Chakrabarti - Review of Riser Analysis Techniques

Review of riser analysis techniques

SUBRATA K. CHAKRABART! and RALPH E. FRAMPTON Chicago Bridge and Iron Co., Plainfield, IL 60544, USA

A state-of-the-art review of the riser analysis is presented. The papers starting with the Mohole project in early 1950 to the present day covering the analysis of a riser are discussed. It is shown how the earlier static analysis of the problem is supercedcd by a more advanccd dynamic analysis with the advent of the modern computer. Several controversial areas in the analysis, e.g., the non- linear drag terms, the effective tension in the system, the buoyant weight of the riser and contents, the non-linearity due to large deformations, etc., are reviewed and the deficiencies of many works in these areas are pointed out. A detailed deriwttion of the horizontal equation of motion of the riser and interpretation of the wtrious terms in the equations have been provided.

INTRODUCTION

A riser is a long slender vertical cylindrical pipe placed at or near the sea surface and extending to the ocean floor (Fig. 1). At the surface it is connected to a surface vessel. At the bottom it may be hanging free or connected to a blow-out prevention (BOP) stack (Fig. 2). Even though the word 'riser' has been used in describing title, the problem addressed here is not limited to risers used in the exploratory offshore drilling operations only. It is equally applicable to production risers, OTEC cold water pipes, single legs of a tension-legged platform, conductors associated with a gravity drilling and production platform, etc. In other words, the review includes any long slender vertical member undergoing deformations due to its weight, vessel motion, waves and current in the deep waters.

The papers reviewed in this report cover the works starting in early 1950 for the Mohole project to the very rccent ones. The review certainly does not include each and every paper published on the subject, but it is intended to give a complete view of how the technology of the riser analysis developed over the years to the present state-of-the-art. How the earlier static analysis progressed to the more complex dynamic analysis with the development of more sophisticated computer procedures is discussed. Several problem areas, e.g., the handling of the non-linear drag term, the calculation of the buoyant weight of the submerged riser, and the effective tension in the riser arc reviewed and the deficiencies of several papers in these areas are pointed out. Different methods of solution adopted by different programs and the advantages and disadvantages of these techniques are discussed. Other areas covercd by this review are the non- linearity of the problem, various end conditions considered, different wave theories, and a brief discussion of the lift force and its effect on the riser motion.

REVIEW OF PREVIOUS WORK

In order to maintain uniformity and ease of comparison among the different works, a uniform set of symbols will be used throughout this paper. A definition sketch of the problem under review here is included in Fig. I. The typical boundary conditions of a riser for both the normal

operating condition and emergency conditions are shown in Fig. 2. The following areas will be discussed in the review of the riser analysis: (1) derivation of riser horizontal equation of motion, (2) effective tension and buoyant weight in the system, (3) static versus dynamic analysis, (4) end conditions, (5) wave theories used, (6) non-linear drag force, (7) lift force, and (8) method of solution. A complete summary of the papers reviewed is presented in Table I. Detailed discussion of the general eqt, ation of motion and the derivation of the individual terms in the equation of motion are given in the next section.

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Applied Ocean Research, 1982, I/ol. 4, No. 2 75

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Applied Ocean Research, 1982, Vol. 4, No. 2 77

Review of riser analysis techniques: S. K. Chakrabarti and R. E. Frampton

"4

Review of riser analysis techniques: S. K. Chakrabarti and R. E. Franlpton

DERIVATION OF RISER HORIZONTAL EQUATION OF MOTION

The equilibrium equations for a bent tubular segment and the statically equivalent loads at its centre resulting from (1) internal and external fluid pressure, and (2) the weight of the segment are derived in Appendices.I, II, and III respectively. To develop a horizontal equation of motion for a marine riser, it is desirable to combine the three equilibrium equations as given in Appendix I into one single equation. There are several approaches to this task and for a better understanding of the problem and the assumptions involved, two approaches will be given here.

In the first derivation, the form of the equations as derived for assumption A (Appendix I) will be combined to show the form of the riser horizontal equation of motion for large deflection and rotation. Then, assumptions B and C (Appendix I) will be used to reduce this equation to a form which is normally used, i.e. small deflection and small rotation.

The equilibrium equations which include assumption A are given in Appendix I by the horizontal, vertical, and moment equations, equations (43)-(45). To develop a horizontal equation of motion for the riser, the vertical equilibrium effects, equation (43), are included in the horizontal equilibrium equation, equation (44). Solving for the coefficient A~ and substituting in equation (44) gives:

B l see 0 + (f.,-f=,)tan 0 +f,, -- m~,.~ =0 (1)

To include the moment equilibrium effects in equation (1), the shear, V, is obtained from equation (45). Using the relationship between the curvature, l/p, and the moment, M, for the case of pure bending,

,,,: _ _ (E, dO) ds\p ] ds\ ds] (2)

The final general form of the horizontal equation of motion is obtained from equations (2) and (47) by substituting the value of B x into equation (1)

d 2 dO [~-~(EI-~s)-T~s]secO-(fw-f:,)tanO+m~'v:=f~

(3)

Including the expressions for the weight of the segment, i.e. equation (86) forfw, and the loads resulting from the internal and external fluid pressures, i.e. equation (67) as part off:, and equation (69) as part off:~, results in:

dO [D( El-~s)-T~s]secO-[y~(Ao-A,)-f:~+f.vsinO] x

tan 0-f .v cos 0 + m~"~ =f.,~ (4)

Introducing equation (70) forf, v into equation (4) and by combining terms and simplifying gives:

d 2 dO _ Aipl)dd~]SecO_ D,(A0_Ai) -

f:s- Ao?o + Awi]tan 0 + nl~.7 =f,, (5)

Equation (5) is the form of the riser horizontal equation of

motion which includes large deflection and large rotations.

If small deflection beam theory (equation 35) is assumed:

-ds2\ as] -(T+A~176 as luz

A ~ dx . . Ao?o+ /?d -~z+mxX=fx, (6)

Further, by assuming small angle deflections (equation 36), equation (6) now takes the usual form as:

d2(d2x) - _ d2x - (T + AoP o [?~(A o - Ai) - - . f . - s - - dz2 EI ~z2 -AiPi)~z2 -

dx . . A oYo + AD'i] dzz + mxx =f,, (7)

To understand better tile above equation and to show the relationship between the coefficients of the slope, dx/dz, and the curvature, d2x/dz 2 terms, another approach to the derivation of the horizontal equation of motion for a riser segment will be given. In this second derivation, we will start with the form of the equilibrium equations given in Appendix I that includes the small deflection and small rotation assumptions, and again combine these three equations into a single horizontal equation of motion.

