No. 246 November 1982 BUCKLING OF RISERS IN TENSION DUE TO INTERNAL PRESSURE by Michael M. Beinitsas Theodore 5ikinis Prepared under contract for the American Bureau of Shipping ^S\tV O Department of Naval Architecture ^~~t-ttJt^*,^ +and Marine Enqineerinq KI^^^ ^ > College of Enqineerinq The University of Michiqan 4 4~* ~ Ann Arbor, Michiqan 48109 II11"'

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ABSTRACT The global Euler buckling loads of marine risers and columns have been presented in an earlier report [5]. In this work we use these results to prove that risers which are in tension along their entire length may globally buckle as Euler columns due to internal static fluid pressure. Numerical examples are used to 1. demonstrate that this phenomenon may occur during normal operating conditions, 2. show the proper distribution of riser supporting forces between top tension and buoyancy, 3. illustrate the effect of boundary conditions on the occurrence of this phenomenon, and 4. study the effect of the riser geometric particulars, internal fluid static pressure and total buoyancy on the riser critical length at which buckling in tension may occur due to internal pressure. -iii

ACKNOWLEDGEMENTS This report is part of the outcome of the research on global elastic stability of risers carried out by the authors for the American Bureau of Shipping at the Department of Naval Architecture and Marine Engineerinq of the University of Michigan in Ann Arbor. The support of the Ocean Fnqineerinq Division of the American Bureau of Shippinq is gratefully acknowledged. -v

CONTENTS paqe ABSTRACT....... ~~~ ~*~...~~~.~~~~~~~~~..~~.~~~~~~~~~~~~~~~~~~~~~~ iii ACKNOWLEDGEMENTS.....~.....~~.....................*.*~.. ~ ~~.**~~~ V LIST OF FIGURES. ~.~~~ ~~~~* ~~~~~... *. ~~.~~ ~*.. *....~..~.~~~~ ix NOMENCLATURE..........................~..........................~. xiii INTRODUCTION AND OUTLINE............ 1 I. PROBLEM FORMULATION AND SOLUTION........................... 3 I. 1. Eigenvalue Problem................................... 3 I.2. Solution............................................. 6 II. BUCKLING OF RISERS IN TENSION DUE TO INTERNAL PRESSURE....... 8 II.1. Buckling in Tension.................................. 8 11.2. Variation of TTR with the Riser Lenth............... 12 II.3. Distribution of Supporting Forces between TTR and Bm......................................... 14 III. CRITICAL LENGTH FOR BUCKLING IN TENSION.................. 19 III.1. Definition of Critical Length..................... 19 111.2. Critical Length for Hinqed-Hinqed Risers........... 20 III.3. Critical Length for Clamped-Hinged Ris ers............ 26 III.4. Critical Length for Hinqed-Clamped Risers.......... 26 III.5. Critical Length for Clamped-Clamped Risers......... 26 II.6. Critical Length for Hinged-Movably Hinqed Risers..... 37 111.7. Critical Length for Clamped-Movably Hinged Risers.... 37 III.8. Critical Length for Hinged-Movably Clamped Risers... 37 III.9. Critical Length for Clamped-Movably Clamped Risers... 37 IV. COMPARISON OF CRITICAL LENGTH FOR RISERS WITH NONMOVABLE BOUNDARIES.................................................. 52 V. COMPARISON OF CRITICAL LENGTH FOR RISERS WITH MOVABLE BOUNDARIES......................................-..........i 56 SUMMARY......................~00~~~~0............0............ 60 REFERENCES.................................................... 61 -vii

LIST OF FIGURES Figure page 1. Stability Boundary and Reqion of Buckling in Tension for a Hinged-Hinqed Riser........................................ 10 2. Stability Boundary and Reqion of Buckling in Tension for a Hinged-Movably Hinged Riser................................ 13 3. Top Tension versus Length for a Hinged-Hinged Riser.......... 15 4. Top Tension versus Length for a Hinged-Movably Hinged Riser.. 16 5. Distribution of Supporting Forces for a Hinqed-Hinqed Riser.. 17 6. Distribution of Supporting Forces for a Hinged-Movably Hinged Riser...................................................... 18 7. * versus n for Risers with Nonmovable Boundaries........ 21 8. L* versus Pm for Hinged-Hinqed Risers for Do = 0.25m and Di = 0.23m................................................. 22 9. L* versus Pm for Hinged-Hinqed Risers for Do = 0.55m and Di = 0.515m................................................ 23 10. L* versus Pm for Hinged-Hinqed Risers for Do = 1.00m and Di = 0. 95m................................................. 24 11. L* versus Pm for Clamped-Hinged Risers for Do = 0.25m and Di = 0.23m................................................. 28 12. L* versus Pm for Clamped-Hinged Risers for Do = 0.55m and Di = ~515m 29 Di = 0.515m..-............................ 29 13. L* versus Pm for Clamped-Hinged Risers for Do = 1.00m and Di = 0.95m........................................... 30 14. L* versus Pm for Hinged-Clamped Risers for Do = 0.25m and Di = 0.23m................................................. 31 15. L* versus Pm for Hinged-Clamped Risers for Do = 0.55m and Di = 0.515m................................................ 32 16. L* versus Pm for Hinged-Clamped Risers for Do = 1.00m and Di = 0.95m................................................. 33 17. L* versus Pm for Clamped-Clamped Risers for Do = 0.25m and Di = 0.23m................................................. 34 18. L* versus Pm for Clamped-Clamped Risers for Do = 0.55m and Di = 0.515m................................................ 35 -ix

Figure paqe 19. L* versus Pm for Clamped-Clamped Risers for Do = 1.00m and Di = 0.95m............................................... 36 20. B* versus n for Risers with Movable Boundaries......... 39 21. L* versus Pm for Hinged-Movably Hinged Risers for Do = 0.25m and Di = 0.23m............................................. 40 22. L* versus Pm for Hinged-Movably Hinged Risers for Do = 0.55m and Di = 0.515m............................................. 41 23. L* versus Pm for Hinged-Movably Hinged Risers for Do = 1.00m and Di =.95m............................................. 42 24. L* versus Pm for Clamped-Movably Hinged Risers for Do = 0.25m and Di = 0.23m............................................. 43 25. L* versus Pm for Clamped-Movably Hinged Risers for Do = 0.55m and Di = 0.515m............................................ 44 26. L* versus Pm for Clamped-Movably Hinged Risers for Do = 1.00m and Di = 0.95m............................................. 45 27. L* versus Pm for Hinged-Movably Clamped Risers for Do = 0.25m and Di = 0.23m............................................. 46 28. L* versus Pm for Hinged-Movably Clamped Risers for Do = 0.55m and Di = 0.515m............................................ 47 29. L* versus Pm for Hinged-Movably Clamped Risers for Do = 1.00m and Di = 0.95m.......................................... 48 30. L* versus Pm for Hinged-Movably Clamped Risers for Do = 0.25m and Di = 0.23m............................................. 49 31. L* versus Pm for Hinged-Movably Clamped Risers for Do = 0.55m and Di = 0.515m............................................ 50 32. L* versus Pm for Hinged-Movably Clamped Risers for Do = 1.00m and Di = 0.95m............................................. 51 33. L* versus Pm for Risers with Nonmovable Boundaries for Do = 0.25m, Di = 0.23m and bm/bw = 0.2.................... 53 34. L* versus Pm for Risers with Nonmovable Boundaries for Do = 0.55m, Di = 0.515m and bm/bw = 0.2................... 54 35. L* versus Pm for Risers with Nonmovable Boundaries for Do = 1.00m, Di = 0.95m and bm/bw = 0.2..................... 55 -x -

