Advanced stochastic analysis program for offshore structures - Theoretical outline and the User's Manual of SAPOS

Theoretical outline and the User's Manual of

SAPOS

TU Delft

DeIft University of Technology

by

Dr. H. KARADENIZ

Report No.: 25.2.89.1.11

August, 1989

FacUlty ofcivilEngineering DMalonof Mechanics and Structures Section of pplIed Mechanics

TECHNISCHE UNIVER$iiit Laboratolium voor Scheepshydromecha Archlef MekeiWeg 2,2628 CD Deift Tol. 015.788813. Faz 015.781858

SAPOS

-CONTENTS

1- INTRODUCTION 1

2- SPECTRAL ANALYSIS 3

2.1 - General Concept and Traditional Modal Analysis Technique 3

2.2 - Modified Modal Analysis Technique 6

2.3 - Transfer functions of Wave Forces 7

2.4 - Direct Formulation of Stress Transfer-function 9

2.5 - Calculation of stress statistical characteristics 11

3- STOCHASTIC FATIGUE DAMAGE CALCULATION 13

4- UNCERTAINTY MODELLING AND RELIABILITY CALCULATION 16

4.1 - Uncertainties in the Fatigue Damage 16

4.1.1 - Structural Category 16

Stiffness matrix 16

Mass matrix 16

Damping matrix 17

4.1.2 - Loading and Environmental Category 18

Wave loading 18

Modelling, of random wave environment 18

4.1.3 - Fatigue Category 20

Stress Concentration Factors 21

S-N model 21

c.) Damage Correction Factor 21

4) Reference Damage . . 22

4.2 - Limit State Function and The Reliability Calculation 22

5- PROGRAM OUTLINE AND GENERAL DESCRIPTION 25

5.1 - Outline of the Program 25

5.2 - General Description . 29

6- PROGRAM INPUTS . . 32

6.1 - Indirect Inputs . 32

6.1.1 - INCidences of submerged members 32

6.1.2 - BOOkkeeping degrees of freedom of submerged joints 33

&1.3 -. COOrdinates of submerged joints 34

6.1.4 - iMSplacements (EIGENMode vectors) of submerged joints 35

61.5 MODal (GENeralized) masses 36

6.1.6 - NATural frequencies (EIGENVa1ues). 36

6.1.7 - PROpertIes of submerged members 37

6.1.8 - FLExibility matrix (stored columnwise from diagonal 'terms) 38

'6.2 - Direct Inputs 40

6.2..1 - DAMping ratios 40

6.2.2 - MATerial constants 40

6.2.3 - LOAding properties .. 41

6.2.4 - WAVe properties F -

i

directional sea 41

L[MULti-]J

6.2.5 - MAIn wave direction r 4

6.2,6 - GAUss points for frequency integration 43

[sPEctral

_]

[STAtic _1.

627-

FATigue

analysis ID'am1c

I contribution 43

LRELiabiUyJ LSTAtic + DYNamicj

[FORCES

i

6.2.8 - LIST STRESses

for the sea state h (and t) 46

[SPECTral moments _J

6.2.9 - SEA properties

[Jswap]] spectrum, SCA (DIA)

47

6.2.10-UNLoaded members 51

6.2.11 - STRess (FORce) locations 51

6.2.12 - LIFe (SERvice) time 52

6.2.13- ITERation number of the reliability analysis 53

6.2.14 - DESign variables 53

PROGRAM OUTPUTS 56

7.1 - Standard Outputs 56

7.2 User contr011ed Outputs 57

DEMONSTRATION - 58

SAPOS

-ADVANCED STOCHASTIC ANALYSIS PROGRAM

FOR OFFSHORE STRUCTURES

Theoretical outline and the User's Manual of

SAPOS

1- INTRODUCTION

This report is intended primarily to describe an update version of the computer

program, SAPOS [1], as being a user's manual, and secondarily to outline the theoretical background of the update. SAPOS stands for Stochastic Analysis

program for Offshore Structures, which is a sophisticated computation-tool for

the spectral, fatigue and fatigue-reliability analyses of offshore structures.

Although it is designed for practical applications, it can also be used for

educational .and research purposes., as well as for checking results of other programmes. It accommodates advanced analysis methods tO put them into practice, giving more confidence in the computation and reliability of results.

SAPOS is a talented software for stochastic analysis of offshore structures, which

has a trend to meet requirements of modern design philosophy. This update

utilizes recent algorithmic developments and improved analysis methods

providing a better performance and more flexibility in the application. New

features of the program are summarized in the following. Effect of marine growths on the wave force calculation.

JONSWAP sea spectrum as additional to Pierson-Moskowitz (P.M.) sea

spectrum.

Two parameters modelling of the sea spectrum as Hs (significant wave height)

and Tz (mean zero crossing period of waves) with scatter diagram for the

long-term distribution of the sea state.

Modification of the modal analysis technique to include the static

contribution of the response.

Con.trbution of the fixed end member forces to the stress calculation. This is

due to the distributed nature of the wave force according to Morison's

equation.

Allowing for an endurance in the S-N fatigue model.

An efficient modelling of uncertainty contents of fatigue damage in the case of reliability assessment to reduce the cOmputation time.

Since SAPOS is arranged in a flexible and independent form it can be easily

implemented into, or run with, an existing structural analysis package. Further, it offers a very friendly and simple usage of inputs and clarity of utputs as will be

explained later in the report. Besides its sophisticatiOn it is, however, subjected to the following restrictions for the time being.

It runs with an analysis package. Thus, it is not completely independent in its

integrity.

Since it has no graphical facilities, direct graphical outputs are not possible Only wave forces are considered. Thus, wave-current interactions and other phenomenal forces are not iÍiludéd.

Structure-water interactions are not yet included.

A code checking, is not implemented.

Only bottom supported structures can be analyzed.

FoundatiOn-structure interactions are not allowed, thus, the underlying

foundation can be represented simply by massless springs.

Contribution of shear deformations is not included in the stress analysis

As indicated above, three calculation items can be distinguished in SAPOS,

which are spectral analysis, fatigue damage calculation and reliability

assessment These items will be outlined first in next chapters and then the

2- SPECTRAL ANALYSIS

2.1- General Concept and Traditional Modal Analysis Technique

In offshore structural engineering, calculation of fatigue damage and finding a safety margin against a probable fatigue failure are of great interest to designers, to insuring and owning companies Calculation of the fatigue damage in a random environment, such as in offshore, is a difficult subject and requires a stochastic treatment. Formulation of a stochastic fatigue damage can be obtained in terms of spectral moments of the stress process [2] in a structural member, which are

calculated by means of a spectral analysis procedure. It i.s meant from the spóctral

analysis that spectral values of the stress process are to be calculated. In SAPOS such an analysis is carried out by using the concept of transfer functions of linear systems in which it is assumed that there are linear relations between inputs and

outputs. An overall calculation scheme of the stress spectrum is shown in

Fig.2.1.1.

WAVE ENVIRONMENT

STRUCTURE ¿ STRUCTURAL TRANSFER FUNCTION SAPOS -w THEORETICAL TREATMENT WAVE FORCE TRANSFER FUNCTION y STRESS TRANSFER FUNCTION WAVE RECORDS V SEA SPECTRUM STRESS SPECTRUM

Fig.2.i.1 - Flow diagram of the calculation of a stress spectrum.

STRUCTURAL TRANSFER

FUNCTION

Fig.2.1.1 gives an implication of the procedure using the cOncept of transfer

functions. It is described as in the following.

Based on wave records from the wave environment, spectrum of the wave

elevation (Sea spectrum) is determined This is usually known in a functional form, and it constitutes the input spectrum of the analysis.

From the theoretical point of view, using the linear wave theory, wave force

transfer functions are calculated. Theseare dependent on the structural

configuration.

On the other hand, some structural transfer functions are calculated as being independent of wave environment and thus, of wave forces.

