OTC 4161 A QUASI-STATIC APPROACHORTRANSPORTATION F ANALYSIS F OFFSHORELATF(IWS O P

by Mohamed F. Zedan, Yildirim O. Bayazitoglu, and Yi-Suang Tein, Brown & Root, Inc.

John

W.

Chianis

@Copyright 1981 Of fshoreTechnology Conference This paper was presented at the 13th Annual OTC in Houston, TX, May 4-7, 1981. The material is subject to correction by the author. Permission to copy is restricted to an abatract of not more than 300 words.

ABSTRACT This paper describes a computer model to predict the maximun probable stress as well as the cumulative fatigue damage at any joint in a jacket or its tiedowns that may occur during the transportation of an offshore platform by a barge. te dynamic transportation process is represented by a series of quasi-static processes in a frequency domain. The barge/jacket interaction is analysed taking into account the barge flexibility. The method utilizes a description of the wave spectra of various anticipated sea states encountered during transportation, their probability of occurrence and the voyages duration. The wave induced forces on the barge and the resulting barge/jacket assembly motions are calculated for unit amplitude waves with frequencies spanning the wave spectrun frequency range. The canputed accelerations are used to determine the inertial forces acting on the The barge/jacket assembly is in jacket members. instantaneous equilibrium under the action of the inertial forces and the pressure forces acting on the barge. These forces are used to calculate the stresses resulting from the unit amplitude waves, RAOS, taking into or account the stress flexibility of the barge. The stress spectrun is :[~:~~ frcm the stress RAOS and the wave . Assuming a Rayleigh distribution for the stress peaks, the maximum probable alternating stress amplitude is computed from the moments of the stress spectrm, the duration of the storm, and the mean period of stress oscillations. A probabilistic procedure is used to derive the cumulative fatigue damage ratio from the cmputed stress spectra, an assumed stress range probability distribution, and an appropriate S-N curve (applying Miners rule). Computed fatigue damage ratios for different sea states are added after being weighted according to their probability of occurrence.

1.0

INTRODUCTION Offshore structures are manufactured in on-shore fabrication yards which may be distant from the installation site. Transportation of the structure from the fabrication yard to the installation site is done by means of a barge, as shown in Figure 1. The time required for the transportation can be as much as a few weeks. During transportation, the barge is subjected to wave forces which, in the event of a storm, can be considerable. The jacket loaded barge assembly responds to wave forces by oscillating in different directions. These oscillations give rise to inertial forces which act on structural These inertial forces as well as members. those from the barge/jacket interaction give rise to stresses that are cyclic by nature, and can reduce the fatigue life of the Thus they should be taken into jacket. account when computing the expected fatigue life of the platform after installation. Maximum values of the stresses developed at any joint in the structure during transportat-ion should not exceed the allowable stress levels, and of particular interest are the forces exerted on the tiedowns, which also should not exceed the allowable forces. The objective of the present paper is to develop a computational procedure which will (1) estimate the maximum stress at every joint in the transported jacket developed during a specified storm sea state of a given duration, and (2) estimate the cumulative fatigue damage ratio for every joint in the jacket due to the cyclic stress of the different sea states The anticipated during the transportation. present procedure solves a major deficiency in previous analyses by accounting for barge flexibility and incorporating barge/jacket interaction in the stress calculations. The procedure is quite general and is not limited to the transportation of jackets. In fact, it can be used for the transportation

References and illustrations at end of paper. 439

analysis of any frame type structure; and therefore, can be applied to deck transportation. The technique developed here is quasiin the frequency static and is applied The details of the computational domain. procedure are given in Section 3, and two example problems are discussed in Section 4. Section 2 discusses sane related studies. 2.0 RELATED STUDIES The importance of analysing the transportation of offshore structures has been recognized by the industry for sane time. This importance has increased in recent years due to the loss of more than one jacket during transportation, but despite this, there reinains very limited published literature on the analysis procedures of offshore structure transportation. In the past, jacket transportation has been primarily handled by a semi-deterministic inertial force approach. Implicit assumptions made in this approach are: (1) the barge is infinitely rigid (i.e. there is no interaction between jacket and barge), and (2) the forces acting on the jacket are primarily caused by A typical motion induced inertial forces. this jacket transportation analysis using approach consists of the followng steps: (1) (2) (3) Define the sea states which simulate the voyage. Predict the maximum motion responses. Compute the inertial force distribution for the jacket based on the maximun motion response in all six degrees of freedom. Compute the stresses in the jacket due te inertial forces.

tion of very large jackets. However, this may lead to substantial errors since it assumes the barge is infinitely rigid. The larger the jacket, the more susceptible the assembly is to hydrodynamic forces from both jacket/barge motion-induced interactions, and inertial One of the most recent studies forces. analyzing jacket transportation by the above The analytical method is reviewed below. section of the study contains some of the deficiencies described earlier. Sekita et al.2 presented an OTC paper that dealt with model tests on the transportation of a large jacket by a launching barge. The 1/60 - scale tests objective was to evaluate barge stability and structural safety during transportation. A Froude modeling scheme was The conducted tests included the used. effects of wind, and both regular and irregBarge motions, jacket accelerular waves. ations, and stresses in the tiedown braces Sekita et al. compared the were measured. experimental results they obtained to the results of a package of computer programs developed for transportation analysis. The computer package used by Sekita et al. started by computing the motion of the barge/ jacket assembly produced by winds and waves. Then the inertial forces and the eccentric loads (due to roll) were calculated and applied statically to the assembly while ostensibly supporting the barge at the mean The stresses developed in the water line. braces were calculated tiedown from the previously mentioned loads. The procedure contains some wide approximations, but principally it fails to adequately consider the pressure force distribution on the barge and assumes its effects will be In effect, the absorbed at the supports. barge has been erroneously assumed to be rigid. This method was also used to compute the stresses on the tiedown braces, and these, showed when compared with measured stresses, some differences. The computational procedure did not consider the fatigue damage from the cyclic stresses produced by the barge oscillations.

