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Development of a Small Cruising-type AUV and Training of Constant Altitude Swimming

by Taku Suto, Member Tamaki Ura,_ Member

Summary A small cruising-type test-bed vehicle named “Manta-Ceresia” is developed to ensure performance of various control architectures for Autonomous Underwater Vehicles (AUVs). Although it is so compact (13 kg in weight) that one researcher can carry and handle the vehicle, it is equipped with sensors and actuators required for prototype AUVs. Pairs of elevators and thrusters make it possible to move in 3 dimensional space. It is realized to swim along walls of a pool keeping constant distance automatically making use of 6 channels of range finder. Taking advantage of these characteristics, adaptive controller which requires comparatively long swimming for adjustment can be examined in a small pool. The performance of the constant-altitude-controller utilizing neural-network, which can accumulate experience by adaptation, is examined with this robot. Switching structure of neural networks is introduced to keep experience which is apt to be forgotten through additional learning. It is demon ­ strated by the developed test-bed vehicle that introduced switching system represents underwater terrain more precisely and the controller is adjusted appropriately.

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Fig. 1 Picture of small cruising-type underwater robot “Manta-Ceresia” DISCLAIMER

Portions of this document may be illegible in electronic image products. Images are produced from the best available original document p y 1- © k afvff ©flSK 207

Table 1 Specification of the robot —Axis of Range Finders asu t-'Xy Kd#7 t LJ—%% £M Manta-Ceresia A* 489 mm A* 634 mm 196 mm ESS* i3.8 e*wtsx 1.5 m

1 m/sec Thruster /Unit 2B#SS30 S'

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State Variables and Environmental Data

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Fig. 6 Neural network for constant altitude swimming

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EI S # * fj* B.A.A.P Balasuriya** za # # #** E# # #* II I E # st*** Visual Feedback Navigation for Cable Tracking by Autonomous Underwater Vehicles

by Motoyuki Takai, Member B.A.A.P Balasuriya Wan Chung Lam, Member Tamaki Ura, Member Yoji Kuroda, Member

Summary This paper proposes and demonstrates a cable tracking system based on a visual feedback navigation as an application for the environmental survey using AUVs (Autonomous Underwater Vehicles). The proposed cable tracking system consists of three levels of controllers : (1) The higher level controller decides the AUVs' steering mode which includes a mode to search the cable in case that the vehicle loses it. ( 2 ) The middle level controller generates target values to realize the steering determined in (1) by fusing vision data and other sensing data. ( 3 ) The lower level controller generates control commands for the actuators according to the target values. This paper also addresses the vision processing techniques including Hough transformation and the transformation from two dimensions to three dimensions which are necessary for the steering decision in(1 land the target generation in( 2 ). The proposed system is implemented on an actual AUV named the “Twin-Burger 2”. The perfor ­ mance of the system was examined though trials carried out at Lake Biwa on October 1996. The vehicle navigated for about fifteen minutes without losing the cable although the cable was sporadically veiled in waterweed and transparency of water was not good. It is shown that the vehicle implemented the proposed cable tracking system can be a practical and general platform for environmental survey in the predetermined area.

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©SSfCtot, T@3 7Fo-5i±f *i?*L©7 7f a t- 7 5###L, nzFy FC#*A&% 5 C kE j:!)y-77©MS'H 2 Descending Ascending Mode Mode

Control Target Command Vehicle Target State Controller * AUV ► Generator i l- V.______ft _ Position 3 Tracking Searching in Vehicle Coodinates 2D to 3D Vision Mode Mode Transforamation r Processing Cable Position in Camera Frame Fig. 1 Steering Mode Transition Diagram of the Cable Tracking System Fig. 2 Schematic Diagram of the Proposed System hg)Billcto*< 215

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Tw(Txw, Ty w> Tzw) Vehicle and Camera Coodinates World Coodinates

Where (Txw Tyyf, Tzw) - (Surge, Sway, Heave) x [pixel] (Rxw Ryw Rzw) = (Roll, Pitch, Yaw)

Fig. 3 Image Plane Fig. 4 Image and Vehicle Coordinate System 216 181 -t

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kTK AUV*:**:^|n|k)mR:iqI^?Ty-y;F$m%L ftV'ifCtSfcft, — 3E© Swaying, Yawing #ST*i# Kci^T, A., y ©#$-%» tQf ak»m^i=itme$'eat©kf a. ©3^0c1##^#5Ck*!pr#k»5. f,©zm#Z,B 4.2 Tti=i> h n — =7 V ftm ti/;S!IBggB* if C ± D ?# 6) ft B#m&%##KiD-5ii%iTaB##&&m-ra%& 6. C@3E#kS&B#FS-C##$W5#:&FS©ga:©2.a ©T&o y h a-9 k Lt, *H'Ci4* lpndf y h KisH Kam-f%kkEZD, *#m#^^©y-y;p©3ik7c a B#@iA%©##©&a PSA (Passive Switch Action) &%& a C k k » a. 3yhn-9""&#m-ra. PSAoyho-y^yyy 4. B j: t/Tfi 3>1p-7 h©yT±S?x:ef/t/&!&fL&!&BkL#V)±, ty 'D-V'(XK:*HTo/fXh#t#L, 4.1 takw^##tKpo. 4.1.1 PSAoy ho-7©#i#&Fig.6K:^1-. kkT, il± ir-y;i/m%t-FK:^^-cAuvK, #mmmsui%e AUVK#*@4t*:ty9-*^^6kLa^0m, %a^B# rate) (Altitude) %K:*MrL, #%Sa#Z D#A 6ft,a B#@, z.(=z„-a) Swaying is =t If Yawing ©fuBfllffll &tf ) f t C J: ^ T SE, ie §• Xe ©HiHikTh a0 Az, A: Z« Xe E%LT#Aa7'f-F/^y9y''f yk-^ak, o ~C%?Stltz'r~7*)V(0 3 ^5ctSI@* 1|y, Swaying £ iff Yawing KOV^-C©&#g##&^%f 6. Fig. 5 ICgrf i. a K, ttitiSil/cttSS AA’ ±tc ggs P(Target Point) a. B#^ p Auv ©y-f f s 7

&a. AITV ©»A##mAG (%'D) disturbances T5 L4©5£ H k U H frZ AA’ ET5 LfcSS HF ©, Force & Rate, Modd PSA Controller

deadzon

-^Target Point Detected Line(A-A')

Fig. 5 Swaying and Yawing Targets Plan View Fig. 6 PSA Controller 217

rze<0, Xe<0 or §@R#E#gtffd-a^6, ad?yh© — KlXer ify \P U\—\ *-Xe>On 0, Xe>0 kSr*fB^^3t©ki-%. y-yvp« 0 otherwise ^*5,-?tf'C|6]A^'TEBSrh-, ## •

fz«<0, %.<0, t,<0 w (6 ) K2.0m, 4.5mT*,5.mi&K:Km%&L, y-y^i± — KlXe if 1 m = i *■ Xe > 0, Xe>0, Xe > 0 k i:5HC3av##iT,-CVr.%. lo otherwise T@3>ha-9-r$SPSA3>ha-9r%m-r'5 %,=«] + % 7-i-p/fy^">'T>'SrTablel©j:^l:^*5. B#@ Surg ­ ing ©*kKB#@S &, X9X^©%#S-#AT%kkt 5. m fsn C Yawing $!] # E S' # /k tk 0.2[m/s] k i$t 3« Fig.7K^T Heaving SBPEdo ty<5 gSKStt CCD is f 7 ©Bft-’f’II I: ds ^ T # ft/k @###4: o #©^%a t#A L, 1.25 [m] k 5 . -y - y t -S y h © Twin-Burger 27)i;^SL> IlIiilMkSnllfflifitC — PT*tt, Surging ©T^TKESilStt 0[m/s] k L, Sway ­ A##KT It -3 fc0 ing do X Zf Yawing © B IS fit If /ID (i S' f: tl tb S fP Twin-Burger 2 K, 0.05[m], 3 [deg] k 6 « * /k AUV a^y - y;PimfT^ lmlkmEI#]a^»v^j:^Ey&/kK'E, 10, 30, 50, 70, 90, 110#©%S-C^:5lRl&aE3-&&. C k 5. n di y h B#m%#l 80 S©/NB i2o^*ry-y;p*!%^,T@»t^*^Ka y$/a>-%^ 7y%/k»E»±t-FE»fzy%. nd?y h k#Ak©B#&a6^kA;W#T$&o Twin- j: &m#%me#d3 z #mm& Fig. g- Buger 2 ©### Sr ftSI Surging 7? 1% ia^6 3E^f. ©**aa t f js i f & ^ Fig. 3 CgSLfc, Hpixel] k 6[deg] S-fflViT^T„ r\y

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Table 1 Gains in PSA Controller

K1 z2

Surge [N/(m/s)] 50 800

Sway [N/m] 110 300

Heave [N/m] 200 1000

Yaw [Nm/rad] 40 100 Fig. 7 The Twin-Burger 2 218 181 •§■

Vision Processing

J. 0.5 ■

Steering Mode

-" Searching 'in.. nn XIIBXmi ~~ Traeking cendi 3g -•+--— Wa

°0 100 200 300 400 500 600 700 800 900 1000 Time[sj Fig. 8-3 Experimental Results (Target and State) Fig. 8-1 Experimental Results (Vision Processing and Steering Mode)