The equilibrium equations which include assumptions A through C are given in Appendix I by equations (49)- (51). Substituting the expression for the shear, V, from equation (51) into the vertical equilibrium equation, equation (49), and neglecting products of differentials, the derivative of the tension, T,, takes the form:

dT =f.,-L, (8)

The indefinite integral of equation (8) relates the tension, T, to the weight and vertical loading as follows~

T= "I~ + Sfwdz - ~f.flz (9)

where T, is an arbitrary cofistant of integration. Now, using the derivative of the shear from equation (51) in tile horizontal equilibrium equation, equation (50), and rearranging terms, the horizontal equation of motion becomes:

822 ~, dz 2) dz T +mx.~=L~ (IO)

where the tension, T, is given by equation (9). An alternate form ofcquation (10) is often used in which

the derivative of the second term is expressed explicitly, i.e. using equation (8) for the derivative of tension, results in

d 2 [ d2x~ d2x dx (11)

When including the expressions for the weight of the segment, i.e. equation (86) for fw, and the loads resulting


Review of riser tmalysis techniques: S. K. Chakrabarti aml R. E. i"rampton

from the internal and external fluid pressures, i.e. equation (73) as part off~ and equation (72) as part off:~, we note that the tension term in equation (10)and the horizontal pressure component, equation (73), are similar and may be combined as follows:

d2 elil i) ~] + m.~.:~ =fx~ dz2(E l~)_d[ (T+, , loPo_ F, d\"

(12)

or, using the alternate expression for horizontal pressure (equation 71), the horizontal equation of motion becomes identical to equation (7), in which

T=T~+I'A(Ao-A3dz-[f: ,dz (13)

For the case of most marine risers, the external loading consists mainly of surface loads due to wave and current which results in horizontal and vertical force intensities, f,~ and f:~. Also, as shown by equation (13), the axial tension, T, at the ccntre of the segment (point s) wlries continuously along the length of the riser due to the weight of the riser itself, or may evcn have abrupt changes caused by buoyancy modules attached to the riser. The internal and external pressures, P~ and Po, are linear functions of the distance from the midpoint, s, of the segment to the free fluid surfaces. Also, because ofchanges in pipe sizes, the quantities El, A o, and A~ may vary along the length of the riser. Thus, a more gcneral form of equation (12) may be written (expressing the derivative of the second term explicitly) as:

[ ] 02 02x - - ~ x Et(Z)Ez 2 - [T(z)+ Ao(z)P(z ) - A,(z)P,(z)]-ff-~ OX

{7,[Ao(z)- A~(z)] -f:,(x,z,t) - Ao(z)7o + A,(z)Ti} ~z +

m~(z).;~ =Z~(x, z, t) (14)

where

"I'(:)= T~+~7~[Ao(z)-A,(z)]dz-~f:~(x, :, t)dz (15)

Note that partial derivatives have been used to reflect that this may be a three-dimensional problem andthat the equation for the horizontal motion in the )'-direction is the same as equations (14) and (I 5) except x is replaced by y and that both x and y are functions of both z and t (elevation and time).

To evaluate equation (15), the axial tension must be known at some point along the riser length. If T-roe is the known tension at the top of the riser ofe[evation zmp, then equation (I 5) may be expressed as:

P

T(z) = "/'To v- 7~[A o(Z ) - A,(z)]dz +

"-r;vf:~(x, :, t)d: (16)

Likewise, if T,o r is the specified tension at the bottom of

the riser of elevation :Boy then equation (15) becomes:

T(z) = Tuor+ f ~,,~[Ao(z ) - A,(z)]dz-

:BOT

f f:~(.\', z, t)dz

:BOT

(17)

DEFINITION OF EFFECTIVE TENSION AND BUOYANT WEIGHT

In the deriwition of the horizontal equation of motion for a marine riser, the vertical wall tension and the horizontal pressure loads were combined into one term as shown in equation (12). As is often done in riser analysis, the term within the brackets in equation (12) is defined as the effective tension, Tr Thus, the horizontal equation of motion may be written as:

0 2 FEI(z)O2x -] 0 [- Ox-] =I,,(x,z,tl

(a) (b) (c) (d) (18)

where

7~.(z) = T(:)+ Ao(z)Po(:)- A,(z)Pi(z) (19)

and T(z) is given by equation (15). Again, an alternate form of equation (18) which is often

used is where the derivative of the second term is expressed explicitly. The derivative of the effective tension becomes:

a~(z ) Oz

- - = 7~[,'1 o (Z) - ,,t,(z)] -L~(x, z, t ) - A o(Z)7o + A,(z)7~

(20)

For the special case when there is no vertical loading, i.e. f.~(x, z, t) = 0, the derivative of the effective tension is often defined as the buoyant weight, w(z), of the riser. Thus, for f:~(x, z, t)= 0

aT,.(z) 0z = w(z) = 7,[Ao(z ) - A/(z)] - Ao(z)7 o +/li(z)7 i

(21) and the horizontal equation of motion is written as:

E1(:)~-~-2 - T~(Z)~zZ -/;(z)b- ? - ~(Z)u_ " +,,~.~ =L,(x.z.0 (22)

It should be noted that the effective tension, ~(z), as given by equation (19) has no real physical significance, but represents only the mathematical combination of the actual wall tension, T(z), as given by equation (15) and the horizontal load rcsuhing from the internal and external field pressure (equation 73). These two distinct force contributions affect the horizontal equation of motion in a similar fashion.



Likewise, when using equation (22) for the horizontal equation of motion, the coefficient of the slope term was defined as the buoyant weight of the riser and contents, w(z), as given by equation (21). Again, this term has no real physical significance, but it is computationally correct to interpret this term as the 'distributed buoyant weight' of the riser.

The different expressions used for these terms in the papers reviewed are listed in Columns 12 and 13 of Table 1. It should be noted that the authors may have used any one of the several forms of the horizontal equation of motion as previously derived.

DISCUSSION OF THE HORIZONTAL EQUATION OF MOTION

The form of the horizontal equation of motion to be discussed is given by equation (18) in which the three terms on the left hand side are referred to as terms (a), (b), and (c) and the right hand side is called term (d). This equation was derived from the consideration of equilibrium of internal forces, i.e. moment, shear, axial tension, body weight, and externally applied loads, i.e. f:,~ andf:~, in addition to the fluid pressure forces acting on a deformed riser segment. It represents the partial differential equation to be solved for the deflection curve in the X direction. In this formulation, axial straining is not considered and the resultant deflections and rotations are assumed small.

The terms affecting the horizontal displacement in eqnation (18) may be interpreted as follows: Term (a) Resistance to lateral loading resulting from

the riser's flexural rigidity, El(z). Term (b) Resultant lateral loading due to the axial

(vertical) force, T(z), in the riser wall area plus the net horizontal loading from the external and internal fluid pressure. Both of these forces may add to or resist the lateral loading, f,~, depending upon the slope and curvature of the riser segment (see Table 2).