(4)

In order to investigate the entire range of possibilities, a number of loading conditions are explored by combining maximum motions in a All the canponent number of different ways. maximum motions are assumed to be either in-phase or 180 degrees out of phase. This approach is often criticized as being too the maximum conservative because vessel motions in all the six modes are not likely to A slightly improved occur simultaneously. approach calls for the computation of participation factors. This approach consists of finding participation factors for each of the motion cmnponents when one of them reaches its maximun in a given sea state for a given durThe estimation of the participation ation. factors is achieved by utilizing the notion of cofactors in random processes (Hutchinson and Bringloel). Although differing in the degree of both of these sophistications approaches are basically of a semi-deterministic nature in that the prediction of stress has not been carried out using the probabilistic notions. Nevertheless, this approach has proven to be a valuable tool in designing sea fastening mechanisms when the jacket is smnewhat small relative to the barge. Recently, there has been a tendency to extrapolate this approach to cover the transporta440

In our approach to transportation analysis, the pressure force distribution acting on the barge is considered while taking into account the flexibility of the barge. Therefore, the effects of the hydrodynamic forces acting on the barge and the loads in the jacket members This avoids the major are accounted for. deficiency of previous transportation analyses. The present method provides an estimation of the maximum probable stress exerted anywhere in the assembly as well as the cumulative fatigue damage incurred during transportation.3.0 3.1 METHOD OF ANALYSIS Outline and General Approach The computational procedure starts by calculating the mass and moment of inertia

characteristics of the barge/jacket assembly. The wave induced forces on the barge are then calculated. This is followed by cwputing the various motions of the barge/jacket assembly. The calculations require linear accelerations, angular velocities and angular accelerations. The accelerations are used to determine the The inertial forces acting on each member. inertial forces acting on the assembly and the pressure (hydrodynamic) forces acting on the barge are used to compute the stresses due to unit amplitude waves which are called the stress Response Amplitude Operators (RAOS). The stress spectrum-is determined fran the RAOS and the input wave spectrun. Once the stress spectrun is determined, the maximum stress that may occur during the period of the specified storm sea state is computed using a probabilistic approach. This is done by assuming a Rayleigh probability distribution for the peaks of the stress record and by using both the storm duration and the mean period of the stress oscillations. For fatigue damage computations, it is assuned that fatigue failure is dependent upon the stress ranges and the number of cycles of In the present each stress range applied. procedure, the calculation of fatigue damage is done by using the Miner/Palmgren hypothesis. The fatigue damage is accumulated linearly over different stress ranges in different sea states. Fatigue failure is assmed to occur when the cumulative fatigue damage ratio achieves unity. To apply the Miner-Palmgren model to the present frequency domain stress responset the stress range probability distribution needs to be known. It has been shown that the stress response spectra are narrow banded and therefore the distribution of the stress peaks and the stress ranges may be approximated using a Rayleigh distribution. The method developed in this paper is quite general, and therefore other stress range probability distributions can be used. Also needed in the fatigue calculations is the choice of an appropriate stress range vs. nmber of cycles to failure curve (S-N curve). The procedure outlined above is presented in the block diagrams shown in Figures 2 and 3. and Notice that DAMS-I, SEALOAD BARMOT, TPFATIGUE are the names of different program modules that constitute the Transportation In the following few secAnalysis System. tions more details are given about different parts of this system. 3.2 Barge/Jacket Structural Model and Computation of structural haracterlstlcs A space frame model of the jacket and the barge are separately derived, then the two are connected via the tiedowns to form the barge/ jacket assembly. From the structural point of view, the need to model the barge using a frame structure results frcm the fact that DAMS-I (structural analysis program) does not have the capability of analyzing plate elements. The aggregate .model of the assembly 441 3.3

