THE DYNAMIC RESPONSE AND INSTABILITY OF ANARTICULATED OFFSHORE LOADING PLATFORM IN WAVES

Transcription

1 THE DYNAMIC RESPONSE AND INSTABILITY OF ANARTICULATED OFFSHORE LOADING PLATFORM IN WAVES P-A luh* M-C Huang** ABSTRACT This paper presents the ana1ytica1 and experimenta1 studies of the dynamic response and the instabi1ity of an articu1ated offshore 10ading p1atform in regu1ar waves without a tanker mooring to it. For eva1uation of the hydrodynamic 10ads,the modified form of Morison's equation which a110ws for the re1ative motion of the structure and waves has been used. A time-domain step by step method has been emp10yed for solving this non1inear prob1em. The axia1 and 1atera1 shear forces at the ALP universa1 joint were computed by Newman's slender body theory for a spar type body. The phenomenon of the ALP instabi1ity is studied by Matieu's equation 且 where the variation of stiffness due to heave force and due to heave force re1ated to the ALP motion in waves are both investigated,respective1y. An extensive mode1 test program with two sca1es 1/150 and 1/50 of a sing1e point mooring articu1ated 10ading p1atform designed to operate out in the North Sea is carried out in wave tanks. An attempt has been made to compare the computed sol utions with the tested rlesu1 ts both in trends and in magnitudes. Thetrends appear to be genera11y simi1iar, except for 1arge discrepancies in magnitude for pitch ang1es and acce1erations for mode1 sca1e The phenomenon of dynamic instabi1ity has not been observed in the experiments. However,when the ratio of weight and buoyancy is greater than 0.95,this phenomenon has been ca1cu1ated ana1yt:ica11y at the second and the fourth harmonics of the fundamental frequency of the structure. The ca1cu1ated axia1 and 1atera1 shear 10adii!lare in genera1 sma11er than the experimenta1 resu1ts. A11 above mentioned discrepancies need to be investigated further,both in ana1ytica1 methods and experimenta1 techniques., 1. INTRODUCTION Since the mid-1950's oi1 production has been moving into deeper and *Professor,Department of Nava1 Architecture and Marine Engineering,Nationa1 Cheng-Kung University,Tainan,Taiwan,R.O.C. * 禽 Associate Professor, Department of Nava1 Architecture and Marine Engineering, Nationa1 Cheng-Kung University, Tainan, Taiwan,R.O.C 一

2 deeper waters,costs of conventional fixed bottom supported structures have increased rapidlyi income from offshore fields have been correspondingly reduced. Increasing interest in use of compliant structures forexploiting energy and mineral resources in deeper waters has led to the consideration of articulated column for certain applications. Typically these structures are supported at or near the seabed by a universal joint,and extended vertically through the water surface. These structures essentially comprise a deck and superstructure which is supported by a single or multiupper support column, which in turn is supported by a buoyancy or floatation chamber. The buoyancy chamber is attached a lower tension leg which is connected to the foundation via an universal joint allowing compliant motion of the structure. An example of the structure designed to operated in the North Sea is shown in Figure 1***. The general arrangement of the above mentioned example with relevant weight of deck and buoyancy chamber are given in table 1. The relevant information of the technical development for the articulated offshore loading platform (ALP) is available in reference [11 through [111. Reference [11 gives a new concept of various use for the articulated column. During the two years test period,the dynamic response of the column has predicted by numerical calculations and model tests. Reference [21 gives an analysis of the dynamic response of an articulated tower to noncoflowing waves and current,where the waves and current loads have been calculated by Morison's equation. The predicted response is a nonlinear whirling motion of the tower about the vertical axis. Reference [31 discusses an integrated system of tower and tanker and describes development,design and construction of two articulated buoyancy steel towers which provide facilities, housing flowlines and control lines from seabed equipment to surface facilities. Reference [41 gives an analysis and model test of a buoyant tower and a tanker as an integrated system in head seas that are collinear with wind and current. Reference [51 furnishes information of an operational experience of an articulated loading column in the Beryl field in the North Sea, which concludes a number of important elements. Experience shows that berthing can be successfully achieved in extremely adverse weather cond 主 tionsi the multiarti ***Figures, tables and references are summaried at the end of paper. 一 598 一