Figure paqe 36. L* versus Pm for risers with Movable Boundaries for Do = 0.25m, Di = 0.23m and bm/bw = 0.2..................... 57 37. L* versus Pm for risers with Movable Boundaries for Do = 0.55m, Di = 0.515m and bm/bw = 0.2................... 58 38. L* versus Pm for risers with Movable Boundaries for Do = 1.00m, Di = 0.95m and bm/bw = 0.2................... 59 -xi

I

NOMENCLATURE Ai Airy Function of the first kind Bi Airy function of the second kind Bm Net buoyancy of buoyancy modules per unit length Bw Displaced weight of water by a unit length of riser CB, CT Linear restoring spring stiffness at bottom and top of riser respectively Di, Do Inner and outer riser diameters respectively E Young's Modulus hm Free surface ordinate of drilling mud hw Free surface ordinate of water I Moment of inertia of cross-sectional area L Length of riser L* Critical L for buckling in tension p Dimensionless vertical coordinate along the riser Pe Effective tension in the riser rg, rT Linear rotational spring stiffness at bottom and top of riser respectively T Actual tension in the riser TTR Tension at the top of the riser U Lateral displacement of riser We Effective weight of riser per unit length Wm Weight per unit length of fluid inside the riser Wst Weight of riser per unit length z Vertical coordinate along the riser -xii

Greek Letters ~B Dimensionless We 8* Critical B for bucklinq in tension 6 Dimensionless TTR 6crit Dimensionless critical TTR In Ratio of apparent to effective riser weiqht Pm Drilling mud mass density Pst Mass density of steel Pw Mass density of water rT Dimensionless effective overpull at the lower end of the riser Tcrit Dimensionless critical effective overpull at the lower end of the riser -xiii

INTRODUCTION AND OUTLINE The objective of this work is to prove that open ended tubular columns may buckle globally as Euler columns due to the action of internal fluid static pressure even while they are in tension along their entire length. Risers, columns, legs of Tension Leg Platforms and hydraulic columns are therefore prone to such instability. The eiqenvalue problem which yields the riser stability boundaries has been formulated and solved in previous work [5]. In this work the results derived in [5] are used to prove this phenomenon and illustrate it numerically for various boundary conditions, geometric configurations and loading conditions. Several aspects of this and related problems have been studied during the past forty years. In chronological order the following work has been done. Willers has calculated the asymptotic behavior of very long columns, with movably hinged lower end and with nonmovable ends, subject to their own weight and compressive end load [22]. Biezeno and Koch studied the stability of closed vertical tubes fully or partially submerged in water with or without internal fluid and concluded that buckling cannot occur [12]. This is expected since they considered closed tubes. Lubinski analyzed the bucklinq of drill strings inside marine drilling risers without end load or internal fluid pressure [17]. Duncan formulated the problem of instability of multiply and continuously loaded columns but scaled all loads in order to reduce the loading variables to one only [13]. Huang and Dareing calculated the natural frequencies and buckling loads of columns submerged in water for various boundary conditions [15,16]. Plunkett computed the asymptotic behavior of buckling loads of drill strings under their own weight and verified Willers' results using a different method [19]. Sugiyama and Ashida solved the problem of -1 -

-2 -buckling of columns under their own weight for various boundary conditions using a power series solution [21]. Sherman and Erzurumlu studied the effect of various loads on the static instability of circular tubular columns [20]. Bernitsas et al have computed the buckling loads, for the general eiqenvalue problem described above, for the entire range of practical interest and eight different sets of boundary conditions that are considered of practical importance [5]. The phenomenon which is studied in this paper is similar in many respects to that observed by Goodman and Breslin namely, that unsupported vertical cables which are heavier than water may be sustained by the external hydrostatic pressure and not collapse [14]. In this report the formulation and solution of the problem is briefly presented in Chapter I and reference to published results is made. In Chapter II it is shown theoretically that risers which are in tension along their entire length may buckle globally as Euler columns due to internal fluid static pressure. This phenomenon is illustrated with numerical examples which show the proper distribution of the riser supporting forces between top tension and buoyancy and that this phenomenon may occur during regular operating conditions. In Chapter III the critical length, for which a riser of given configuration and loading condition may buckle in tension, is defined, calculated and plotted versus the loading condition for the eight sets of boundary conditions which are considered of practical importance [5]. In Chapter IV the riser critical length is compared for the four sets of boundary condition of risers with nonmovable supports. Finally in Chapter V similar comparisons are carried out for risers with movable supports.

I. PROBLEM FORMULATION AND SOLUTION The eigenvalue problem described in the Introduction has been formulated and solved in reference [5]. In this chapter we briefly present the formulation and solution for the sake of completeness of this report. I. 1. Eigenvalue Problem In dimensionless form the differential equation for the Euler global buckling analysis for a uniform riser is [5]: d4U d2U dU - (Bp + T) -- - = 0, (I-1) dp4 dp2 dp where WeL3 = (1-2) EI is the dimensionless effective weight per unit length, Pe(0)L3 T =: (1-3) EI is the dimensionless effective tension at the lower end of the riser, z p =- (1-4) L is the dimensionless coordinate along the riser, We = Wst + Wm - Bw - Bm (I-5) Wst = Pstg - (D 2 - Di2), (-6) 4 Wm = Pmg - Di2 (-7) 4 B,= pwg- D2, (1-8) -3 -

-4 -dPe(z) = We (I-9) dz rDo2 1rDi Pe(Z) E T(z) + Pwq (hw - z) - Pmq (h - z) (I-10) 4 4 and I = (Do4-Di4) 64 The rest of the symbols in the above equations are defined in the Nomenclature. The general boundary conditions for a riser with rotational and linear springs at both ends are: at the lower end where p = 0 EI d3U(0) T(0) dU(0) ^ ~~ - - + CB U(0) = 0 (I-11) L~ dp3 L dp and EI d2U(0) rB dU(0) + _- = 0 (1-12) L2 dp2 L dp where CB and rB are the linear and rotational sprinq constants at the riser bottom respectively, and at the top of the riser where p = 1 EI d3U(1) T(1) dU(1). - - ~ CT U(1(1) =0 (-13) L3 dp3 L dp and EI d2U(1) rT dU(1)..-. = 0 (1-14) L2 dp2 L dp where CT, rT are the linear and rotational sprinq constants at the riser top respectively. Equations (I-1) and (I-11) to (1-14) constitute an eiqenvalue problem the eigenvalues of which define the riser stability boundary in the B-T plane.