should reflect (1) primary Structural properties such as torsion and bending stiffnesses, and (2) the mass properties of the actual barge-jacket assembly, such as the total mass and the mass distribution. A space frame model of the jacket is easily derived because of its prismatic tubular A barge, on the other member construction. hand, is of multi-celled construction, and a space frame model of this structure is not as For structural considerastraightforward. tions, the barge model should reflect the overall bending and torsional rigidity of the real barge. In the vertical and lateral bending modes, the model is given stiffness by its The cross-secmain longitudinal members. tional size of these members and their location depend entirely on the actual rigidity of the real barge. The goal is to reproduce the stiffness of the main longitudinal members of the real barge. The torsional rigidity of the real barge, due to the plate structure, can be estimated from shear flow methods and is reproduced in the model by the addition of lateral cross members. Once the entire barge model is constructed, an iterative procedure is performed to further calibrate the model in the bending and torsion modes. The models of the barge and jacket are joined together by the tiedowns (Figure 4) and a preliminary DAMS-I run is made to compute the total mass and mass moments of inertia, and the location of the center of gravity. A detailed check should be made before the motion analysis to ensure the compatibility of the model properties and the actual assembly. First, the total weight of the model calculated by DAMS-I must be equal to the barge displacement used in the motion analysis for Secondly, the assembly vertical equilibrium. center of gravity must be on the same vertical axis as the center of buoyancy so that there In most instances, will be no net moments. both of the above checks will not be within a close tolerance. A simple and effective way to correct the problem is to add ballast at In the selected locations on the barge. model, this can be accomplished by adding The concentrated loads to selected joints. mass corresponding to these should be included DAMS-I program is in the dynamic analysis. then run again with the modified model and the values of total mass and mass moments of inertia, and the location of the center of If these computed gravity are checked again. values are acceptable, the motion analysis can be performed. Computation of Excitation Forces and Assembly ~otlon Responses Among all the environmental forces acting on ocean-going marine vessels, wave forces are the most important source of loading. Similar to the frequency domain approach conroonly employed in seakeeping analysis, the structural response of a marine vessel in random seaways can also be studied based on the concept of linear of the super-position Each sea state is response to regular waves.

The wave represented by its wave spectrum. spectrum is subdivided into a number (nf) of For each slice, a unit frequency slices. amplitude wave with a frequency equal to the central frequency of the slice is applied to The response of the barge/jacket the barge. assembly to these unit amplitude waves, motion RAOS, is then obtained for each of these regular waves via the BARMOT motion analysis Using the notation of seakeeping program. Kim3, suggested by the motion analysis response of the vessel in regular sinusoidal waves can be obtained through solving a system of coupled linear equations of the form, 6 zj=l

i@ (Mij + Aij) ~j + B..;. + Cijnj = Fie lJ J j = 1,2,3,4,5, & 6

to these linear terms, it is roll motion exhibits strong nonlinear behavior near resonance due to the presence of viscous roll damping. In view of the significance of roll motion on the structure, a special technique has been developed to account for the roll viscous damping. The technique calls for the introduction of an additional term: B i ]fi4\to the equation of The coeffic! ent B, primarily a funcmotion. tion of hull geometry and the amplitude of roll motion, can be evaluated using Tanakas An iteraempirical formula (Salvensen5). tive scheme based on Tanakass method has been developed for predicting the non-linear viscExtensive model ous damping and roll motion. testing has been conducted to verify this Good correlations were obtained scheme. between model tests and theoretical. predictions.

In

addition

observed

that

where, are the elements of ij mass matrix of the vessel, are l%ix, the 5 ~~~fficl~tm~~rixeements are the elements %!toring force matrix 9 of of the hydrostatic damping the elements of the generalized

the

added

mass

are the elements of ~~!ce and moment vector,are the j vector, and elements of

the wave

exciting

Once the motion responses of the. vessel are forces calculated, wave and hydrodynamic forces due to vessel motions can be readily Furthermore, accepting the fact calculated. that the natural. frequencies of the structure are much higher than those of the dominant waves, there is little, if any, dynamic amplification effect. Therefore, the structural response can be obtained from an equivalent static structural analysis. In this analysis, the loads acting on the structure consist of the inertial forces induced by vessel motions and the hydrodynamic forces due to wave and It is worthwhile to point out vessel motion. that the structure is a free-free system and that the net inertial forces and moments are balanced by those due to hydrodynamic forces,i.e. there are no net forces or moments acting

the

6-D

motion

on the system. frequency in radians per The computation of inertial forces is a straightforward process which involves multiplying the lumped nodal masses of the structure by the corresponding accelerations. The computation of hydrodynamic forces is also conceptually simple. However, one must exercise care to ensure that proper force lumping is achieved. The best guideline is that the lumping of hydrodynamic forces must be consistent with that of the motion analysis, me ~ indicator for such consistency is how forces balance the wel1 the hydrodynamic inertia forces. 3.4 Computation of Stress Spectra For a barge in a given sea state, in a given direction, angular and linear accelerations and angular velocities of the center of gravity of the assembly, as well as the distribution of the pressure forces on the barge surface, were computed for unit wave amplitudes as discussed in the previous section. At this point, it should be emphasized that each of the response quantities (velocity, acceleration and pressure forces) is represented by a complex variable which has real and imaginary parts. This is to preserve the phase different forces relations between acting on the assembly.

is the wave s~cond.

In the above equation, the 6-D motions (Vj, ~ea~el,2~;~;,5, and 6) refer to surge, sway, pitch, and yaw, respectively. The generali~ed mass matrix is a function of total mass and the special distribution of the mass. The wave forces and hydrodynamic coefficients are primarily functions of the Methods for hull geometry of the barge. computing the wave forces and hydrodynamic coefficients are based on the well-known strip method technique 5)(Ogilvie al Salvensen et . . Specifil%ly~c% computations involve the following steps:(1) Canpute sectional velocity potentials Frank using the close-fit method (Frank6). Compute sectional wave forces and hydroc(ynamic coefficents fran contour integration of hydrodynamic pressure. Canpute three-dimensional wave forces and hydrodynamic coefficients for the equation of motion from longitudinal integration of sectional two-dimensional values.