3 relative motion of water particles with respect to the oscillating structure. Reference [71 describes the development,design and construction of a concrete articulated tower CONATand a large scale test program,this paper also presents a method of analytical investigation. Reference [81 discusses the features of an articulated loading tower and a chained articulated tower, and concludes that in the hostile environments the articulated loading tower with buoyancy alone is less favorable than the chained articulated tower in shallow waters, both in view of performance and economy. Reference [91 includes more test results of motions and loads of an articulated loading platform, both with and without a tanker moored to it,in the non-collinear wi~d,current and random seas. Reference [101 presents the motions,the axial and lateral shear loads at universal joint which have been computed within the frame work or linearization by Newman's theory of a slender yertical body of revolution. Reference [111 presents the results of the full scale measurements of motions and loads of an articulated tower designed to operate in the North Sèa. 11. DESCR1PT10N OF THE PROBLEM 1n this study,the platform is symmetrical about a vertical axis, on which also lies its center of gravity and a principle axis of moment of inertia, as shown in Fig.2. The platform is connected to the bottom by an universal joint, therefore,one degree of freedom motion equation can be written for the platform pitch angle. Firstly we consider that the rotational structual damping at the universal joint is proportional to the pitch velocity. Secondly the effect of viscosity can be ëlccounted for by the use of modified Morison's equation where the drag force is proportional to the square of relative velocity. Thirdly the pitch moment froßi the vertical hydrodynamic load on the platform is small compared with the effect from the excess huoyancy and therefore can be omitted. Thus the equation of motion, considering these simplification can be expressed in terms of the rotation about the vertical axis as follows: 工 ^ ë + 自由自 C n å + K" 8 = M (t) (1) where 工 8 is the mass moment of inertia,c 8 is the rotational structural damping,k 8 is the restoring coefficient,and M(t) is the exciting moment about the universal joint which will be derived from the Morison's equation. Thé natural frequency of the ALP in pitch is therefore given by ω8 = L [ -:;:- ]ι二 -8 (2) 111. DYNAM1C ANALYS1S 一 599 一

4 (a) Motion Analysis According to the previous description of the problem,a modified form of Morison's equation which allows for relative motion of the structure and wave can be employed,and the right hand side of Eq. (1) can be expressed as rd+n M(t) = Iι[.5pC n A(u-rå) lu-rëlr + pvc M 這 (-r8)r+pvrsr]dr (3) where d+n is the water surface elevation. The first term in Eq. (3) is the nonlinear drag term,the second term is the Froude-krylov component and the third term is the added mass term. Rewritting the above equation,we have M(t}=(1+n[pvrZ(V)5+ 刊 CMur + 門 A(ufh} ud r]dr (4) The added mass moment of inertia term can be transferred to the left hand side of the motion equation and we have,d+n ( 工 + 工 A) 由 +Ce8+V=jo[ 叫 lr +.5pC n A( 叫 )1\ 日 å\r] dr where (5) rd+n where 工 A=\ pvr 2 (C M -1ldr Owing to the discontinous conf~guration of the outer dimension, the integrals in Eq. (5) should be calculated piecewisely and then summed together. The nonlinear drag term renders the solution of this problem more suited to a time incremental solution. A computer program was developed to solve the nonlinear equation of motion. The moment exert on the structure has been calculated to the free surface level,so that the variation in buoyancy as a consequence of the wave profile variation is allowed for in the calculations. A trial and error procedure was used to adjust the hydrodynamic coefficients in the numerical solutions and then compared with the results from model tests. (b) Dynamic Instability It has been suggested in Ref. [121 that in certain conditions the vertical force on the ALP may be significantly reduced and thus lead to dynamic instability. This instability is commonly referred to as Mathieu's instability 一