Term (c) Riser's inertial resistance to lateral loading. Term (d) Applied horizontal force intensity.

Using equation (19) in term (b) and expressing the partial derivative explicitly, further observations may be made.

~I " Ox] OZx OT(z)Ox z T~(Z)~z = T(Z)~z2-t Oz Oz 'erm(b) Term(e) Term(O

"~2 - - - O X

4- [ A o(Z) Po(Z) -- A i(z ) P i(z)] ~z z

Term (g)

EA o(Z):r - A ~(z)Ti] ~-z

Term (h) (23)

Term (e)

T~rln (f)

Term (9)

Term (h)

The axial tension, T(z), in the riser causes lateral loading due to the curvature, (catenary effect). The rate of change of axial tension causes lateral loading due to the slope of the riser. External and internal fluid pressure causes a net lateral loading due to the curvature of the riser. The rate of change of external and internal fluid pressure, i.e. the fluid densities 70 and ?i,

causes a net latcr~al load due to the slope of the riser.

The axial tension, T(z), may be evaluated from equations (16) or (17) and the rate of change of axial tension, ~T(z)/~3z, is given by equation (8) which indicates that these parameters are a function of the weight of the riser, f,., and any vertical load intensity,f:~. However, as shown by equation (72), these terms are not affected by fluid pressure forces.

The riser segment in its deflected position experiences the external forces due to waves which act normal to the segment. If the normal component of this force is resolved into two orthogonal axes, then one of these components contributes to the vertical load intensity,f:~. The analyses of St. Denis and Armijo I and of Bennett and Metcalf 18 include such a force component in the equilibrium equations. It should be noted that these force components are of second order under small angle assumption. The internal fluid flow in the riser may also produce a vertical force component contributing to f_-~.

The resulting force from internal and external fluid pressure is horizontal assuming small deflections and rotations. Therefore, 'buoyancy' has no effect on the axial tension except at the ends of the riser. The fact that this force is a function of the riser curvature (term g) and a function of the riser slope (term h) allows it to be mathematically combined with the resulting lateral load from axial wall tension, terms (e) and (1) respectively. The interpretation of term (b) in equation (18) appears to be a source of some confusion in several of the papers reviewed.

Table 2 illustrates how the various components of the effective tension Te(z), in term (b) as given by equation (23) may either add to or resist lateral loading depending on the riser's slope and curvature. This Table shows the direction of the resultant horizontal load in heavy arrows assuming that the various coefficients of the slope and curvature terms are positive. Observations about the effects of both 'heavy mud' and axial tension may be concluded by examining this Table.

STATIC VERSUS DYNAMIC ANALYSIS

The horizontal component of the external loading is included on the right-hand side of equation (18). The resulting equation may, then, be used to solve both static and dynamic riser problems.

For static riser problems, the inertia term (c) is not included and the right-hand side of equation (18) is the horizontal load due to drag from asteady current profile as follows:

~2x c3 ~x ~ , ~ [El(Z) ~zZ ] -~z [ T~(z)-~z ]=-2pwCo(z)Do( ")U(z)lU(z)l

(24)

where p,.=mass density of water, CD(z)=drag force coefficient (varies with z), U(z)= horizontal component of the current velocity.

The solution of equation (24) requires additional constraints or boundary conditions. Typically, deflections and rotations at the top and bottom of the riser are specified in order to model the end restraints (e.g., fixed, pinned, free, or non-linear) or to specify top horizontal offset.


Review of riser analysis techniques: S. K. Chakrabarti aml R. E. Frampton

Table 2. Components r cffective tet,sion vs. slope and curvature

CURVATURE SLOPE TERM (b) AXIAL TENSION HORIZONTAL PRESSURE

[ ] [ _ a=x ax a ax _ , ,a 'x aT(z ) 3x ]Ts z --az ~ Te(Z)-~- { = "rlzJ~-~-z + az az + Aotz (z)Pi(z) -[Ao(Z)'fo ,o, "rl .0

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Term (e) Term (f) Term (g) Term (h)

The initial effort in solving a riser problem has been expended in writing computer programs based on static analysis of the riser. Fisher and Ludwig 4 and Gosse and Barksdale 6, among others, performed a static analysis of the riser as a variable tension beam with current loading. Some of the later papers e.g., Tidwell and Iifrey 7 and Kopecky 8, included both the static current loading and a 'dynamic' time slice for wave loading. These are summarized in Table I, Column 5.

For dynamic riser problems, the right-hand side of equation (18) nmy include the effects of waves, drag forces due to relative water velocity, and vessel motions. Note that the inertia of the system includes the added mass effect. In equation (18)f~(z, t) is the forcing function representing the above mentioned loads. For waves it is represented by a modified form of the Morison equation in terms of an inertia and a drag term linearly added together. The inertia term is proportional to the normal component of the water particle acceleration, ~,,, on a bent riser segment and depends on an inertia coefficient. The drag term depends on a drag coefficient and the normal relative velocity component which comprise the water particle velocity and the velocity of the riser segment.

f~(z,t) = pwCM(z)~Dow(Z,t ) + 89 ) x

(25)

The hydrodynamic coefficients, C , and Co depend on several non-dimensional quantities, e.g. Reynolds number and the Keulegan-Carpenter parameter (KC) as well as the riser configuration. Note that KC should depend on the relative velocity, [w(z,t)-Ox/Ot]. This would require an iterative solution since the drag force depends on KC which depends on the relative velocity. Recent laboratory tests of different riser geometries have provided values of C , and Co that will be useful in the choice of these values in a riser analysis 27-29.

Sometimes the static and dynamic analyses are combined into one equation in which the forcing function includes both the current and the wave force. In this case, the non-linear drag force is written in terms of a relative velocity as in equation (25), but including the effect of the current, U as [w(z,t)-(Ox/O0-U(z)].

The dynamic analysis of a riser problem has been the subject of study for the last seven or eight years with the exception ofone of the earliest papers in 1955 by St. Denis and Armijo t. Numerous papers have been published in this area. The dynamic analysis generally includes waves and current e.g., in one of the earlier works by Tucker and Murtha ~~ Sometimes the dynamic analysis due to waves is done separately and superimposed onto the static solution due to current only, e.g., as was done by Burke ~ 1, Gnone et al) 4, and Kirk et al. 26. Among others who provided solutions of the dynamic equation of motion are Heuze 13, Sexton et al. t6, Gardner and Kotch 17, Bennett and Metcalf t8, Maison and Lea ~9, Paulling 2~176 Chou et al. 2t, Young et al. 22, Dareing and Huang 24, and finally,


Review of riser analysis techniques: S. K. Chakrabarti aml R. E. Frampton

Krolikowski and Gay -'5. The works of Paulling and Chou et al. arc dircctcd towards the cold water pipe of an Occan Thermal Energy Conversion system (OTEC). For a complete listing of static vs. dynamic analyscs for the papers reviewed, see Column 5 of Table 1.

