Dynamic Analysis

Dynamic Analysis Prof. Dr Kurian V. John

Outline of contents • Overview: Introduction • Single Degree of Freedom System • DAF for Fixed Platforms: Examples • Floating Structure Dynamics

Dynamic Analysis

1

Overview: Introduction •As the offshore platforms are always subjected to the dynamic wave loads, it is essential that the engineers responsible for the design, construction and maintenance of these platforms, have a fairly good idea of the dynamic behavior of these structures. This presentation will supplement your knowledge and explain some basic ideas regarding the types of dynamic analysis used for fixed and floating 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 dynamic analysis • Differentiate between coupled and uncoupled dynamic analysis

Dynamic Analysis

2

Lesson 1 Single Degree of Freedom System

Single Degree of Freedom System D

H 15 m

C

G

22.5 m

15 m

B

•The analysis of an offshore structure using stiffness matrix methods and joint loadings based on extreme environmental conditions necessarily neglects any dynamic effects associated with the wave-induced periodic motion of the structure. •Such a static analysis can, therefore, only be applied when the dynamic loadings are small in comparison with the maximum static loadings.

F 15 m A

E

•Let us consider an approximate analysis of a typical platform as shown in the Figure.

Dynamic Analysis

3

Single Degree of Freedom System •Regular sinusoidal water waves are assumed and the forces on the structure are represented approximately by a single concentrated force F acting at the top of the structure and of the form • F = F0 sin ωt •where ω is the frequency of the wave, t is the time & F0 is the amplitude of the idealized wave force, chosen so as to give the same static deck deflection as that found from the actual distributed wave force acting on the structure.

•It is assumed that one-half of the mass of the support structure is lumped into the deck mass to give an effective deck mass M given by • M = MD + MS/2 •where MD is the deck mass & MS is the total virtual mass of the support structure (actual mass + added mass resulting from its motion in water).

Dynamic Analysis

Single Degree of Freedom System •

• •

With this simplification, support structure itself may regarded as mass less and response calculated using equilibrium methods.

the be its the

Because there remains only one movable mass (the effective mass at the top of the structure) and only one direction of sensible motion (the horizontal direction), the analysis for dynamic response in this case is known as singledegree-of-freedom dynamic analysis.

Dynamic Analysis

•

4

Single Degree of Freedom System • Now, the total horizontal force FT acting at the top of the structure can be regarded as the sum of the applied force F, the inertia force 

M x

• • and a resistive damping force represented approximately by

•



C x

Dynamic Analysis

Single Degree of Freedom System • where C denotes a constant damping coefficient and x is the response so that we have the total force acting at the top of the structure given by 



F  F  M x C x T

• From the static equilibrium methods, the total force FT can be related to the horizontal displacement x at the top of the structure by the equation • FT = K x • where K denotes the stiffness of the structure. Dynamic Analysis

5

Single Degree of Freedom System •Thus, the above equations,  on combining  we get 0

M x  C x  Kx  F sin t

• • • • • • •

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

Dynamic Analysis

Single Degree of Freedom System • The complete solution consists of the free oscillation known as the complementary function and the forced oscillation known as the particular solution. • However, the damped motion of the transient oscillation disappears after a few initial oscillations following the start of the motion. • The number of cycles of the transient oscillations depends on the amount of damping in the system. • The damping values for offshore structures typically range from about 5% to 10% of critical damping.

OFFSHORE ENGINERING: ADVANCES AND SUSTAINABILITY

6

Single Degree of Freedom System • Only the steady-state oscillations at the frequency of the forcing function remain. • The damping in waves is usually higher than the damping in the free oscillation of the system.

• where X is the amplitude of oscillation and β the (lagging) phase angle between the motion and the external force. •

The valuesF0of X and β X can be obtained as 1/ 2 2 2 K  M 2  C 



tan  

• We get the steady-state solution as • • x = X sin (ωt-β)





C K  M 2

Dynamic Analysis

Single Degree of Freedom System • Defining the static deflection of the spring-mass system XS • XS = F0/K 1 •X The solutions may be  1/ 2 2 in non-dimensional X Swritten 2 2          as  form 1      2       n      n    2 tan  

 n

 1    n

  

2

• This constant X/XS is called the dynamic amplification factor (DAF). • It can be observed that the DAF will be very high when the natural frequency is close to the wave frequency. • • If the DAF is less than 1.1, it is enough that the design is based on a regular design wave and static methods of analysis.