(2)

(3)

442

The distributed pressure force on the barge surface is lmped appropriately at a nunber of joints on the barge frame model by the SEALOAD program (Figure 2). The velocities and accelerations of the center of gravity of the assembly are used to compute the accelerations and consequently the inertial forces at different joints in the jacket. These inertial and pressure forces are applied statically to the barge/jacket assembly. The DAMS-I structural analysis program is used to compute the resulting stresses at different joints and for different member ends in the jacket, as shown in the block diagram of Figure 3. It should be noted that such stresses are produced by Al SO unit wave amplitudes and are canplex. these stresses have bending and axial contriAppropriate stress amplification butions. factors are applied to these stresses. Additional stresses caused by static weight eccentricity due to the roll of the assembly have to be considered in obtaining the, total stress spectrum. Correction of Stress Amplitudes 17fect The real (R) and imaginary for the Roll

amplitude resulting from the application of a unit amplitude wave at this frequency to the assembly. Therefore,

MO(W)

= [(OT(U))2+

(J-(U))*]*

(4)

The stress spectrum So (u) is obtained from the wave spectrum S H~w) and the response amplitude operator RAOU), using the relation ? Sea(w) = ]RAO(~)]2 SHH(W) The stress spectrum is obtained using the above procedure for all sea states, in all directions of incidence, at all joints and member ends in the jacket. The cmnputation of these stress spectra and subsequent calculations of maximum probable stresses and the cumulative fatigue damage, are done by the TPFATIGUE program (Figure 3). 3.5 Computation of Cumulative Fatigue Damage Ratio The fatigue damage is cmnputed using the wellknown Miners rule which assumes that fatigue damage on a structure accumulates linearly In the present case, (Clough and Penzien7). the structure is subjected to stresses that are continuously varying with time in a someTherefore, a probabiwhat random fashion. listic approach is used to calculate the cumulative damage (CDR). According to Miners rule, fatigue failure is expected to occur Though when CDR equals or exceeds unity. widely used for the lack of a better criterion, the validity of applying Miners rule to random is questionable. stress conditions Based on other studies, Vughts and Kinras indicated that a linear accumulation of damage

(I) parts of the

stress amplitude (axial + bending) for a unit(*) amplitude wave, excluding the effects of roll, are designated by o;(w) and u:(w) The stress that results from the static weight of the barge/jacket assembly and the balancing pressure forces (buoyancy), without the presence of hydrodynamic excitation (no waves), is designated by o for a zero roll angle and bycru for a unit ro?l angle (1 degree). These

stresses are computed using the DAMS-I structural analysis progra described earlier. The incremental change in the stress due to a unit roll angle is given by Ao:=ou-o 9 (1)

(as in Miners rule) is approximatelytrue incases where fatigue is predominantly a result of the propagation of initial cracks that are Values of CDR at a given already present. point,on a certain joint caused by dif- ferent sea states will be added to obtain a total cumulative damage ratio at that point. For a discrete application of sinusoidal stress signals, Miners rule gives CDR as

Therefore, the stress correction due to a roll. angle of @(uJ) (produced by a unit amplitude wave of a frequency w) is

ACT*r((t))Acr~ =

q

6*(w)

(2) CDR = X

n(ri) ~ (6)

The total stress amplitude produced by a unit amplitude wave (including static roll effect) is given by (J*r(W)= o*(u) + AIS*r(U) (3) The above procedure is applied to both the real and imaginary parts of the the stress to obtain cry(w) and o~r(u)

where: n (rf) = Number of stress cycles at a StreSS range ri. N (ri) = Average number of stress cycles to failure at a stress range ri. For a continuous stationar a~+J, a#dom stress and Wallis et process addox Wildenstein\ 0), CDR is given by,

Response Amplitude Operators and Stress Spectra CDR=~ The stress response amplitude operator at a given frequency, RAO (u), is simply the stress 443 av Jrn;(r) dr o T (7)

where: T = Time duration of the random stress process. Average period for stress variation in the random process. density function of Probability stress range. Average number of cycles to failure at a stress range r. (7),

(Wallis et and the stress maximum function is considered such a adequate for narrow band spectra with small values of E. ::$:. The Rayleigh approximation was improved using the distribution of the maxima for an arbitrary spectral band width by Wallis et al.9 by assuming that the stress range r is equal The reader is to twice the stress peak. referred to Reference (9) for more information about the improved expressions for P(r). Once P(r) and N(r) are defined, the integration in Equation (7) is carried out numeriSpecial care should be taken in the tally. since the integrand numerical integration converges slowly. As for the average period ~av (in Equation (7) ), it is computed from the expression 0 av = Zn[--(l #)31/2

