisymmetry.

Variation in Hydrodynamic Damping with DifferentParameters

The hydrodynamic damping is a major source of damping,especially with higher sea states. It is derived from the motionof the tower in the water, and it is affected by this motion. Assuch, the behavior of the tower foundation affects the hydro­dynamic damping. For all the results presented in this section,the deck mass is assumed to be 59.21 % of the total mass ofthe tower.

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• - - Linear-- Nonlinear

0.5 1.0 1.5 '12.0 2.5Frequency II (s )

15 20 25 30 35 40Wind lpeedU (m II)

"I

: u..._~......._ ......._ ........_"'_____'

0.0

2

6

-- Intenctlon• - - No Interaction

4

10

2

4

0'---'--'---'--'----'----'10 15 20 25 _ 30 35 40

Wind lpeed U (mil)

2.0

..- 8NS_ 6

'=::'4

2.50.5 1.0 1.5.1

2.0Frequency II (I )

15 20 25 30 35 40Wind lpeed n(m I I )

15 20 25 30 35 40Wind lpeedli (m II)

4

oL-L--l_-=_"""----"'_---'0.0

4

6

8

2

4

2

8

6

15

(c)10 ...-...,....-,--...,....--,,....--r---, 8 ...-...,....-,--...,....-,---,---,

(d)10 .-----r--r---r-...,..---.,.-.., 10 ...---r--,....---,--...,..---.,.--,

(b)

(a)

~

.~

JI~

FIG. 5. Effect of Plle-5oll-Plle Interaction on HydrodynamicDamping Ratio [V.(L) = 100 mis, Nonlinear Foundation]: (a) Ho­mogeneous Nonlinear; (b) Parabolic Nonlinear; (c) Homoge­neous Linear; (d) Parabolic Linear

FIG. 6. Effect of Foundation Nonlinearity on Tower-ResponseSpectrum for Six Wind Speeds [Homogeneous 5011 Profile,V.(L) = 100 mis, Interaction Considered]: (a) 0= 10 m/s; (b) 0=20 m/s; (c) 0 =25 m/s; Cd) 0 =30 m/s; (e) 0 =35 m/s; (f) 0 =40m/s

g 00 00I 1.2 r-----r--,---.-...---. 1.0 ,....----r---,--,---,----.

i 0.9 0.8

:. • ~o.6l:l NS06 Nil

'0 '" .

i go, -~: \I 0.0 '-----'-""---.........-"-----'---"-... 0.0 L...L.---L.~""__I.Ao-.J._...J

0.0 0.5 1.0 1.5'1 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5(b) Frequency AI (s ) (e) Frequency AI (1,1)j 5,---.--.--....-.--.., 2.5,-......,...-.--....-.--.,

-8i.. ..f Ne'0" 2II 8e ~

I 0.5 ~I 0 L..-.L....J_--l..'=....l..._"""-.....I 0.0 u...---'''''=O-...._......._-L....--l

0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5(c) Frequency AI (S,I) (0 Frequency AI (I-I)j 20 r----,--,---.-...---. 12 ...----r---,--,---,----.

-8i.. ..! N S 10o "I1~ 5

I

40

40

35

35

-NonlinearLinear

20 25 30Wind lpeedU (mil)

15

10

8- V,(L) - 100 mil ,

~- - V,(L) - 200 mil

I

'i I

f 6 II

II

4 II

::c I2 ""

010 15 20 25 30 35 40

Wind lpeed U ( mil )

(a)

-- 10.------,-----r---,---r---.----,

'f 8

j 6

1:1 ~---I ~----= 0 L..-_--l.__....l...__...l...-__L...-_--l..__....J

(b)10 15 20 25_ 30 35 40

Wind lpeed U (mil)

-- 10.------,-----r---,---r---.----,.gl! 8

j 6

14

1 2L-_-­l. L..__..J....__...I..-__.l...-__L...-_--1__.....J= 010

Effect of Foundation Nonlinearity

Response of the tower to wave forces is evaluated for thethree soil profiles with V.(L) = 100 mls and six different windspeeds ranging from 10 to 40 m/s. Hydrodynamic ratios areobtained by both considering and neglecting the nonlinear be­havior of the supporting piles. Fig. 3 shows the variation ofthe hydrodynamic damping with the wind speed for the threesoil profiles for both cases. The hydrodynamic damping in­creases as the wind speed increases. This is understood be-

