dynamic analysis

1Dynamic AnalysisProf. Dr Kurian V. John

Outline of contents

Overview: Introduction

Single Degree of Freedom System

DAF for Fixed Platforms: Examples

Floating Structure Dynamics

Dynamic Analysis

2As the offshore platforms are alwayssubjected to the dynamic wave loads,it is essential that the engineersresponsible for the design,construction and maintenance of theseplatforms, have a fairly good idea ofthe dynamic behavior of thesestructures.

Overview: Introduction

This presentation will supplementyour knowledge and explain somebasic ideas regarding the types ofdynamic analysis used for fixed andfloating types of platforms.

Dynamic Analysis

Course goals The participants shall be able to

Formulate the basic equation for SDOF system

Draw and interpret the frequency Vs DAF graph

Differentiate between frequency & time domain dynamicanalysis

Differentiate between coupled and uncoupled dynamicanalysis

Dynamic Analysis

3Lesson 1


Single Degree of Freedom SystemThe analysis of an offshore structure usingstiffness matrix methods and joint loadingsbased on extreme environmental conditionsnecessarily neglects any dynamic effectsassociated with the wave-induced periodicmotion of the structure.

Such a static analysis can, therefore, only beapplied when the dynamic loadings are smallin comparison with the maximum staticloadings.

Let us consider an approximate analysis of atypical platform as shown in the Figure.A

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15m22.5m

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Dynamic Analysis

4Single Degree of Freedom SystemRegular sinusoidal water wavesare assumed and the forces on thestructure are representedapproximately by a singleconcentrated force F acting at thetop of the structure and of the form F = F0 sin twhere is the frequency of thewave, t is the time & F0 is theamplitude of the idealized waveforce, chosen so as to give thesame static deck deflection as thatfound from the actual distributedwave force acting on the structure.

It is assumed that one-half of themass of the support structure islumped into the deck mass togive an effective deck mass Mgiven by

M = MD + MS/2

where MD is the deck mass & MSis the total virtual mass of thesupport structure (actual mass +added mass resulting from itsmotion in water).

Dynamic Analysis

Single Degree of Freedom System With this simplification, the

support structure itself may beregarded as mass less and itsresponse calculated using theequilibrium methods.

Because there remains only

one movable mass (the effectivemass at the top of the structure) andonly one direction of sensible motion(the horizontal direction), theanalysis for dynamic response inthis case is known as single-degree-of-freedom dynamicanalysis.

Dynamic Analysis

5Single Degree of Freedom System Now, the total horizontal force FT

acting at the top of the structurecan be regarded as the sum of theapplied force F, the inertia force

and a resistive damping force

represented approximately by

xM

xC

Dynamic Analysis

Single Degree of Freedom System where C denotes a constant damping

coefficient and x is the response so thatwe have the total force acting at the topof the structure given by

From the static equilibrium methods,the total force FT can be related to thehorizontal displacement x at the top ofthe structure by the equation

FT = K x where K denotes the stiffness of the

structure.

xCxMFFT

Dynamic Analysis

6Single Degree of Freedom System

Thus, on combining the above equations,we get

Let us use the parameters: natural frequency n = (K/M)1/2 critical damping Cc = 2(KM)1/2 = 2Mn damping ratio = C/Cc

tFKxxCxM sin0

Dynamic Analysis


OFFSHOREENGINERING:ADVANCESANDSUSTAINABILITY

The complete solution consists of the freeoscillation known as the complementaryfunction and the forced oscillation knownas the particular solution.

However, the damped motion of the transientoscillation disappears after a few initialoscillations following the start of the motion.

The number of cycles of the transientoscillations depends on the amount ofdamping in the system.

The damping values for offshore structurestypically range from about 5% to 10% ofcritical damping.


Only the steady-stateoscillations at the frequency ofthe forcing function remain.

The damping in waves isusually higher than thedamping in the free oscillationof the system.

We get the steady-statesolution as

x = X sin (t-)

where X is the amplitude ofoscillation and the (lagging)phase angle between themotion and the external force.

The values of X and can be obtained as 2/1222 0 CMK FX

2tan MK

C

Dynamic Analysis


Defining the static deflection ofthe spring-mass system XS

XS = F0/K The solutions may be

written in non-dimensionalform as

This constant X/XS is calledthe dynamic amplificationfactor (DAF).

It can be observed that the DAFwill be very high when thenatural frequency is close to thewave frequency.