5 Consider a p1atform subject to unidirectiona1 waves in the surge direction as shown in Fíg. 2. The equation of motion invo1ves a sinusoida1 variation of stiffness due to the varying tension caused by the vertica1 component of wave force and the associated 1atera1 motion of the structure. Writing the tension as fo11ows: T = Zo + Z cos ωt (6) If neg1ecting damping,the equation of mot~ion due to a force X sin ωt may be written as Zo Z (M + Ma)X + [τ+ 互 cos ωt]x = x sin ωt (7) if we consider the effect of position in waves and 1inearized the effect of position,the equation of motion may be written as. Zo,Z 一一 (M + M_)x + [ 一一 +( 一 +KX) cos wt]x = X s 土 n ωt d. 'd (8) The secord part of the above equation contains an additiona1 term as a consequence of the variation in the heave force and this is in phase with the position of the structure in wave. This is Mathieu equation for which there are regimes of instabi1ity. To investigate the phenomenon of the instabi1ity,a computer program is deve10ped to solve the above equation for the fo11owing two conditions:(a) The variation in heave force on1y,and (b) The variations in heave and in the instantaneous position of the structure in wave. (c) Ana1ysis of the Axia1 and Latera1 Loads Newman [131 deve10ped a 1inearized slender body theory for eva1uating the wave exciting forces and moments on a spar buoy,which is a body of revo1ution with vertica1 axis. This theory was emp10yed in this study to compute the 1atera1 force,,and vertica1 force, per wave height at the universa1 joint in the fo11owing forms: T' rd+η 計划志是 YE coshkz s(z)dz (9) F T' rd+η 百三 = 內 [s (d) - 古拉布 J 0 sinh kz s (z) dz] (10) 一

6 In the above equations,s(z) is a cross section area, S(d) is the cross section area at the free surface. The lateral load in Eq. (9) is identical to the corresponding Morison formula for unit added mass coefficient with zero damping. The axial force in Eq. (10) is different from Morison's formula; in fact,it can be derived from the incident wave velocity potential Eq. (10), strictly speaking, is not applicable when the distribution of the cross sectional area changes abruptly along the vertical axis of the platform. Undp.r present consideration,such sharp changes near the end of the ballast and buoyancy chamber can be avoided by smoothing the distribution with slight change of the sectional area near the end to overcome this difficulity. IV. MODEL DESCRIPTION AND INSTRUMENTATION (a) The Model Data Two model of the ALP were built with linear scale 1/50 and 1/150 for a single point mooring articulated loading platform designed to operate out in the North Sea. The former model was constructed by steel, the later was constructed by PVC tube to give the correct outer-dimensions and weight distribution. Table 2 gives the weight and buoyancy distribution of the model and full scale ALPS,and the corresponding computed geometrical values, such as KG,KB and 1,etc. (b) Instumentations and Measurements (1) The wave elevation was measured by a capacitance type wave gauge mounted about two meter ahead of the m6del in order to avoid wave excited by the model and wave reflected from the model. (2) A grid of pitch angle was mounted on the tank side. The maximum pitch angle of the ALP was measured by observation. (3) The axial force and the lateral force at the universal joint were measured by a ring gauge and a cantilever plate. Owing to the size restriction, only a right gauge was mounted on the universal joint of the 1/150 scale model,hence only the axial force was measured. Both,models were equipped with an accelerometer at the top of upper shaft for measuring the acceleration. (4) The test program was carried out in two different wave tanks, both with regular wave generators. They have dimensions 120 m x 8 m x 3.8 m (length x breadth x depth) and 15 m x 1.2 m x 1m, respectively. The 1/50 scale modelwas tested in the larger tank and the 1/ 一

7 sca1e mode1 was' tested 主 n the sma11er one. (5) The signa1s from the different measuring points of the system were fed into a persona1 computer through amp1ifiers,fi1ters and an A/D converter. The resu1ts were shown on the PC screen and a1so stored. (c) Test Conditions The comp1ete test program consists of twenty different tests for each sca1e mode1. An examp1e of the measured resu1ts for the 1/150 sca1e mode1 is shown in Figure 3,whi1e an examp1e of the measured resu1ts for the 1/50 sca1e mode1 is shown in Figure 4. ~uring the tests regu1ar waves were generated with periods ranged from 1.3 to 4.3 second for 1/150 sca1e mode1,and with periods from 1.1 to 2.4 second for 1/50 sca1e mode1. The wave heights ranged between 3 to 19 cm for 1/150 sca1e mode1,and between 5 to 17 cm for 1/50 sca1e mode1,respective1y. v. PRF.SENTATION OF RESULTS Using the given weight and buoyancy distribution a10ng the vertica1 axis of the ALP in the previous section,we have computed the mass moment of inertia and restoring moment.. The structura1 data used in the computations is representative of the typica1 10ading tower used in the North 8ea,such that D/L =0.03,d/L=0.73,dB/L=0.42. A comprehensive study of response time histroy has been carried out, by using the fo11owing parameters taken from Ref. [ 工 4-]: (for 1/150 sca1e) C e = 0.1,C D = 0.6,C M = 1.8,Lw/L = , H/L w = (for 1/50 sca1e) Ce = 0.1,C D = 0.33,C,-M M 一 = 1.8,Lw/L = , H/L W = 一曲 3 一