(c)-- 12 .------,...----.,.----,-----r----,-----.,

'f 10

J :14I :L.-..................~

10 15 20 25 30Wind lpeed U (mil)

FIG. 3. Effect of Foundation Nonlinearity on HydrodynamicDamping for Three 5011 Profiles [V.(L) = 100 mis, InteractionConsidered]: (a) Homogeneous; (b) Parabolic; (c) Linear

FIG. 4. Effect of 5011 Stiffness on Hydrodynamic Damping Ra­tio (Parabolic Profile, Interaction Considered)

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cause higher wind speeds mean larger tower response. Also,considering the nonlinear behavior of the foundation leads tothe prediction of higher values of the hydrodynamic dampingparticularly for higher wind speeds. This is because the as­sumption of linear foundation overpredicts the tower stiffnessand consequently underpredicts the tower response leading tothe evaluation of lower hydrodynamic values.

Effect of Soil Stiffness

The tower is analyzed for two different shear-wave veloci­ties, V.(L) = 100 and 200 mls assuming a parabolic soil profile.The nonlinear behavior of the soil is considered, and the in­teraction between piles is taken into account. Fig. 4 shows thevariation of the hydrodynamic damping ratio with the windspeed for the two shear-wave velocities. As expected, the fig­ure shows that the hydrodynamic ratios for the stiffer soil areless than those of the softer soil, especially for higher windspeeds.

Effect of Pile-Soil-Pile Interaction

The hydrodynamic damping is calculated for two soil pro­files, homogeneous and parabolic, with V.(L) = 100 m/s. Fig.S presents the hydrodynamic damping ratios calculated, con­sidering and neglecting the pile-soil-pile interaction for bothlinear and nonlinear foundations. It may be observed from thefigure that the effect of the interaction on the hydrodynamicdamping is such that the interaction decreases the hydrody­namic damping ratios for lower wind speeds and increases itfor higher wind speeds. This is because the interaction be-

tween piles increases the structural flexibility leading to largerhydrodynamic damping, but it also increases the total dampingsignificantly.

Variation in Tower Response with Different SoilParameters

A parametric study is conducted to illustrate the effect ofdifferent foundation parameters on the tower response to waveforces. In this study, three different soil profiles are considered.lhey are characterized by constant, parabolic, or linear vari­ation of soil shear-wave velocity with depth.

Effect of Foundation Nonlinearity

The tower response is evaluated for the three different soilprofiles with the shear-wave velocity Vs(L) = 100 mls and sixdifferent wind speeds ranging from 10 to 40 m/s. For all cases,the pile-soil-pile interaction is considered. Figs. 6, 7, and 8show the power spectra of top node deflection for differentwind speeds for homogeneous, parabolic, and linear soil pro­files, respectively. In Fig. 6, the power spectrum of top nodedeflection for the homogeneous soil profile evaluated for bothlinear and nonlinear pile behavior is presented. The figureshows that for wind speed, 0, equal to 10 mIs, the responseis linear and the spectrum has a pronounced resonance peak,and a smaller one, the background response peak, is at wave­energy spectral-peak frequency w. As 0 increases, the resonantpeak is suppressed and the background peak increases. It maybe noticed from the figure that the resonance frequency andresonant peak evaluated from the nonlinear analysis are less

2.5

2.5

2.5

• - - Linear-- Nonlinear

0.5 1.0 1.5.,2.0Frequency II (, )

0.5 1.0 1.5 .,2.0Frequency.. (, )

.",,

\~OU-.......:l---'-_...._-J----J

0.0

10 f\

10

40

12

."I,

II

2 ~\.oL.-.L...J._~_""_...L---J 0 '-'----'..=.a: ...._-J----J

0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0(c) Frequency II (,.1) (f) Frequency II (s·l)

.~ 70 r-----r---r--,--..,---, 50 r-r---r---r--,--..,----,

~ 60

i 50" M

.t "s 40...~ "I:! 30a ~ 20

§. 10

j 0 l-L..--'--.....:;!I........~--J"--...J0.0 0.5 1.0 1.5 2.0 2.5

Frequency II (s")

~0.0 '----l.~_'__....L....__::""""""--l 0 I-I.---I-=....._ ...._-J----J

0.0 0.5 1.0 1.5., 2.0 2.5 0.0.~ (b) Frequency II (s ) (e)Ii 15 ...--~-__r_-r__~-., 12 r---;--__r_-r__~-.,