Auv kti&o —Jj Surging is J: If Heaving ©SJfll

520#$TK:^^T«7-7/P A60[m]oE#

0#^(, 200#, 520#WKK:ot)T«, 7-7/P

Twin-Burger 2 7 — 7/1/1 M.% o X 7 — 7/P@ 8re-FK:#frfakkt,K:, 7-7/pe%^.LR(f7-

6. # W

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y-f, j3ZifB#*%sa#-?'T@3 7t'0-7k»a Time [s] PSA 3 > h n-7©#A&e-3A:.*%Xa7-7/P0@ Fig. 8-2 Experimental Results (Control Command) $ y h

— ~?)V\z. J: O XSilfe # % X IK 1) mm, #m, /i\&, min, a# rma-y-yA- b ? y ^ >y@&mm*#j# (f ® 2 SSNamCA#N: f-7;i/ b 7 y 4f >/#{$)J, B*)#iB#^^#, Tgg#4:o%Ky h»mv^->--y/ym%m^K:amL#E Vol. 175, pp. 219-226, (1994). 2) mo#*, rnmr^fb/pmis^n^vbom^k &f5S%. y-y;i/K:j:cT#@m#3aK:mS #W#BWRJ, Vol. 178, pp. 657 -665, (1995.12). 2fL%. H±, y i TA&#RLA AUV TT-;l/7>"Od;7 bj #B8 ~7 % —A h & 9 tho as^essitiRj, m«im, PP. 113-118, (1997.2). m »' 4) B. A. A. P Balasuriya and T. Ura, ’’Vision Based Tracking for Unmanned Underwater Vehicles”, Proc. of SICE’ 96, (1996.7). m»6^6oT*,o. 5) B. A. A. P Balasuriya and T. Ura, ’’Vision Based Object Following for Underwater Vehicles”, @*6#^#:%*, Vol. 180, pp. 663-668, (1996.11). Hh^/aLSfo 6) ###,#*, r#$LfT#m*o^yh®3)^Ch7

#%*, Vol. 180, pp. 677-684, (1996.11). 7) W. C. Lam and T. Ura, "Non-Linear Controller with Switched Control Law for Tracking Control of Non-Cruising AUV”, Proc. of AUV ’96 pp. 78 Bern -85, (1996.6). 8) T. Fujii, T. Ura and Y. Kuroda, ’’Development of a Versatile Test-Bed ’Twin-Burger ’toward In­ telligent Behaviors of Autonomous Underwater Vehicles”, Proc. of OCEANS ’93, Vol. 1, pp. 186- 191, Victoria, (1993). 221 2-10 7 Ab 7 —9 &WI1I

7 — h' 0

2# % # tr iEi

Preliminary Estimation Tool of Propulsive Performance for High Speed Craft based on Artificial Neural Networks

by Taketsune Matsumura, Member Tamaki Ura, Member

Summary In preliminary designs of high speed craft, it is often that main engines, reduction gears, and propellers are specified referring to accumulated trial data of actual craft, which is usually converted to design charts. These stern components should satisfy the requirements of both propulsive perfor ­ mance and practicable stern arrangement. The design knowhow of compromise between these requirements builds up by collecting the trial data. This paper proposes a trial database system for high speed craft, which consists of a collection of trial data, mapping neural networks (what we call a memory model) , and a descriptive neural network. The memory model, whose input is a design condition (length, displacement, target speed) , and outputs are the required horse power and the standard propeller specification, is generated by learning of the trial data. The descriptive neural network, which denotes the frequency of actual craft with the similar design condition, indicates the designer how conservative the given design condition is. The proposed database system is based on the trial data of 36 craft. The constructed neural networks set standard for the stern components, quickly estimate the propulsive performance, and reduce the number of iteration of the design spiral. When additional trial data is available, it is easy to modify the constructed networks, taking advantage of learning ability of neural network.

l. (i D A tc 0#w CFD##r#&ff p

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(m), Et (kts), BHP(PS) k LT7 E S 9

BHP/W/r), BHP/J ¥58 9 IF 1 H 10 H Fig.l j; o $36 %0 b 9 -f ^T E t 222 mm-#

225 200 a 175 6 150 1 125 ts < 100 75 50 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 Fn r = 0.83 Fig. 1 Cadm of high speed craft

100 125 150 175 200 225 a'&9©/? Estimation by Niwa chart

Fig. 2 Correlation between Actual Cadm & Estimation by Niwa chart

* o, amemf-©##?#; < #m 2 an: w a. Fig. 2 - TRIAL DATABASE SYSTEM - i— NEURAL NETWORKS . X- MEMORY MODEL ^03^" KS"^wA#^@k^#@k©itR% C^. ©B C Manning Net 0 (1) SPEED POWER CHARACTERISTIC T^L-C^aas, j=cTi±20%j^±©K^at*,9, C Mapping Net ) (2) STANDARD PROPELLER DESIGN POINT r « 0.83 C Descriptive Net 0 (3) FREQUENCY OF ACTUAL CRAFT WITH Learning THE SIMILAR DESIGN CONDITION a c kamm±#L <, a#imm##ekam COLLECTION OF __ TRIAL DATA 0§mB&BB&W&a#LA:_k-r, %*7mA%»Kj:a &S&L%9, ff-hmg@k0fBMt#^L-Cjsh, (3)m#(m*©ma, 7i A*^m^t#iii-&a#^kfa. ##K# #^©Km-#*e@i±, #%a#m#»§s#A©#emeUkUT, ^##»5#©§W(f-7%gK:, %*fa kLT®f-7^-xa;#gK*»aa:, s/%f A©#s@g7it#@Kf a. 7 )V^y 17-?( 1°) ©KH £fBfSfa^E i. V, XEfWIB ##, ^#Ti±/\-Ff4"f>'jl6mK:Za#Bya-t7 4--rL-D yfEoS^lES^raST*, 5SiEto»:9-* 2*##R^*#kLTt^aat, f©#m##K:MLTH $'to#e,a^^#7'-7"<-xi/XTA©#ie&M» Appendix l(A.l) K, S$tS$#tt^MLTti Appendix a. %*faS/XTA©#^(±Fig.3E#7j:^E, h? 2(A.2) Kf #Lf kL#g&asf #, — a —y h 7 — 7 -Y7;i/f-f&k, y h<'»Ej:aB«t7;K ©^tf/l/CRLT# Appendix 3(A.3)Kf ©#g& y h<"'a^a^9ao. it. ASr#efa7k»E, 2. h7-r7Arf?-^^<-7.'>XTA<7)S@ #f-Ejs^af-f-fyx/<'f9;Fti^*L, 2.i ^sSUmticfcttSTV-f >x/Hv;u

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Trim & Stability Check Preferable Trim Condition Intact & Damage Stability

Balance OK Speed Power Curve Main Engine Specification Neural Network Propeller Specification Model Basic Design Accomplished | Reduction Gear Ratio

Fig. 4 Design procedure of high speed craft Fig. 5 Memory model of present trial database system 224 181# k0@A m#@^%0 60K@m(L7W5^ yT/;FX&B, cmmcT, y%f %O-10##T#^7a^ y F 9-7 : y F A0A#»#df-< > F $rS«7 y F7-7A##$n.&. w F8 y F 8, Bg k a* 0#^ F 7-7 (*M#0-a-n>%4) k, AR r*0%m0@w^#A^#&4) k gt^#f^»M'=/-Mig0#AT*'%mmL, T.k#-wr K#MLT#^L7W%. ZfUaBZ k 1&- gEm#76—d0^y &k#@04%$ T#@$#3^kKj:D#Mg*&^k^T#5o ##T8, 1.2mT^LAM=36 0F7^7;kf-f& *#L< —#, @Ai#**%(Cad.) #7 5##77 ny|£8 5 kf 6 kK, R#F*4=#0#Fma# y* k L-C 100%±%m# %. Fig. ? cam r r* 6c #=G,C8AT0@#0#&S. F7-fr##75# (1) M. 0/|\g v^#*T8, B. *:* < T B. 0iSv\ w e, -=.-9/F$'y F7-70T@#%A^%/fk^y F & %o & /j\m%)mR0^*A^gg L -c w t. 7-70m#@KM73#R%*^a%»C:kE»D, R (2) M.AiA#<»ak, ^#0##4]'D«4:ea0 B.TB, 0A#V\ Wyi»5Am*aKK:$/7 F k ^ *#6 f, mmcKtr < mf*!##»0*\ LT^3. LA^Lf0&M:j&K:M6l'L-Ct^. flik6M%A%#%©^, a%F#AF*%76lT/»v^k (3) 7^F/\>y@[#0m77#tR#FAkL^ K%3. CCn±#.5#A*mRLTv^|@#$v F'"'& mBteb, M,=6.0—7.0 0@#cse?##73. #A7&CkEj:DZ0RlieK:*0!7&. —^,y^y 7799> F k LTH 0.4^B,^1.2, lOO^B.a: mem#? smm* y* k*#c, 600, 5.5<:Mn