The stress transfer function is calculated by a suitable combination of the

structural and wave force transfer functions. This is shown in Fig.2.1.2, when using the traditjonal calculation procedure.

The stress spectrum is calculated from the product of the stress transfer

function and the sea spectrum as indicated in ig.2.1.1.

WAVE FORCE TRANSFER FUNCTION

L

DISPLACEMENT TRANSFER FUNCTION

Fig.2.1.2- Calculation of stress transfer functions using the traditional procedure

As it can be realized from above, a great deal of efforts are spent on the

calculation of the stress transfer function which constitutes the hearth of the

spectral analysis. In thIs calculation, a special care needs to be given to a correct

determination of structural tranSfer functions since they may affect the

calculation dramatically These functions can be derived directly from the overall

dynamic equilibrium equation of the system [3] written as in the matrix nOtation,

[K] {U} + [C] {Ú} + [M] {Ü} = {F(t)} (2.1.1)

4

STRESS TRANSFER

in which, [K], [C] and [M] are respectively the overall stiffness, damping and mass

matrices, {U} and {F(t)} are respectively the overall displacement and applied force vectors, a dot (.) denotes a time derivative. Taking the Fourier transform of both sides of Eq.(2.1.l), the frequency-domain equation can be written as,

([K] + iw[C] -w2 [M]) {U(w)} = {F(w)} (2.1.2)

where i=V-1, and w denotes the angular frequency. From Eq.(2 1.2) the

structural transfer functions can be stated in the matrix notation as,,

[HUF(w)] ([K] +iw[C]

-Then the displacement vector will be written from Eq.(2.1.2) as,,

{U(w)} = [HUF(w)] {F(w)} (2.1.4)

Here it is worth noting that E4.(2.1.3) is the exact form of the structural transfer

functions. As it might be realized, their evaluations

take excessive

computation-times Therefore, using Eq (2 1 3) is not suitable in practical

applications. Instead, an approximate formulation is widely used in practice,

which is obtained by employiñg the well known modal analysis technique leading

to,

[HUF(w)]

i

hr(CV) {cIr} {(br}' ' (2.1.5)

In this statement, q is the total number of natural modes considered, {r}

denotes an eigenmode vector,, the superscript (T) indicates a matrix transposition

and hr(w) is a frequency fünction corresponding to 'the natural mode r. This

function is termed here as a modal transfer function anddefined by,

i W

kr (Wr2- w2) +2irWrW

SAPOS

-(2.1.3)

(2.1.6)

where Wr, r are respectively the natural frequency and the damping ratio to the

critical, kr is the generalized stiffness.

This traditional analysis technique was used in the previous version of SAPOS. A

concept of Spectral Participation Factors (SPF) [4], which is based on this

technique, was introduced to ease the analysis. Although it was elaborately used

in SAPOS the traditional technique has been known to be inadequate to produce

satisfactory results in the stress analysis [5]. Satisfactoty results can only be

obtained if a large number of natural modes are included in the analysis in

Page :

general. However, it may also depend on the structural cOnfiguration. For highly

sensitive structures to dynamic excitations few natural modes may suffice to produce good results whereas for insensitive structures good results cannot be

achieved by considering only few modes. In such a case, the quasi-static

contribution of the response becomes very important. Even it dominates the

whole response behaviour when the fundamental natural frequency of the

structure is far from the excitation frequency of the forcing phenomenon.

Therefore, in this case, it cannot be truncated by using the traditional modal analysis technique unless a large number of natural modes are included in the

analysis. This latter option is obviously computationally-awkward and

unpractical. In order to eliminate this unpleasant situation and to improve the

performance of the modal analysis technique, an alternative procedure, which is used in the update version of SAPOS, is presented below.

2.2- Modified Modal Analysis Technique

As outlined above, the traditional modal analysis technique fails to produce a

complete structural response. The quasi-static part of the response is not

sufficiently taken into account in the analysis. Alternatively, a pure quasi-static

analysis does not include the dynamic contribution. It seems however that an

analysis technique which takes into account both the quasi-static and dynamic contributions, which is also computationally practical, is likely the most Suitable and desirable in the calculation of spectral response stresses. Starting from this point of view of thinking, a structural transfer function matrix can be defined as,

[HUF(w)]

[K]' +ar(w) {} {r}T

(2.2.1)

where the first term on the right hand side corresponds to the quasi-static

solution and the second one is included to represent the dynamic contribution. In

Eq.(2.2.1) the parameter ar(w) is a complex frequency function which determines

the participation of the natural mode r. This parameter is obtained [6] to be,

ar(w) hr(W) - (2.2.2)

where hr(w) is defined by Eq.(2.1.6) and kr is the generalized stiffness for the

mode r

This modified modal analysis technique is implemented into SAPOS. It reduces computation-time considerably and produces accurate results. Additionally, it provides flexibility in usage as probably to apply a pure quasi-static analysis or

the traditional modal analysis technique. These options are also available in SAPOS, and can be activated by user's command. An efficient calculation

algorithm is developed, to manipulate this advanced technique in the program.

The algorithm is based on eliminating the displacement field from the

determination of stress transfer functions, thus direct relations between applied

SAPOS

-forces and resulting stresses are constituted. This is a remarkable achievement that prevents excessive calculations in the spectral analysis. This subjèct will be explained in section 2.4. First, transfer functions of wave forces are presented in the following section.

2.3- Transfer functions of Wave Forces

Wave forces acting on offshore structural members are calculated from Morison's

equation. These forces are distributed along member length.s and being normal

to member axes as shown in Fig.2.3.1.

J4PL

À4w.

Wave direCtbO, X t, Normal water velocity Morison 's p = CD k Member axis equation w +

a) structural system under wave b) wave loading on a member Fig.2.3.1 A schematic offshore structure and wave loading.

Morisón's equation is Writtén in the time domain as,

p = CD Iwl. w + M'

(2.3.1)

where w and * are respectively water velocity and acceleration being normal to the member, Ci pCdDh/ 2 and CM =JrpcmDh2/ 4 in which p is the mass density

of water, Cd and Cm are respectively the drag and inertia force coefficients, Dh denotes an increased member diameter due to marine growth. As it is seen from

Eq (2 3 1) the drag force term in the Morison's equation is non- linear with

respect to the water velocity w. In order to employ the concept of transfer

functions, this term needs to be linearized somehow In SAPOS, a linearization technique is used assuming that the linearized and non-linear forces produce the

same variance. The linearized Morison's equation can be written in vectorial

notation as,

{p} = f(0) CD Ou {w} + CM {''} (2.3.2)

where f(0) is a function of wave direction angle, see Fig.2.3.1 (a), and °u is the standard deviation of horizontal water velocity in the wave direction Using the

linear wave theOry and Eq.(2.3.2) the wave force vector can be written in the

frequency domain [4] as,

{p(w)} = {H,7(w)} (w) (2.3.3)

where ij(w) denotes water elevation, which is considered as the input of the analysis, and {Hp,1(w)} denotes transfer function vector of the wave force

according to the Morison's equation. This vector can be stated as,

{Hp,7(w)} = (f(0) CD au + i(I) CM) {Hw,1(w)} (2.3.4)

in which {H,7(w)} denotes transfer function vector between the normal water

velocity and the watet elevation. The transfer functiOn vector of the wave force,

given by Eq.(2.3.4), is written per unit member length. Its distribution is

exponential in vertical direction and harmonic in horizontal direction. Using principles of external work, transfer functions of equivalent wave forces at

member ends will be calculated, assuming that cubic polynomial fui ctions are used to represent member deformations. These transfer functions can. be written

in vectorial notation as, e.g. fr member i,

{HpL(w)}i

tN]i {H(w)} dx

(2.3.5)

where [N] denotes shape function matrix ofmember deformations, l denotes the

member length and x denotes a variable along the member axis. The subscript (L)

in above statement indicates local coordinates. For the assemblage this vector

will be transformed to the global coordinates, see Fig.2.3.2 for local and global coordinates.

z

X

a) Member local coordinates b) Member global coordinates

SAPOS

-The transformation is wr1tten as,

{HF,7(w)}i = rT]iT{HFL(w)}i (2.3.6)

where {HF,7(w)}i is the transfer function vector of consistent member forces in global coordinates and [T]i is a transformation matrix between member degrees

of freedom in two còordinate systems. The subscript i denOtes a member. Having

calculated transfer functions of member forces, an assembly process will be

carried out to obtain transfer functions Of overall wave forces. This process is formulated as,

{HF)} =:

{HF(w)}i (2.3.7)

where [B] is a matrix which determines locations of member degrees of freedom in the overall system (its elements are either i or O), n denotes total number of loaded members. This corresponds to the traditional analysis and may be time

consuming in the dynamic stress calculation which is largely applied in this

report. An alternative procedure, which seems to be more efficient and

promising, is used in SAPOS Basic formulations of this attractive procedure is presented in the following section.