~"8C M

! "5 9" "IS :! 6e ~

Ij

.! (a) (d)c!! 0.8 r-----r---r--,--..,---, 4 r----;--.-..,--,..---,

-81 -;;-0.6

g- N'

oS "IE 0.4

~ g! 0.2

I2.5

2.5

2.5

• - - Linear-- Nonlinear

0.5 1.0 1.5 2.0Frequency .. (S·I)

0.5 1.0 1.5 .,2.0Frequency II (s )

5

"

\~O,-,--...;:Q-....L...._.J..---II...---l0.0

10

1.5

0.5

-;;- 4

"E 3-:'Q

;:. 2

2.50.5 1.0 1.5 2.0Frequency II (s'l)

2

8

oL.-.L--'-_'----'--=-'---' 0 u......_\.........",,-........----''---'----'0.0 0.5 1.0 1.5 2.0 2.5 0.0

(c) Frequency II (s.1) (f)j :r---rn~--r-_,_-...,....__,:

"8~ ~ 30

=Ne

~ '%: 201~ 10

1

c (~ (~j 1.2...-----r---r--,--..,---, 2.0

1 -;;-0.9

'" .~ "s 060" .s ~e ~J 0.3

j ~,0.0 '-----'.L---'-_-'-_-=""--' 0.0 I-I.---'-..::"'"'"'_...._-J----J

0.0 0.5 1.0 1.5., 2.0 2.5 0.0 0.5 1.0 1.5 .12.0.! (b) Frequency .. (s ) ( ) Frequency .. (s ).!l 10 ...-----r---r--,--..,---, e 6 r-----r---r--,---,----,

-8"8C M

S ME 6'0 "s :! 4e ~

J1

FIG. 7. Effect of Foundation Nonlinearity on Tower-ResponseSpectrum for Six Wind Speeds [Parabolic Soil Profile, V.(L) =100 mis, Interaction Considered]: (a) 0 =10 m/s; (b) 0 =20 mls;(c) 0= 25 m/si (d) 0= 30 m/s; (e) 0= 35 m/s; (f) 0= 40 m/s

FIG. 8. Effect of Foundation Linearity on Tower-ResponseSpectrum for Six Wind Speeds [Linear Soil Profile, V.(L) =100'!l's, Interactlon.Consldered]: ta) 0= 10 m/siJb) 0= 20 mi.; (c)U = 25 m/s; (d) U = 30 m/s; (e) U = 35 m/si (f) U = 40 m/s

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2.S

2.5

2.5

• - - Linear--Nonlinear

3S 40

0.5 1.0 1.5 12.0Frequenc)' II (I' )

2.0

.... 8

:S 6'co::. 4

2.50.5 1.0 1.5.1

2.0Frequency II (I )

6

oL...L.....L.--::::t:o.._-===:...L.-l.....J

0.0

1.0

S-- Nonlinear

c V,(L) - 100 mil • Linear.S!

~ 0.8 - - Nonlinear-!I V,(L) - 200 mls . Linear

'3c2

0.6.~

.c:

ic~ 0.4...0c.2

l 0.2i!.. •'8.. .c?J

0.010 15 20 25 30

Wind speedU( mls )

is much higher than that of the resonance component. Thebackground response is quasi-static with little or no dynamicamplification. Thus, the magnitude of the background responseis controlled by the stiffness of the system. Also, the nonlineareffects are more pronounced for higher wind speeds.

0.5 ,"l0.0 L.-..I....J'----'==_-'----L....J 0.0

0.0 0.5 1.0 1.5 2.0 2.S 0.0 0.5 1.0 1.5 2.0.~ (e) 8 Frequenc)' II (s·l) (f) . Frequenc)' II (S·I)

c!! r-----,--.--r---r--., 12 r-----,--.--.----r---,

-!I 10

~.. ..!"s 4o ':>i g 2

I

-;;0.4

"Ii_ 0.3'co •

~: '\,0.0 L----l_....._..c::.-...l-~ 0.0 L...L--J....;::,;;......_ ..........,..................J

0.0 O.S 1.0 1.S., 2.0 2.5 0.0 0.5 1.0 1.5 2.0" (b) Frequenc)' .. (s ) (e) Frequency .. (1,1)J 2.S r---..-----r----r--,.--, 2.5 r-----rr----r----r--,.--,

-!I 2.0

1 ..t .. ' 1.5... So ':>S g 1.0

~~ O.S

J

FIG. 10. Effect of Foundation Nonlinearity on Tower-Re­sponse Spectrum for Six Wind Speeds [Parabolic 5011 Profile,V.(L) = 200 mIs, Interaction Considered]: (a) 0= 10 m/s; (b) 0=20 m/s; (c) 0 =25 m/s; (d) 0 =30 m/s; (e) 0 =35 m/s; (f) 0 =40mls

Effect of Soil Stiffness

To examine the effect of the soil stiffness on the tower re­sponse to wave forces, a parabolic soil profile with V.(L) =

.1 (a) (d)o!! 3.0 0.6,---,r----,---,.--,.--.