Descriptive ^=6'=216 0A$mmL-CV^.# Neural Network #KBLTa;=6 0m%a^mv^T^=M' kLt. Cadm Mapping Neural Network Fig. 8 8, zme,0#af-7»6kK##g$lFA|»# *-y F©m#fi0#SHT, #@@m« 8,6406 = 20,000% Mn -» 432 7sT v ~7~C'h6c Table 1 C-a-n >P^0|g^-)ffi

k — 3. — n y©[li{B$'7ir'4"0@8 Fn-Rn VH$■—"O Rn* -► k L, M,

'Propeller Design Point k 0@g, %^0**0## K $ 6 ^ 0—1 ©m# &&-?-[ Mapping Neural Network C k#W#k»a„ S 6 f ;kC138 Fig. 6 Neural networks for present trial database &##$ y F 9-9 0m##K:*f system 225 3.2 mm^TMztnizmtZ'v h ±TR#*f4=k@K:D, y b K, y b 0#@ k 1.2 m #@0IRme*&Fig.9-12K:^fA*, BA^*,AB& ®36%Ob7-f7;Ff-f$-&kCfr^. If C^, fl0.95, 0.93, 0.97, 0.97 kL-n±ioo%±*mAm^img&Rof-f & k»o-CV)&. #KC^»K:MLTa, Fig. 2 k Fig. 12 $r BA AB%If Cm* 0@0#G lt#f6k, #f-f^-%y%TA(Df8MIW9#f + - filf jl.5—16, 28—55, 0.50—1.25, &tf50—220 KR%L, #/r0-l #C7° Fn

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& z k t#a. <=,#.&. Fig. 13 Kyo-tvfif-f > b0#:#^y b 7-

Table 1 Weight and threshhold values of D. N. N.

W0(1)= 0.800 KD = Mn hl(l) = 5.400 W0(2)= 2.500 1(2) = Fn hl(2) = -2.000 W0(3)= 0.004 1(3) = Rn hl(3) = -1.400 witi.i) 1 = 1 1 = 2 1 = 3 i = 1 3.868 -38.657 -17,705 h2(l) = -7.026 i = 2 -37.490 -0.948 -25.683 h2(2) = 0.245 i = 3 -8.811 -4.723 7.898 h2(3) = 4.748 Fn i = 4 -4.949 -0.199 3.974 h2(4) = 0.361 W2(i.i) 1 = 1 1 = 2 1 = 3 .1 = 4 Fig. 7 Distribution of trial data (M„, Ft, R%) i = 1 -50.909 45.562 74.515 -80.819 h3(l) = -46.521

Mn = 5.5 Mn = 6.0 Mn — 6.5 Mn - 7.0 Mn = 8.0

e 3

60 80 100 120 40 60 80 100 120 40 60 SO 100 120 40 60 80 100 120 40 60 80 100 120

Fn * 100 Fn * 100 Fn * 100 Fn * 100 Fn * 100

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Output of Descriptive Network

Fig. 8 Descriptive Neural Network 1.4

r = 0.95 r = 0.97

0 5 10 15 20 Network output Network output

Fig. 9 Cerrelation between Actual Bf and network Fig. 11 Correlation between Actual AR and network output output

Cadm

O 150

r = 0.93 r = 0.97

50 75 100 125 150 175 200 225 20 30 40 50 60 Network output Network output Fig. 12 Correlation between Actual Cadm and network Fig. 10 Correlation between Actual 8* and network output output

7 Horn LXG&kkk, to E 7° □ ^ 7$ < &5l®|iW;£>$ k k

%>o Fig. 14 E Cadm ©¥B* 7 h V- 7 ©tH^lS^A?^ am##E»5EdlfC31^k6 f. 36* L(i )# #. a#, LTV1< AR t$Elf-%© k»%E%cTC^.©#l±±#T6. EK-WXKM. Eoft-T Cadm (DM t> ±Wi~ 6. (3)ARA^#<»5E%W(#E1.0j^±), C.^© a:, m©aa-fk, :&3HkE #7 < » 6*l=l^mmftoE*A 6 fret) k ^ 6 y b K, % E t -I.-?y h ij/B 227

Bp* mapping 8* mapping AR mapping Descriptive net

Fn * 100 Fn *100 Fn * 100 Fn * 100

Output of ^ ^ Output of 55 0.5 Output of ^ ® ® Output of * ® Bp* mapping net 5* mapping net AR mapping net descriptive net

Fig. 13 Propeller design point mapping neural network

Table 4 K, 3-v ymoftnaffiAk-^ — u „ HftKiH Fig. 13 ®.t/Fig. 14 CO

-ylV^y J-9-7 ©ifi/fblg* C J: D, 513 &$R9, ## fUf, mi##Kj:9, McMLTH h7-?&##BT#a;:kB996a:'T&&. Table 2 Table 3 K, C^. y h KMLT B 228 181#

AR = 0.6 AR = 0.8 AR = 1.0 AR = 1.2 Descriptive Net

Fn*100

50 75 100 125 150 175 200 0.0 1.0 Output of Cadm mapping net Output of descriptive net

Fig. 14 Speed power {Cadm) mapping neural network

4. m m m

**:, Fig. 15KKC«d.0fBM&'^L-Ct

f/!/ K: j: 5 A &, 3t#"'""<.4xi.x:.) ®% ■So Cadm 0^##. y b Cll AR y b (A-E) Table 5 0 SLfco y A, B, D, ~BlXI E K©BSt 0, c % c t A* y h v—>; 229

&

tiS*C js»5KIRS* B//P*(PS) ©fiSBiStt, * y h y h 7 - 7 © $ fL & . 15%J%FSK:W:i|%&^Tw&a!, CjgTIi 3%T$6. @@ Table 4 Weight and threshhold values of Cadm Dpi m), SHB»Jt AR ©JfSBgElHCTfc, A, B, M. N. N.

%^ERTHef±#»©K^a*±i;-cv^a*, CRT# W0(1)= 0.800 1(0 = Mn hl(l) = 5.400 ffLffb, 3%, 2%, 5%k»-pTV^. Fig.l6K#*©K W0(2)= 2.500 1(2) = Fn hi (2) = -2.000 W0(3)= 0.004 1(3) = Rn hi (3) = -1.400 W0(4)= 2.667 1(4) = AR hl(4) = -2.333 wia.i) i = i 1 = 2 1 = 3 1=4 i = 1 -5.704 -1.734 3.290 2.216 52(1) = 0.521 Table 2 Weight and threshhold values of B* and 5* i = 2 1.464 0.013 -3.407 2.134 52(2) = -2.750 i = 3 5.005 0.143 0.294 -1.789 52(3) = 1.151 M. N. N. i = 4 0.714 -0.526 -1.338 -1.271 52(4) = -0.757 i = 5 -4.486 -0.554 -2.429 3.424 52(5) = -2.598 W0(1)= 0.800 Mn hl(l) = 5.400 W2(iJ) i = i 1 = 2 1 = 3 j =4 .1-5 W0(2)= 2.500 Fn hl(2) = -2.000 i = 1 -3.255 -2.473 -3.831 -3.166 -1.717 53(1) = 5.790

W0(3)= 0.004 s i c Rn hl(3) = -1.400 Wl(i,i) 1 = 1 1 = 2 1 = 3 i = 1 -3.636 5.971 5.696 52(1) = 7.871 i = 2 0.376 -3.642 -3.824 52(2) = -2.419 i = 3 0.905 4.425 0.920 52(3) = -1.077 Cadm i = 4 1.244 9.783 2.299 52(4) = -3.940 W2(U) j = l 1 = 2 1 = 3 1 = 4 i = 1 -5.569 -3.358 -2.792 -1.156 53(1) = 6.613 i = 2 -7.069 -2.154 2.259 -4.750 53(2) = 6.200 ® 150 (i = 1 for Bp* i = 2 for 6 *)

Table 3 Weight and threshhold values of AR M.N. N. -1 100

W0(1)= 0.800 1(1) = Mn 51(1)= 5.400 W0(2)= 2.500 1(2) = Fn 51(2) = -2.000 r = 0.93 W0(3)= 0.004 1(3) = Rn 51(3) = -1.400 Wl(ij) 1 = 1 1=2 1 = 3 i = 1 -5.205 0.933 4.941 52(1) = -2.786 50 75 100 125 150 175 200 225 \ = 2 -3.411 8.833 5.980 52(2) = 1.199 i = 3 5.294 -11.988 -12.012 52(3) = -3.005 Network output i = 4 1.274 -7.370 -0.565 52(4) = -3.126 W2(U) 1 = 1 1=2 1 = 3 J* = 4 Fig. 15 Correlation between Actual Cadm and network i = 1 5.164 -8.790 -6.685 -3.028 53(1) = 7.437 output for non-learned craft

Tale 5 Comparison of actual trial data of non-learned craft and the estimated by present database systen