(av) = P(r) N(r) = =

To evaluate the integration in Equation

~~~dfunctions p(r) and N(r) have to be specfThe choice of N(r) is fairly easy. The func{ion N(r) is simply the equation of the curve of the stress range versus the number of cycles to failure (known as S-N curve). Such a function is empirical and depends on the Of prime interest here is type of material. the fatigue in the welded joints between members in the jacket. These are the weakest points as far as fatigue is concerned. Laboratory and full scale tests have been conducted for different types of joints. Various organizations such as the Pmerican Welding Society and the Welding Institute have proThe duced S-N curves for different welds. curves usually differ due to different philo-

-

(11)

where a spectral moment mi is defined as

sophies of

data presentation.

Gurney and

mi =~mui

Sea(w)

dw

(12)

S-N curves that have Maddoxll published represent the mean of test data and therefore indicate 50% survival rate. Also, they provide curves representing mean minus two stan95% survival indicating dard deviations rate. The choice of an appropriate S-N curve is the job of the designer. In his choice, he should keep in mind what survival rate the structure is designed for. It should be noted that it is customary to represent the S-N curve (on a log-log plot) either by a straight A straight line or straight line portions. line S-N curve may be represented by (8)

with i = 0,1,2, ... etc. Special Case When the S-N curve is a straight line (on a log-log plot) as represented by Equation (8) and when P(r) is given by a Rayleigh distribution (Equation (9)), a closed form exPression for the integration in Equation (7) is possible. The cumulative fatigue damage ratio under such conditions is, thus, given by (8mo)y/2 r(l+~) A

N(r) = A/ry

CDR = (+)

(13)

where A and yare empirical constants obtained from fitting fatigue experimental data. As for the probability distribution of the stress range, it has been found that it can be approximated by a Rayleigh distribution in the form -r2/8mo

where r( ) is the Gamma function and T is the duration of the application of the cyclic It is interesting to note that stress. similar expressions to Equation (13) were obtained by ~olt~ and Hansford13 and by Wirsching and Light 4. Wirsching and Light focused on fatigue under wide band random stresses by using an equivalent narrow band process that has both the same m. and rate of zero, crossing fo. The CDR obtained frcm this equivalent narrow band process is then corrected by a factork whose statistical characteristics are provided as a function of the bandwidth parameters. Accumulation of Fatigue Damage

P(r) =*e

(9)

This relation is exact for a stress spectral bandwidth parameter (e) of zero. s is defined by $ E mom4 -m~)/(mom4) ~

(lo)The value of CDR obtained as discussed earlier covers only one sea state in one given direction of incidence at a point on a joint at a member end. The values of CDR are added algebraically for different sea states in different directions at the same point in the structure to obtain the total CDR at this point during the voyage.

m., whereo s s s 1 and M2 , are and m4 spectral moments defined later. The function P(r) of Equation (9) can be obtained frcm the Rayleigh distribution of the ma ima derived by Cartwright and Longuet-Higginsf 2 by applying a simple transformation between the stress 444

Comments on the Fatigue Procedure The procedure just described to compute the fatigue damage ignores the effect of the mean Such an assumption is cmnonly made stress. in fatigue analyses of welded joints and is design by codes (Wirsching and ~;;;~fl. The choice of an S-N curve is another source of uncertainty in the fatigue Most available data in literature analysis. were obtained by subjecting the test speciments to sinusoidal stress loading with In the present case of constant amplitude. randcm loading, such amplitudes vary widely. Also in laboratory tests the frequency of the cyclic stress is often very high compared to the range of stress frequencies encountered This is because of during transportation. practical limitations on the time duration of The rate of load experiment. any laboratory application in a laboratory is much higher and this may have an effect on the fatigue life. 3.6 Canputation of the Maximum Stress The maximum probable stress is obtained for the storm sea state by assuming that the probability distribution of the stress peaks P(s) follows a Rayleigh distribution in the form 4.1 P(s) =~e where m. = ~w Sea(w) doo

% The stress amax is the maximum alternating stress around some mean stress. The omean in the present case is simply the static stress o due to the weight of the assembly and the %alancing pressure forces (buoyancy) acting on the barge surface. Therefore

amax = ICrg] tJm

(18)

The above described procedure to compute the maximum stress is applied at all joints at all member ends. The maximum force at one member end can be estimated from these stresses. The maximum stress at any point in the structure should not exceed the allowable design value, otherwise failure may occur. 4.0

EXAMPLEROBLEMS PTwo example problems are discussed in this The section. first overutilizes an simplified structural representation of the barge and the jacket. This simplification was intended to check the program computations by hand. The second example represents a more realistic test case where the transportation of a small jacket is considered. Example 1: Simplified Beam Model