-!I 2.5 0.5

1 .... 2.0t .. ''0 "a 1.5S coi ::'1.0

O.S

I

FIG. 11. Effect of Stiffness on Tower-Response Amplitude(Parabolic 5011 Profile, Interaction Considered)

40

40

40

35

35

-Nonlinear• - - Linear

20 25_ 30 35Wind lpood U (mls)

20 25 30Wind lpood IT (mls)

15

15

15

0.0 b====::r:=..::---L__.l-_--L-_-.J10

(ll)0.5 r----,---,-----r--..---,----,

~0.4i.!ld O•3

.H-8i 0.2

lL~ 01).8 .0.0 I:===::L::::::~~...I....__...I....__...l...-_._J

10

(b)1.0 ,-----,---.--.----.,----,-----,

i~"Sg-..... 0.8-"~J 0.6

:~-!I-8i 0.4

~.~ 02],g .t?i

(e)

~~ 1.0cS!i0.80'-

dO.6.i-!I

UO.4

I.E 0.2

0.0 L....====::i::=-----L_-l._---I_--.J10 20 25 30

Wind speed U (m/s)

FIG. 9. Effect of Foundation Nonlinearity on Tower-ResponseAmplitude for Three 5011 Profiles [V.(L) = 100 mIs, InteractionConsidered]: (a) Homogeneous; (b) Parabolic; (c) Linear

than those evaluated from the linear analysis. On the otherhand, the background response derived from the nonlinearanalysis is almost 30% higher than that obtained from the lin­ear analysis. Fig. 7 displays the power spectrum of the topnode deflection for the parabolic case. Similar observationscan be made with the exception that the resonant and back­ground peaks are closer to each other, and the nonlinear effectsfor higher wind speeds are more pronounced. Fig. 8 presentsthe power spectrum of the top node deflection for a linear soilprofile. It may be noticed from the figure that the backgroundpeak is larger than the resonant one, and the response spectrumis somewhat broad banded due to the closeness of the domi­nant natural frequency of the tower, wo, and w. As a increases,the background response increases with the response evaluatedfrom the nonlinear analysis being higher than the linear re­sponse until it becomes almost twice the linear response for[j = 40 m/s. To further illustrate the nonlinearity effects onthe tower response to wave forces, the variation in the totalresponse amplitude (standard deviation) of the top node de­flection is presented in Fig. 9 with the wind speed for the threesoil profiles. The amplitudes for homogeneous soil profile arehigher than that of the parabolic one, and the linear is thelowest for lower wind speeds with no difference between theresults obtained from linear or nonlinear foundation. This isbecause the resonance component of the response, which islarger for stiffer soils with lower damping ratios, is higher thanthat of the background component for lower wind speeds. As[j increases, the response amplitude for the linear case be­comes the larger, and the response amplitude for the homo­geneous case becomes the smaller. This is because the contri­bution to the total response from the background component

722/ JOURNAL OF GEOTECHNICAL ENGINEERING / SEPTEMBER 1996

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The following conclusions are drawn. Hydrodynamic damp­ing ratios increase due to the soil nonlinearity. They also in­crease as the soil shear-wave velocity decreases. Pile-soil-pileinteraction decreases the hydrodynamic damping ratios forlower wind speeds while it increases the hydrodynamic damp­ing ratios for higher wind speeds. The nonlinear behavior ofthe foundation tends to slightly reduce the resonant peak andfrequency while it significantly increases the background partof the response. The total response increases due to nonlin­earity for all wind speeds and all soil profiles. The increase insoil shear-wave velocity may decrease or increase the towerresponse at low wind speed, but it decreases the tower re-