2.4' Direct Formulation of Stress Transfer-function

When the straightforward procedure is applied, the usual way of determining _a stress transfer fúnction is to calculate displacement transfer-functions first, and then, transfer functions of member internal forces. In the alternative procedure, it is so formulated that the stress transfer function can be obtained directly from those of applied forces This is achieved by eliminating displacements from the calculation of the stress. The formulation of this powerful procedure is actually derived from the traditional procedure, and therefore, it is presented first. Similar to Eq.(2.1.4), transfer functions of system displacements can be written in matrix notation as,

{Hu,7(w)} [HuF(w)] {HF7J(w)} (2.4.1)

in which [HuF(w)] and {HF,(a)} are calculated respectively from Eqs.(2.2.1)

and (2.3.7). Transfer functions of member displacements will now be extracted from Eq.(24.1). This extraction is formulated as, e.g. for a stress member j,

{Hu,7(w)}j [BlT {Hu,7(w)} _(2.4.2)

where [B] is the same matrix as in Eq.(2.3.7.), but this time it is for the stress

member _j. The vector given by Eq.(2.4.2) will be transformed to member local coordinates first, and then, transfer-functions óf member internal forces will be calculated from,

{Ho,7(w)} = [KL]j {HuL(w)}j - {HFL,7(w)}j (2.4.3)

where the subscript L denotes local coordinates, {Ho(w)}j is the vector of

transfer functions òf the member internal forces, [KL]j is the member stiffness matrix, {HUL(w)}j is the transfer function vector of member displacements in local coordinates, and {HFL,7(w)}j is calculated from Eq.(2.3.5), for the stress member j,

which represents the contribution of fixed-end member-forces.

Transfer function of a stress at a given location can now be calculated using the linear relation between internal forces and the stress. This is formulated as,

H7(co)J

{L}j' {H(w)}

(2.4.4)

where {L} is a vector containing member properties (cross- sectional area and

inertia moments), and H,7(w) is the stress transfer function we would like to

calculate.

The alternative procedure is formulated by introducing transfer functions of

displacements, given by Eqs.(2.4.1) and (2.4.2), into Eq.(2.4.3), and by making

use of Eqs.(2.3.7) and (2.2.1). In this procedure, contributions of transfer

functions of member wave forces to the stress transfer function are calculated

first. Then, the total stress transfer function is obtained by superimposing

individual contributions of member forces, which is written as,

Hs(w)j {Z(w)}

{HFL)}i -

{L}jT{HFL(w)}j

(4.5)

where n denotes the number of loaded members. In Eq.(2.4.5), the vector,

{Z(w)}, determines the participation of a loaded member in thestress transfer function, the vectors {HFL,7(w)}i and {HFL,7(w)}j are respectively transfer

function vectors of consistent forces of a loaded member i and the stress member

j. These vectors will be calculated from Eq.(2.3.5). The participation vector,

{Z(w)}, is obtained to be,

{Z(w)} = + {V(w)}ji (2.4.6)

where the vectors {U}i and {V(w)} represent respectively the quasi-static and

dynamic parts of the response. As it might be realized from Eq.(2.4.6) the

quasi-static part is a frequency independent vector unlike the dynamic part which is a function of the frequency, but independent of the wave loading. The quasi-static part is formulated as,

-SAPOS.

[T]i [A]Ji[T]jT [KL]j {L} (2.4.7a)

where the matrix [A]ï determines the participation of a loaded member in

quasi-static displacements of the stress member j This matrix is defined as,

[BIIT [K]4 [B]j (2.4.7b)

in which [K]' is the system flexibility matrix. The dynamic part in Eq.(2.4.6) is

simply calculated from,

q

{V(w)} =)

(ar(W) sjr) {bLr}i (2.4.8)

r=1

where Sjr is a modal stress corresponding to the eigenmode r, ar(w) is as defined

by Eq.(2.2.2), and {Lr}i is the eigenmode vector of a loaded member in local

coordinates.

Calculation of the quasi-static part of the response from Eq.(2.4.7) looks like

complicated. As a matter of fact, it is very simple and needs to be calculated once

for all frequency variations Although the calculation of the matrix [A]JL from

E.(2.4.7b) seems to require matrix multiplications, in fàct, it is a matter of

extraction from the system flexibility matrix.

2.5- Calculation of stress statistical characteùistics

As it will be explained in the next chapter, a cumulative fatigue damage is formulated depén4ently on some stress statistical characteristics which are

calculated in terms of spectral moments of the stress. In this calculation, a

narrow-band stationary Gaussian process is assumed for the stress variation.

Stress spectral moments are calculated from the frequency integration,

00

mn

Jwn'

Sss(co) dw (2.5.1)

where n shows the degree of the moment and Sss(w) is the stress spectrum. In

practice,, the frequently used. spectral moments (zero, second and fourth degrees)

are obtained by using n 0, 2 and 4. The stress spectrum is calculated in terms

of the stress transfer function and a sea spectrum as written by, for a stress

member j,

S(w) =

Hs.*,l(w)j _{H,7(w) S,7(w) (2.5.2)}

where H,(w) is the stress transfer function calculated from Eq.(2.4.5), the (*)

denotes a complex conjugate, and S,7(w) is an assumed sea spectrum. If,

however, a directional distribütiön of the sea spectrum is taken into account, Eq (2 5 1) will also be integrated over the distribution region which is mostly defined between -r/2 and +r/2 in practice An automatic integration procedure [7] is applied in SAPOS to calculate the stress spectral moments The stress

statistical characteristics which are used in. the damage förmulation. are defined

as,

Standard deviation (m0 (2.5.3a)

Mean zero crossing period : T0

= 2r \1iii.

(2.5.3b)

Mean period of maxima

Ïm =

2.ir (2.5.3c)

Calculation of the ftigue damage which is implemented in SAPOS is explained briefly in the next chapter

SAPOS

-3- STOCHASTIC FATIGUE DAMAGE CALCULATION

Calculation of the stochastic fatigue damage is carried ou in SAPOS assuming that,

an experimentally determined S-N fatigue model is known, the stress process is a narrow-band stationary Gaussian, the well known Palmgren-Miner's rule is applied.