Design Condition D.N.N. B p* 6 ^ AR Cadm* Mn Fn* Rn* Output Actual Estimated Actual Estimated Actual Estimated Actual Estimated A 5.73 1.17 330.1 0.00 7.80 5.74 31.68 28.79 1.20 1.22 159 147 B 5.07 0.97 194.1 0.00 9.41 10.08 36.68 36.71 0.75 1.21 140 122 C 6.33 1.17 308.8 0.95 5.75 5.69 28.93 28.81 0.94 0.89 184 179 D 6.93 1.02 263.6 0.03 6.55 6.30 33.83 30.08 0.70 0.69 158 142 E 7.53 0.77 356.1 0.00 8.48 9.91 33.85 38.27 0.90 0.99 137 139 Design Condition Graph BHP* (PS) Np* (rpm) Dp (m) L (m) A (t) V* (kts) Symbol Actual Estimated Actual Estimated Actual Estimated A 22.50 61.90 33.89 o 3820 4155 1193 842 0.90 1.16 B 17.90 45.20 25.05 A 1425 1637 1107 1106 0.83 0.83 C 21.55 40.47 33.10 ▲ 2320 2388 1064 1037 0.90 0.92 D 21.31 29.81 28.57 □ 1415 1577 1074 979 0.90 0.88 E 31.25 73.40 26.32 • 2335 2299 882 1039 1.01 0.97 230 mm-#-

5000 TRIAL DATA ESTIMATION %©l%k^9 F 9-96#^?# 5. A O ------B A ------4000 to # C A ------D □ ------E # ------— lra, ^#i%#±#©B#m^#emc- 3000 ~v)V^v h7-^KM1-5#a@S)W%S^rfc*D, m < 33lifpL$>if£to $fc, *im"e##KLtz&W$M,

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Vk (kts) 1) #m-#:^MiKK#m#(20), m©## Vol. 30 (1977), pp. 78-86 Fig. 16 Estimated speed power curves and actual trial 2) /M&ES, mom#: #mmk#m, mmmmiz data for non-learned craft mm## (1989), pp. 75-118 3) g*mmm#as: ## 5. |p Em mm), B^mm#m&@ dose)

9-x(2)3o^-h/Emi@immmeK:ok)T, m m, Vol. 51, No. 7 (1978), pp. 39-63 s) m±^)rmmmwK#K:m±*^)T@fmgy F9T7/Ff-^©@mK:#BL, 9 -xdo)iso F>mmmm "t&e", mm, voi. 53, No. 587 (1980), pp. 9-24 -^.-9,p^.yF9 6) Mmgm@#:mmmgBK Mmmm##B%/i/, — 9 4-fEfflLT, —f"<~ No. 019-4 (1993), pp. 47-48 7) msmmm#: #mm@B^, Mmmm«##e. A,, Aii, *aa, &tfXo No. 021-10 (1993), pp. 48-49 8 ) fEBWt- : HiSEE#, (1971) 9 ) #R5m:*-#me©-EM-md3), mo## %o-c, Vol. 47, No. 2 (1994), pp. 51-53 m^AK^v^T, *me#±0g$hEm±og^o/f7 10) ati%j5:3^^^3-XAk#@, gm ##©#§g 4 (1991), pp. 51-77 #ifF*©lR^^#»e#^W#Ek U^k. 11) ##' ^ ^ - 9 ;p^. y h #@# K#^yh&me@-e-CW5#R:j:0, #FmKa* %©K*k%#R##, B#mm###3i:#, Vol. 171 (1992), pp. 587-593 EM-W##%»© 0a^*Rl:%l 6 12) msmmm#:#mm@B*, M@mmm#*6A/, T#, ##e0Sg/<7>x&#-f AE#rg&«,%^-C, No. 032-7 (1996), pp. 66-67 ErHEK©$###cbtt*), H 13) Rammm#: grmmg B*, M@mm%#m g. A,, k T, T t K»&k V^9 No. 032-7 (1996), pp. 68-69 14) Mmmmm#:m@m#B^, Mmmmm#B('A„ f©#, -a- No. 019-4 (1993), pp. 49-50 9;E^.yF?-9©#@##K:Z0, f-f/t-XBBC is) #m?*ep = f^g%MnW34mrnt-f-9> 9 “CUSTOMS 1101”, mm, Vol. 53, No. 5o 591 (1980), pp. 55-63 i6) zmmm^r: mmm ‘t-*-^ 2” ©es, mm, SfcibU, f~? SSSEIfiityBOK^SS/cA 5, $#m Vol. 54, No. 601 (1981), pp. 50-66 TarLZrf—?/<—%S/XfAB, -a—9;>4"yF7 — Appendix ^KJ:6E%t7',y»#K.-Cj30, &RK:j:T)T#k,fLA A. 1 fcOB^KmibL-C^-Cwt. mEPBTktaaeWkf #-a©#K*ammd v m/\- F f ^ T >mm-r F 9 231

5.0 4.0 ., 3.0 g ■ 5 b d a c 0.6 - a'_TYPE CP ° 2.0 a 0.69 43.5 b 0.77 43.8 1.0 <» 0.4 c 0.81 42.9 0.0 d 0.78 40.6 4.5 5.0 5.5 6.0 LCB %LWL from AE B/d Fig. A-2 Csw of typical high speed craft AE 1 5 6 8 9 FP Station r - kc-ii , 4)

Fig. A-l Cp curves of typical high speed craft 4h9iaeB-7?n%,<7/-f k#^@Bt#OMK:me*l

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k C«d. # h o, k #K*1K: £., £„ M„ &# A£ ©M&k LT, L, ^©AAKSy-JE-TSSS Lv^i±j*% U k LT, £= mm, y2 (h-o,)'T-#tGfL&ai2#S,E &#&, £ e#/J\ ^ 6 gf gm* &# < # a. (#%K:«@/J\) k#a@Hk wSomt^fb^ta. @L# r kc-VT-VR-VH(Fn)-Vo(AR) , r \

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x\ = wj = w°Ii + h\ (7) Bilge Hulls, Second Symposium on Small Fast Warships and Security Vessels No. 18 RINA Xi=f(uf) {n> 2) (8) (1982), pp. 239-252 /(«) = l/(l+exp (-«)) (9) A 6) fmm- : SfiWi$EBM<19>, mm Vol. 56, No. 621 iic, uf: H w HIS z # @ © — j. — n y ©IHtli, (1983), pp. 62-73 tyg™ 1 : S » —1 ESS/ #g@—zz. —□ % If SI z # a 7) ARs# • AiEffia:: /j\g#mm©*m*m©*# g©-a.-nyv X©s-B'SS, )!?:inli!#|c-a m, No. 75 (m?), pp. 36-si A8) am@#-##:-:i-7/R&yhCi5#WK© mm#j@-*#m?©$mw©mm-, aAitm# z'#g©—i —n y©m^i, /,: AA, w°: AtlfrbAtiM #H;%W, Vol. 166 (1989), pp. 503-511 C^a-ny^®g^fS, h\ : AtlM/K:j:?-CfTi. taA* 581-586 £rirM-ifc5} 233 2-11 a ^ v h (RTV-SHIP) ©|$■

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Development of the Hull Inspection Robot (RTV-SHIP)

by Yoshihisa Nakata, Member Masatake Otsuka Hiroomi Ozawa, Member Makoto Konosu

Summary Recently, demands for increased safe operation and ocean environment protection, especially preventing oil pollution in the sea are remarkable as the international tendency. On the other hand, the trend of the shipping world indicates clearly a strong demands to reduce the cost for operations and maintenance of ships. From the viewpoint of survey and inspection of ship hull structure, the improve ­ ment of reliability of inspection, safety of work and reduction of costs for inspections are one of the most required subjects today. In order to correspond to the social requirements for surveys and inspections of hull structures, the authors are studying and developing the new monitoring system by the underwater RTV robot (MITSUI RTV-SHIP). The advantages of the RTV-SHIP are as follows : 1) All the tank walls can be inspected and easily recorded on video tapes. 2) All the operations can be controlled on the upper deck by minimum operators. 3) All the surveys can be performed on voyage. The basic tests of RTV-SHIP with regard to positioning in the tank, measurement of plate thickness and large diflection of panels were carryed out and satisfactory results were obtained. The utility of the RTV-SHIP was proved in this study and further tests for the actual ships are now planned with the aim of realizing this system. The RTV-SHIP is expected to make a major contribution to the safe operations and ocean environment protection.

i. at w

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241 2-12 A Study on Unified Automatic Control System for Longitudinal Motion of Jetfoil Based on Optimal Servo Theory

by Hiroyuki Yamato*, Member Takeo Koyama*, Member Akira Fushimi*, Member Sang-hyun Kim**, Member

Summary Current Jetfoil control has been made by platform and contour mode selection and by manual input of fore-foil depth according to sea conditions by human operator, which cause overload and may be difficult for operators. This requires the further automation in the control system. In this paper, authors propose unified automatic control system (ACS) which is based on optimal servo theory and eliminates human operation in ACS. The reference input signal in the servo system was modified to the sea conditions to achieve unification of platforming and contouring. First, optimal servo system (OSS) was designed to keep the fore and aft foil depths in waves to find that the sole OSS was still sluggish. Therefore, authors added proportional and differential element in inner feedback loop in addition to the modification of reference input. The MOSS+MRI, which represents modified optimal servo system (MOSS) and modification of reference input (MRI), was confirmed the good control performance in the wide range of waves through simulation.