END CONDITIONS

Various end conditions have been incorporated in the analyses. In several analyses, the top of the riser is free, while others assumed the top end pinned or fixed to a floating object. Likcwise the bottom end of the riser could be free or fixed or pinned to the bottom. Both cnds of the riser wcre considercd pinned by Fischer and Ludwig 'L, Jones tS, Kirk et al. 23, and Dareing and Huang 2"*. The bottom end is free for the OTEC cold water pipe analysed by Paulling 2~176 and Chou et a/. 2t and for mining application by Bennett and MetcalP 8. Graham et t.t[. 3 considered bottom end to be fixed and top end to experience a harnlonic translation and rolling. A variety of end boundary conditions are assumed by Burke l' Heuze 13, Gnone et al. ''t, Sexton eta[. 16, Gardner and Kotch 'v, Bennett and Metcalf '8, Maison and Lea '9, Young et al. 22 and Kirk et al. '3"26. Addilional comments about the boundary conditions are listed in Column 21 of Table !.

WAVE THEORIES

Usually linear Airy wave theory has been used in the dynamic analysis of the riser. This is particularly true for a frequency domain analysis. The analyses of Gardner and Kotch 17 and Maison and Lea ~9 include Stokes' fifth order theory. A summary of the wave theories used is listed in Column 20 of Table I. Several people have considered random waves in their analysis including Tucker and Murtha I~ Sexton et al. ~6, Young et al. 22 and Krolikowski and Gay 25. Unlike Tucker and Murtha who have used a random wave system for computing riser response spectra, Sexton et al. used the random wave input in a time domain. Column 17 of Table I indicates those papers that have included wave force computation.

NON-LINEAR DRAG FORCE

One of the most difficult areas in tile equation of motion has been the fluid dynamic drag force. Being non-linear, the drag force can only be handled without modification in a numerical time domain analysis. In terms of the relative velocity it is given as,

i)x ] ~x[ L,~:, t)=89 t)- ~ 31"(:, t)--S{ (26)

in which w=normal component of the water particle velocity. In the case of static analysis, w will become the current velocity, U. Somctimcs, U and w are combined as relative velocity to describe the complete riser motion analysis, e.g. Kirk et al. z6 and Krolikowski and Gay 25. In a simplificd analysis, particularly in a frequency domain where harmonic solutions are sought, the drag force is linearizcd. An approximate equivalent form in this case is:

f~o(z, t)=~p,,.Co(z)Do(z)ff[w(z, t ) - ~-~]

(27)

in which ~.=mean value of the rclative velocity. For example, assuming

w= Re lWo(z)e""} (28)

and

x = Re [Xo(Z)e "'''+ "~} (29)

in which wo=amplitude of w, Xo=amplitude of x and = its phase angle, it can be shown by Fourier analysis

that:

8 I~'= ~- Iwo - itoxoei'l (30)

Another equivalent linearization approach generally used in the frequency domain analysis was introduced by Borgman 3~ It consists of writing the (relative) velocity squared term as the (relative) velocity term multiplied by a factor x/8/r~ and the rms value of the velocity. Tung and Huang 3t cxtended this technique to include current and wrote expressions for the linearized drag force as well as modified wave energy density spectrum in the presence of current.

LIFT FORCE

In the dynamic analysis, the motion of the riser in the presence of tile waves (and current) will introduce the formation of a wake region near the riser and shedding of vortices from the surface of the riser. This, in effect, will introduce a force at right angles to the plane ofthe motion of the riser. In a three-dimensional problem e.g., with multidirectional waves and current, it is quite difficult to introduce this effect, in a two-dimensional motion, however, this transverse forcc can be computed and introduced as a transverse vibration. The force on a unit length of the structure due to a normal water particle vclocity, w, is given by:

f}. = 89 (31)

depending on a lift coefficient, Ce Owing to the motion of the structure, a lift force on the unit length of the form:

f~. = }p,,.C t~z, t)Do(z)5:2(z) (32)

is generated where 5.(= ~3xlg;t)= velocity of the centre of tile segment in tile x direction. In general, these two forces depend on differcnt values of Ct: I lowcvcr, considering an eqnivalent single value of C~, a lift force in terms of the relative velocity may be written as:

fr = 89 dz, t)Do(z)(w-.;;)2 (33)

in which CL, now, depends on the relative velocity, w-.~. Since the lift force is, gcncrally, irregular in nature, the value of Ct.will vary in a wave cycle. Howevcr, the lift force may be approximatcd in terms of the most predominant frcqucncy of the lift force in which case, Ct. may bc



SPATIAL .r TECHNIQUE TEI~PQRAL r TECHNIQUE I I I

DYNAMIC -OL'N TYPE I I

ANALYTICAL

R D E ~ (A.D.) roi~ i~lsk R

DISPLACEMENT NUMERICAL (FD, NI,FE)

STATIC ANALYSIS

TEMPORAL $OL'N.

DIRECT INTEGRATION TIHE HISTORY , SSOLU' | ION ~ / (FD.N,) - - (T ' , )

rN~.r ~ . /2 - - DE T E ST IC - -~_ E . . . . . ~OD~T.~9 . ELIMINATE

1" .'TA 'ISF() - M AT I0 N --I~ TiME FREQUENCY 0r A IN (M T.) \ RM (SS.TF) (FD)

NON'DETE INISTIC--SPECTRAL RANEK~IVI VIBRATIONS ANALYSIS (RV)

DYNAMIC A NALY~.I S _1 (NOTE: NOTATION IS GIVEN IN TABLE 1)

Figure 3. Methods of static aml dynamic analysis

- I

assumed constant over the cycle. Once the form of the lift force is known, a vibration analysis transverse to the wave direction may be set up. Thus, the two equations (transverse and inline) may be solved simultaneously and the resulting riser forces may be used for a more complete stress analysis. None of the papers have included the effect of the lift force in the stress analysis of the riser. Young et al. 22 make a note ofthis effect, but do not show an analysis including this phenomenon.

METHOD OF SOLUTION

The basic form of the differential equation to be used for dynamic riser problems is given by equation (18). The static analysis involves finding the riser deflected shape for a constant load distribution on the riser, (spatial solution to equation (24)). The dynamic analysis requires finding the deflected shape (spatial solution of equation (18)) for a time varying load distribution acting on the riser (temporal solution to equation (18)). Table 1 shows the techniques used by the references listed for both the spatial solutions (column 8) and the temporal solutions (column 9). The various methods used for the static and dynamic analysis of risers are summarized in Fig. 3.