Under these conditions, a closed form damage-formulation can be obtained [8]. Experimental results of fatigue tests display always a scattered diagram A linear S-N model, which is defined in logarithmic scales, is usually fitted to data of the

fatigue tests in practice. This model dòes not allow for fàtigue endurance at lower

stress ranges which may be required sometimes to be considered in the analysis. In order to meet such requirements, a linear model with an endurance limit and also a bilinear model ( See Fig 3.1) are implemented into SAPOS in addition to a commonly used linear S-N model. It is, however, obvious that data of the fatigue

tests can be better represented in a functional form by using a bilinear S-N

model, which surely reduces uncertainties in the fatigue model when a reliability analysis is carried out.

in Sb

in S

ln Sb

inC

inN

mCi

lnC2

in N

a) A S-N model With endurance limit b) A bilinear S-N model

Fig.3.1- Additional S-N models implemented intò SAPOS

The assumption of using a narrow-band Gaussian process for the Stress variation

leads to a Rayleigh distribution for the stress range so that a mean fatigue damage

caused by one stress cycle can analytically be calculated [8]. Mean fatigue

damages, under one stress cycle, corresponding to the S-N models which _are

implemented into SAPOS are written below,

a) A linear S-N model:

E[dD] ₌

(2v'2 X9 us)k r(1±--)

_(3.1)

where C and k are the fatigue parameters, X9 is a random variable with u(X9) = i

representing uncertainties in the stress concentration factors (SCF) of the

normal force and bending moments, as is the standard deviation of the stress at a hot- spot, and r(.) denotes the Gamma function.

b) A linear S-N model With an endurance limit:

E[dD]

=c

(2V2X9Os)kF(l+

y)

where T(.,.) denotes the incomplete Gamma function [9], and the parameter y is

defined as,

i

S _Z X9U

in which Si is the endurance limit shown as in Fig.3.1.a.

e) A bilinear S-N model:

i

k1 kl-k2. k2 k2

E[dD] = -

(2V2X9as)kl(T(1±_,y)+y 2

_{(I'(l+ --)-F(1+ ,y)))}

(3.3a)

C1 2 2

where Ci and ki, C2 and k2 are the parameters of the bilinear S-N model, see

Fig.3.lb. The parameter, y, is defined as,

C2 _{2/k2-kP -2}

y

(-)

'

(2v'2 X9 as) _(3.3b)

Ci

The damage caused by one stress cycle will be accumulated during a sea state and

in a long-term period The cumulative damage is simply calculated by

superimposing the damages of one cycle in an assumed period of time Since sea states in the long-term are usually described to be probabilistic, the mean of the cumulative damage in. the long-term can be stated as,

Dtot T

JJ

E[dD] fHs,Tz(h,t) dh dt (3.4)

where T is an assumed life-time, To is the mean zero-crossing period calculated from Eq.(2.5.3), and fHs,Tz(h,t) is a joint probability density function of H _and

T which represent the sea state in the long-term If, however, the _{sea state is}

represented by H5 only, the density function fHs,Tz(h,t) will be replaced by the

marginal density function of H and this will be denoted by, fHs(h) The

parameter, ..%, in Eq.(3.4) is a damage correctiOn factor due to the fact that the damage is formulated by using a narrow-band stress process But, in practice, the

stress process iS usually non narrow-banded and displays a bimódal spectral

(3.2a)

SAPOS

-shape so that the damage due to the ñarrow-band stress process is corrected by the parameter, . This parameter is a function of the spectral bandwidth of the

stress process as well as slope(s) of the S-N fatigue model used.

As it will be. explained later, a cOntinuous joint density function of H and Tz

which is based on a provided scatter diagram. is also implçmented into SAPOS, in

addition to using a discrete scatter diagram If the sea state is represented by Hs,

in the long-term, a three parameters

Weibull and a log-normal probability

functions are provided in .the program

fr the marginal distribution of H. If,

however, multiple main-wave directions are to be considered, Eq (3 4) will also

be integrated over O23r, assuming that the main-wave direction is uniformly

distributed in O-2x..

n the above equatiOns of the àtigue damage there are some inevitable

uncertainties that must be taken into account when a reliability analysis is carried

out These uncertainties and the reliability calculation, which are treated in

SAPOS, are explained in the following chapter.

4- UNCERTAINTY MODELLING AND RELIABILITY CALCULATION

4.1- Uncertainties in the Fatigue Damage

Uncertainties in the cumulative fatigue damage presented by Eq.(3.4), may be. classified in three categories as:

structural,

loading and environméntal,

fatigue related.

These uncertainty groups are. discussed below separately. 4.1.1- Structural Category

Uncertainties reflected in the system stiffness, mass and damping matrices are considered in this category as explained in the following.

Stiffness matrix : Member thicknesses and foundation parameters may be

considered as major uncertainty origins. The total stiffness matrix can be stated in the uncertainty space as,

Xk 4u([K]) = Xt 4u([Kt]) + Xg 4u([Kg]) (4.1.1.1)

where Xk, Xt and Xg are respectively uncertainty parameters in the system,

structural and foundation stiffness matrices, p([K]), 4u([Kt]) and 1u([Kg]) are the

corresponding mean matrices. As it may be realized from Eq.(4.1.1.1) the mean values of the uncertainty parameters are all equal to 1. In SAPOS, Xk is used to

represent the total uncertainty related to the system stiffness matrix. As

mentioned, its mean value equals 1. Its variance can be estimated from,

Xk = Xt (r + Xg (-)r

(4.1.1.2)

where k, kt and kg are the generalized stiffnesses calculated respectively by using the matrices 1u([K]), 4u([Kt]) and 4u([Kg]), the subscript r denotes an eigenmode

It is likely that Eq.(4.1.1.2) produces different values of Xk for different

eigenmodes in which case an average value of the variance can be used.

Mass matrix : Added masses due to structure-water interactions, the mass of the deck and structural masses due to member thicknesses are considered to be

substantial uncertainty origins. Similar to the stiffness matrix, the total mass

matrix can be stated in the uncertainty space as,

SAPOS

-where Xm, Xt (same as in thé stiffness matrix), Xa, and Xd are respectively

uncertainty parameters of the system, structural, added, and deck mass matrices, u([M]), u([Mt]), i«[Ma]) and u([Md]) are the corresponding mean matrices. The

matrix [Mw] in Eq.(4.1.Í.3) is a deterministic matrix due to water included in members. Mean values of these uncertainty parameters are all eqúal to 1. The

parameter Xm is used in SAPOS to represent all uncertainties related to the

system mass matrix. Given statistical characteristics of Xt, Xa and Xd, the

variance of Xm can be calculated from,

Xrn = X ( )r + Xa (

-)r + Xd (

-)r +(

(4.1.1.4)

where m, mt, ma, m and mw are the mean generalized masses. Since the

thickness parameter,, Xt, is defined in both the stiffness and mass matrices in

common, Xk and Xm Will be correlated variables to some degree. Thécorrelation

coefficient can be obtained by using Eqs.(4.1.1.2) and (4.1.1.4). Then, these two dependent variables Will be made independent by an orthogonal transformation [4]. The independent variables, which are denoted by Xi and X2, are used in the reliability calculation as to be basic random variables.

c) Damping matrix : Because the modal analysis technique is used in SAPOS,

critical damping ratios are considered to be uncertain. A full correlation is

assumed between thése ratios. Thus, it can bé written that,

= X3 4U(r) (4.1.1.5)

where X3 is the uncertainty variable and 4u(4r) denotes the mean of the critical damping ratio for the eigenmode r.

It is Worth noting here that, since uncertainties in the system stiffness and mass matrices are represented by single random variables, eigenmode vectors will be independent of the idealized uncertainty variables. This results in a considerable

shortcut in the total computation-time since the eigenvalue solution will be

carried out once for all reliability iterations. This can be verified from the

eigenvalue equation written in the uncertainty space as,

Xk i«[K])

Xm u([M]) {}

(4.1.1.6)

where {1} denotes an eigenmode vector. From this equation, it may be seen that natural frequencies are well dependent on the uncertainty variables Xk and Xm unlike eigenmode shapes. They are calculated from,

Wr2 () u(Wr2)

(4.1.1.7)

where U(Wr) corresponds to the mean natural frequency for the eigenmode r.

4.1.2- Loading and Environmental Category

Uncertainties arising from, wave loading and modelling of random

wave

environment are considered in this category as outlined below.