-5 -In this work eight sets of boundary conditions which are considered of practical importance are studied. Movable and nonmovable top supports are considered. The former are more appropriate for modelling risers and the latter for modelling regular columns. All conditions can be derived from the general equations (I-11) to (I-14) by setting the linear and rotational spring stiffness equal to 0 or. For risers (or columns) with nonmovable lower and upper boundaries the conditions become U(0) = 0 (1-15) and TU(1) = 0 (1-16) in all cases and 1. for hinged-hinged risers d2U(O) d2U(1) = 0 (1-17) and = 0 (1-18) dp2 dp2 2. for clamped-hinged risers dU(0) d2U(1) = 0 (I-19) and = 0 (1-20) dp dp2 3. for hinged-clamped risers d2U(O) dU(1) - = 0 (1-21) and = 0 (1-22) dp2 dp 4. for clamped-clamped risers dU(0) dU(1 ) = 0 (1-23) and = 0 (1-24) dp dp For risers (or columns) with nonmovable lower boundary and movable upper boundary we have in all cases U(0) = 0 (1-25)

-6 -and 5. for hinged-movably hinged risers d2U(O) d2U(1) = 0 (1-26) 2 = 0 (1-27) dp2 dp d3U(1) dU(1) and - ( + T) = 0 (1-28) dp3 dp 6. for clamped-movably hinged risers dU(0) d2U(1) = 0 (1-29) = 0 (1-30) dp dp2 d3U(1) dU(1) and - (B + T) - = 0 (1-31) dp3 dp 7. for hinged-movably clamped risers d2U(O) dU(1) d ~ = 0 (1-32) = 0 (1-33) dp2 dp d3U(1) and = 0 (1-34) dp3 8. and for clamped-movably clamped risers d2U(O) dU(1) = 0 (I-35) = 0 (I-36) dp2 dp d3U(1) and = 0 (1-37) dp3 I.2. Solution Two methods of numerical solution have been developed for these eiqenvalue problems [5]. The first method is based on a closed form solution in terms of Airy functions of the first and second kind [4] which is given by equation (1-38)

-7 -U(p) = -1/3 {c;U1(x) + c'U2(x) + c'U3(x) + c'} (1-38) where c; c r c; and c' are constants of integration x x Ul(x) = f Ai(i) di (I-39), U2(x) = f Bi(i) di, (1-40) x C x C U3(x) = - Ai()) f Bi(n) dndi + f Bi()) f Ai(n) dndC, (1-41) and x = B1/3p + B-2/3T (1-42) The second method is based on a power series solution of the form [4] U(p) = I anPn1 (1-43) n=1 Numerically neither method is stable over the entire domain of practical interest. The combined results of the two methods yield the eiqenvalues in the O-T plane up to the range of high B values where the asymptotic behavior of the stability boundaries can be derived analytically [9]. The results of the numerical implementation of these two methods are presented in report [5] in the form of stability boundaries in the B-T plane for the first six buckling modes for each one of the eight sets of boundary conditions. Both linear and logarithmic scale graphs are used since the former better depict the low B riser behavior and the latter better do so for the high 6 behavior. Some of these results are used in this work to show that risers which are in tension along their entire length may buckle globally as Euler columns due to internal pressure.

II. BUCKLING OF RISERS IN TENSION DUE TO INTERNAL PRESSURE The results derived in the previous chapter and published in references [5] and [9] are used in this chapter to prove that internal pressure may destabilize even short risers in tension. II.1. Buckling in Tension Buckling of a riser may occur if T ( Tcrit (II-1) or equivalently if Pe(0) Pe (0) (11-2) crit Using the definition of Pe(O), given by equation (I-10), we can write (II-2) as T(0) + (Bw - Wm)L 4 Pe (0). (11-3) crit For T(0) > 0, and a partially buoyed riser, which is usually the case in practice, T(z) is greater than zero for all values of z. Consequently, a riser may buckle even if the actual tension is positive along its entire length, if the following inequality holds (Bw - Wm)L < T(0) + (Bw - Wm)L Pe (0) (11-4) crit This inequality can be satisfied even for moderate values of Wm, the weight of the drilling fluid per unit length of riser. This observation is very important for the computation of TTR. Equations (1-9) and (I-10) can be used to compute TTR for risers with properties independent of z * Since TTR = T(L), (11-5) T(z) = T(0) + (Wst - Bm)z, (II-6) -8 -

-9 -and by definition Pe(0) = T(0) + (Bw - Wm)L (11-7) we have TTR = Pe(0) + WeL. (11-8) In dimensionless form equation (II-8) becomes 6 = t + (11-9) TTR L2 where 6 = -- (II-10) EI More general expressions for TTR, for risers with properties varying with z, are derived in reference [7]. According to equations (II-5) and (11-6) to achieve positive T(0) we must satisfy inequality (II-11) TTR ) (Wst - Bm)L. (II-11) In dimensionless form (II-11) becomes 6 > no (11-12) where Wst - Bm n =, (11-13) We n is the ratio of the apparent riser weiqht in water to the effective riser weight. Consequently buckling in tension will occur if no < 6 < Tcrit + B = 6crit * (11-14) This region is marked by R in Figure 1 and is confined between line [DCR] which represents the dimensionless critical top tension, 6crit, and line [1] which represents the first inequality in (11II-14).