8 Use of the foregoing informations and Eq. (5) with initial conditions (9)=0, (9)=0 gives the time histories for the pitch motion amplitude, velocity and acceleration. Some computed exarnples for 1/150 日 cale model and 1/50 scale model are shown in Figure 5 and Figure 6,respectively. The natural pitch period was computed by Eq. (2). The results are nearly the same as the test results : 4.7 second for 1/150 scale model and 8.2 second for 1/50 scale model. The experimental results and the corresponding calculated reuslts for the pitch motion of both scale models were analyzed by a sixth-order regression model and the regression curves are shown in Figures The experimental results for the axial and shear forces at the universal joint are illustrated in Figures The phenomena of the instability was studied by Eqs. (7)-(8). The following two conditions were simulated (a) Basic model with. no additional mass on deck or in buoyancy chamber. Ratio of weight/buoyancy is 0.93 for both 1/50 1/150 scale models. This basic model corresponds to the model employed in the dynamic analysis. The exciting wave frequencies were choosen to be approximately twice and fourth the natural frequency of the model. The simulated time historied are shown in Figure 14. Experimental observations for the response of the ALP models were conducted during the dynamic tests and the phenomenon of instability was not found. (b) Basic model with mass 2 Kg and 0.23 Kg added on the buoyancy chamber of 1/50 and 1/150 scale models,respectively. Ratio of weight/buoyancy = 0.95 for both model scales. The exciting wave frequencies were choosen as (a). The simulated times histories are shown in Figure 15. Figure 15 clearly indicates the phenomenon of dynamic instability where the stiffness of the system becomes negative. The axial and lateral loads at the universal joint were calculated by Eqs. (9)-(10) for 1/50 and 1/150 scale models,respectively. These numerical values are very small. VI. CONCLUSTIONS AND RECOMMENDATIONS According to the measured and calculated results,a comparison has been made both in trend and in magnitude. We conclude as follows: The pitch motion response as predicted by the modified Morison's equation displays a harmonic oscillation with wave frequency. There are both ingood agreement in trend and in magnitude between the analytical results and those obtained experimentally,except for 1/50 scale model. The discrepancy in magnitude for the 1/50 scale model may be caused by the improper scaling for the rotational structural damping,wave scattering,etc.. However,the trends -604 一

9 appear to be generally similiar. It is important to conduct a careful model test which simulates closely to the condition of the analytical model in regular waves to predict the dynamic response and related coefficient used in present study. Theoretical studies of the dynamics of ALP have shown that there are conditions in which instabilities describable in terms of Mathieu's equation may develop. When the ratio of weight to buoyancy is equal to,or greater than 0.95,and the wave frequency is twice or fourth of the natural frequenqy of the structure, the phenomenon of dynamic instability has been calculated i 品, the present study. When the ratio of weight to buoyancy is smaller than 0.93, the phenomenon of damping term should be included in the equation of equation, thus this limits the dynamic instability. Regarding practical applicat 主 ons, the present authors are not aware of any examples of such instability affecting real structure in the sea. The axial and lateral shear loads at universal joint have been computed by Newman's linear 主 zed slender body theory,where the numerical results are very small. These are not in good agreement with experimental data. It is recommended to develop a refined analytical technique for solving such problem. It is also important to conduct extensive model test. ACKNOWLEDGEMENTS Financial support provided by the National Science Council under grant No E ~ gratefully acknowledged. The anthors also express their deepest appreciation to Mr. W-L Hsu,Mr. W-Y Shyy and Mr. Z-S Lin for their assistance in conducting the model test and developing the computer programs required to complete the results presented in this paper. NOMENGLATURES ^ = scale factor o = diameter of the ALP x = horizontal Displacement M = mass of the ALP g = gravitational constant v = volume of the buoyancy chamber U = horizontal water particle velocity w = vertical water particle velocity u = horizontal water particle acceleration w = vertical water particle acceleration '" = water density -605 一