Wave loading: Wave loading is calculated according to the Morison's equation

given by Eqs.(2.3.1) and (2.3.2). The parameters of the drag and inertia fOrce

terms are defined by,

CD p cd Dh/ 2

(4.1.2.la)

CM = 2rp Cm Dh2 /4 (4.1.2. lb)

in which c and Cm are respectively drag and inertia force coefficients, and Dh is the increased member diameter due to marine growths. Uncertainties in thewave

loading are substantially introduced by cd, Cm and the marine growth inctements

on members.. However, there are also some other uncertainties introduced by the stochastic linearization of the drag force term. These uncertainties are telated to the sea spectrum and will be outlined later in this section.

The increased member diameter can be written as,

Dh= D + 2 X4 (hm) (4.1.2.2)

where D is the structural diameter of the member, u(hm) is the mean thickness

of the marine growths on the member, and X4 is an uncertainty variable. Here, it

is assumed that marine growths on all members are fully correlated so that X4 represents all uncertainties related to the marine growths. The mean value of this

variable is given as u(X4) = 1, and its variance will be estimated on the basis of observations in the field. In a similar way, the inertia and drag forceterms can be

stated as,

Cm = Xi /i(Cm) _(4J.2.3a)

Cd = Xg 4u(cd) _(4.l.2.3a)

where X7 and X8 are random variables with 1u(Xi)= i and u(X8) = 1, u(cm) and

u(cd) are the mean inertia and drag force coefficients.. Eqs.(4.1.2.2) and (4.1.2.3) will be used in Eq.(.4.i.2.1) to idealize the uncertainties in the wave loading.

Modelling of random wave environment : In practice, the _{wave elevation, 'i, is} used to idealize the random wave environment. It is assumed that this parameter is a stationary Gaussian process with _{zero mean, for each short term sea state.} This random sea is reasonably characterized by_{a spectral function of ,j, which is}

a,1g2 Wp 42r H2 Tz4 4 _i14 2.t Tz 5 W,4 exp(-=

exp(-- m--) "

SAPOS

-:pjerson..Moskowjtz (P.M.) and JONSWAP [10] spectral fUnctions are

implemented. These are the most commonly used sea spectra in practice.

Uncertainties in these spectral functions are separately discussed below. I-) The P.M. sea spectrum : This spectrum is defined as,

ag2

w4

S,7(w) = --

exp(-

_{j- -}

₎ (4.1.2.4)

where a,

is the shape parameter obtained from data, g is the gravitational

acceleratión, andWpis the peak wave-frequency.

When the severity of the sea state is represented by H, the peak frequency can

be stated in terms of a and i{ assuming that a narrowband process is used for

the wavê elevation. The pak frequency is calculated from,

Wp4 a, (4.1.2.5)

As it can be seen from Eqs.(4.1.2.4) .and (4.1.2.5), the only uncertain parameter in this spectral function for a given H value is a, which is assumed to represent uncertainties arising from the idealization of the random wave phenomenon This uncertain parameter is represented by X5 in the program.

If, however, the severity of the sea state is represented by both H and Tz, the

parameters of the sea spectrum can be obtained in terms of H and T as written

by,

(4.1.2.6a)

(4. 1.2.6b)

In the light of these statements it can be said that fr a given sea state, i.e. for known H and T values, the spectral function is fully detetmined, and thus, no

uncertainty is introduced.

II-) The JONSWAP sea spectrum : This spectrum is defined as,

( cu-w,)2

2a2w2 _(4.L27)

where y is a peak enhancement factOr and a is defined as a = cia if w<Wp, and

a=ai, if

W>Wp This spectral function is a generalized form of the PM spectrum

which is obtained by using y = 1 in Eq.(4.1.2.7). The mean values of the

parameters y, Ga, and ob of the peak enhancement functïon are reported to be

y = 3.3, Ga = 0.07, and ai, = 0.09, see e.g. Ref.[10]. Since the peak enhancement

function is defined in a narrow frequency-band around the peak frequency w1,, it

is assumed that variations in

Ga

and ab do not affect the fatigue damage

considerably. Besidés, théir uncertainties may be included in y. Therefore, these

two parameters are considered as being deterministic in SAPOS. Using their

mean values written above, the following relations can be obtained.

a,7g2

42

(i-0.286 ln y) f2(y) (4.1.2.8a)

wp2 =(;)

_.98255 (4.1.2.8b)

in which thé function f(y) is defined as,

- i

- 0,13763587 in (4.1.2.8c)

If the severity of the sea state is represented by H only, the peak frequency wp can be written in terms of H, a and y, from above equations. For a given H, the

parameters a and y are considered to be uncertain variables. In SAPOS, these

variables are assumed to be correlated. The independent variables obtained from

an orthogonal transformation are shown by Xs and X6 which are used in the

program as being basic variables.

If, howevér, the severity of the sea state is represented by both H and T, the

parameters a,7 and cop can be stated in terms of y as written above in Eqs.(4.2.1.8)

so that the parameter y is assumed to represent inherent uncertainties. In this

case, the random variable y is denoted by X5 in the program as a basic variable. The uncertainties outlined above correspond to the short-term description of the sea state. In the long-term, the probability distribution of the sea state is used in

the damage formulation, see Eq.(3.4), so that there may be some extra

uncertainties introduced by the modelling of the probability distributions. From

numerical experiences [1], it is revealed that these uncertainties have minor

contributions to the reliability format. This fact can also be verified analytically.

Based on this reason, no extra uncertainties related to the idealization of the

long-term probability distribution of the sea state are implemented in the update

of SAPOS.

4.1.3- Fatigue Category

Uncertainties in stress concentration factors (SCF), S-N model, damage

SAPOS

-occurs, mostly taken as unity) are considered in this category. Uncertainties in each item is outlined below.

Stress Concentration Factors : Fatigue damage is formulated in terms of the

normal stress at a hot-spot. The hot-spot stress is usually calculated from the

nominal normal-stress in practice, using stress concentration factors (SCF) which

are often obtained from some

semi-empirical expressions. Therefore, these

factors always include uncertainties. Since the normal stress is calculated from the axial (normal) force and the bending moments, a different SCF is likely to use

for each term of the stress components. It is assumed here however that

uncertainties in the hot-spot stress due to SCF are represented by a single

variable, denoted by X9, see also chapter 3. It means that SCF for all stress terms are fully correlated. The random hot-spot stress can now be written as,

Sh = X9 4u(Sh) (4.1.3.1)

where 4u(Sh) is the mean value of the hot-spot stress. As it can be realized from

Eq.(4.1.3.1) the mean value of the random variable X9 is unity, i.e. 4u(X9) = 1.

S-N model : Three SN models are implemented into SAPOS as to be a linear

model, a linear model with an endurance limit, and finally a bilinear model.

Uncertainties in the S-N model, to be used in practice, are introduced by

experimental data obtained from laboratory setups which cannot exactly

represent the real situation wherein the loading case, structural

joint

configurations and environmental conditions are much more complicated than those simulated in a laboratory. The experimental data are also widely scattered so that a large deviation from an idealized model, which is mostly a line, cannot be avoided. The uncertainties in a fatigue model can be best represented by the parameters, lnC and k, see section 3. In the case of using a linear S-N model, lnC

and k are assumed to be correlated random variables. The independent basic

variables are denoted by Xio and Xii. If a linear S-N model with an endurance

limit is used, the endurance limit Si, see Fig.3.la, is also taken as a random variable and denoted by X12. If, however, a bilinear S-N model is used, it is

assumed that the lines of the S-N model are statistically independent from each

other. The inherent uncertainties are represented by (mCi, ki) and (lnC2, k2)

respectively for each line, see Fig.3.lb for definitions. These parameters of each line are assumed to be correlated. The independent basic variables are denoted by (Xio, Xii) for the first line, and (X12, Xi3) for the second line.

e) Damage Correction Factor : The fatigue damage is formulated on the basis of

a narrow-band stress process. However, in practice, it proves that the stress

process is not exactly narrow banded [8]. Usually, it displays a bimodal spectral

shape. The fatigue damage under a non-narrow-band stress process can be obtained by modifying the damage due to the narrow band assumption by a

factor, see Eq.(3.4). The correction factor, A., is very complicated since it depends

on not only the stress spectral shape but also the S-N model used. An

approximate formulation for practical purposes can be stated as[81,

A = i

- X14 ( - 1) (4.1.3.2) where T0 and Tm are respectively periods Of zero-crossings and maxima of the

stress process given by E4s.(2.5.3b) and (2.5.3c). X14

is a random variable

representing inherent uncertainties. The mean value of this

variable is

determined dependently on the slope of the S-N line (k). In the program, this

random variable is treated as to be independent.