Dynamic Analysis

7

Single Degree of Freedom System

Dynamic Analysis

Suggestions for practice

• Determine the natural frequency of the offshore platforms in your Jurisdiction for the predominant degrees of freedom.

Dynamic Analysis

8

Lesson 2 DAF for Fixed Platforms: Worked Examples

DAF For Fixed Platforms: Worked Examples

D

H 15 m

C

G

22.5 m

15 m

B

F 15 m A

E

• Example 1 • Consider the steel offshore structure with side face as shown in the Figure and determine if a static analysis is appropriate for a design wave having height of 12 m and a period of 6 s. All four sides of the structure are identical. •Vertical members have outside diameter of 1.2 m and wall thickness of 38 mm. Horizontal and diagonal members have outside diameter of 600 mm and wall thickness 13 mm. •When nodal loads of 100 kN each were applied at joints D & H, the resulting horizontal displacement was obtained as 26 mm by matrix methods. •The deck weighs 2220 kN & the support structure weighs 2160 kN in air. The value of CM may be assumed as 2. ρ may be taken as 1.025 t/m3. Assume a damping ratio of 5%. Dynamic Analysis

9

DAF For Fixed Platforms: Worked Examples • Solution: D

H 15 m

C

G

22.5 m

15 m

B

F 15 m E

A

• 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 up to MSL. • The water mass is 4*1.025(π/4)*1.1242*22.5 = 91.54 t • The total actual mass of support structure is 220.25 + 91.54 = 311.79 t.

Dynamic Analysis

DAF For Fixed Platforms: Worked Examples

D

C 22.5 m

B 15 m A

•The added mass of support structure is calculated as follows. • 2 verticals = 2*1.025*(2-1)*(π/4)*1.22*22.5 = 52.17 t 2 lower diagonals = 2*1.025*(2-1)*(π/4)*0.62*15 H • = 8.69 t 15 m • 2 upper diagonals = 2*1.025*(2-1)*(π/4)*0.62*7.5 • = 4.35 t = 65.21 t G • Doubling this for other side, we get = 130.42 t 15 m Front face •Lower diagonals = 2*1.025*(2-1)*(π/4)*0.62*21.21 = 12.3 t F • Upper diagonals = 2*1.025*(2-1)*(π/4)*0.62*10.61 = 6.15 t E • Horizontal = 1.025*(2-1)*(π/4)*0.62*15 = 4.35 t Dynamic Analysis

10

DAF For Fixed Platforms: Worked Examples

D

H 15 m

C

G

22.5 m

15 m

B

F 15 m

• Adding & doubling for back face, total for faces • = 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 ωn = (15384/470.17)1/2 • = 5.72 rad/s

E

A

Dynamic Analysis

DAF For Fixed Platforms: Worked Examples

D

H 15 m

C

G

22.5 m

15 m

B

F 15 m A

E

• Damping ratio ζ = 0.05 • ω = 2*π/6 = 1.0472 rad/s • ω/ωn = 0.1831, (ω/ωn)2 = 0.0335, • [1-(ω/ ωn)2]2 = 0.9341 • [2ζ*(ω/ ωn)]2 = 0.000335 X 1 1  X  1/ 2 1/ 2 2 2 X 2 • DAF= S XS  2  2  2                1  1    2  2      n   n    n   n     • 1/(0.9341+0.000335)1/2 = 1.0345 • The dynamic response is only 3.45% above the static response. • Hence a static analysis is appropriate. Dynamic Analysis

11

DAF For Fixed Platforms: Worked Examples

• Example 2 • •Figure gives the details of a gravity platform. Determine the dynamic amplification factor for the horizontal response of the deck when acted upon by a wave of 8 s period, if the damping ratio is 2%. • •The value of CM may be assumed as 2. ρ may be taken as 1.025 t/m3 for sea water and 2.4 for concrete.