10 H = wave height K = wave number C = wave velocity MCi;)= exciting pitch moment I~ = virtual moment of inertia of ALP Ce = damping coefficient of the ALP Ke = restoring moment of the ALP C= = inertia coefficient Cd = drag coefficient I~ = added mass moment of inertia zo = vertical restoring force Z = variation of vertical restoring force 文 = horizontal maximum exiciting force T = tension of the ALP τ= shear force at universal joint F= = axial force at universal joint e = displacement of pitch L...= wave length L = length of ALP d = depth of buoyancy chamber REFERENCES 1. de Chasse, C.B.,"Various Uses for the Articulated Column Elfocean,A New Concept, "Proceedings of the Third Offshore Technology Conference, OTC 1392,1971,pp Kirk, C.L. and Jain,R.K.,"Response of Articulated Towers Waves and Current,"Proceedings of the Ninth Offshore Technology Conference, OTC 2798,1977,pp Burns,G.E. and D'Amorim,G.C.,"Buoyant Towers for Phase 1 Development of Garoupa Field,"Proceedings of the Ninth Offshore Technology Conference, OTC 2828,1977,pp Chakrabarti, S.K. and Cotter,D. C.,"Analysis of a Tower-Tanker System, "Proceedings of the Tenth Offshore Technology Conference,OTC 3202, 1978, pp O. 5. Hays, D.L.,McSwiggen,M.,and Vilain,R.,"Operation of an Articulated Oil Loading Column in the Beryl Field in the North Sea, "Proceedings of the Eleventh Offshore Technology Conference~ OTC 3563,1979, pp Kokkinowrachos, K. "Dynamic Analysis of One and Multi-colum Articulated Structures, " Hydrodynamics in Ocean Engineering,Trodheim, Butt,H.G., Salewski,J.,Wagner, P., et al., "A Large-Scale Test with the -606 一

11 Concrete Articulated Tower CONATin the Vicinity of the Research Platform NORDSEE," IMT /01, International Conference on Marine Sciences and Ocean Engineering,Sept. 1980,Hamburg,Germany. 8. Pollack, J., "Hybrid Articulated Tower Systegm,"Proceedings of Marine Technology,1980,pp Naess, A., "Loads and Motions of an Articulated Loading Platform with Moored Tanker,"Proceedings of the Twelfth Offshore Technology Conference, OTC 3841,1980,pp Kim, C.H. and Luh,P.A.,"Prediction of Pitching Motions and Loads of an Articulated Loading Platform in Waves,"Proceedings of the Fourteenth Offshore Technology Conference,OTC 4247,1982 pp Brathang, H.P., "Measured Motions of a North Sea Articulated Loading Platform, "Proceedings of the fifteenth Offshore Technology Conference, OTC 4639,1983,pp Richardson,J.R., "Mathieu Instabilities and Response of Compliant Offshore Structures,"NMI,R49,Feb Newman, J.N., "The Motions of a Spar Buoy in Regular Waves, "David W. Taylor Navel Ship Research and Development Center,Report 1499,May Ewing, J.A. "Environmental Conditions,"Proceedings ISSC 1988, Denmark, pp 一 607 一

12 電 Table 1 We1ght and buoyancy distribution of the prototype ALP PAARTLP OF Head 273 Upper Shaft Haln JlEIGHT BUOtoyAnM. CY tons TB 凹 anykan 叮 L~er Shaft ll i y 第二 8alla5t Tank 岫 14 2 句 70 TOTAl Figure 2. Sketch of an articulated load!ng platform (ALP) Table 2 Char 電 cterlstlcs of" the protot:,ope and scale roodels of" AlP I\B p, 可 tot.:ype 74.7J'tl..59. TlI t t-j'tl. 25OOOo00t-J'tl. 1/50 sca.le 14Q.35clU cIll 4.811{g Qkg-cllJ kg-cIt1 2 一一 ---- 一一 Zo"'8-\o1 1/150 scale 4Q.68clU 38.97cl1l O.181{g rm "e= 8K8-\01 喂 Z 41.01kg-cm kg-cTll:~ 一一一一一一一一一一 包, 0.06 叫 0.04 s j -OOoJP 咱們 呵 o ' 吭, J 一一一一一一一一 A 1\ 1\ 1\ o 2.f- a e 叮叮,. 間 扭扭扭扭.J~.Je.J~ 刊 42 村一一 τrme(5) Flgure 3. Example record of acceleration for scale model 1.00 CASE 8 戶戶 ( / 7 斗!!!.. --?- Z.3 m Z O - 臣 00 w 一 J w ιj 足一 50 - Figure 1. Outer d1mension.s of ALP at water depth m TIME (SEC) Figure 4. Example record of acceleration for 1/50 scale model -608 一