may be impossible. 1. Introduction In this paper, authors propose the unified automatic Jetfoil is essentially configured to provide good ride control system for eliminating human operation in ACS quality and speed performance since it is free from based on the optimal servo theory. In the optimal servo water, and the control system is efficient to stable or to system, the reference input is used as the signal to augment stability of the vehicle* 1,~* 4'. follow and this may suitably be deformed to platform ­ The present automatic control system (ACS) of Jet­ ing and contouring according to the sea conditions. foil is designed by optimal feedback gain theory which First, optimal servo system (OSS) to keep fore and use inertial and ultrasonic sensor signals. And this aft foil depth in waves is designed. And the modified includes platform and contour mode selection and optimal servo system (MOSS) which has proportional manual input of fore-foil depth by human operator. and differential element in inner feedback loop is The Jetfoil copes with the wide range of waves by designed to improve control performance. And also the human operation in ACS. modification of reference input (MRI) is performed to The platforming is the normal foilborne mode of Jetfoil achieve unification of platforming and contouring. and regulates heave and pitch motion. This gives the Finally the MOSS + MRI, which consists of MOSS and best ride quality in wave heights up to strut length. MRI, is proposed as the unified ACS of Jetfoil. But among long and high swells, the Jetfoil needs 2. Jetfoil contouring the wave. In this case, contour mode is selected in ACS to avoid cresting and broaching. This 2.1 Configuration* 3''* 41 contour mode is keeping fore foil depth in waves'1''14'. The configuration of Jetfoil which becomes simula­ However, an suitable mode selection and manual tion model in this paper is shown in Fig. 2-1. input of foil depth may be difficult and overload for The principal dimension of Jetfoil is shown in Table operators. Especially very poor visibility or night, this 2-1. But the detailed fore and aft foil and strut type are not open to public, the type of foil and strut is deter­ mined by authors in this paper. And flaps were * Dept, of Naval Architecture and Ocean Engi ­ equipped in fore and aft foil to control the motion. neering, University of Tokyo. 2. 2 Coordinate system and equation of motion ** Graduate Student, Dept, of Naval Architecture and Ocean Engineering, University of Tokyo. The coordinate system is shown in Fig. 2-2. Authors origin at the center of gravity and positive direction of Received 10th Jan. 1997 translation and rotation are in the direction of the Read at the Spring meeting 15th May 1997 arrows in Fig. 2-2. 242 Journal of The Society of Naval Architects of Japan, Vol. 181

Fig. 2-3 The flap deflection

Wave Automatic Control System Selection

Fig. 2-1 Three view of the Jetfoil • Platform Mode Control |----- • Contour Mode Control Control Input

Table 2-1 The principal dimension of Jetfoil Sensor Signal

Length over all Loa 27.4 m Plant( Jetfoil ) Model

Breadth B 9.5 m Wave drift (hull borne) d 5.2 m Fig. 3-1 The concept of current Jetfoil controller drift(foil borne) d 1.7 m

Displacement Disp. 115 Lt Mrotal Force __ Mlirl Heave Force i M,surge Force 6 = (2.3) Depth (hull borne) 12.8 m Ivy Depth (foil borne) 15.5 m where z '■ have acceleration Ship Speed ^ship.speed 43 kn z '■ heave velocity Fore/Aft Foil Type NACA631-412 z : heave displacement Fore/Aft Foil Type NACA66-006 6 : pitch angle acceleration 8 : pitch angle velocity 8 : pitch angle X : total surge force

Z : total heave force Pitch Angle m ." mass of hull Mrotai Force ’■ total force moment around y axis lyy '■ moment of inertia around y axis 2.3 Flap-controIt5H6) The flap-control is used to control Jetfoil motion in this paper. The configuration of flap-control is shown in Fig. 2-3. In case of flap-control, lift is shown by

Fig. 2-2 The coordinate system Foiluft =-^P‘V2‘^Cl + 8 j• Sfou (2.4)

where S: flap deflection angle The longitudinal equation of motion is given by 1) Surge motion 3. Automatic control system of Jetfoil 3.1 Current Jetfoil controller 111-'21 Detailed design method of current Jetfoil controller is Hulhc Foil * = -z8 + L+ not clearly states in references. The concept of Jetfoil controller is shown in Fig. 3-1. , Strut* Hullth (2.1) In this paper, platform mode and contour mode con ­ trol system of Jetfoil are designed by tuning weight 2) Heave motion function in optimal regulator. And we define Zm (relative distance bow and wave), dZ (heave velocity) , Ftullweie Foilb-, 8 (pitch angle) , d8 (pitch angle velocity) as state = £9 + L+ m m variable of Jetfoil. The configuration of ACS of Jetfoil Stmtbuoy | Foiluft by optimal regulator is shown in Fig. 3-2. + ( . ) m 2 2 This optimal regulator controller was used as a base 3) Pitch motion controller to be referenced as present one. The primary A Study on Unified Automatic Control System for Longitudinal Motion of Jetfoil Based on Optimal Servo Theory 243

Automatic Control System

Modification of Optimal Servo System Reference Input Desired Zrb

Error signal

Calculation of Error signal Plant( Jetfoil ) Model

Desired Fore/Aft Fig. 3-2 ACS of Jetfoil by optimal regulator Foil Depth

Fig. 3-3 The concept of proposed ACS of Jetfoil problem of current Jetfoil controller may be considered that the contour mode does not work so well, since using only fore foil depth. In particular, it becomes serious in head sea. 3. 2 Proposed ACS of Jetfoil in this paper In this paper, authors propose ACS of Jetfoil which is based on optimal servo theory for keeping fore and aft A--[i ?] M”J we % Mil foil depths in waves. The features of this ACS may be (4.4) considered as follows : And in infinite time, x{t) and u(t) become xs, us which 1) The reference input in optimal servo system can is given as be modified to the sea conditions. 2) The unified ACS of Jetfoil which has no platform From the previous results, we can define dx(t), Su(t) as and contour mode selection and manual input of fore dx(t)~x(t) — xs depth may be realized by using reference input. du(t) — u(t) — us (4.6) 3) Improvement of performance of contouring to Let wave profile, since using fore and aft foil depths in ACS. &c.(f)=[&r(fy 5u(<)T]T- And the concept of proposed ACS of Jetfoil in this Equation (4.3) can be written paper is shown in Fig. 3-3. 8Xe(t) = AeSxe(t) + Bev(t) + Ded(t) . . 4. Optimal Servo Theory (7)~ai) y(t)—r=CeSxe(t) Consequently, we can suggest that the control prob ­ Optimal servo theory is one of the method to design lem of y{t)~^r as replace the control problem of linear multi-input-output optimal tracking system. It $xe(t)—>Q as f-*o° which may be achieved by using is common to associate an integral compensator with optimal regulator theory. Namely, if the performance the given plant and to apply a stabilizing control law to index may be expressed as the resulting augmented system. And the optimal regu ­ lator problem is extensively used for the stabilization. Je=fa (||&re(f)|||.+]|r(f)||*«)aif (4.8) In this chapter, the abstract of optimal servo theory where and modified optimal servo theory is explained. Qe(n+ m x n + m). = n + m X n + m) > 0 4.1 Mathematical formulation Re(mXm)=Re{mXm)> 0 The linearized equation of motion written in state The control problem of y(t)^>r can be replaced the equation form are optimal regulator problem for minimizing equation x(t)=Ax(t) + Bu(t) + d(t) : A(nXn), B(nXm) (4.8). y{t)=Cx{t) : C{pxn) The solution to equation (4.8) is given as (4.1) v(t) = ~Fe8x e(t) (4.9) where, ,r(0) = -ro, d(t) is disturbance, m=p and plant is where controllable and observable Fe = Rl1BePe (4.10) Here, we consider servo system that output y(t) tracks Pe=PI>Q satisfies Riccati equation reference input r(=#=0) in infinite time. In this case, Substituting feedback form of state variable x(t) and control input of servo system becomes constant value ye(t)=i—y(t), equation (4.9) is given as for infinite time and also satisfies limii(<)=0. M) = -[f K^Jc^^-Fxifi + Kir-yit)) If we can use v(t)=u(t) (4.2) (4.11) as new control input, the equation (4.1) is given as From integration of equation (4.11), we can get Xe(t) = AeXe(t) + Bev(t) + Ded(t) , . control input as g(f)=C.a.(f) ^ u(t) = — Fx(t) + K£ ye{t)dt + Fx(0) (4.12) where 244 Journal of The Society of Naval Architects of Japan. Vol. 181 where initial condition is .r(0) = 0 x(t) = Ax(t) + Bu(t) : A(4X4), £(4x2) . , This provides the control law. The block diagram of y{t)=Cx{t) '. C(2X4) optimal servo system (OSS) is shown in Fig. 4-1. z 0 0 1 0 z 4. 2 Modified optimal servo system Q 0 0 0 1 Q In simulation, OSS was found sluggish. To improve z -0.80 -94.65 -4.74 15.94 z performance of OSS, authors tried tuning of weight 0 -0.001 -0.39 -0.019 -2.20 e function in OSS. But this needs excessive deploy of flaps. 0 0 Therefore, it is not theoretical, authors attempt to add o o ra/i + proportional and derivative element in inner feedback -7.141 -20.567 LdJ loop of optimal servo system to improve control perfor ­ 0.637 -0.767 mance. This modified optimal servo system (MOSS) is shown in Fig. 4-2. 2 [Z/Ziri -13 0 01 8 (5.2) 5. Design of Unified Automatic Control System Izafi Ll 5.8 0 0J i 5.1 State equation 6 The longitudinal equations of motion are where 8/ : fore-foil flap deflection angle Sa : aft-foil flap deflection angle Disturbance Zff : heave motion in fore-foil Servo Compensator Stabilization Compensator Zaf : heave motion in aft-foil

Reference Input lYe^« 5. 2 Reference input x=Ax+Bu+d The reference input is a error signal between desired foil depths and actual foil depths in waves. This refer­ ence input is equal to the necessary heave displacement in position of fore and aft foil for keeping desired foil depths. And this reference input is deformed to cope Fig. 4-1 Optimal servo system (OSS) with sea state. The reference input in waves is shown in Fig. 5-1.