Dynamic Analysis

DAF For Fixed Platforms: Worked Examples •For this simple structure, the stiffness K relating horizontal force and displacement at the top of the structure is expressible as • K = 3EI/L3 • E is Young’s Modulus •= 27500* 106 N/m2 = 27.5*106 kN/m2 •I = Moment of inertia = (π/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

Dynamic Analysis

12

DAF For Fixed Platforms: Worked Examples •Deck mass = 14 * 103/9.807 = 1427.55 t •Column mass =2.4*(π/4) * (52-42) *50*= 848.23 t •Added mass = (2-1)*1.025*(π/4)*52*40 = 805.03 t • Total mass M = 1427.55 + (848.23+805.03)/2 = 2254.18 t • Natural frequency ωn = (11953/2254.18)1/2 •= 2.30 rad/s • Damping ratio ζ = 0.02 • ω = 2*π/8 = 0.7854 rad/s

Dynamic Analysis

DAF For Fixed Platforms: Worked Examples • ω/ωn = 0.3415, (ω/ωn)2 = 0.1166 • [1-(ω/ωn)2]2 = 0.7804 • [2ζ*(ω/ ωn)]2 = 0.000187 11 XX  1 / 12 / 2 XXS S   2 22 2 2 2           1       22    1  n  n           n n  

• = 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. Dynamic Analysis

13

Provisions in PTS 20.073 • 4.8 Fatigue Analysis

• 4.9 Dynamic Analysis

•A dynamic spectral fatigue analysis will be required during detailed design.

•The fundamental natural modes of vibration in each of the primary orthogonal direction shall be determined. If fundamental mode natural periods exceed 2.5 s, additional inertia loads due to dynamic response effects shall be considered for all in-place analyses using the method documented in •PTS 20.061 Practice for the Dynamic Analysis of Fixed Offshore Platforms For Extreme Storm Conditions

 Linearized Foundation  2% damping  Frequency Domain  Linear Airy Wave Theory  Transfer Functions Wave Spectra SCF

Dynamic Analysis

Suggestions for practice 1. Determine DAF for the horizontal vibration of a Jacket Platform under your Jurisdiction using this approximate method and compare with the values given in the design calculations. 2. Determine DAF for the horizontal vibration of any GBS that you have come across or read about .

Dynamic Analysis

14

Lesson 3 Floating Structure Dynamics

Floating Structure Dynamics  Frequency domain analysis has been applied extensively to problems of floating structure dynamics and is particularly useful for long term response prediction.  It can estimate random wave responses through spectral formulation.  Simpler than time domain computation and the results are simpler to interpret and apply.  Preferred at the preliminary design stage  The significant limitation is that all nonlinearities in the equation of motion must be replaced by linear approximations.

Time domain analysis utilizes the direct numerical integration of equations of motion allowing the inclusion of all system nonlinearities such as: •Fluid drag force •Mooring line force •Viscous damping etc. The significant disadvantages are increased computer time and increased complexity in the computed results making it difficult to interpret and apply.

Dynamic Analysis

15



Frequency Domain Formulations

Dynamic Analysis

TLP : Surge & Heave

Dynamic Analysis

16

TLP: Pitch & Tether Tension

Dynamic Analysis

Triangular TLP: Model Tests 0.35 RAO Surge (m/m)

0.3 0.25

Graph of Surge RAO Vs Frequency Analyt… Experi…

0.2

0.15 0.1

0.05 0 0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Frequency (Hz)

Dynamic Analysis

17

Spar: Model Tests

Dynamic Analysis

Truss Spar : Surge & Heave

Dynamic Analysis

18

Semisubmersible : Surge

Dynamic Analysis

Semisubmersible : Heave

Dynamic Analysis

19

Time Domain Procedure •In time domain, the equation of motion is solved using numerical integration technique incorporating all the time dependent nonlinearities such as •stiffness coefficient changes due to mooring line tension variation with time, added mass from Morison equation, viscous damping and evaluation of wave forces at the instantaneous displaced position of the structure. • •At each step, the force vector is updated to take into account the change in the mooring line tension. The equation of motion is solved by an iterative procedure using unconditionally stable Newmark Beta method or Wilson Theta Method. Dynamic Analysis

Time Domain Procedure

Dynamic Analysis

20

Coupled/Uncoupled Analysis •Fully integrated analysis is a comprehensive analysis applying simultaneous analysis of the platform and the mooring lines after dividing them into various types of finite elements. It is very complicated, consumes large amount of time and the software available are very costly. Also, the technology regarding this analysis has not yet been completely developed. •Uncoupled analysis assumes the platform as a rigid body and the mooring lines as linear spring supports. •Coupled analysis considers a) 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. These types of analysis are quiteDynamic common and preferred now. Analysis •

Dynamic Analysis

21