If the DAF is less than 1.1, it is

enough that the design is basedon a regular design wave andstatic methods of analysis.

2/1222

21

1

nn

SXX

2

1

2tan

n

n

Dynamic Analysis


Dynamic Analysis

Suggestions for practice

Determine the natural frequency of the offshore platformsin your Jurisdiction for the predominant degrees offreedom.

Dynamic Analysis

9Lesson 2

DAF for Fixed Platforms: Worked Examples

DAF For Fixed Platforms: Worked Examples

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Dynamic Analysis

Example 1 Consider the steel offshore structure with sideface as shown in the Figure and determine if astatic analysis is appropriate for a design wavehaving height of 12 m and a period of 6 s. All foursides of the structure are identical.Vertical members have outside diameter of 1.2 mand wall thickness of 38 mm. Horizontal anddiagonal members have outside diameter of 600mm and wall thickness 13 mm.When nodal loads of 100 kN each were applied atjoints D & H, the resulting horizontal displacementwas obtained as 26 mm by matrix methods.The deck weighs 2220 kN & the support structureweighs 2160 kN in air. The value of CM may beassumed as 2. may be taken as 1.025 t/m3.Assume a damping ratio of 5%.

10


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Dynamic Analysis

Solution: Stiffness for the shown side frame = 2*100 /0.026

= 7692.3 kN/m. The remaining side frame also has same stiffness. Hence the total stiffness of the structure = 2*7692.3= 15385 kN/m.

Deck mass = 2220/9.807 = 226.37 t. Mass of support structure = 2160/9.807 = 220.25 t. Vertical legs are assumed to be filled with water upto MSL.

The water mass is 4*1.025(/4)*1.1242*22.5 = 91.54 t

The total actual mass of support structure is 220.25 + 91.54 = 311.79 t.

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Dynamic Analysis

The added mass of support structure is calculated asfollows. 2 verticals = 2*1.025*(2-1)*(/4)*1.22*22.5 = 52.17 t2 lower diagonals = 2*1.025*(2-1)*(/4)*0.62*15

= 8.69 t 2 upper diagonals = 2*1.025*(2-1)*(/4)*0.62*7.5 = 4.35 t = 65.21 t Doubling this for other side, we get = 130.42 tFront faceLower diagonals = 2*1.025*(2-1)*(/4)*0.62*21.21 =12.3 tUpper diagonals = 2*1.025*(2-1)*(/4)*0.62*10.61 =6.15 t

Horizontal = 1.025*(2-1)*(/4)*0.62*15 = 4.35 t


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Dynamic Analysis

Adding & doubling for back face, total forfaces

= 45.6 t Total added mass of support structure = 130.42 + 45.6 = 175.8 t Total mass M of the structure = MD + (1/2)MS = 226.37 + (1/2)(311.79+175.8) = 470.17 t Natural frequency n = (15384/470.17)1/2 = 5.72 rad/s

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2/1222

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nn

SXX

Dynamic Analysis

Damping ratio = 0.05 = 2*/6 = 1.0472 rad/s /n = 0.1831, (/n)2 = 0.0335, [1-(/ n)2]2 = 0.9341 [2*(/ n)]2 = 0.000335

DAF=

1/(0.9341+0.000335)1/2 = 1.0345The dynamic response is only 3.45% above the static response.

Hence a static analysis is appropriate.


2/1222

21

1

nn

SXX

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Example 2Figure gives the details of a gravityplatform. Determine the dynamicamplification factor for the horizontalresponse of the deck when acted uponby a wave of 8 s period, if the dampingratio is 2%.The value of CM may be assumed as 2. may be taken as 1.025 t/m3 for seawater and 2.4 for concrete.

Dynamic Analysis


Dynamic Analysis

For this simple structure, the stiffness Krelating horizontal force and displacementat the top of the structure is expressible as K = 3EI/L3

E is Youngs Modulus= 27500* 106 N/m2 = 27.5*106 kN/m2

I = Moment of inertia = (/64)*(54-44) =18.11 m4

L = Effective length of structure = 40 + 10= 50 m K = 3* 27.5*106* 18.11/503 = 11953

kN/m


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Dynamic Analysis

Deck mass = 14 * 103/9.807 = 1427.55 tColumn mass =2.4*(/4) * (52-42) *50*= 848.23 tAdded mass = (2-1)*1.025*(/4)*52*40 = 805.03 t Total mass M = 1427.55 + (848.23+805.03)/2 = 2254.18 t Natural frequency n = (11953/2254.18)1/2