The techniques for the spatial solution of either the static equation, equation (24), or the dynamic equation, equation (18), are numerous but may be placed in two major categories; analytical methods and numerical methods.

The analytical methods are limited in that the geometry of the riser must generally be uniform along its length. One procedure, often used, is to assume a mathematical form. for the deflection solution of the riser. This procedure was used,by Fischer and Ludwig* and Dareing and Huang 24 who assumed the deflection to be in a form of an infinite series and by Kirk et al. z3 who assumed the deflection to be a sinusoidal series. Another approach is to assume that deflection of the riser is similar to structural components for which solutions exist. This was done by Jones I s who assumed the deflection to be that ofan elastic catenary. Also, for their dynamic riser analysis Graham et al. 3 assumed that the deflection behaves as a constant tension beam at either end and a variable tension cable in the middle. A normal mode method (modalsuperposition) has been used by Kirk et al. 23'26 and Dareing and Huang 2"~.

The numerical methods for the spatial solution of equation (18) are more general in that they allow for changes in geometry, i.e. riser size, buoyancy modules, etc. and non-linear end restraints. The two main approaches

often encountered are (1) direct solution of the partial differential equation by numerical approximations, and (2) finite element idealization resulting in matrix solution methods.

One numerical procedure for the direct solution is the finite difference technique. This approach leads to a set of simultaneous equations which are banded but not necessarily symmetric. Often the boundary conditions require iterative procedures. Butler et al. 5, Gosse and Barksdale 6, and Tidweil and Iifrey 7, and Kopecky s use this approach for their static analysis, while Tucker and Murtha 1~ Sexton and Agbezuge 16, Bennett and Metcalf Is, Chou et al. 21, and Young et al. 22 use the same procedure, i.e. finite difference, for the spatial solution in their dynamic analyses. Another procedure for the direct solution is numerical integration as was done by Burke t 1. Both the finite difference and the numerical integration procedures are better suited to solve two-dimensional riser problems.

The second numerical method is the finite element procedures. In this approach, the riser is modelled by variable tensioned beam elements with discrete degrees of freedom as unknowns at the connection points of the elements. This results in a set of simultaneous equations that is both banded and symmetric9 Computer procedures are well established to solve these equations as well as to determine the natural periods of the system. This method is well suited to three-dimensional problems and may be used to include large deformations (see Column 7, Table 1) of the riser. Bathe and Wilson 12, Heuze 13, Gnome et al. 1., Gardner and Kotch aT, Maison and Lea 19, and Paulling 2~ use this approach in their analysis.

The solution of the dynamic riser problem with time dependent loads falls into two main categories: deterministic (time and frequency domains) and nondeterministic solutions (stochastic). These dynamic solution types are shown in Column 10 of Table 1. The method of solution (temporal solution technique) is given in Column 9 of Table 1.

Deterministic solutions include both the time domain (time history analysis) and the frequency domain analysis. The time history analysis is more general in that the time dependent loads may be non-linear (as is drag) and variation in buoyancy near the surface may be included. The numerical procedure to step through time may be the finite difference technique as used by Sexton and Agbezugc ~6 and Maison and Lea 19, or the numerical integration technique as used by Heuze 13, Gnone et al. ~4, Gardner and Kotch ~7, Bennett and Metcalf IB, and Chou et al. 21. The disadvantage of time history analysis is that


Review of riser amdysis techniques: S. K. Chakrabarti aml R. E. Frampton

usually long computer runs are necessary. For frequency domain solutions, wave loading and

vessel motion are assumcd to be harmonic which allows for a direct sinusoidal steady-state solution. This approach has the advantages of being more suitable for fatigue analyses and involving shorter computer rnns. However, non-linear loading (drag) must be linearizcd and the response appears to be sensitive to minor changes in the wave spectra. Graham et al. 3, Burke ~ ~, Paulling 2~ Young et al. 22, and Kirk et al. 23'26 solve the dynamic riser problem in the frequency domain.

The non-deterministic approaches to solving the dynamic riser problem include the procedure of modal superposition (random vibration). Time dcpcndent loads (including drag) must again be linearized and the interpretation of the results is a more difficult task. Tucker and Murtha t~ have used this approach.

CONCLUSIONS A state-of-the-art review of the riser analysis techniques has been made. The papers included in this review are not claimed to be a complete list, but illustrates how the earlier static analysis progressed to the more sophisticated and complex dynamic analyses with the advent of modern computers. A general equation of motion for a riser is derivcd from a basic bent riser element and it is illustratcd how the controversial tcrms, e.g., the effective tension and buoyant weight are derived. Various non-linearities in the problem are discussed. Expressions appropriate for large angle of deflcction and large deformation have been shown. Different analysis techniques employed by various investigators in a riser analysis have been discussed.

A comparison of six different analysis methods including time and frcquency domain analysis of a riser has been made by Egeland and Solli 32. They have provided results on the sensitivity of these different techniques. In a recent API report 33, numerical results from eleven diffcrcnt computer pi'ograms on dynamic riser analysis have been compared. The results are found to be in general agreement among the w~rious programs. These programs, however, have not been tested against field measurement. It is recommended that such test programs be undertaken in an ocean environment so that more confidence may be gained in these and other computer programs.

NOMENCLATURE

A = nD~/4 [B] = damping matrix Co=drag coefficient CL= lift coefficient

C.~ = inertia coefficient Do =diameter of riser

EA(z)=axial rigidity of riser (varies with Z)

El =flexural rigidity of riser F=external force on riser f= force density

E(z)l(z), El(z)=flexural rigidity of riser (varies with Z)

[tr}=nodal point force vcctor equivalent to the element stresses at time t

[K,.] =elastic stiffness matrix

At= Tlo p =

"r,.(z), T(z) [Table 1] =

[Ko] = geometric stiffness matrix (differential matrix) which takes into account the tension of the structure in the static cqnilibriunl configuration

[tK] = tangent stiffiless matrix at time t re=constant mass including riscr

and contcnts re(z) = total mass of riser and contents

(varies with Z) m*(z) = total mass of riser and contents

including virtual mass (varies with Z)

M = moment of force [M] =constant mass matrix [A/'~] = structural mass matrix [M~.] = virtual mass matrix

P=pressure on the riser from fluid ~tt +AtR} = external load vector applied at

time t + At time incremental time stcp top riser tension effective tension = Ttop- buoyant wcight of riser and contents (7" not defined)

T(z, t)=same as T(Z) but also varies with time, t

T~(z, t) = cffective tension = Top- buoyant weight of riser and contents+vertical wave inertia forces (varies with Z and t)