Disturbance

Servo Compensator Stabilization Compensator

Reference Input lYe^tV x = Ax+Bu+d-

Fig. 4-2 Modified optimal servo system (MOSS) Fig. 5-1 The reference input in waves

Disturbance

Servo Compensator Stabi lization Compensator Modifying Error signal Desired Error signal | 1 Reference I Input foil depth x=Ax+Bu

Foil depth in wave Zff, Zaf l di, 9, de Observer

Distance from jerfoil to wave

Fig. 5-2 MOSS + MRI A Study on Unified Automatic Control System for Longitudinal Motion of Jetfoil Based on Optimal Servo Theory 245

5.3 MOSS+MRI 6. Simulation The MOSS + MRI, which consists of MOSS and modification of reference input (MRI), is designed to 6.1 Simulation tool improve control performance and to achieve unification In simulation, we used MATRIXx design tool which of platforming and contouring. The MOSS + MRI is had been developed by Integrated System Incorporated shown in Fig. 5-2. Authors propose the MOSS + MRI as (ISI)(12). The simulation model of Jetfoil which was unified automatic control system (ACS) of Jetfoil in constructed by MATRIXx is shown in Fig. 61. this paper. 6.2 ACS And also, state variables, which are estimated by The ACS which is shown in Table 6-1 was simulated observer is used, because of all state variables of plant in this paper. can not be observable in general. In simulation, the control-limit of ACS is defined as 5.4 Modification of reference input broaching and cresting. In ACS of current Jetfoil, fundamentally, the contour And also, authors use dominant wave component of mode is selected for long and high wave height, and the irregular waves to estimate control performance of platform mode is selected for short and low wave ACS. Namely, the criterion to estimate ACS becomes height <1H4). whether ACS is effective or not in dominant wave of Authors refer to these operation in modification of irregular waves. The Pierson-Moskowitz (PM) spec­ reference input. In this paper, authors use gain and low trum of irregular waves with each significant wave -pass filter in modification of reference input. The aim height is shown in Fig. 6-2. of using gain is to reduce reference input for platform ­ 6. 3 Simulation results of ACS ing. The example of gain is shown in Fig. 5-3. In case In simulation, Jetfoil speed is 20 m/s. of low wave height, the gain becomes small and reduces 1) Platform and contour mode control reference input to strengthen platforming. The control-limit of platform and contour mode The low-pass filter is used to cut high frequency compo ­ control in regular waves and dominant wave of irregu ­ nent from reference input for platforming. The cut-off lar waves is shown in Fig. 6-3. frequency of low-pass filter is determined by dynamic Fig. 6-3 shows that platform and Contour mode control characteristics of plant and characteristics of encounter is satisfied by platform and contour mode selection in waves. In case of cut-off frequency 0.5 rad/sec, 1.0 rad/ following sea, but it is not satisfied by platform and sec, 1.5 rad/sec, the modification of reference input by low-pass filter is shown in Fig.5-3. Table 6-1 Automatic control system of Jetfoil

ACS of Jetfoil System Purpose Contouring Keeping constant heave height Platforming I Platform Optimal regulator Contour Optimal regulator Keeping foil depth in wave Platforming 0.5 1.0 1.5 Wave Height OSS Optimal servo Keeping foil depth in wave

Cut-off frequency of low-; To improve control performance MOSS Modified optimal servo against OSS

Fig. 5-3 The concept of modification of reference Modified optimal servo and To unification of platforming and MOSS+MRI input Modification of Reference Input contouring in MOSS

JETFOIL 3A7A rrrfESD

Continuous

WAVE PROFILE iiJ WAVE DATA ill

Continuous Continuous

Fig. 6-1 Simulation model 246 Journal of The Society of Naval Architects of Japan, Vol. 181

Encounter Ffequency(rad/sec)

Fig. 6-4 Control-limit of OSS and MOSS

Heave Acc. in FP(Max. & Min.) in Wave Height 1.0m

Frequency (radfeec)

Fig. 6-2 Pierson-Moskowitz (PM) spectrum

• - Dominant Wave(Follow Encounter Frequency (rad/sec)

Fig. 6-5 Heave acceleration

Encounter Frequency(rad/sec)

Fig. 6-3 Control-limit of platform and contour mode control contour mode selection in head sea. This result Encounter Frequency!rad/sec) confirms necessity of control system which is improved performance of keeping fore and aft foil depths in Fig. 6-6 Control-limit of MOSS + MRI and MOSS waves. 2) OSS and MOSS The control-limit of OSS and MOSS is shown in Fig. ing is strengthened by modification of reference input. 6~4. And the control-limit of MOSS + MRI and MOSS is Fig. 6-4 shows that OSS and MOSS are more effective shown in Fig. 6-6. than platform and contour mode control in the wide Fig. 6-6 shows that the control-limit of MOSS + MRI range of waves. It is because OSS and MOSS may be becomes narrow against MOSS by modification of refer­ improved performance of keeping foil depth in waves ence input. And MOSS + MRI is more effective than by using fore and aft foil depths in ACS. platform and contour mode control in wide range of And also, Fig. 6-4 shows that control-limit of MOSS is waves, even if it has no platform and contour mode wider than OSS and is satisfied with in following and selection in ACS. head sea. These result confirms validity of MOSS. But MOSS+MRI is not sufficient in head sea. This 3) MOSS + MRI result confirms the necessity of more effective The heave acceleration is investigated in terms of modification of reference input, in terms of ride quality ride quality. The max-min heave acceleration of plat­ and range of control-limit. form mode control, MOSS and MOSS + MRI is shown in 4) Simulation result Fig. 6-5. Fig. 6-7 shows the simulation result of MOSS in Fig. 6-5 shows that ride quality of MOSS+MRI is much regular wave (wave height 1 m, encounter frequency 5. better than MOSS. This result confirms that platform ­ 4 rad/sec). And Fig. 6-8 shows the simulation result of A Study on Unified Automatic Control System for Longitudinal Motion of Jetfoil Based on Optimal Servo Theory 947

fled optimal Se.^-e Sy

- n n fri n n n n i 1 n n n n i n n n n inn jjjjj

Fig. 6-7 Simulation result of MOSS in regular wave Fig. 6-8 Simulation result of MOSS in regular wave

MOSS in regular wave (wave height 2.5 m, encounter 4) Weist, W. R. and Mitchell, W. I.: “The Auto ­ frequency 1.0 rad/sec). These result shows that MOSS matic Control System for Boeing Commercial is achieved unification of platforming and contouring, “JETFOIL””, IEEE, NAECON, pp. 366-375(1976) 5) Ira, H. Abbott and Albert, E. Von Doenhoff : even if it has no platform and contour mode selection in THEORY OF WING SECTION, Dover Publica­ ACS. tion, New York 7. Conclusions 6) John, D. Anderson : Fundamentals of Aer­ odynamics, McGRAW-HILL In this paper, authors develop unified ACS of Jetfoil 7) Pak, P. S. and Suzuki, Y. and Fujii, K.: ’’Synthe ­ to eliminate human operation in ACS. And the validity sis of Multivariable Linear Optimal Servo-Sys ­ of the proposed system was confirmed by simulation. tem Incorporating Integral-Type Controllers” , Transactions of the Society of Instrument and 1) The unified ACS was effective in the wide range Control Engineers, Vol. 10, No. 1, pp. 1-5 (1974), of waves without mode selection in ACS. (in Japanese) 2) The design method of reference input in optimal 8) Takeda, T. and Kitamori, T.: “A Design Method servo system was proposed. The unified ACS could of Linear Multi-Input-Output Tracking Sys ­ provide better ride quality, however, worse control- tems”, Transactions of the Society of Instrument limit. and Control Engineers, Vol. 14, No. 4, pp. 13-18 In this regard, effective modification on reference (1978), (in Japanese) 9) Fujii, T. and Mizushima, N.: “A New Approach input or tuning is necessary. And, in general, heave to LQ Design —Application to the Design of displacement and foil depth measurement system Optimal Servo Systems —”, Transactions of the becomes essential concern for the actual application of Society of Instrument and Control Engineers, this system. Vol. 23, No.2, pp. 129-135 (1987), (in Japanese) 10) keda, M. and Suda, N.: “Synthesis of Optimal References Servosystems ”, Transactions of the Society of 1) Saito, Y. and Kuroi, A.: “Active Control System Instrument and Control Engineers, Vol. 24, No. 1, of Hydrofoils ”, Prediction of Seakeeping Qual­ pp. 40-46 (1988), (in Japanese) ities of High-Speed Craft, The 6th Symposium of 11) Fujii, T. and Simomura, T.: “Generalization of The Society of Naval Architects of Japan, pp. 74 ILQ Method for the Design of Optimal Servo -110 (1989), (in Japanese) Systems ”, Transactions of the Institute of Sys ­ 2) Saito, Y. and Ikebuchi, T.: “Fully Submerged tems, Control and Information Engineers, Vol. 24, Hydrofoil Craft”, Prediction of Seakeeping Qual­ No. 1, pp. 40-46 (1988), (in Japanese) ities of High-Speed Craft, The 7th Marine 12) Integrated System Incorporated : MATRIXx Dynamics Symposium of The Society of Naval V 4.0, ISI (1994) Architects of Japan, pp. 107-141 (1990), (in 13) Kim, S. H.: “A Study on Longitudinal Motion Japanese) Control System of Jetfoil to cope with Sea- 3) Imamura, H. and Saito, Y. et al.: “Automatic state”, A master’s degree thesis (in Japanese), Control System for Jetfoil ”, KAWASAKI TECH­ Dept, of Naval Architecture and Ocean Engineer ­ NICAL REVIEW, Vol. 107, pp. 1-8 (1991), (in ing, University of Tokyo. (1996) Japanese)