2500. - HHINGED-HINGED RISER _I CDCRJ CRITICAL DELTA CDC 2000. L CTCRJ CRITICAL TAU CD C13 DELTA FOR RISER IN TENSION R REGION OF BUCKLING IN TENSION 1500. F1000. 500. _BETAw BETA 0, 1 nn~n - _ 1500. 2000 2500. -5 0. Figure I. Stability Boundary and Region of Bucklinq in 18nsion for a Hinged-Hinqed Riser

-11 -We define 3* (BETA*) as the abscissa of the intersection of lines [DCR] and [1] in Figure 1. This point indicates the beginning of the dimensionless region of buckling of risers in tension. Curve [DCR] is convex, monotonically increasing and starts below point (B=0, 6=0). Curve [1] is a straight line through the origin with a positive slope for a partially buoyed riser. This is true for all boundary conditions considered in this paper. Consequently the two curves will either intersect at one point or not intersect at all. The necessary and sufficient condition for their intersection is that n < Tcrit + = 6crit, (11-15) or B(1-n) > - Tcrit (11-16) Using regression analysis we can show that the variation of crit with 8, for high B values, is approximately geometric and has the form Tcrit = -c2B2/3. (11-17) where c is a real constant depending on the boundary conditions. Actually for B + ~ the asymptotic behavior of the stability boundaries is geometric with exponent 2/3 [9]. Therefore 6crit = B - c2B2/3 (II-18) Combining equations (11-15) and (11-17) or (II-16) and (11-18) we get for high B values B(1-n) > c2B2/3 (11-19) or 1/3(1-n) > c2. (11-20) Inequality (11-20) can be satisfied as lonq as n < 1. (11-21) that is when

-12 -Wst - Bm < 1 (11-22) Wst + Wm - Bw - Bm or equivalently when Wm > Bw (II-23) This indicates that if the destabilizinq effect from the drilling fluid is more significant than the stabilizing effect of the external hydrostatic pressure there will be a value of 3 beyond which the riser will buckle in tension. The value of B* is a function of n and the boundary conditions. Fiqure 7 shows the value of 3* for the first buckling mode of risers with nonmovable boundaries. It should be noticed that theoretically the value of wm may become high enough to cause buckling of risers in tension in higher modes. 11.2. Variation of TTR with the Riser Length The phenomenon described in the previous section may occur for realistic operating conditions and even for relatively short risers. The following examples demonstrate the case. For a riser with Do = 0.506 m (11-24) Di = 0.476 m (11-25) Pm = 1.5 Pw (II-26), Bm =.3 Bw (11-27) and E = 30.106 psi = 2.07.1011 Nt (11-28) -7 we can transform the stability boundary from the dimensionless form in Fiqures 1 and 2 to the dimensional one in Figures 3 and 4 respectively. Using equations (1-10) and (1-2). Similarly we can transform the first inequality of (11-14), that is lines [1] in Figures 1 and 2 to lines [1] in Figures 3 and 4 respectively.

2500. H- HINGED-MOVABLY HINGED RISER _J CDCR) CRITICAL DELTA cD 2000. Li CTCR) CRITICAL TAU n ( C1) DELTA FOR RISER IN TENSION R REGION FOR BUCKLING IN TENSION 1500..- R 1000. 500. BE 2 Stability Boundary and Reion of Buckling in nsion for a ingedMovably ined Riser BETA U~ 5 ~. _ TT.,.,__, UO. (TCR I -500. - Fiqure 2. Stability Boundary and Reion of Bucklin in Tension for a Hined-Movably Hined Riser

-14 -Fiqure 3 shows, for this particular example, the region of buckling in tension for hinged-hinged risers. The intersection of lines [1] and [TTCR] occurs for relatively short risers. That is for L > 265 m (11-29) a hinged-hinged riser with the particulars shown in Figure 3 may buckle in tension. Figure 4 is similar to Figure 3 for hinqed-movably hinged risers. In this case buckling of risers in tension due to internal pressure may occur for L > 160m (11-30) 11.3. Distribution of Supporting Forces between TTR and Bm For a 500 m riser with the same geometric particulars as the one in the previous example we can transform the dimensionless stability boundaries in Figures 1 and 2 to the dimensional ones in Figures 5 and 6 respectively. Each riser is supported by top tension TTR and buoyancy modules measured by the ratio Bm/Bw. Figure 5 shows the stability boundary and the buckling in tension region for hinged-hinged risers in the TTR, Bm/Bw plane. Further Figure 5 shows the proper distribution of the riser supporting forces between top tension and buoyancy in order to prevent buckling. In a similar manner we can transform Figure 2 to Figure 6 for a 500 m hinged-movably hinged riser with the same geometric particulars.

12000000 HINGED-HINGED RISER z CTTCR) CRITICAL TTR 1000o. Cl ci3 TTR FOR RISER IN TENSION CTTCR u~ R REGION OF BUCKLING IN TENSION z 800000. _ DoI= O5O6 M D:=0.,476 M m: P=:1.5 Pw w B= 0.3 B /=, 3 R I" 600000.- / / 400000. im200000. iL* Lp RISER LENGTH CM) 200000. L\ 100. 200. 300. 400. 500. 600. Figure 3. Top Tension versus Lenqth for a Hinqed-Hinqed Riser

1200000. HINGED-MOVABLY HINGED RISER Z CTTCRJ CRITICAL TTR CTTCR] 1000000.o C 1 ] TTR FOR RISER IN TENSION O1 R REGION OF BUCKLING IN TENSION z H800000. Do =. 506 M D, =0. 476 M,1: PM =1.5 P / w BB=0.3 B, / 1" 600000.- 400000. nnn200000. -_ _LY L, RISER LENGTH CM] 100. 200. 300. 400. 500. 600. Fiqure 4. Top Tension versus Lenqth for a Hinqed-Movably Hinqed Riser

1400000. H1 HINGED-HINGED RISER 1200000. z CTTCR) CRITICAL TTR Z [13 TTR FOR RISER IN TENSION 1000000. K R REGION OF BUCKLING IN TENSION 1000000.-^ ^ 1 JC TTCR) 800000. t0 L =500.0 M C 1) DO=0.506 M D=0.476 M v) Pm=1.5 Pv 600000.- \ \ \ R m 400000. zH 200000. — \ \ B, /Bwy MODULE BUOYANCY R TUBE BU CY I 0.0 0.2 0.4 0.6 0.8 1.0 1.2 wFiure 5. Distribution of Supporting Forces for a Hinged-Hined Riser^ Figure 5. Distribution of Supporting Forces for a Hinqed-Hinqed Riser ^ ^ C;'"',..*',

1400000.1 HINGED-MOVABLY HINGED RISER 1200000. t CTTCR3 CRITICAL TTR [ C1) TTR FOR RISER IN TENSION 0^ R REGION OF BUCKLING IN TENSION 1000000.t V TTCR) 800000.- X L =500. 0 M c 1 ) o =0. 506 M D = \ Di 0. -0476 M ( LO^ Pm =1.5 Pv 600000 400000. \' 200000. O.___ BM /Bw, MODULE BUOYANCY R TUBE BUO CY 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Figure 6. Distribution of Supporting Forces for a Hinqed-Movably Hinqed Riser