CONCLUSION

Effect of Pile-Soil-Pile Interaction

The interaction between piles affects the response of thetower as it alters the values of supporting piles stiffness anddamping. Fig. 12 shows the power spectrum of top node de­flection for a homogeneous soil profile with V.(L) = 100 m/sfor different wind speeds, considering and neglecting the pile­soil-pile interaction, respectively. It may be noticed from thefigure that, for lower wind speeds, neglecting the interactionleads to significant overprediction of the resonance peak be­cause it reduces the damping dramatically, and it also over­predicts the resonant frequency due to the overprediction ofthe stiffness. On the other hand, the background peak increasesdramatically due to the interaction between piles. This behav­ior is further illustrated in Fig. 13, which shows the variationin the standard deviation of top node horizontal deflection withthe wind speed for both the homogeneous and parabolic soilprofiles with V.(L) =100 m/s. It is evident from the figure thatthe interaction between piles reduces the response amplitudesfor lower wind speeds because it increases the damping in thesystem. For higher wind speeds, the interaction increases theresponse amplitudes as the contribution to the total responseamplitude from the background component, which depends onthe stiffness, is much higher than the contribution from theresonance component.

200 m/s was considered in the analysis, and the results ob­tained are compared to those for the parabolic soil profile withV.(L) =100 m/s. The power spectra of the top node horizontaldeflection for V.(L) = 200 m/s for different wind speeds areshown in Fig. 10. The figure shows that the response is es­sentially linear for lower wind speeds, and rather significantpeaks are distinguished at 000 and wcompared to the case withV.(L) = 100 m/s shown in Fig. 7. At 0= 25 mis, the resonancepeak for the linear case becomes slightly less than the nonlin­ear case while the background peak for the linear case be­comes slightly higher than the nonlinear case. This may beattributed to the drop of the interaction between piles beingmuch more than the decrease in single-pile stiffness for stiffersoils in the intermediate range of loading. This drop in inter­action leads to a decrease in the system damping and an in­crease in the tower stiffness. These two factors, the drop inthe damping and the increase in the stiffness, cause the in­crease in the resonant peak and the decrease in the backgroundpeak for the nonlinear analysis. Also, it may be noticed fromthe figure that the increase in the response at higher windspeeds is much greater in the nonlinear case than that of thelinear case compared to the profile with V.(L) = 100 m/sshown in Fig. 7. The standard deviations of top node horizon­tal deflection for V.(L) = 100 and 200 m/s are presented inFig. 11. It may be concluded from the figure that the responseamplitudes for the stiffer soil are less than that for the softersoil for all wind speeds, and the nonlinear response amplitudesare higher than the linear ones for all wind speeds.

1.11

-- Interaction• - - No Interaction

0.11 1.0 1.11 1.0 1.11Frequency .. <il

)

15 20 25 30 35 40Wind IpeedU <m I I)

"

10

0.1

0.11

1.0

0.1

-- Interaction0.2 • - - No Interaction

0.3

0.1

'::1.11

(d) 0.6

O.S

0.4

0.3I I

I 0.2 ,I ,, ,

~,I"I', ,, ,It, I, II I, I, I, II ,

0.0 L--..L._.......-..l._..J---J'----'

15 20 25_ 30 3S 40 10 IS 20 25 30 3S 40Wind lpeed U (ml I) Wind IpeedU (m I I)

0.0 l:...-....J...._.l..-....J...._.l..--l..~

15 20 25 30 35 40 10

Wind lpeedU <m I I)

II

",,- \ 1 ~~

o L.....I.......JL-;:..,iI_.....o.L-:"..,--.'..... 0 L.L.--::L3Ioo.--J_--'-_---'-_--'0.0 0.11 1.0 1.8 1.0 1.8 0.0

FrequellCY" <1.1)

II

:\~a L...........JI...- O"='-~.-.~ 0.0 L...L_L.::.............._ ...._..l:..a---'0.0 0.11 1.0 1.11 1.0 1.11 0.0 0.11 1.0 1.11 1.0 1.11

Frequency .. (1.1) (e) Frequency .. (i

l)

II ...--,--....,-......,.---r--, 1.11 ,-----,-......,.---r---,--..,

II

(d), ...--,-----,----,---r--, 1.0 .------,-......,.---r---,--..,

(a) 0.' (e)J ....---,--..-......,.-.....-r--, 0.4 ...---r-..-......,.-........,,--..,

-§~~~0.4-II'0.2g~0.3

:h-8 '3 0.2

'd].i O.1

'"