-s2/2mo

(14)

(15)

In this problem, the barge is represented (structurally) by a beam and so is the These two beams are connected by 10 jacket. the short members simulating tiedown The upper beam has a moment of connections. inertia (in bending) and a mass distribution of the same order as the jacket of Example 2.The lower beam has the same mass distribution

The distribution given by Equation (14) hasbeen shown by Longuet-Higgins15 to hold for peak wave amplitudes in a randcm sea which has

and moment of inertia (in bending) as that ofthe barge. Correct mass distribution was

obtained

by artificially

adjusting

the

weight

a narrow band spectrm.

The highest of%the

density of different members.

It should be

stress peaks (maximum probable stress) ~max is obtained by solving the integral equation % max NJo

P(s) ds=N-1

(16)

where N is the number of stress cycles during Substituting the the storm sea state. expression for P(s) frcm Equation (14) into the left hand side of Equation (16) and integrating one obtains . N(l-e-~~ax/2mo) = N - 1 which gives

m~x=-

(17)

noted that wave forces as well as motion responses were computed for the real (physical) barge. Figure 5 shows a sketch of the A small cantistructural models geometry. lever member defined by joints 4 and 16, with a mass (10 Kips) at its free end (joint 16), was added to the previously described model. The stresses due to unit amplitude waves in this member, which are mainly produced by the inertial forces, can be estimated approximately by hand from the computed acceleration. The stress spectrum and, consequently, the cumulative fatigue damage ratio can also be computed by hand. The stresses computed by hand agreed with the program results. Because of the artificial nature of this example problem, stress and fatigue results are not presented. Although the main purpose of this example was to check the program, some information about the effect of barge flexibility on the tiedown forces was obtained. As mentioned earlier, the force system consists of inertial forces acting on the entire structure and hydrodynamic forces acting on These hydrodynamic forces were the barge.

Longuet-Higgins15 derived a similar equation for the maximun wave amplitude in terms of half the root mean square of the wave height and the nunber of cycles. It can be shown that both expressions are equivalent.

445

derived from the motion analysis, and were lunped, at twenty-one equ#:y spaced stations, with Simpsons in compliance same the motion integration rule employed in analysis. It is worthwhile to point out that forces balanced hydrodynamic the these inertial forces, thus, the structure was In This was equilibrium under these forces. indicated by near zero reactions at the supports in Figure 5. This result implied the consistency of the calculations up to this point. Figures 6 and 7 illustrate the structural response of the jacket-barge system to two typical unit amplitude sinusoidal waves. Both waves are in head seas and the wave frequencies are 0.47 radian/second and 0.68 Using canplex radian/second, respectively. domain representation, Figures 6a and 6b show the real part and imaginary part of the structural responses to the first wave, while Figures 7a and 7b show the responses to the second wave. There are five curves on each of the figures. Fran top to bottcm, the curves respectively illustrate the bending moment distribution along the structure, the deformation pattern of the jacket, the deformation pattern of the barge, and finally, the comparison between tiedown forces and the vertical motion pattern (due to heave and pitch) of the structure. The bending moment distributions illustrated in these figures were obtained frcinthe longiintegration of tudinal sectional dynamic loads, which was computed as the difference between the sectional hydrodynamic load and sectional inertial load. Notice that although the spatial distribution of the sectional hydrodynamic load is quite different fran that of the inertial load, the overall net force and moment due to these loads are negligible. Frcm these figures, one can make the following observations: 1. The free-free boundary condition was satisfied by the present analysis, evidenced by the bending mmnent distribution. The barge and the jacket deform together and their deformation patterns are highly bending correlated to the moment distribution. The tiedown forces do not correlate well with the motion pattern of the structure.

Therefore, the forces in ally different. individual tiedowns obtained by the inertial force may be approach substantially underestimated. 4.2 Example 2: Jacket Transportation Analysis of a Small

The present transportation procedure (system) was also used for the analysis of a four leg jacket. Figure 1 shows the barge carrying the jacket. The length of the jacket is approximately 260 feet. The diameter of the legs is about 53 inches, however, the wall thickness varies from 1.25 inches to 0.75 inches. The jacket is braced at five elevations. The bottom bracing section is 60 x 101 feet, while the top section is 60 x 51 feet. The barge is 377 feet in length, and has a beam of 100 feet. The draft used in the analysis is 14.5 feet. The barge/jacket assembly model is shown in Figure 4. The model is devised so that it reflects the primary stiffness and mass properties of the actual barge/jacket assembly of Figure 1. The preliminary DAMS-I run gave a of the assembly m,(mx=m =mZ) total mass of 1031 kip*sec2/ft. This mass inc{ udes the ballast water which was used to realize an appropriate draft (such that the center of buoyancy and the center of gravity lie along the same vertical axis). The computed moments of inertia are: Ixx = 1.207 x 106 kipftsec2 Iyy = 1.208 x 107 kip.ft.sec2