249 2-13

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Basic Studies on Accuracy Management System Based on Estimating of Weld Deformations

by Toshiharu Nomoto, Member Shoji Takechi, Member Kazuhiro Aoyama, Member

Summary Accuracy management of products is one of the most important issues in industries. There are two significant ways for accuracy management. One way is accuracy control by measuring products in assembling stage. The other way is “accuracy planning ” that is accuracy management in production design stage. Authors have already reported about the former, and the latter is discussed in this paper. Authors have been engaged in development of SODAS for shipbuilding CIM. Therefore authors developed accuracy planning system considered weld deformation based on SODAS. For accuracy planning, it is important to estimate deformation. Authors practice modeling of weld deformation of products and make it possible to estimate easily welding deformation. By using them, the accuracy management system by estimating of weld deformations is implemented.

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Development of Computer Aided Design and Manufacturing System for Advanced Composite Marine Structures (4 th Report : Investigation of Computer Aided Manufacturing System in Fitability)

by Takeshi Takatoya, Member Isao Kimpara, Member Kazuro Kageyama, Member

Summary The present paper aims at developing a computer aided design and manufacturing system for advanced composite marine structures, that is, shell structures in which stiffness and strength for out- plane pressure and light-weight are achieved by making efficient use of advanced composite materials. In the previous reports, some specialized prototypes of CAD/CAE system for laminated materials were developed by means of object oriented language. It showed that the developed system is effective in designing and analyzing laminated panels. In this report, specifications of the system are discussed in the point of developing an effective system. Based on the specifications, two points are important in the systems aided for manufacturing, one point is making tool mold with easiness and accuracy, and the other is layup simulation on a curved surface. The system aided for making tool mold was developed by means of generating surfaces which are apart thickness from the base surface. By using a numerical cutting machine with cutting paths on the offset surfaces, it is available to obtain an accurate male mold, which is achieved smooth outer surfaces. In order to simulate of prepreg layup process on a curved surface, three deformation mechanisms of laminated materials are investigated in the point of fitability, plastic deformation, shear deformation, and apparently deformation induced to out-plane deformation. According to observations, unidir­ ectional fiber yarn is discrepancy from the curved surfaces without some points, and it induced that fiber wrinkles in a vacuuming process. Then a criterion of fitability is proposed by means of calculating apparently in-plane deformations. Some examples are demonstrated, and it shows that the developed system is effective in manufactur­ ing simulations of laminated panels.

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Developed Systems CAD Shape Design Laminate Design Product Design Lamina Laminated Structure X \ r...... CAE ...... : ...... CAM ...... ; Mechanical Analysis i Mold Manufacturing i Thermal Analysis Weight Estimation fiber direction ! Optimization Laminated Simulation Layup Plan Automatic Layup Fig. 2 Stratum of laminated composite structures

Fig. 1 Flowchart of laminate design and process design for composite structures Fig. 2

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Fig. 4 Graphical User Interface of the developed CAD systems

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271 3-1

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5

A Consideration on the Elastic Response and Design of Deepwater Riser

by Hideyuki Suzuki, Member Koichiro Yoshida, Member Tomonari Ishizaka, Student Member

Summary Research and development of deepwater risers for scientific drilling and offshore oil development are now under way in Japan. Depending on the purpose of riser, target water depth varies from 2000 m of offshore oil development to 4000 m of scientific drilling. Water depth of 4000 m is far deeper than the world record of deepwater drilling with riser, and the technology to be developed in the research and development will be one of the most advanced one. This technology will be also applicable for other fields such as sequestration system of carbon dioxide into deepwater and so on. This paper discusses overview of dynamic elastic response of a deepwater riser and design of the response. It becomes clear that a lighter and more rigid riser is generally desirable for deepwater from a viewpoint of elastic response. It also becomes clear that the distribution of buoyancy material must be designed so that the initial tension distribution is optimized and compressive axial force becomes hard to excite under dynamic tension fluctuation in hung-off condition. Two types of coupled response of riser, riser-mud coupled response and longitudinal-lateral coupled response of riser, which might have significant effects on the design and operation of a deepwater riser are chosen and examined. From a series of calculation, it is understood that dynamic response of mud has significant effects on the dynamic tension induced in the riser. Natural frequency of mud column is around 12 seconds for 4000 m riser case and a sharp peak is observed in the frequency response of riser tension. The peak height is still significant even when highly viscous mud is used. In a hung-off condition LMRP will need to be opened under severe weather condition. It is shown that the lateral response, so called parametric oscillation induced by tension fluctuation under longitudinal vibration, will not be a serious problems for real 4000 m riser case. Deepwater risers are designed so that the compressive axial force will not be induced even in hung-off condition. Under this condition, some amount of lateral drag force which is usually expected for normal configuration of riser will suppress the lateral response. But it is also shown that if a large longitudinal response is applied or lateral drag is very small, lateral large response can be induced.

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0 ' "C1 + C2 " U~ Wlr ~ U — c p©K#©#Bri±tf-7a#<»a»v^ 77P©@w + 0 W-m_ . V. . —Cz Cz . . V. )@%©tf-^i±&7.^^P/l/#BC^$f|,-Cj30, c©^ 'EAr 0 ' u"' >7-5* kKLMRP$rMCT7-fif-^^K:7 7 + =0 0 EAm, .v" _ .{J-mQ. ]|BT/\yy^7L7k#e, 7 7P&Ztf7-fif-K:#'»D 7-fif-&K^I=IK:)gK&(uTlm@Lft#-g-K:3^T, (5 *@»#;%a!±Uaam#&aTLTwa«, %R©&*TK: j= ^ c 'u "zfdW mmf&ckai^gTto, 77F&m#L%ms/\:/y + . V. _yd(%)_ ^-7fac:kamL< LMRP Ck*ii2'Bk»ak 275

no relative friction viscosity=120cp 1.46+07 depth at 3600m — depth at 3600m —

3e+07

2e+07

4e+06 le+07 2e+06

0 1------'— ' ------1 1 1 15 0 5 10 15 20 25 30 period(sec) period(sec) Fig. 4 Frequency response of tension of riner induced Fig. 7 Frequency response of tension of riser induced by riser-mud coupled response (mud: no fric­ tion) . by riser-mud coupled response (mud: 120 cp) .

Table 1 Priincipal properties of a riser used in the mud viscosity=20cp 2.5e+07 numerical calculation. depth at 3600m — Upper Half Lower Half O.D. (m) 0.4064 0.4064 I.D. (m) 0.3556 0.3734 Weight of pipe (kg/m) 239.0. 158.9 Auxiliary pipe (kg/m) 107.2 107.2 Buoyancy (kg/m) 120 (density 0.45) 120 (denslty0.63) Weightofriser Wm) 466.1 409.4 Weight of mud (kg/m) 143.0 157.7 Weight of riser in water (kg/m) 179.9 198.2

15 4. mmm-tzt> period(sec)

Fig. 5 Frequency response of tension of riser induced by riser-mud coupled response (mud : 20 cp) .