T*(z) = effective tension = 7~o o - buoyant wcight of riser-inertia loading of the drilling mud velocity (varies with Z)

T~ =constant riser tension- inertia loading of the drilling mud velocity

[u} = vector of nodal point displacement increments from time t to time t+At (total) displacement for linear analysis)

'd't} = vector of nodal point velocities It + At,.,} = vector of nodal point velocities

at time t + At [ii} = vcctor of nodal point

accelerations It +Ate} = vector of nodal point

accelerations at time t+At U =current velocity

U: = axial riser displacement V= shear force w=constant buoyant weight of

riser and contents or normal water particle velocity

w(z)=buoyant weight of riser and contents (varies with Z)

w(z, t)=buoyant weight of riser and contents (varies with Z and time)

X =global horizontal axis (used for 2-D analyses)

Y=global horizontal axis (used for


Review of riser analysis techniques: S. K. Chakrabttrti aml R. E. Frampton

Subscripts

3-D analyses with X axis) Z=globa l vcrtical axis with

assumed origin at the mud line and being posit ive upward

p = radius of curvature p . . . . mass density of water 70 = weight densi ty of outer fluid ;,'~ = weight density of inner fluid ;,'~ = weight density of riser mater ia l

0 = refers to outer region of riser i= refers to inner region of riser

w= refers to weight s = refers to segment of riser n = refers to normal d i rect ion

REFERENCES

I St. Denis, M. and Armijo, L. On the dynamic anal)sis of the Mohole riser, Proc. Ocean Sci. Ocean Eng. Cm~, Oslo, 1955

2 NESCO Report: Structural Dynamic Analysis oJ the Riser and Drill Striml for Project Mohole, National Engineering Science Company, Pasadena, December 1965

3 Graham, R. D., Frost, M. A. and Wilhoit, J. C., Anal)sis of the motion of deep-water drill strings - - Part 1: Forced lateral motion --and Part 2: Forced rolling motion, J. Eng. Ind. Trans. ASME, 1965, (May), p. 137

4 Fischer, W. and Ludwig, M. Design of floating vessel drilling riser, J. Petrol. Tedmol. 1966. (March), p. 272

5 Butler, II. L., Delfosse, C., Galef, A. and Thorn, B. J. Numerical analysis of a beam under tension, J. Struct. Die., Proc. ,4SCE, 1967, (October), p. 165

6 Gosse, C. G. and Barksdalc, G. L. The marine riser - - a procedure for analysis, Offshore Technol. Cm~, IIouston, 1969, Paper no. OTC 1080

7 Tidwcll, D. R. and Ilfrey, W. T. Developments in marine drilling riser technology, ASME Paper no. 69-PET-14, September 1969

8 Kopccky, J. A. Drilling riser stress measurements, ASME Paper no. 71-PET-I, 1971

9 Morgan, G. W. Riser Dynamic Analysis, Sun Oil Co., Production Research Laboratory Report 7320-71-14, 1972

10 Tucker, T. C. and Murtha, J. P. Nondeterministic analysis of a marine riser, Offshore Technol. Conf, tlouston, 1973, Paper no. OTC 1770

I I Burke, B G. An analysis of marine risers for deep water, oJJ~hore TechnoL Co~, ttouston, 1973, Paper no. OTC 1771

12 Bathe, K., Wilson, E. L. and Iding, R. !I. NONSAP -- a structural analysis program for static and dynamic response of nonlinear systems, Report no. UC SESM 74-3, University of California, Berkeley, February 1974

13 lleuze, L. A. A 4000 ft riser, Offshore Teclmol. Col~, ltouston, 1975, Paper No. OTC 2325

14 Gnone, E., Signorclli, P. and Giuliano, V. Three-dimensional static and dynamic analysis of deep-water seelines and risers, Offshore Technol. Cot~, Ilouston, 1975, Paper no. OTC 2326

15 Jones, M. R. Problems affecting the design of drilling risers, SPE paper 5268, London, April 1975

16 Sexton, R. M. and Agbezugc, L. K. Random wave and vessel motion effects on drilling riser dynamics, Offshore Teclmol. Col~, llo,~ston, 1976, Paper no. OTC 2650

17 Gardner, T. N. and Kotch, M. A. Dynamic anal)sis of risers and caissons by the element method, Offshore T~,chnol. Col~, llousto,, 1976, Paper no. OTC 2651

18 Bennett, B. E. and Mctcalf, M. F. Nonlinear dynamic anal~,sis of coupled axial and lateral motions of marine risers, Offihore Technol. Cot~, Ihmston, 1977, Paper no. OTC 2776

19 Maison, J. R. and Lea, J. F. Sensitivity analysis of parameters affecting riser performance, Offshore Technol. Cm~, Ilouston, 1977, Paper no. OTC 2918

20a Paulling, J. R. ,-I Lineari:ed Dynamic Analysis of the Coupled OTEC Cold-Water Pipe and IIAfB-I Barge System Morris Guralnick Associates, Inc., August 1977

20b |'aulling, J. R. Frequency domain analysis of OTEC CW pipe and platform dynamics, Proe. Ilth OJf~hore Teclmol. Cot~, Ihmston, 1979, Paper no. OTC 3543, III, 1641

21 Chou, D. Y., Minner, W. F., Ragusa, L. and Ho, R. T. Dynamic analysis of coupled OTEC platform - - cold water pipe system, Offshore Technol. Cot~, ilouston, 1978, Paper no. OTC 3338

22 Young, R. D., Fo~der, J. R., Fisher, E. A. and Luke. R. R. Dynamic analysis as an aid to the design of marine risers, J. Pressure Vessel Teclmol., Trans. ASME, 1978, (May], 200

23 Kirk, C. L, Etok, E. U. and Cooper, M. T. Dynamic and static analysis of a marine riser, Appl. Ocean Res., 1979, I, 125

24 Dareing, D. W. and lluang,T. Marine riser vibration response by modal analysis, J. Energy Resource Technol., AS,tIE, 1979, 101, 159

25 Krolikowski, L. P. and Gay, T. A. An improved linearization technique for frequency domain riser analysis, Proc. Twelfth Off'shore Technol. Cot~, Houston, 1980, Paper no. OTC 3777, II, 341

26 Etok, E. U. and Kirk, C. L. Random d)namic response of a tethered buoyant platform production riser, Appl. Oeean Res., 1981, 3, 73

27 Loken, A. B. et al. Aspects of hydrodynamic loading in design of production risers, Proe. Elercnth Offshore Technol. Co~, tlouston 1979, Paper no. OTC 3538, III, 1591