Izz = 1.246x

107 kipftsec2

2.

The axis system in which these computations were made is the standard axis system Other moments of (x, Y, z) of Figure 4. inertia (off-diagonal terms) were negligible because of lateral symmetry and near fore and aft symmetry. The mass of the assembly was lumped at 21 equally spaced nodes along the x-axis for the dynamic analysis to obtain the dynamic force distribution. Notice that these lumped masses include both can and ballast water mass. The duration of the voyage in this example was taken as thirty days. Two sea states were The wave spectra for applied to the barge. these two sea states are shown in Figure 8. Sea state 1 has a significant wave height of 20 feet, its probability of occurrence is 30 percent, while sea state 2 has a significant wave height of 8 feet and a probability of occurrence of 70 percent. Two directions of wave incidence with respect to the barge were considered; direction 1 represents head seas (V= 180) and direction 2 represents oblique seas ( p= 2250). For the first sea state the probability of occurrence in direction 1 is 60 percent and in direction 2 is 40 percent. As for sea state 2 the probability of occurrence in direction 1 is 55 percent and in direction 2 is 45 percent. Each of ttre ave w spectra was sliced into 5 equal slices 446

3.

The fact that the jacket and the barge deform together suggests that there is significant structural interaction between the two structures. Thus, the barge cannot be assumed to be infinitely rigid when cc+npared to the Furthermore, the rigidity of the jacket. structural interactions between the two structures transmitted via tiedown connections greatly affect the ,magnitude of individual While the overall resultant tiedown forces. force and moment of the tiedown forces correlate well with those predicted by applying the inertial force approach, the distribution of such forces in the tiedowns is substanti-

with AU = 0.25 radian/see; the central frequencies of these slices are 0.425, 0.675, 0.925, 1.175, and 1.425 radian/see. The motion analysis of the barge/jacket assembly was performed for unit wave amplitudes with the frequencies mentioned earlier. The motion RAOS of the system center of gravity are plotted versus circular frequency in Figures 9 and 10; Figure 9 shows the response for the head seas ( u= 1800) and Figure 10 for the quartering seas ( M= 2250). For head seas, the system responds primarily in only heave and pitch while for quartering seas it responds primarily in heave, sway, pitch, The response for all degreesroll and yaw. of-freedcm (in both cases) is higher at low These responses were used to frequencies. compute the accelerations and angular velocities for inertial force computations later. It shall be noted that five points (frequencies) are not enough to define the motion A larger number of response accurately. frequency points were used to plot the motion responses in Figures 9 and 10. However, only five frequency points were used in all the subsequent calculations to conserve computer time in this illustrative example. Fatigue maximum stress results are and presented for the member connecting joints 111020 and 121020 at joint 121020. The stress RAOS were computed at eight circumferential Figure 11 shows the stress response points. amplitude operators versus frequency for the two wave directions (heading and quartering These stress RAOS were computed seas ). discussed in according to the procedure spectra (at this point) Section 3. The stress were obtained from the- stress RAOS and the Figure 12 shows two stress wave spectra. spectra for sea state 1, which also represents the storm, when applied in the two directions discussed earlier. The maximum (probable) stress for the 100 hour storm sea state which occurs at p= 1800 (head seas), was computed from the stress spectrum represented by the solid line in A Rayleigh distribution was Figure 12. assumed for the stress peaks probability distribution. The results are as follows: Omean = % ~max,alt. = ~max = -1.25 KSI :6.26 -7.51

Numerical experiments on present approach. simplified transportation problems have inciicated the important effect of the barge/jacket interaction on the forces in the tiedowns. While the overall resultant force and moment of the tiedowns forces correlate well with those of the inertial forces on the jacket, the distribution of the tiedown forces is substantially different from that of the inertial forces. Results further show that the barge and jacket deform together indicating significant interaction. Therefore, some of the individual tiedown forces can be substantially larger than those obtained from the widely used inertial force approach, especially when the moment of inertia of the jacket is not substantially smaller than that of the barge. Acknowledgements The authors wish to acknowledge the technical contributions of Drs. D. Karsan, A. Mangiavacchi, and F. Chou of the Marine Division, Brown & Root, Inc. They also would like to thank Messers. Bob Henry and Doug King of the Computer Services of Brown & Root, and Pat Moore, a Computer Consultant, for their excellent work in the system development. REFERENCES