IHE UNIVERS!TY OF1c ENGINEERIiNG lI&?Ai III. CRITICAL LENGTH FOR BUCKLING IN TENSION An important dimensional quantity that can shed light into this counterintuitive phenomenon is the critical length for which a riser, which is in tensiop along its entire length and has given geometric and loading particulars, may buckle globally as Euler column. III.1. Definition of Critical Length Using the definition of 6* from Chapter II and equation (1-2) we can define the critical length as 3B*EI L* =s ~ (III-1) s We Using equation (11-13) we can rewrite (III-1) as 3B*EI - L* =... (III-2) Wst-Bm B* is a dimensionless quantity and, as shown in Chapter II, depends only on the boundary condition and n. Figure 7 depicts B* versus n for the four boundary conditions for risers with nonmovable boundaries studied in this work. Similarly Figure 20 shows the dependence of B* on n and the boundary conditions for risers with movable top support. A better physical understanding of this phenomenon can be achieved by studying Figures 8 to 19 and 21 to 32 which show the critical length L* versus dimensional quantities for a systematic variation of the riser qeometric and loading particulars. The systematic variation includes the following variations of variables and parameters. -19 -

-20 -1. Three different riser cross sections Do = 0.25 m (III-3), Di = 0.23 m (III-4) 0 1 1 Do = 0.55 m (III-5), Di = 0.515 m (III-6) o2 2 Do = 1.05 m (III-7), Di = 0.95 m (III-8) 3 3 2. Mud density varying in the ranqe 1,200 kg < Pm < 2,250 kg (III-9) 3. Amount of buoyancy measured by the ratio Bm/Bw which takes four values 0.00, 0.20, 0.40, 0.60. 4. Finally in these fiqures the dependence of L* on n is also shown. However, it should be emphasized that n and Bm/Bw are not independent variables and for all the numerical cases in the figures the value of the latter defines the former through equation (III-1 0) Wst-Bm Wst/Bw-Bm/Bw n. = -...= - (III-10) Wst+wm-Bw-Bm (Wst+Wm-Bw) /B-Bm/Bw 111.2. Critical Length for Hinged-Hinged Risers For the three risers defined by equations (III-3) to (III-8) the value of the critical length is plotted in Fiqures 8, 9 and 10 versus the drillinq mud density and for four different values of Bm/BW. We can draw the following conclusions from these figures 1. The critical length decreases with increasing mud density. This was expected since it is the mud-static pressure inside the riser that reduces the effective riser tension and causes buckling.

2500. 2000. < COLUMNS WITH =L NONMOVABLE n BOUNDAR IES 1500. 1000.- / 500. CH HH 0. 0.0 0.2 0.4 0.6 0.8 1.0 Figure 7. 3* versus n for Risers with Nonmovable Boundaries

10001 M DTA - s0.2 0.54 0.23 -sonn^ /0,:^9 80.83 -i o 79 0.~ ~ ~~. r00. /a 2 i~ 0.4 qOO.~~~~~~~~~0 20001 0i/M3 MUD DENST 60 0.~~~~~~~~20 180 0-7 2000 0 24a ~ 0 lzO~ 16001 luu i200~ ^, n -0.25 and Oi - 0 pisers for Do o f Hinred-Hine, eversus Pm fo -Fiqure ~

1000, ETA HH 0.54 800. - 0' 8/B Do0. 55 M 0.79 0 0.83 0.2 DE 0.52 M 0.2 2 4O0. -0 200.MUD DENS I TY C KG/M] 1000. 1200. 1400. 1600. 1800. 2000. 2200. 2400. 2600. Figure 9. L* versus pm for Hinged-Hinged Risers for Do = 0.55m and Di = 0.515m

1 000. ETA HH 0.54 0.70 D1. 800. 0. 79 B/B 0\\83 0/ 1 D0 0.95 M 0.82 O\. 86 4 D. 4 600o 400. 200. MUD DENSITY CKG/M'3 0. i i 1000. 1200. 1400. 1600. 1800. 2000. 2200. 2400. 2600. Figure 10. L* versus pm for Hinqed-Hinqed Risers for Do = 1.00m and Di = 0.95m

-25 -2. As the mud density decreases tendinq to the water density the critical length tends to infinity since the internal pressure destabilizing effect is counterbalanced by the external hydrostatic pressure stabilizing effect. 3. The critical length L* increases with n, that is the ratio of the apparent weight of the riser to the effective riser weight. It should be noted that the apparent weight is that of the riser in vacuum minus the net buoyancy of the modules only as indicated by equation (III-10). 4. L* decreases with increasing buoyancy support. At first this appears to be counter-intuitive. However, it should be noticed that for a given riser as the buoyancy provided by the modules increases n decreases. 5. For a given value of Bm/BW, L* decreases with increasing mud density. 6. L* decreases faster as the mud density increases for a given Bm/Bw than for a constant value of n ~ This can be explained by the same argument used in item 4 above. 7. The critical length L* increases with increasing riser size for the three risers considered in Figures 8, 9 and 10. However, this conclusion cannot be generalized since no specific criterion, other than experience, was used to compute the internal riser diameter, Di, for a given Do.

-26 -III.3. Critical Lenqth for Clamped-Hinqed Risers Figures 11, 12 and 13 present results similar to those in 8, 9 and 10 respectively for clamped-hinged risers. The same conclusions as those in Section 111.2. can be drawn for this set of boundary conditions, III.4. Critical Lenqth for Hinged-Clamped Risers Figures 14, 15 and 16 present the results of this analysis for hinqedclamped risers. Obviously the numerical answers are different hut the general conclusions are the same. 111.5. Critical Length for Clamped-Clamped Risers Figures 17, 18 and 19 show the results of this systematic analysis for clamped-hinged risers and show similar trends as all the other nonmovable boundary cases. Finally by comparing all Figures 8 to 19 we can derive the following conclusions 1. For given mud density, Pm, n or Bm/BW, and riser size, the critical length L* changes with the riser boundary conditions and increases with increasing restriction imposed by the boundary conditions that is, in the following order (a) hinqed-hinged riser, (b) hinged-clamped riser, (c) clamped-hinqed riser and (d) clamped-clamped riser (see also Chapter IV). 2. L* is of the order of a few hundred meters only, even for relatively low values of the drilling mud density.

-27 -3. Figures 8 to 19 indicate that even for hiqh values of n, buckling in tension may occur for relatively low values of mud density and L*. This indicates that the exact 6crit values must be used to compute the minimum values of the tension required at the top of the riser.