(b) 0.7 r--r-.....-.,..--,--r-"1I

-§~0.6"e~o.S'0.2.§~ 0.41il".- ... 0 3.5i .~.§0.2... ~~,g 0.1

0.0 =---'-_.1..----'-_.1..---'-----'10

,,., .0.11 ' \\

o 1...-.L...J1...-.....;l"""..&..::l:I...---"'.......... 0.00.0 0.11 1.0 1.11 1.0 1.11 0.0 0.11 1.0 1.11 2.0

j (e) 10 ...-_....-Freq....,Ur-eIl_cy---,.._<_I·I_)..----,(1) 11 ...----,_F_re'TqU_en_c_yT.._(_I·I_).----,

-8-§" 'MC\, •

~ "'e 10" .,e ~2 ~

~ 8

!FIG. 12. Effect of Pile-Soil-Pile Interaction on Power Spectrumof Tower Response [Homogeneous Soil Profile, V.(L) =100 mIs,NonJlnea! Foundation]: ia) 0 =10 m/s;Jb) 0 =20 m/s; (c) 0 =25m/s; (d) U=30 m/s; (e) U=35 m/s; (f) U=40 mls

FIG. 13. Effect of Plle-Soll-Plle Interaction on Tower-ResponseAmplitude for 1\vo Soli Profiles [V.(L) = 100 m/s]: (a) Homoge­neous Nonlinear; (b) Parabolic Nonlinear; (c) HomogeneousLinear; (d) Parabolic Linear

" (b)

j-8-§" ..C\, • 1I~ 4'4e" .,E ~ 22

~

!

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sponse at high wind speeds. Soil shear-wave velocity profilehas a significant effect on the tower response. The resonantresponse for homogeneous profile is higher than the resonantresponse of the parabolic and linear profiles, with the linearprofile being the lowest. The background part of the responsefollows the opposite trend. At low wind speeds, the total re­sponse for stiffer soils is higher than that for softer soils, butfor higher wind speeds, the total response of the softer soilsis much greater than that of the stiffer soils. Pile-soil-pile in­teraction decreases the resonant response dramatically and sig­nificantly increases the background part of the response. Pile­soil-pile interaction decreases the total response of the towerto wave forces at lower wind speeds, but it increases the totalresponse significantly for higher wind speeds.

ACKNOWLEDGMENT

This research was supported by a grant in aid of research from theNatural Sciences and Engineering Research Council of Canada.

APPENDIX. REFERENCES

Bea, R. G. (1991). "Earthquake geotechnology in offshore structures."Proc., 2nd Int. Con/. on Recent Advances in Geotech. EarthquakeEngrg. and Soil Dyn., No. SOAI3, Univ. of Missouri-Rolla, Mo.

724/ JOURNAL OF GEOTECHNICAL ENGINEERING / SEPTEMBER 1996

Davenport, A. G. (1964). "The distribution of largest values of randomfunction with application to gust loading." Proc., 28, Instn. of Civ.Engrs., London, U.K.

EI-Marsafawi, H., Kaynia, A. M., and Novak, M. (1992). "The super­position approach to pile group dynamics. Piles under dynamic loads."J. Geotech. Engrg., 34, 114-135.

EI Naggar, M. H., and Novak, M. (I 994a). "Nonlinear model for dynamicaxial pile response." J. Geotech. Engrg., ASCE, 120(2), 308-329.

EI Naggar, M. H., and Novak, M. (1994b). "Nonlinear axial interactionin pile dynamics." J. Geotech. Engrg., ASCE, 120(4),678-696.

EI Naggar, M. H., and Novak, M. (1995a). "Nonlinear analysis for dy­namic lateral pile response." J. Soil Dyn. and Earthquake Engrg., Ac­cepted for publication.

EI Naggar, M. H., and Novak, M. (1995b). "Nonlinear lateral interactionin pile dynamics." J. Soil Dyn. and Earthquake Engrg., 14(2), 141­157.

EI Naggar, M. H., and Novak, M. (1995c). "Effect of foundation non­linearity on modal properties of offshore towers." J. Geotech. Engrg.,ASCE, 121(9), 660-668.

Foster, E. T. (1970). "Model for nonlinear dynamics of offshore towers."J. Engrg. Mech. Div., ASCE, 96(1), 41 -67.

Mitwally, H., and Novak, M. (1989). "Wave forces on fixed offshorestructures in short-crested seas." J. Engrg. Mech., ASCE, 115(3),636-655.

Novak, M. (1977). "Vertical vibration of floating piles." J. Engrg. Mech.Div., ASCE, 103(1), 153- 167.

Novak, M., and EI Hifnawy, L. (1983). "Effect of soil structure inter­action on damping of structures." J. Earthquake Engrg. and Struct.Dyn., 11, 595-621.

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