13 團 同 / 叭 1~ 戶叫 白... 0, tj.:;!e " t 0.08 ' ~ 0,0= -i e a 10 m 叫自 } [,,.., 7 " ", " Flgure 5. Exarnple of computed pitch dlsplacement for 1/150 :J::(:; T 一一一 --, 1 L 斗 孔 建 l ;;. : 三五三三! 于三 三二 Figure B. C':lmparlson of measured and computed nondlmensl 叫.1.' -... :: 于 " 一一 一一一一 ( 一 - ~ < t ~1.! r,q I!,~ J _',~ I '(1!' 內 U 一一 - 一一一一一一一 Flgu 玄 e 9, 耳 omparison of rneasured and computed pitch acceleratlon for 1/150 seale model Io/H. ' " ~l"'''''' 也 '10 旬 '" e \ 比 '---...! 一 月半 :L:::::;d:L::::=r---", 一 -r 一 -, 一一 O! 1,0 l' 雪 i! 一 1 土一一土上一 ~ Flgure 7. Comparison of rneasured and computed nondimensional pltch aligle for 1/150 scale model,01c"-h'.d ~ 一一 ~ 于 \ /// 1-- 仔一 4 吋 I---r-~-----~----- 一 --, 句 一 - 9 俏一一一一 I.! 宜 '.1 J!.',.! ~~!! 'Q M Figure 10. Comp, 主 r!son of measured and comp 叫 ted pitch acceleration for 1/50 scale model -609 一

14 悟 " SCALE 11150,.~ B~.88D42e:m.T. Zo""2.1Sk 還 <>.0005 u 一一一 風 h 白白 1 " 1.! L~ r! Flgure 11. Measured axlal 安 rce at universal-jolnt for 1/150 scale model ~.OOl <>.002 YDa tn Sl:COND F 土 gure 14. Instabil1 ty at ratio of weight/buoyancy '" 0.95, wave perlod '" 4.3 sec. 1/150 seale model r'tai 一一一一一一一一一一一 (13 SCALE 1/50 g'::!;i1.805 cn 7 '" s.c Zo=63. Ic!l ~ h 1 心 \ 包 ~...~- ~ 一.-_ Ot6 斗 0.t122 斗 " 0.0% i ~ 才 ~ 斗 i:: f 的 0,0" -! 斗 1 一一 一一 τ- 一一,- 情 -, 一 - -r 一 一一一一一一!-'T- 一一 --,-, 一一一 一一 " 街 " 舟 I! ~n : 晶!! ~~ 11 Figure 12. Measured axial f~rce at universal joint for 1/50 scale rnodel % TllCE JN SECONÞ Figure 15. Stability at ratio of weight Ibuoyancy '" 0.93, wave period '" sec,1/50 scale rnodel I~_'},-l11. / /':- 十 /: 六三士了一三七一 Figure 13. Measured shear force at unlversal joint for 1/50 acale model -610 一

15 陸揖安黃明志 不文提供間接構架在無油輪繫泊情況下, 其在規則按中之動力反應及穩定之解析及實驗研究 流體動力負荷之計算, 係應用結構物與法浪問相對運動之修正莫利遜公式, 解非線性運勤, 採用逐步法 動力不穩定之研究, 於解馬遜 (Mathieu 中相關位置改變, 分別予以處崖研究 ) 方程式時, 其勁度變化包括由起伏力與波浪 萬向接頭處之軸向及側向負荷, 應用紐壘 ( Newman ) 細長垂直旋轉體理論計算之 為北海油田設計及使用之間接構架起原型, 以 1/50 及 1/1 印兩模型, 在柚槽中試驗, 其 結果與前述解析值就趨勢攻大小比較之, 結果顛示, 除 1/50 之模型之縱 1 搖及加速度太小外, 其趨勢皮大小均非常一致 動力不穩定現象在各試驗過期中均未發現, 解析時當增加重力 / 浮力比大於 0095 時, 不論就起伏力或起伏力灰位置同時變化考慮時, 於結構物自然頻率兩倍說四倍時, 均發現勤刀不穩定現象 利用紐壘理論計算所得之軸向及側向負荷, 均遠小於試驗值, 前述各封析沒試驗值間之差異, 猶待進一步就分析方法灰試驗技巧上予以研究改進 ' 國立成功大學造船工程學系教授 國立成功大學造船工程學系副教授 一 -611 一 -