= 2.30 rad/s Damping ratio = 0.02 = 2*/8 = 0.7854 rad/s


2/1222

21

1

nn

SXX

Dynamic Analysis

/n = 0.3415, (/n)2 = 0.1166 [1-(/n)2]2 = 0.7804 [2*(/ n)]2 = 0.000187

= 1/(0.7804+0.000187)1/2 = 1.132 The dynamic response is 13.2% above the

static response. Hence a static analysis is not appropriate

and dynamic analysis is required.


2/1222

21

1

nn

SXX

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Provisions in PTS 20.073 4.8 Fatigue AnalysisA dynamic spectral fatigueanalysis will be required duringdetailed design.

Linearized Foundation2% dampingFrequency DomainLinear Airy Wave TheoryTransfer FunctionsWave SpectraSCF

4.9 Dynamic AnalysisThe fundamental natural modesof vibration in each of the primaryorthogonal direction shall bedetermined. If fundamental modenatural periods exceed 2.5 s,additional inertia loads due todynamic response effects shall beconsidered for all in-placeanalyses using the methoddocumented inPTS 20.061 Practice for theDynamic Analysis of FixedOffshore Platforms For ExtremeStorm Conditions

Dynamic Analysis

Suggestions for practice1. Determine DAF for the horizontal vibration of a Jacket

Platform under your Jurisdiction using this approximatemethod and compare with the values given in the designcalculations.

2. Determine DAF for the horizontal vibration of any GBSthat you have come across or read about .

Dynamic Analysis

15

Lesson 3

Floating Structure Dynamics

Floating Structure Dynamics Frequency domain analysis has beenapplied extensively to problems offloating structure dynamics and isparticularly useful for long term responseprediction. It can estimate random waveresponses through spectral formulation. Simpler than time domain computationand the results are simpler to interpretand apply. Preferred at the preliminary designstage The significant limitation is that allnonlinearities in the equation of motionmust be replaced by linearapproximations.

Time domain analysis utilizesthe direct numerical integration ofequations of motion allowing theinclusion of all systemnonlinearities such as:Fluid drag forceMooring line forceViscous damping etc.

The significant disadvantagesare increased computer time andincreased complexity in thecomputed results making itdifficult to interpret and apply.

Dynamic Analysis

16

Frequency Domain Formulations

Dynamic Analysis

TLP : Surge & Heave

Dynamic Analysis

17

TLP: Pitch & Tether Tension

Dynamic Analysis

Triangular TLP: Model Tests

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

RAO

Sur

ge (

m/m

)

Frequency (Hz)

Graph of Surge RAO Vs Frequency

AnalytExperi

Dynamic Analysis

18

Spar: Model Tests

Dynamic Analysis

Truss Spar : Surge & Heave

Dynamic Analysis

19

Semisubmersible : Surge

Dynamic Analysis

Semisubmersible : Heave

Dynamic Analysis

20

Time Domain Procedure

In time domain, the equation of motion is solved using numericalintegration technique incorporating all the time dependent nonlinearitiessuch asstiffness coefficient changes due to mooring line tension variation withtime, added mass from Morison equation, viscous damping andevaluation of wave forces at the instantaneous displaced position ofthe structure.At each step, the force vector is updated to take into account thechange in the mooring line tension. The equation of motion is solved byan iterative procedure using unconditionally stable Newmark Betamethod or Wilson Theta Method.

Dynamic Analysis

Time Domain Procedure

Dynamic Analysis

21

Coupled/Uncoupled AnalysisFully integrated analysis is a comprehensive analysis applyingsimultaneous analysis of the platform and the mooring lines afterdividing them into various types of finite elements. It is verycomplicated, consumes large amount of time and the software availableare very costly. Also, the technology regarding this analysis has not yetbeen completely developed.

Uncoupled analysis assumes the platform as a rigid body and themooring lines as linear spring supports.

Coupled analysis considersa) Platform as rigid body and mooring line inputs given based on a

separate analysis done on mooring lines.b) Mooring Lines made up of elements and platform motion inputs

given based on a separate analysis done on the platform. Thesetypes of analysis are quite common and preferred now.

Dynamic Analysis

Dynamic Analysis

dynamic analysis

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