Reference Damage : As a criterion it is assumed that failure occurs when the cumulative fatigue damage Dtot, see Eq.(3.2), reaches a level, say Df. This limit

value of the damage is termed here as the reference damage, and displays a large variation in practice. Therefore, it is treated as an uncertain variable in SAPOS. The corresponding basic variable is denoted by Xi5. A mean value which equals

unity, i.e. 4u(Xis) = 1, is commonly accepted in practicç.

4.2- Limit State Function and The Reliability Calculation

Uncertainties modelled in above sections are incorporated in a reliability

calculation to determine a safety, or conversely a failure, probability for an

assumed life-time The reliability calculation is carried out On the basis of a limit state function, g(X), which is defined jn the uncertainty space, where X denotes a vector of basic variables representing the uncertainties. As outlined in above

sections, a number of fifteen basic variables (Xi to X15) are introduced into

SAPOS. It is assumed that if g(X) O, a failure domain, and if g(X) > O, a safe domain are defined. These two domains are separated by the limit state surface defined as g(X) O. Satisfying these conditions, the limit state function, g(.), is defined on a logarithmic basis as,

g(X) = ln X15 - in Dtt (4.2.1)

where X15 represents the reference damage as outlined above and Dtot is the

cumulative damage in the uncertainty space containing the basic variables Xi to

X14.

In SAPOS, the reliability calculation is performed by using First Order Second Moment (FOSM) reliability techniques, see e.g. RefS. [11], [12] and [13]. The basic principles of FOSM are

Transformation of dependent non-Normal basic variables into independent

Normal form, see e.g. Ref.[14]

Linearization of the g(.) function in the independent Normal variables space. Calculation of the design point on the limit state surface.

This calculation requires an iterative procedure. The point on the limit state surface corresponds to the highest probability that is intented to find out. In

i=1

where u(Xj) and UXj are the mean and standard deviation of the design variable

X1 Since the go function is linearized in the independent Normal variables

space, its probability distribution will also be Normal. Then, the failure

probability will be calculated from,

r°

PF[ZOJ

= J

fz(z) dz. . (2.4.5)

-where fz(z) denotes the normal density of the linearized g(.) function. ThIs

statement can be written in terms of the standard normal distribution function as,

PF _{= (-ß) (2.4.6)}

where_fi is the well known reliability index defined by,

fi=z/az

. (2.4.7)

The statistical measùres of the g(.) function, uz and oz, will be calculated from Eqs (4 2 3) and (4 2 4) respectively at the design point on the limit state surface,

i.e. when g(X ) =O. A useful definition in this calculation technique is the

Page : 23

-SAPOS

-is the closest point to the origin. The d-istance between th-is point and the origin

gives the reliability index. In SAPOS, non-Normal variables are treated

according to Ref [15J to find out equivalent Normal distributions The principle

of finding the equivalent Normal distribution is to use values of probability distributions and denÍty functions of 4ormaL and non-Normal functions at

design points, i.e. FN(X )= F(x ) and fN(x )=f(x ) where the subscript N denotes

the Normal distribution and the superscript

denotes a design point. The

linearized g(.) function is written as,

g(X) = g(X*)

+i(Xi-

XÏ*) aX)

(4.2.2)

i:=1

where the denotes a design point. From here, the mean value and the variance of the g(.) function Which is also denoted by Z can be written as,

m * 1. * ag(X)

uz

= _{g(X )} (4t(X)- X . (4.2.3) ) äg(X) ₂ ) * (4.2.4.) 2

az=,)

(OXi aX

sensitivity factor which is,

ög(X)

Xi

y

'aj =

(4.2.8)

az

The square of this factor (ai2) determinçs the relative contribution of the design

variable Xi to thé variancé öf the g(.) functiOn, ¿z. This provides a valuable

information about the importance of an uncertainty field in the reliability term

In this way, dominant uncertainty fields can b.e identified so that further attention

would be given to a better treatment.

As it can be seen from above equations, the partial derivatjves of the g(.) function

are evaluated at design points These derivatives are calculated numerically in

SAPOS

-S- PROGRAM OUTLINE AND GENERAL DESCRIPTION

5.1- Outline of the Program

SAPOS is designed in such a way that it can be easily implemented into a general

structural computing environment. In this case, it will be a part of the

environment, and the data transmission occurs automatically between them. SAPOS can alsO be used as an independent program W carry out stochastic analyses of offshore structures provided that the eigenvalue solution and the

flexibility matrix of the system are readily available. An analysis package will be used necessarily to provide the eigenvalue solution and the flexibility matrix for SAPOS. In this case, the package also generates some other data for' SAPOS, so

that the input environment of 'SAPOS can be classified into two categories as

being Indirect and Direct Inputs, see Fig.5.1.1.

DIRECT INPUT

Supplied by users

i

Fig.5.11 - Input environment of SAPOS

The indirect inputs are generated and supplied by the analysis package while the

direct inputs are provided directly by users. These input categories will be

explained in chapter 6 The main role of the analysis package within the SAPOS environment is to supply the flexibility matrix and natural characteristics (natural frequencies and eigenmode vectors) of the system. It is assumed henceforth that they are readily available, and no attention will be paid to their calculations in this report.

SAPOS offers three distinct analysis options as to be (see Fig.5.1.2): Spectral analysis (chapter 2),

Fatigue damage, calculation (chapter 3),

Reliability assessment (chapter 4),

Which are calculated in the order written. Among these options, it is obvious that the spectral analysis is the most complex and time consuming part of the whole

Page : 25

INDIRECT INPUT

Supplied by an analysis package

program. A special algorithm is .thérefore developed and used. in SAPOS to

optimize the calculation of the spectral analysis which is outlined mainly in the

following paragraphs.

SPECTRAL ANALYSIS

FATIGUE DAMAGE CALCULATION

RELIABILITY CALCULATION

Fig.5.1.2 - Analysis options of SAPOS in the calculation order.

The essence of this algorithm is suchthat the stress transfer function is calculated directly from transfer functions of member wave fórces without calculating the

displacements. The flow diagram of the calculation procedure is shown in

Fig.5.1.3 where the eigenvalue solution is assumed to be available using an

analysis package.

As it might be realized from Fig.5.1.3, the algorithm is based on calculating contributions of all loaded members. The contribution of a member is merely

calculated as a vector product of transfer functions of member-wave-forces and some total response terms (dynamic and quasi-static) that explained in chapter 2.

The final stress transfer function is then obtained by superimposing the

contribution of the fixed-end member-forces, see Eq.(2.4.4), to the cumulative contributions of members. This is formulated by Eq.(2.4.5).

The quasi-static part of the response is calculated from the syste.m flexibility

matrix according to Eq.(.2.4.7a). This part of the response is frequency

independent. Therefore, it is calculated once, and used whenever needed. In the

case of the reliability calculation, the quasi-static response contains

uncertainties regarding only the stiffness matrix. This property is considered in the numerical evaluation of the partial derivatives of the limit state function g(), see chapter 4, to make a shortcut in the. overall computation.