1000~T002 0.74 0.25 GOO O 0.2 400. 06 200o U E N 20^-1 20l 00 6N00 2^ ^'00.~ ~~~~220 1 2 0 0 - ^ o o - c ^ ^ or 'Do tor jaP-a-incxed Risers Ve vrsus PM Fiure *

1000. ETA CH 0.22 800. 100 1800. 2 Fiure 12. L* versus Pm for Clamped-Hined Risers for D =.55m and D = O.515m 000. ~~~~~~~~40 ~~0. 200. MUD DENSITY CKG/M3J 0. 1000. 1200. 1400. 1600. 1800. 2000. 2200. 2400. 2600. Figure 12. L* versus Pm for Clamped-Hinged Risers for Do = 0.55m and Di = 0.515m

1000. ETA CH 0.22 0. 48 800. -' 0.B/B 1 G M 0.69 0. 0. o. 2 0.74 600 -400.- 200. MUD DENS ITY C K/MG 3 0. 1000. 1200. 1400. 1600. 1800. 2000. 2200. 2400. 2600. Figure 13. L* versus Pm for Clamped-Hinged Risers for Do = 1.00m and Di = 0.95m

1000. ETA HC 0.51 0.70 D= 0. 2 800. - 0780 25 M 0.83 D=0.23 M 0.86 600. "\// ^B/B 600, 0K 0It~~~~~~, 0. 0 0. 4 0 6 400. 200. MUD DENSITY CKG/MJ 1200. 1400. 1600. 1800. 2000. 2200. 2400. 2600. Figure 14. L* versus Pm for Hinqed-Clamped Risers for Do 0.25m and Di = o.23m

1000. HC ETA 0ET PM o.5D. 0.70 0.52 m ~~~~~~~0.78 00 D= 800oo.- 0.83 0. 0. 86 ^ 45a - 400.- 4000~~~~~~~~~~~20 20O., MUD DENSITY 1800- 2000- 2200 -1000. 1200., 140006001. ifHinqe-Clamped Risers for Do = 0.55m and Di Ficure 15 LT* vets'us PM o

1000. ETA HC 0. 5 0. 78 800.- t /^0. B/B, DF0.OOM 0. 86 ^\\\ / ^^ ^^^ Dr0.-o. 95 M 0. 4 400. 400, 200. MUD DENSITY CK6/MJ 0.~\~~~~ 1000. 1200. 1400. 1600. 1800. 2000. 2200. 2400. 2600. Figure 16. L* versus pm for Hinged-Clamped Risers for Do = 1.00m and Di = o.95m

10008,;" l o o o - T ~ ~ r ^ ^ ^ ^ ^ ^ II E TA D 0O- ^0.14 D 0.23 M 0.47 soon ~0.61 800.^ ^^69 0.74~~~~~ 4000~ ~ ~~~~. 2000 mu^ Di ~ TYno zou 400.~~~~~~~~~20 l200~ 2000 0.~~~~,^0^ ^ 0 0.~160 000 1400~0 00G260V ^ clamp qed fsr or 'Do for Clamped-ClamPe-a qure 17* versus Pm o

1000. ETA CC 0800. \~ 47 B/BV D- 0.55 M 0.~2~ 400. 200. 1'00 140MUD DENS I TY C IKG/MJ O. t! ' I I I! I ' 1000. 1200. 1400. 1600. 1800. 2000. 2200. 2400. 2600. Figure 18. L* versus Pm for Clamped-Clamped Risers for Do = 0.55m and Di = 0.515m

1000. lOOGOyN~ ^TSi ETA cc 0.800 14 0. 47 BB, ~l.o I M 800. 0^ 0.61. O. 69 D,= 0 95 M 0.74 0. 2 0. 6 200. MUD DENSITY CKG/M3] 0. i 1000. 1200. 1400. 1600. 1800. 2000. 2200. 2400. 2600. Figure 19. L* versus Pm for Clamped-Clamped Risers for Do = 1.00m and Di = 0.95m

-37 -III.6. Critical Length for Hinged-Movably Hinqed Risers For this set of boundary conditions, Fiqures 21, 22 and 23 show L* versus Pm for the same systematic variation of parameters as the one for risers with nonmovable boundaries. The conclusions listed in Section III.2. hold in this case and all the following cases of risers with movable boundaries. III.7. Critical Length for Clamped-Movably Hinged Risers Figures 24, 25 and 26 depict similar numerical results for the critical length of clamped-movably hinged risers. II.8. Critical Length for Hinged-Movably Clamped Risers The results for this set of boundary conditions are presented in Fiqures 27, 28, 29 and show the same type of dependence of L* on Pm, Bm/Bw and riser size. III.9. Critical Length for Clamped-Movably Clamped Risers For this last set of boundary conditions the results are presented in Figures 30, 31 and 32. Comparing all four cases of risers with movable top support we can draw the following conclusions 1. For given Pm, n or Bm/Bw, and riser size the critical length L* changes with the riser boundary conditions and increases with increasing restriction imposed by the boundary conditions, that is in the following order (a) hinged-movably hinqed riser, (b) hinqedmovably clamped riser, (c) clamped-movably hinged riser and (d) clamped-movably clamped riser (see also Chapter V).

-38 -2. L* is of the order of a few hundred meters only, even for relatively low values of the drilling mud density. 3. Figures 21 to 32 show that even for high n values, buckling in tension may occur for relatively low values of Pm and L*.

2500. < 1 2000.1 m 2000. m COLUMNS WITH MOVABLE BOUNDAR IES 1500. CMC 1000.- /MH HMC HMH 500. TA 0.0 0.2 0.4 0.6 0.8 1.0 Fiqure 20. B* versus n for Risers with Movable Roundaries

1000~T; 02 Z H^ ETA ^ 0.78 D- \ 800. 0*89 091 0192~~~BB 0.0 r0000 0.2 4000 2000~ ~ ~~~~. ^ ^ISO-^^ MUD DENS TYY 1200e ~~1~~-66 O18,0 2000020~20~.2m60 'Do.2Sm and Oi 'iad RiSetS 'for o for Hinled-MovablY Fiq-urp_ 2,* versus Pm ^

1000. ETA HMH 0.78 O..85 B/lD 0,5 800.- 0.89 B/B0 0. 0 / 0.91 0.2 D=0.52 M 0.92 0.4 800. 400. 7 200. rn ^ reMUD DENSITY CKO/M3 0. 1000. 1200. 1400. 1600. 1800. 2000. 2200. 2400. 2600C2fRa Pv Figure 22. L* versus pm for Hinged-Movably Hinged Risers for % = 0.55m and Di = 0.515m "^ ^^ 1c)

1000. HM, z ~~~~~ETA 0.78 B/Bw Do 0.85 8 D-l.oo M ^ /0.83 0^ Dr 800.soo 0.91 0.2 0.92 0. 4 4000 200, MUD DENS TY4~ 0.~ ~ ~ ~ ~ ~ ~~~~20 1000~; 2~4 ~1$0018000 2000. 20 40~2 ined Risers for Do 1.00m and n. -0. ^ versus PM fo Hin^ Figure 23. L*