28 Sarpkaya, T. tlydrodynamic forces on various multiple tube riser configurations, Proc. Elerenth Offshore Tecl, nol. Co~, tlouslon, 1979, Paper no. OTC 3539, I!1, 1603

29 Hanscn, N. E., Jacobsen, V. and Lundgren, H. H)drodynamic forces on composite risers and individual cylinders, Proc. Eleventh Offshore Technol. Col~, llouston, 1979, Paper no. OTC 3541, I!!, 1607

30 Borgman, L. E. Spectral analysis ofocean x~a~e forces of piling, J. Waterways Ilarbors Dir., ASCE, 1967, 93, (WW2), 129

31 Tung, C. C. and Iluang, N. E. Combined effects of current and waves on fluid force, Ocean Eng., 1973, 2, 183

32 Egeland, O. and Solli, L. P. Some approaches to the comparison of riser analysis methods against full-scale data, Proc. Twelfth Offshore Technol. Cot~, tlouston, 1980, Paper no. OTC 3778, I!, 355

33 APt, Comparison of Marlin, Drilling Analyses, American Petroleum Institute, Bulletin 2J, 1977

APPENDIX I

Eqttilibritmt equations Jbr a bent tubular segment

Most mar ine structures being built today have tubular members as some or all of its structural components . These members are often used in the convent iona l sense to resist the appl ied forces and to t ransmit loads in the structure by their bending, shear, and axial stiffness. However , in many appl icat ions the buoyancy force created f rom a difference between internal and external pressure is used for overal l structure stabil ity or to reduce the loads in the individnal members. It is impor tant , therefore to understand the equi l ibr ium of a curved (bent) tubular segment as shown in Fig. 4.

For the purpose of this der ivat ion, it will be assumed that the segment is bent in one plane only (the X-Z plane) and that mot ion occurs only in the +X direct ion. It can be shown that the equat ions of mot ion may bc der ived independent ly for the two or thogona l vert ical p lanes and that coup l ing is int roduced by means of the external forces only.

The general form of the equat ions in the X and Z direct ions and [he moment cquat ion will be der ived first, then approx imat ions will be made in succession using the fo l lowing assumpt ions: (A) The length of segment is small so that

cosd0- ~ ! and s ind0"- -d0 (34)



/

vo~ "2"

.* F/S" * ~,~

FI~ ' W - - _ - rx$ FIs dO

aV 2

" "~.."-....] ae ~ \ e.d--- e

T_! . T ~' e .de r

~ X

Similarly, the horizontal equilibrium equation becomes:

A sin 0 + B cos 0 + F~ - m~.~ds = 0 (41)

in which m~, = mass of the segment including added mass per unit length acting in the X direction; '~ = acceleration of point s in the X dircction [.~ =(02x/i}t2)]. The moment equilibrium eqt, ation is:

dT( , - cos d~O-) + d M + 2 Vsin d-if- = 0 (42)

Considering the length of the segment to be small, equation (34), the three equilibrivm cquations respectively are :

Figure 4. Free body diagram of a bent tubuhu" segment A~ cos0-Bt s in0- f , .+f : ,=0 (43)

(B) Small deflection beam theory is applicable. Then

sin0 ~-dx cos0-~ ds and (35)

(C) Angle of deflection, 0, is small, i.c.

- dv ds-~dz, cos 0" 1, sin 0" 0"--'"

dz

dO d2x and dzz "" d---2 ~2- (36)

As shown in Fig. 4, the length ofthe segment is ds with a radius ofcurvature p and a mid-point slope of 0. At both ends of the segment, the internal member forces, i.e. the shear, moment, and axial forces are shown iV, AI, T respectively). The resultant of all the external loads are shown at the midpoint of the segment (point s) as F.,~ and F_.,. A portion of these resultant loads is from the distributed internal and external pressure forces. Also, acting at point s is the resultant member weight, F,.. The cquilibrium of the internal member forces and the resultant loads acting on the segment at point s provides the following cquation for the vertical equilibrium:

At sin0+B, cos0+f,s-m~,.~=0 (44)

dA! d--~-~ + V = 0 (45)

in which

dT vdO (46) AI = ~s- ds

dV ,,dO Bt = ds + 1 ~ss (47)

while f,.,f:s and f~s are the weight and force intensities (force per unit length).

Next, if the small deflection beam theory, equation (35), is applied, then the quantities sin0 and cos0 in the horizontal and vertical equilibrium cquations are replaced by dx/ds and d-/ds respectively. Finally, assuming small angle deflection, equation (36) and using the following relation between the moment and curwtture for pure bending,

M=E1 = EIdO d2x (48)

and neglecting products of differentials as second order, the following final forms of equilibrium equations are derived.

(37) Fw+F.s=0

which reduces to:

A cos0- B sin 0 - F,,.+ F:, =0 (38)

where,

A = dTcos d~- 2 Vsin d~-~ (39)

Vertical Jbrce equilibrium

dTd_ dzd (v~z)-f"+f:~=O

llori-ontal force equilibrium

d[ dx\ dV ..

Moment equilibrium

(49)

(50)

d [ . d2x\ I" ~ t " :1~) + =0 (511


Review of riser analysis techniques: S. K. Chakrabarti amt R. E. Frampton

47

m x

Figure 5. Extermd pressure distribution on a bent tubular segment

APPENDIX 11

Interred aml extermd fluid pressure on a bent tubular seyment - - statically equivalent loads

The statistically equivalent loads due to external pressure will be derived first, and then the results for internal pressure become obvious. Consider the bent tubular segmcnt shown in Fig. 4 to be exposed to external fluid pressure as shown in Fig. 5. The variation of pressure is linear with z. A small incremental area, d,,l o, upon which the external pressure, Po, may be assumed constant, will be used to derive an expression for the incremental force, dFo.

The exposed surface area, dA o, upon which external pressure acts, is:

Do[ Do dAo =-~-(p + -~-cos (p)d~(d(p (52)

in which Do=outside diameter of the tube; tp=angle around the cross-section of the tube measured counter- clockwise from the negative X-axis. The pressure, Pc, at the centre of the tube where the normal to the centre of the area, dA o, intersects the axis of the tube (point C) is:

P, = Po - 7oP(sin ~ - sin 0) (53)

in which /5o=external pressure at the elevation of the centre of the cross-section (point S); 7o=density of the external fluid. The pressure, P0, at the outer surh~ce of the incremental areas is:

/5 Do 1"o= ~-7o-rCOS~osin~ (54)

or, using equation (53)

Po = Po - 7op(sin = -s in 0)- 89 cos ~o sin (55)

The differential force, dFo=PodA0, on the area then becomes (using equations (521 and (55))

dFo = {~pDo[Po - 7op(sin a - sin 0)] +

]D~[Po - 7op(2 sin ~- sin 0)]

cos r sin o~ cos 2 ~p}d~dq~ (56)

The global components of this differential force are:

dF.,o=dFocos~p cos~ (57)

dF,.o=dFosinq~ (58)

dF.o= -" IF o cos ~p sin ct (59)

Integrating dF~o in the tp and ct directions gives the total force acting in the global X direction, Fxo, due to external pressure

O+ldO,'21 2n

~ =0- [d0 ,21 4 ,= 0

(60)

Using equations (56) and (57), and integrating with respect to

Review of riser analysis techniques: S. K. Chak,'aba,'ti and R. E. Frampton

Fri = 0 (65)

(66)

in which Ai=nD~/4=inner cross-sectional area of the tube; D~=inside diameter of the tnbe; P~=internal pressure at the elevation of the centre of the cross-section (point s); "/~ =density of the internal fluid.