The dynamic part of the response is calculated according to Eq.(2.4.8). Unlike

the quasi-static part, it is frequency dependent and requires the eigenvalue

solution. The flow diagram of the calculation is shown in Fig.5.1.4.

As written in Eq.(2.4.8), and also illustrated in Fig.5.1.4, the dynamic part of the

response is calculated by superimposing modal contributions. A modal

contribution is obtained from a triple combination of the member eigenmode

,vector, a modal stress and the modified modal transfer function given by

Eq (2 2 2) When the reliability analysis is carried out, the dynamic response is stated in the uncertainty space that concerns only the stiffness, mass and damping

matrices This fact is also accounted in the numerical calculation of the partial

derivatives of the g(.) function.

I-I

CONTRIBUTION TO STRESS TRANSFER FUNCTION I STRESS TRANSFER FUNCTION MODAL TRANSFER FUNCTION SAPOS -EIG EN VALU E SOLUTION 4

Fig.5.L3 - Flow diagram for the calcûlation of a stress transfer function.

y MODAL STRESS 4 MODAL CONTRIBUTION I DYNAMIC CONTRIBUTION TRANS FER FUNCTION OF MEMBER WAVE FORCE CONTRIBUTION OF FIXED-END MEMBER FORCES I MEMBER EIGENMÖDE

Fig.5. 1.4- Flow diagram for the càlculation of the dynamic part of the response.

Page : 27

f

DYNAMIC CONTRIBUTION STATIC CONTRIBUTION G) E G) E -c '1 SUPERPOSITION a) -c o

As the response components are concerned, it is worth noting in this context that thé total response term (quasi-staticand dynamic), see Eq.(2.4.6), is independent of wave loadings This leads to a shortcut in the computation time when a

multi-directional wave climate is taken intO account in the analysis since the total

response term remains unchanged by the variation of the wave direction. This is especially remarkable in the reliability analysis.

Having calculated the stress transfer function, s outlined above, the spectrum and spectral moments of the stress process, the fatigue damage and the reliability

calculations are carried out consequently. The flow diagram Of the calculations is

shown in Fig.51.5. STRESS TRANSFER FUNCTION RELIABILITY CALCULATION NO

STRESS SPECTRUM _{I SPECtRUM}

STRESS SPECTRAL MOMENtS FATIGUE DAMAGE YES RELIABILITY INDEX fi YES NO SEA

Fig.5.1.5 - Flow diagram for the fatigue damage and the reliability calculations.

For a given sea spectrum, either P.M. or JONSWAP, the stress spectrum is

calculated using Eq (2 5 2) The stress spectral moments are calculated from the

frequency intégration given by Eq.(2.5.1). Then, the calculation of the

cumulative fatigue damage is carried out following the procedure explained in

SAPOS

-chapter 3. If, however, the relIability analysis is concerned ,it is carried out using

the procedure explained in section 4 2 In this case, the calculation is repeated

until a required convergence is obtained.. 5.2- General Description

SAPOS, as a whole, consists of one main program and 29 subprograms which are

coded in FORTRAN 77 The main program controls the flow of execution in

general as depending on input commands. Calculations in the subprograms are arranged in a way that an optimal runtime is reached It is especially remarkable in the case of the reliability calculation. The philosophy behind this organization

is such that repetiti.ve calculations are prevented essentially in the program.

However, simple calculations which. can be performed in very Short time, are

repeated in some subprograms to optimize the storage as well The routines of the whole program are shown in Fig 5 2 1, which are written in the execution

order.

Fig.5.2. i - Câlculation routines of SAPOS

Page :

29

SAPOS

DESINP

PINPUT SEASTA GASIN

HSCAL MOWAS PAWED TRAVAR BACTRA LÍNDRA . NICOF INFOPA - - 'EXTRAC FROGEN MODALF

Wave direction DIRECT

Frequency TETFUN FIXEND REDFUN WSPECT REFOPA MEMFOR STRESS SPEMÖM OUTPUT FATREL GASIN TWOSNL FATOUT RELOUT

In the program, some simplifications (some precalculated values) are also made

using SI-UNITS (METER for length, SECOND for time, kg for mass) so that this

unit system must be maintained throughout an analysis option.

Since calculations. in the whole program are .arranged in an optimal way, it is very

difficult to explain specific tasks of the program routines. Nevertheless, their

principle duties are shortly described below to give a general idea.

SAPOS PINPUT DESINP SEASTA HSCAL GASIN MO WAS PAWED MODALF: DIRECT TETFUN Main program.

Reads input commands and data. Calculates member diameters from member properties and member direction cosines.

It is activated only in the case of the reliability analysis to read

information about design variables.

Reads data of the sea spectrum and the sea state.

It is activated only when using a continuous long-term distribution

of the sea state. It evaluates H, and also T in the case of a joint

distribution, at integration points in the long-term.

Contains abscissa and weight factors of Gauss integrations.

Calculates moments of conditional and marginal distributions of I-Is

and T from a provided scatter diagram. They are used to find out

continuous distribution functions for these measures.

It is used to calculate the parameters of the Weibull distribution

given that the mean, variance and the third central moment are

known.

It is activated only in the casé of the

reliability analysis. It calculates independent pairs of the correlated random variables.

It is also activated only in the case of the reliability analysis. It

calculates equivalent normal distributions of non-normal variables, and evàluates original (correlated) variables at the increments of the corresponding basic variables.

Calculates linearized wave force constants of loaded members.

Contains coefficients of a special numerical integration process to

calculate variance of the water velocity in the wave direction.

Calculates coordinate transformation-matrices of stress members. It also calculates partly the static part of the response.

Extracts coefficient matrices, [A], of member displacements from the flexibility matrix, see Eq.(2.4.7b)

Generates frequency integration points to be dependent on natural

frequencies. Here, an automatic generation is applied, which

produces minimal frequency points resulting in

accurate

integrations.

Calculates member modal forces (member internal forces under

eigenmode configurations), modal transfer functions, and, the

products of these two items.

It is activated oniy in the case of multi-directional sea states. It

calculates abscissas and weight factors of the numerical integration in the wave direction.

Evaluates wave direction-functions of linearized drag forces of

TRAVAR BACTRA: LINDRA NICOF INFOPA EXTRAC FRQGEN:

SAPOS

-loaded members.

FIXEND Calculates partly transfer functions òf member consistent forces,

which are resulted from the integration given by Eq.(2.3.5).

REDFUN: Calculates frequency reduction-function fthe inertia force-term in the Morison's equation.

WSPECT Evaluates sea spectrum at the freqûency integration points.

.REFOPA Calculates contributions of loaded members to transfer functions Of

the internal forces of stress members.

MEMFOR: Calculates transfer functions of internal forces of stress members

STRESS Calculates transfer functions and stress spectra of stress members.

SPEMOM: Calculates spectral moments of stresses using a piecewise numerical integration process.

OUTPUT: Prints out transfer functions of member internal forces,

stress

transfer functions and stress spectral moments at request.

FATREL It is activated fOr both fatigue and reliability analyses to calculate

stochastic fatigue damage for a given S-N model.

TWO SNL Carries out calculations for bilinear SN fatigue model.

FATOUT Prints out results of the fatigue damage and life-time calculations

RELOUT: Completes the reliability calculation for one iteration, resets design variables for further iteration and prints out the results.