1000., ETA CMH 0.50 8000:0. 65 D- 025 M 800.- 0.74 \. 79 D= 0.23 M 0. 82 600. B 0. 0 0.2 0. 4 0.6 400. 200. ' ' Figure 24. L* versus Pm for Clamped-Movably Hinqed Risers for Do = 0. 25m and Di = 0.23,;.~ ' 2z

2000 cmH 0.50 DD 0.65 n- == 0 1 m ~~~~~~~0. 74 0.0 D. Boof 0. 79 0D.24 0.82 0. r 6000 4000 200a MUD DE s I TY 1000 for lamed-~ova-)I Mned Risers for 'Do ==O Fiq-ure L* versus Pm ^

1000. — I ETA CMH 0.50 X 0.65D 1.00.65 800. _, 0.74 BBW Do1 00 0o 82 0.4 0. 4 800. 400. — 200. MUD DENS I TY CKG/M) 1000. 1200. 1400. 1600. 1800. 2000. 2200. 2400. 2600. Figure 26. L* versus Pm for Clamped-Movably Hinqed Risers for Do = 1.00m and Di = 0.95m

1000\ -^ n 25 I Z ET^- D^ 0.78 0.0- M C ^0.85 800. 0.92 ~ ~ 0. 600'. 2K l~~~~u _^OnNO 2^^ ~260 4000~~~~~~~~~20 200,~~~~~~~00 180000~,2201 O's400 16000 ~0 1200, 1400,, pisers for 'D',ginqed-Mo-vably ClamPe-d Fiqlure 27. *vr

1000. 1I ETA HMC 0.78 800.- /.85 B/B Do-055 M 0. 1 0 D 0. 52 M 0. 2 0. 9 200. 2 0Fiure 28. L* versus m for HnedMoabl Clamped Risers for Do.55m and.515m 200. MUD DENSITY CKG/M'] 1000. 1200. 14000 1600. 1800. 2000. 2200. 2400. 26000 Figure 28. L* versus Pm for Hinqed-Movably Clamped Risers for Do = 0.55m and Di =

1000~ HM(IM z ~~~~~ETA ~-, C 78.1.00o 0.85 0. 0 D m /^89 00 D-.95 9 Boo. ~0. 91 0.2 80. U /0.92 0. 4 0. " \~t\\~ / ^ ^ \ 400. 200. MUD DENS^ ITY 01140 GOI 1000. 120O.-$0 10-200 20~240 6 ly C amp d R ser00m and fi 0.9 f or Hinged-rovabs fr D Figure 29. L* versus Pm

1000. ETA C0C 0.50 0. 65 800. 0.74 D 0.25 M 0.79 Dj0.23 M 0. 82 800. BIBV 0. 0 0 2 0. 4 400. I'D 200. MUD DENSITY CKG/M'D 0.~ ~1~ 1200. 1400. 1600. 1800. 2000. 2200. 2400. 2600. Figure 30. L* versus pm for Hinqed-Movably Clamped Risers for D = 0.25m and Di = 0.23m

CM\ \^-'\^^^ ETA n0 55 M I \ /5 ^ DF^ 065 D05 0.74 0. 0 goo, IL 0 a79 0. 4 0.82. F,00. 400. 0 UDo, DENS I 0.~~~~~~~~~20 gj~u~-?~U-~1600, 1 800 200~ 2000L600 CaeaRisers for n 'Rincled-MloVablYCame i* versus Pm for Fiqure 31 L

1000. ETA CMC 0.50 o.7s MUD DENSITY C95 K/M 0.5 82 0.4 600. 400. 200. MUD DENS I TY [ KG/MJ 1000. 1200. 1400. 1600. 1800. 2000. 2200. 2400. 2600. Figure 32. L* versus Pm for Hinqed-Movably Clamped Risers for Do = 1.00m and Di = 0.95m

IV. COMPARISON OF CRITICAL LENGTH FOR RISERS WITH NONMOVABLE BOUNDARIES To make the comparison of the critical length curves derived in Chapter III easier, in this Chapter we plot the critical length versus Pm curves in Figures 33, 34 and 35 for all three riser configurations, defined by equations (11-3) to (III-8), and for Bm/Bw = 0.2. Each figure includes one curve for each set of boundary conditions for risers with nonmovable boundaries. These figures verify the conclusions derived in Section 111.5. and in addition show that the effect of the lower boundary condition is more important than that of the upper one. Actually the critical length curves for hinged-hinged and hinged-clamped risers almost coincide and so do the corresponding curves for clamped-hinged and clampedclamped risers. In fact for high values of L* the curves are identical as can be deduced from the asymptotic behavior of riser stability boundaries derived in reference [9]. -52 -

1000.-^ ^ COLUMNS VITH ONMOV A BLE 800.s BOUNO^ I ES D: 0. 25 0 m 0.0.230 2000 ~ 400,2 HC~~~C 200~ r\D DENSIT"Y CKG/WSW ^^^ ^ 400. ^0^~,nd ^ ana ^ - 100. 1400,w 1rO00~ 80 00 2002m i-020~3 60 1200 ov~~~~wt~ onoable Boundaries for Do. 5 f or piserswit 4n Fr 3L* versus Pm Figure 33~

100010 COOLUMNS m NONM^OV \BL 800S BOUND\ IES 0a: o.55 0 =0.51 r, 1^ ^ ^r~~~nQ louu~ v.1 BonaIe 600. a 400a" C 200, ~ MUD DENS I T\600 0. -L* vesu P Figure 1400- 16~0 3800^*20003 * 120U, t~~~~~~~~~~niaie or 'Do with o""oable Wuaritor Riserswih40m Figure 34. 'vrssP

1000. 800.- COLUMNS WITH NONMOVABLE BOUNDARIES D\ 1.D000 M 600. D 00. 950 M ~\ \^^ B/B= 0.2 400. 400 -C I 200. MUD DENSITY CKG/M3m 0. I 1200. 1400. 1600. 1800. 2000. 2200. 2400. 2600. Figure 35. L* versus pm for Risers with Nonmovable Boundaries for Do = 1.00m, Di = 0.95m and bm/bw = 0.2

V. COMPARISON OF CRITICAL LENGTH FOR RISERS WITH MOVABLE BOUNDARIES Analysis similar to the one done in Chapter IV is repeated in this Chapter for risers with movable top supports and specifically for the four sets of boundary conditions considered in this work. The results are shown in Fiqures 36, 37 and 38 and verify the conclusions derived in Section III.9. In addition, as in the case of risers with nonmovable boundaries it is obvious that the effect of the lower end boundary condition is more important than that of the upper end condition. In all three fiqures, curves corresponding to risers with the same lower end boundary condition are identical. This can also be explained theoretically on the results of reference [9] which deals with the asymptotic behavior of riser stability boundaries. -56 -