1.

J.T.: Bringloe, Hutchinson, B.L. and Application of Seakeeping Analysis, Marine Technology, SNAME (October 1978) 416-431.Sekita,Model

2.

K.,Tests

Sawada,on the

Y.,

and

Kimura,of

T.:a

Transportation

Large Offshore Structure Launching by Barge, Offshore Technology Conference Proceedings, OTC Paper No. 3517 (May 1979), Houston, Tx. 3. Kim, C.H.: Wave Exciting Forces and Moments on a Ship Running in Oblique Seas, Davidson Laboratory, Stevens Institute of Technology Technical Memorandum (1974). Ogilvie, T.F. and Tuck, T.O.: A Rational Strip Theory of Ship Motion, Part I, Department of Naval Architecture and Marine Engineering; The University of Michigan, Report No. 013 (1969). Salvensen, Ship actions of and Marine

4.

KSI KSI

5.

o .:

N., Tuck, E.O., and Faltinsen, Motions and Sea Loads, Transthe Society of Naval Architects Engineers (1970) 250-287.

The cumulative fatigue damage ratio (CDR) at the same point due to both sea states, during the whole duration of voyage was found to be 5.66 x 10-6, which is fairly small. The calculations for this illustrative example were made using stress concentration factors of unity for both axial and bending stresses. With the use of appropriate stress concentration factors the value of CDR should increase dramatically. It should be noted that the results presented in this example illustrate the use of the 447

6.

Frank, W.: The Frank Close-Fit Ship Motion Computer Program, Naval Ship Research and Development Center, Report 3289 (1970). Clough, R.W. and Penzien, J.: Dynamics of Structures, McGraw-Hill Book Company, New York, N.Y. (1975) 502-504. Vughts, J.H. and Kinra, R.K.: Probabilistic Fatigue Analysis of Fixed Offshore Structures, Offshore Technology Conference Proceeding, OTC Paper No. 2608 (May 1976),

7.

8.

Houston, Tx. 9. Wallis, J.R., Bayazitoglu, Y.O., Chapman, F.M., and Mangiavacchi, A.: Fatigue Analysis of Offshore Structures, Offshore Technoloqy Conference Proceedings, OTC Paper No. 3379 (May 1979), Houston, Tx. Maddox, N.R., and Widenstein, A.W.: A for Fatigue Analysis Offshore Spectral Structures, Offshore Technology Conference Proceedings, OTC Paper No. 2261 (1975), Houston, Tx. A Maddox, S.J.: Gurney, T.R. and for Welded Re-analysis of Fatigue Data Joints in Steel, Welding Institute Report E/44/72 (1972). Cartwright, D.E., and Longuet-Higgins,M.S.:

The Statistical Distribution of the Maxima of a Random Function, Proceedings of Royal Society, Series A (1956) 212-232. 13. Nolte, K.G., and Hansford, J.E.: ClosedForm Expressions for Determining the Fatigue Oamage of Structures Due to Ocean Waves, Journal of Society of Petroleum Engineers (Dee. 1977), 431-440. Wirsching, P.H., and Light, M.C., Fatigue Under Wide Band Random Stresses, Journal of (July 1980), ASCE Oivision, Structural 1593-1607. Longuet-Higgins, M.S.: On the Statistical Distribution of the Heights of Sea Waves, Journal of Marine Research, Sears Foundation for Marine Research, Bingham Oceanographic

10.

14.

11.

15.

12.

Laboratory,Yale University,(1952).

-

\

,

Figure 1

Jacket Transportation by a Barge (Example Problem ,2)

ACCELERATIONS, BARQE JACKET

VELOCITIES STRUCTURAL

ANO

PRESSURE

LOADS

REPRESENTATION

0DAMS - I [OAMSGL) FOR UNIT WAVE AMPLITUDES

~

wDAMS - I

I

STRESSES

DUE

TO UNIT WAVE AMPLITUDES

I

I

I

v(!1 [2) MAXIMUM aTRESSES FATIGuE CUMULATIVE

I

I

I

I OAMAGE

Figure 2

Flow Chart for Inertialand Pressure Force Calculations

Figure 3

Flow Chart for Maximum Stress and Cumulative Fatigue Damage Calculations

STRUCTU AXES

Figure 4

Structural Model of Barge-Jacket Assembly (Example Problem 2)

Figure 5

Geometryof Example Problem 1

L ~u !1:~;~ _________________ % ~.[ E;;*I . .

6

$~qg

b

6

Lll

9

7

0

i

:

0.r!

-.

X=o

LENGTH

OF BARGE IS 3770 FT HEAO SEAS FREOUENCY 0,47 8AD./SEC, REAL RESPONSE

x:l-

Figure 6

StructuralResponseof Barge-JacketSystem to a Unit ilnpl itude Wave, m = 0.47 rad. /see

h F=--Q@x,0

LENGTH

OF BARGE IS 37%0 HEAO SEAS FREOUENCY O,Sf! RAD/SEC REAL RESPONSE

x,-2

X,o

LENGTH

OF

BARGE HEAo

IS 3770

FT

X =L

SEAS 0.68 RAD/SEC

FREQUENCY

IMAGINARY

RESPONSE

Figure 7

StructuralResponseof Barge-JacketSystem to a Unit Pmplitude Wave, to= 0.68 rad. /see

,70;) II 1 +---H 60[l : .: 50l\ ~1 \ )1 : q II : d ~ g ~o11 \ : / 20 \ 40 h \ I I

!

!

1

!

1

I

I

I

I

1 HEADINO

I ANGLE 1

p = 180

l,oSEA STATE ,20ft s NO, I (STORM)

2 ~gF

o,s-

k~ 0 s HEAVE 5 ~ 0.4 0,6-

0.2 sEA STATE NO. 2 I \ If Y 1 0,6 I 0s FREQUENCY, I III u ( RAO./SEC.) I 1.2 1 1.4 I

i

0,4

I o; I i 0 0,5 , 1.0 1 1.5 J.o I 25 Figure 3.0 9 System

Motion

Response

knpl

itude Operators;

Head Seas

FREQUENCY,

w ( RAD./SEC.l

Figure

8

Wave

Spectra

for

Sea

States

1

and 2

1

!

I

1

! HEADlffi ANSLE

I

L4-

-0.7

/b = 225

L2~

0,6

Lo

0.5 j > \ ~ 1 % \YAW \ \ [ 0,4 ; 2 x Q 0,3 t & = ~ g

0.8-

0,6

o.4-

SWAY

ro.2

2

0,2-

-0,1

0 03 a4

I C16

, 0.s

1:0

1 12

I I ,4

0

FREQuENCY,

w

(

RAO.I SEC)

Figure 10

System

Motion

Response

Amplitude

Operators;

QuarteringSeas

1

1

!

t

I

I

HEAOING o +

ANGLE

p = 180 /4. 225

0:4

0:6

ok

Lo

12

1:4

FREQUENCY,

u (RAD, /SEC.)

Figure 1

Stress Response Amplitude DifferentFrequencies

Operators

at

II/ /I

/--\ \

,\

0.8

-1/II

///

/; I

\\

i

\ \ \ \ \ \\ \\ \ ., , I

0

A

/10.4

0,6

1

I

I

0,8

LO

12

1,4

FREQUENCY,

d ( RAC1/SEC,l

Figure

12

Stress

Spectra

at

Joint

No.

121020