6- PROGRAM. INPUTS

As mentioned in the previous chapter, SAPOS is used in co-operation with an analysis package which provides mainly eigenvalue solution and the system

flexibility matrix. Although it may improve the runtime efficiency,

implementation of SAPOS into an analysis package is not essentially needed. In this case, SAPOS receives some commands and data from a file which will be

produced by the package used. This kind of input to SAPOS is named here as

Indirect Inputs, because they are generated by another program. However, some extra inputs, which are needed for the stochastic analysis, must also be supplied

directly by users. Such inputs are named here as Direct Inputs. Inputs to the analysis package are not considered in this manual since they may vary from

package to package. Here, attention will be paid only to the inputs of SAPOS. Although not necessary, it may be recommended that the same data file is used

for both the indirect and direct inputs. In this case, the former precedes the latter, because the program requires the indirect inputs first, then the direct

inputs. These input items are explained in this chapter in detail. 6.1- Indirect Inputs

As noted above, the indirect inputs are defined to be those generated by an

analysis package. These are not only the results of the eigenvalue solution and

the system flexibility matrix but also some other data concerning submerged members and joints, as well as generalized masses. These data and related

information will be written to a data file by the package used. Forms of the data and information will be explained below in this section in detail. However, it is worth noting in this context that the package to be used must be adapted to meet

requirements of these inputs. In the following, instructions (commands) and

forms of data are presented, where the uppercase characters need .to be specified while the rest with lowercase characters are optional. It is also important that all

commands must be specified starting from first columns of relevant records. Text

written in brackets, if stated, can also be used as an alternative command

statement. Data inputs are specified succeeding the command specification.

Unless it is stated otherwise, the order of input items (command +. data) can be

changed if re4uired. First characters in all data records must be kept empty

(blank). This is especially needed to identify data and command records.

Following each input item examples are presented to demonstrate how the inputs are specified.

6.1.1- INCidences of submerged members

mijk

where,

m : an integer denoting a submerged member,

i : an integer denoting the number of member ends (i = 2),

SAPOS

-k: an integer denoting the joint number of the second end of the

member m.

This command is used to input submerged members. One data record is used for orte member. Data will be repeated for all submergéd members.

The format specification is : FORMAT(415).

Example:

INCIDENCES OF SUBMERGED MEMBERS / ELEMENTS

and So on

6.1.2- BOOkkeeping degrees of freedOm of submerged joints

m3 ml m2

where,

ml: an integer denoting a joint number,

an integer denoting a local degree of freedom at the joint ml (it is

a number between 1 and 6),

an integer denoting the global degree of freedom corresponding to

m2.

This command is used to keep track of local and global degrees of

freedom of submerged joints It provides the list of connections between

local and global degrees of freedom of joints as demonstrated by the

following example which corresponds to Fig.6.l.2. Data will be repeated

for all lOcal degrees of freedom of all submerged joints. The format

specification is : FORMAT(3I5).

Example:

BOOKKEEPING DEG. OF FREE. OF SUBMERGED JOINTS

Page : 33

721

822

923

10 2 11 2 .5

U 2

6 13 3 1 14 3 2 15 3 3 16 3 4 17 3 5 18 3 6

92109

10 2 11 10 11 2 12 11

6.1.3- COOrdinates olsubmerged joints

n x y z

where,

n an integer denoting a joint in water, x : X coordinate of the joint n,

y : Y coordinate of the joint n,

z : Z coordinate of the joint n, measured from the still water level, and assumed () in upward direct jon.

This command is used to input global coordinates of submerged joints. Here, X, y and z are floating point constants. One record is used for one

joint Data will be repeated for all submerged Joints The format is

FORMAT(15,3F13.3).

Local degrees of freedom at a joint

Fig.6.1.2- Demonstration of bookkeeping degrees of freedom of submerged

joints. m3 ml rn2 7 2 I 8 2 2 9 2 3 10 2 4 11 2 5 12 2 6 13 3 1 14 3 2 15 3 3 16 3 4 17 3 5. 18 3 6

Example:

COORDINATES OF SUBMERGED JOINTS

and so on

6.1.4- DiSplacements (EIGENMode vectors) of submerged joints

m

i

ux uy uz

j

rx ry rz where,

m: an integer denoting an eigenmode,

j : an integer denoting a joint in water,

ux, uy, uz : floating point constants denoting components of normalized translational eigenvector in the global X, Y and Z directions

at the jo!ntj,

rx, ry, rz : floating point constants denoting components of normalized

rotational eigenvector in the global X, Y and Z directions at

the joint j,

This command must succeed the command, COORDINATES OF

SUBMERGED JOINTS presented in 6.1.3. It is used to input eigenmode vectors of submerged joints, which must be normalized to i. before they

are specified. At a joint, first the translational and then the rotational

eigenmode vectors have tobe specified in two subsequent data records.

This process of one eigenmode specification will be repeated until the

data of all submerged joints are input. The same process, as a whole, will also be repeated until all eigenmode vectors are input in th.e increasing

order. The data are Specified in the free format, but must be floating

point. It is important, however, that eigenmode vectors need to be

specified with sufficiently significant digits to obtain accurate modal

stresses. Example:

EIGENMODE VECTORS OF SUBMERGED JOINTS

1 SAPOS -Page : 35 16 1.852 -1.070 -9.106 17 -1.852 -1.070 -9.106 18 0.000 2.139 -9.106 19 5.373 -3.102 -16.146 20 5.373 -3.102 -16.146

9 0.1806886E + 00 0.1029443E +00 -0.8314867E-06

9 -0.1487340E-01 0.2610410E-01 -0i160778E-01 10 0.1272312E +00 0.7246580E-01 -0.8448677E-06

6.1.5 MODal (GENeralized) masses

n g

where,

n an integer denoting an eigenmode,

g a floating point constant denoting the generalized (modal) mass for

the eigenmode n.

This command is used to input generalized masses of the structure. One

record is used for one eigenmode Data will be repeated until all

eigenmodes, which are taken into accöunt, are specified. The format

specification is : FORMAT(I5,E15.8). Example GENERALIZED MASSES I 0.24017E±06 2 0.29896E ±06 and so on

6.1.6- NATural frequencies (EIGENVa1ues)

n w

where,

n : an integer denoting an eigenmode,

w :a floating point constant denOting the natural frequency of the mode n in (rad/sec).

This command is used t input natural frequencies of the structure. One

record is defined for one eigenmode. Data will be repeated until all

modes are specified The format specification is FORMAT(15,E15 8)

Example: NATURAL FREQUENCIES 1 3.22863 2 3.26338

andsoon

2 and so on

9 -0.1179547E±QQ 0.2Ú69789E+00 -0.5567765E-03

9 -0 2995057E-01 -0 1706761E-01 -0 1379644E-04

10 -0.8300674E-01 0.1456519E +00 -0.5443799E-03

SAPOS

-6.1.7- PROperties of submerged members

ma xy z

t

where,

m: an integer denoting a submerged member,

a : cross-sectional area of the member m,

x : Ix, torsional constant

Of the member cross-section (polar inertia

moment for circular cross-section),

y ly, flexural inertia moment of the member cross- section about the

local Y axis,

z : Iz, flexural inertia moment

of the member cross section about the

local Z axis,

t : average thickness of marine growths on the member in as shown in

fig.6.1.7.

PROPERTIES OF SUBMERGED MEMBERS

9 O.2100E+O0 02321E+0O 0.1160E+00

lo 0.2100E+00 0.2321E+00 0.1160E+00

11 0.2529E +00 0.2756E +00 0.1378E + 00 12 0.2529E +00 0.2756E +00 0.1378E +00 13 O.1683E+OO 0A871E+O0 0.9353E-01

4 0.1683E + 00 0.187 lE + 00 0.9353E-01

and so on

(thickness of marine growth)

Fig.6.1.7 - A tubular member cross-section and the average thickness of marine growths

This command is used to input seçtional properties of, submerged

members, as well as average thicknesses of marine growths on members

Ïn the general form written above, a, x, y, z and t aré floating point

constants. This command and corresponding data are produced by an

analysis package, except thicknesses of the marine growths which must be later externally added on the data records. One record is used for one

member so that the number of data records are equal to the number of

submerged members. The format specification is : FORMAT(15,5E12.4).

Example: Page 37 0.1160E + 00 0.20 0.1160E +00 0.20 0.1378E + 00 0.20 0.1378E + OÙ 0.20 0.9353E-01 0.20 0.9353E-0 i 0.20