1000. 800. - COLUMNS WITH MOVABLE BOUNDAR I ES D O0. 250 M 600.t D00.230 M B/B= 0. 2 M V 400. I 200. HM MUD DENSITY C KG/M3I 1200. 1400. 1600. 1800. 2000. 2200. 2400. 2600. Figure 36. L* versus Pm for risers with Movable Boundaries for Do = 0.25m, Di = 0.23m and bm/bw = 0.2

1000. g80oo. COLUMNS WITH MOVABLE BOUNDARIES D,- 0. 550 M 600.- \ g.515 M B/Bw= 0.2 400. HM\ HMC C ICMC 200. MUD DENSITY CKG/M'D 0.1200. 1400. 1600. 1800. 2000. 2200. 2400. 2600. Figure 37. L* versus pm for risers with Movable Boundaries for Do = 0.55m, Di = 0.515m and hm/b - 0.2

1000. z: 800. - \ COLUMNS WITH MOVABLE BOUNDARIES D=1. 000 M 600. D=0.950 M B/B= 0.2 400.- \ HMH HMC CMH MC 200. MUD DENSITY CKG/MJ 1200. 1400. 1600. 1800. 2000. 2200. 2400. 2600. Figure 38. L* versus pm for risers with Movable Boundaries for Do = 1.00m, Di = 0.95m and bm/bw = 0.2

SUMMARY The stability boundaries of risers for eight different sets of boundary conditions have been derived in previous work [5,9]. In this report these results were used to prove that risers which are in tension along their entire length may buckle globally as Euler columns due to the action of internal fluid static pressure force. Critical B value, B*, which if exceeded may result in buckling in tension has been defined and computed for the eight sets of boundary conditions considered in this work. Critical values of the riser length, L*, corresponding to 3*, have been derived for typical risers and over wide ranges of the drillinq mud density Pm and buoyancy supporting forces, Bm/BW, showing that for typical designs, L* is of the order of a few hundred meters. The dependence of L* on the major riser properties and boundary conditions has also been studied. It has been shown that for a given type of top support, movable or nonmovable, the critical riser length, L*, strongly depends on the lower end boundary condition and only weakly on the upper end one. -60 -

REFERENCES 1. Abramowitz, M. and Stegun, I.A., Handbook of Mathematical Functions, Dover, New York, 1970. 2. Bender, C.M., and Orszag, S.A., Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill Book Co., 1978. 3. Bernitsas, M.M., "Riser Top Tension and Riser Buckling Loads," ASME AMD, Vol. 37, November 1980, pp. 101-109. 4. Bernitsas, M.M., "Static Analysis of Marine Risers," Department of Naval Architecture and Marine Engineering, The University of Michigan, Report No. 234, June 1981. 5. Bernitsas, M.M., Kokkinis, T., and Faller, W., "Buckling of Slender Columns under Distributed Load," Department of Naval Architecture and Marine Engineering, The University of Michigan, Report No. 240, December 1981. 6. Bernitsas, M.M., "Problems in Marine Riser Design," Marine Technology, Vol. 19, Number 1, January 1982, pp. 73-82. 7. Bernitsas, M.M., "Three Dimensional, Non-Linear Large Deflection Model of Dynamic Response of Risers, Pipelines and Cables," Journal of Ship Research, Vol. 26, No. 1, March 1982, pp. 59-64. 8. Bernitsas, M.M. and Kokkinis, T., "Global Buckling Design Criteria for Risers," Proceedings of Conference on the Behavior of Offshore Structures, BOSS '82, Boston, Massachusetts, August 1982, pp. 705-715. -61 -

-62 -9. Bernitsas, M.M., and Kokkinis, T., "Asymptotic Behavior of Riser Stability Boundaries," Department of Naval Architecture and Marine Engineerinq, The University of Michigan, Report No. 255, December 1982. 10. Bernitsas, M.M., and Kokkinis, T., "Bucklinq of Risers in Tension due to Internal Pressure: Nonmovable Boundaries," Proceedings of the Second Offshore Mechanics and Arctic Engineering Symposium, ASME, February 1983, and Journal of Energy Resources Technology, (in press, 1983). 11. Bernitsas, M.M., and Kokkinis, T., "Buckling of Columns with Movable Boundaries," Journal of Structural Mechanics, (in press, 1983). 12. Biezeno, C.B. and Koch, J.J., "Note on the Buckling of a Vertically Submerged Tube," Applied Scientific Research, Sec. A., Vol. 1, No. 2, 1948, pp. 131-138. 13. Duncan, W.J., "Multiply and Continuously Loaded Struts," Engineering, Vol. 174, 1952, pp. 180-182. 14. Goodman, T.R. and J.S. Breslin, "Static and Dynamics of Anchoring Cables in Wakes," Journal of Hydronautics, Volume 10, No. 4, pp. 113-120. 15. Huang, T. and Dareing, D.W., "Buckling and Frequency of Lonq Vertical Pipes," Journal of Engineering Mechanics Division, Proceedings of the ASCE, Vol. 95, Feb. 1967, pp. 167-181. 16. Huang, T., and Dareing, D.W., "Buckling and Lateral Vibration of Drill Pipe," Journal of Engineering for Industry, Transactions ASME, Vol. 90, Nov. 1968, pp. 613-619.

-63 -17. Lubinski, A., "A Study of the Bucklinq of Rotary Drilling Strings," Drilling and Production Practice, API, 1950, pp. 178-214. 18. Morgan, G.W., "Force Systems Acting on Arbitrarily Directed Tubular Members in the Sea," Energy Technology Conference and Exhibit, Paper 77-Pet-38, Houston, Texas, Sept. 1977. 19. Plunkett, R., "Static Bending Stresses in Catenaries and Drill Strings," Journal of Engineering for Industry, Transactions ASME, Series B, Vol. 89, No. 1., February 1967, pp. 31-36. 20. Sherman, D.R. and Erzurumlu, H., "Behavioral Study on Circular Tubular Beam Columns," Journal of Structural Division, Proceedings of the ASCE, Vol. 105, STG, June 1979, pp. 1055-1068. 21. Sugiyama, Y. and Ashida, K., "Buckling of Long Columns under Their Own Weight," Transactions, Japanese Society of Mechanical Engineers, Vol. 21, No. 158, August 1978, pp. 1228-1235. 22. Willers, F.A., "Das knicken Schwerer Gestaenge," Zeitunq fuer Anqewandte Mathematik und Mechanik, Vol. 21, No. 1, 1941, pp. 43-51.

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