The total global load components due to internal and external fluid pressure may be expressed by combining equations (61) and (64) for F,p, equations (62) and (65) for Frp, and equations (63) and (66) for F._ v. By considering the length of the segment to be small, i.e. dO small, the approximations ofcquation (34) may be made. With these approximations, the global components of the statically equivalent forces due to internal and external pressure may be expressed as the components of the force intensity, f,v,f~v, and f.-p, or

fxp Lv = ~- =f,p cos 0 (67)

F).p f r , - ~ = 0 (68)

fzp f--v = ds - -f.v sin 0 (69)

respect to z are equal to (the negative of) the corresponding fluid densities, the expression for the horizontal component of the pressure may be written as

d ~ ~ dx f~p=dmz[(Ao ! , , -A i l i)~:~ (73)

Next, to examine the moment due to extcrna[ pressure, the global components, Xo, Yo, Zo, of the distance from point s to the centre of the differential area, dAo, are nccdcd. F'igure 5 shows a new coordinate system (x~, A, z~) with origin at point s and parallel to the original coordinate system (X, K Z). With reference to tiffs new coordinate system, the global components (Xo, Yo, Zo) may bc written as:

Do Xo =/,(cos 0 - cos ~) - T cos ~ cos q~ (74)

O 0 . Yo = - ~- sm ~p (75)

Do z o = p(sin ~ -- sin 0) + ~ sin ~ cos (76)

Now, using the global components (dF ..... dFro, dF:,,) of the differential force acting on the area, dA o, the moment may be expressed as follows:

Differential moment about the x, axis

in which p was replaced by ds/dO and dM.,~o = J'odF:o- zodFro (77)

- - dO f.v = (A o!'o - AiPi) ~ -(AoTo - AiTi) sin 0 (70)

DiJferential moment about the y~ axis

dM ~.~,, = - xodF:o + zodF.~o (78)

Examining Fig. 5 and equation (70) it is observed that the hydrostatic load contribution resulting from the positive curvature, dO]ds, opposes the hydrostatic load contribution resulting from the positive mid-point slope, 0, of the segment for both the internal and external fluid. If the curvature of the segment is negative (bent in the opposite direction as that shown in Fig. 5) while still maintaining a positive mid-point slope, the hydrostatic load contribution from either internal or external fluid pressure are additive. In all cases the hydrostatic load due to the internal fluid opposes the hydrostatic load from the external fluid.

If small dellection beam theory and small angle of deflection are assumed and if the terms with the products of differentials are neglected, the statically equivalent pressure loads become:

Differential moment about the z~ axis

dM:~o = - xodFro + 3'odF~o (79)

The expressions for the differential moments due to internal pressure are similar to equations (77)-(79) except for a change in sign. The total moment about each of the coordinate axcs (x~, 3'~, z~) at point s is then found by integrating the above expressions in the cp and directions. The final expressions combining the moments due to external and internal pressures are

M~ v =0 (80)

Mysp = (/1o7o- Ai7i)t'2( sin d0- dO) cos 0 (81)

d2x dx f~p=(AoPo-- AiPi) ~z_ --(Ao;.o-- AiTi)-~z (71)

f_.p = 0 (72)

Noting that tile pressures Po and Pi are linear functions of z and therefore the derivatives of these pressures with

Al :~p=0 (82)

By considering the length of the segment to be small, i.e. dO small, the approximations of equation (34) may be used. Then,

M~.,p=0 (83)

Applied Ocean Research, 1982, l~l. 4, No. 2 89


/ Figure 6.

,x4 .-:/-. _~.t( II

( 4 '1co~,Co~r ~. "I" o

x

Weight of a bent tubular segment

APPENDIX III

Weight of a bent tubular segment - - statically equivalent loads

For the statically equivalent weight of the bent tubular segment, consider the tube as shown in Fig. 6. The differential volume, dV, is again bounded by the angles d~ and dtp on the wall of the tubular segment. However, because the tube is bent and the elements on the convex side suffer extension and those on the concave side compression, the wall thickness of each differential volume is different (increasing in the direction of the positive local X axis). As in the case of pure bending, each transverse section of the tube, originally plane, is assumed to remain plane and normal to the longitudinal fibres of the tube after bending. Then, the elements lying on the neutral axis (local ~'axis) do not undergo straining during bending. Thus, the volume computed for a differential element lying on the neutral axis (local Yaxis) will be the same for all of the elements around the cross section.

The differential volume, d V, as shown in Fig. 6 is then

1 2 d V= ~(Do - D2~)pd~td9 (84)

The incremental weight, dFw, is:

in which 7~ = weight density of the tubular wall. The total statically equivalent weight, Fw, is then found

by integrating dFw in the tp and ct directions, which gives

f,.= ~s =7,(Ao- A,) (86)

Next, to determine the statically equivalent moment due to weight of the tube, the global components, x,,, and Yw, of the distance from point s to the centre of the differential volume, d V, are needed. Again referencing the (x,, 3'5, z,) coordinate system with origin at point s, the global components, xw and Yw, may be written as (see Fig. 6):

, {Do+D, 'X xw=p(cos 0 -cos cq -k~)cos 0c cos tp (87)

['Do + Di'~ 9 y, . . . . k~ js 'n ~o (88)

The moment of the weight, dF,,,, of the differential volume may be expressed as -y ,dFw about the x~ axis, as xwdF w about the y~ axis and as

dM .... =M .... =0 (89)

about the z~ axis. The total moment about the x~ axis is:

O+dO]2 2n

:~=O-IdO/2) 4~ = 0

which on integration with respect to tp results in

Mx~,.=0 (91)

Similarly, on integration the total moment about the ),~ axis is:

Mr~.,='A(Ao-A,)p2(dO-2sind~O2)cosO (92)

By considering the length of the segment to be small, i.e. dO small, the approximations of equation (34) give:

dFw=?~dV (85) Mr~w = 0 (93)


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