4.1 depth at 3600m — 1.8e+07

(mp + mm-\-rnu}u —cdu +EAu"

+-^~EA(w'2)' + y7<7=0 (10) 1e+07 (nip + m, + mw) w — EIw""+EA{ u' w')' —IzpCdD — (w — v)=0 6e+06 (11)

, mP= y 4SBSEfrS VYlu 2e+06 SKlalfiftEite:, ma,=@S‘|tofTSD*e, % = £A=7-ff-»iI 15 w=?4^-

k, c: k**T#, 0 *KSK«HA:k»fE#kBLa:K:#< CkA4T#6c#(±#

500

u=£^-x (l—+ Cix + C2 (12) 1000 #Tt 9 %© Z ^ K##fb

Jo K=e^g*" 1500

'e «> \ ^ 2000 ■J co2+d2) I 2500

d=____ Si____ cz=----- EA___ (14) mp + mm + mu' mp + mm + mu 3000 LAAicT, u = [eMcos(y?x + cot) e“sin(@x + cot) e~axcos( — 0x

Ci 3500 02 + wt) e “sin( — ffx + a)t)\ (15) Os 4000 .<2:4- -1 . -0.5 0 0.5 1 deflection(m)

£ ti% (11) & Ctt\-f JiiCiOftfc Fig. 8 Lateral parametric oscillation induced by longi ­ S* 5f#5>#i5io tudinal oscillation (longitudinal oshillation : 4.2 h U -y amp. = 8m, angular frequency=0.6rad/sec ; lat­ K@*©#u t Tkk) »©)g#^@^;(11)3%KRA LT eral response : angular frequency=0.3rad/sec ; plot interval = 2sec).

W*#iB3Wkk-r& #&9. 7tk»(6#®m6^KgK:mgg^TW5,'.'% w = {x)a{t) (17) fU&Fig.SKarf. C©BD±y^-y-±%E@ Cjbt3tB^@^;(16)K:M;AL, t-Fm»k©m#^% 118m, JffflSfc o = 0.6 rad/sec ©±T^tR]A[l#^7H]A7k# & k#%#©&5-7y^-m©3ESS:5@3Sa4#f,fL6. ■aOl&MflflltiS. ISESl&IOMCJ: Dfcfo&TJlRl© a(t)+ Ca(t) + {(F+ G) — 2Hcoscot}a(t)=(> jbTV)% z kasbi&'&c ;©i^, ESirI© ■ K, r^'dk fE/fdr o/2T*>5. Ld>L%*3&, y/f-f-©^#©##^^? r> J 0______zr ->Q______f'—JS*______V— fL > r — fL » l~r— ri » HE*A©#±AWg^jbf. 3(,K:, i§*^©**^© Jo/ m2dx Jo/ m dx Jo / mcfrdx fNBAi/fy/ b 1; 6 Ck*) H=i>d- ’dx 6, ^!#©9d-f-©a^K:gA:^-n±A;6»)S#©#e (18) ©#aa«)%g»V^kA*^©j:^K^-&6. $f, 7k 2 mtp 2dx fc^.©SE^®5$(ll) kKIEEi©® U &RAWz±T, a

m#fk G>2Hk»6. LTk^-^T. %^^9Ao. kEj: 9f5gt3^Ef &±5»/t)w'Y=0 (16) z©A#^, Fig. 9 ©#aai»t>##"7fa.-m3E@a:5@ :;e, m(x)=WmWMb^mz ;7d^-^^(D aK b 9 **, T.W=*4:m*K: T„W=*&# kjtR-f 4 k, -FSSl£l:$-St'RllE'l4l4^& 9/h$ © = 277

> tension limit — F+G > 2H divergence when c=1000or ♦ ♦ divergence when c=300o» 4- divergence when c=100tv ■

Stable Unstable

Stable Unstable

Stable, Unstable

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Unstable vertical oscillation frequency w (rad/sec) Fig. 10 Calculated stability of lateral parametric responses due to longitudinal oscillation (lat­ eral drag c=1000to, 300to, lOOcu).

Unstable 1 c©#e, Unstable AL»w$)g&fHic>250kg/(msec)k#&. C©#lf ^m©»gk aim c#k »&©-?, 5^ Fig. 9 Stability judgment of parametric oscillation of deepwater riser.

ka%M3. 26K:, », f6#a#K±©%#b:KS&-f 9 $ y f:lkk»%. 3#^e^(lG)Km^-C8/9-Xgf-#*- (20) e^^n@*&Fig. locarf. C^j;9±T«9'f'f-K:EmilA!±l:-C^6 ck&#L-cw6. tifcJiftliDffitl c=1000to SfSL'fcS'B'KoV'T, ±T 13 IrJ ©MIlStM SliLTvxo tzMi-& E fc t> frfc&imWc? V^iirn So "T b F©SMS* 5#! ;k SSnffiSfiSrTjxLTdo 9 , tz k %.i£, cu=l rad/sec ©So",

©#a-KA c i±##mm© 2#Kitg!iL^RA»ie#fkL %%&a&&wE##T5#k»^Tir>6,, dOBEfft if, omrat$ft.&a%©Ax*9/|\g» c^iooto ©#-e, C ^ 2k7 > (21) a&w;e±L&< -c&/<9/ b 9 y9##E f 9 2a>o ^ 4(F+G) 1SJ1 zSErtinS$ n sS-n f tivjx$ h;F©#@TK/< kLTF'&##f 6. k if-0%^l=I 9 / b 9 y?m#B&U»WS, (5E@%E**ai/j\^» oambtmak7'ff'-*4'%TwmgojtK»5. #^-E » Z k ^ S. 0.5 rad/sec WTT K: 9 -f & k f & k C j: -) KK

ZfWf, )@&%E*#L (22) /k%BMSa*#@E/J\g<»5CkEj:6t©TtS. Z Z.Z.T, —$Jk LX m = W}aWMb^^tz^-^i ’Smim ©a&E#%K"3V^-C*##R9lt#aLT(10), (11)5% kg/m, « = $6^|n]jra$EiE5m, to = 6to|R]MIM$lB 1 rad/sec, /ug=9d" If—©SSSTSlKI® 5 m/sec2©/®^ m@@©9 t y blf^f 9;t/»^Tk 278 B #§###%&%* #181#

5.i# m Fig. 10 Kjmg#@$#LTt^A##K, k(Dhtz D fr Flg.llI4c = 1000(U (T)Xi>Z>0 ElG^f 4 9 C 1.25rad/sec ®MMK T**%7d'-9-*-©#ttfBl :®Etf-6^'OV1T#S&*D£M t 0.3 rad/sec DlT©@^&»@r±Tjm# amRk|g|U^&#rr^k»g:#A#Tv^A*, THA^MKIK® l/2#©@KRT/tti*fG:#A#-Cw% Zk&bfrZc ttz, 7-f -9-*-±^SC*TS'[6]KlFfitilipI #5„ fcr, c©#m&Rff-±^©j:^K:%af5 ©MIS-SUi-feB®, ±1&frt> 3100m (D&IZiSlif&fzt) »#@^0K:LAt©*!Fig. 12T&6. 2) /\>y;f7##©4000m#rW:7-'f-f'-F'3K:##3 nfew y k 17 -f -9-'- ©issisg © giwiiSE ® t°-

tension Iimit — osci 1iation at w ♦ osci i Iat ion at w/2 + 3) ESS) b tz ;b friggKDMl$ tc ov > r tt, SH±ttEiig

-f k©*#©TTC ©JSS^HJliPrSM k % G & © £ b Ltzo

# # £ m 1) M. Ozaki, Y. Fujioka, K. Takeuchi, K. Sonoda and O. Tsukamoto, “Length of Vertical Pipes for Deep-Ocean Sequestration of CO 2 in Rough Seas”, International J. of Energy, Vol. 22, No. 2/3,

0.2 0.4 0.6 0. 1.2 1.4 1.6 1. pp. 229-237, 1997. vertical oscillation frequency oi (rad/sec) 2) C-T. Kwan, T. L. Marion and T. N. Gardner, “Storm Disconnect of Deepwater Drilling Risers”, Fig. 11 Frequency of lateral oscillation due to longitu ­ OTC 3586, 1979. dinal oscillation co (lateral drag c = 1000tv). 3) “Recommended Practice for Design, Selection, Operation and Maintenance of .Marine Drilling Riser Systems ”, API. Recommended Practice 16 Q (RP 16 Q), First Edition, November 1, 1993 (Fo rmerly RP 2 Q and RP 2 K) 4) C. P. Sparks, “The Influence of Tension, Pressure and Weight on Pipe and Riser Deformations and Stresses”, J. of Energy Resources Technology, ~ 1.2 Vol. 106, pp. 46-54, 1984. 5) ±##, o 0.8 voi. 167, mo, pp. 137-145. 6) #*3% WEES-S : “zk ip © IB### mkf©mmEowT', voi. 168, 1990, pp.381-389. 7) &M3E2:, WBB&-GF, mm, ##&m

±, : “T 7 t 4 7*®J#PC -f 0.2 0.4 0.6 0.8 . 1.2.... 1.4...... 1.6 1.8 horizontal oscillation frequency(rad/sec) UlScS, Vol. 174, 1993, pp.865-874. Fig. 12 Natural frequencies of lateral response of dee­ s) smeE#: pwater riser (riser length = 4000m, at mid point eufiie h 5 **i$ 7 -i v-*-©#SBttiRi± citsi of riser). ^B", voi. 175,1994, pp.223- 279

232. Long Cylindrical Marine Structures under 9) J. E. Miller and R. D. Young, "Influence of Mud Different Excitation ”, ISOPE’94, pp. 231-237, Column Dynamics on Top Tension of Suspended 1994.

Deepwater Drilling Riser’’, OTC5015, 1985. 11) ms#, 86,1982. 10) H. 1. Park, “The Response Characteristics of 1^1 ^ lj' 3-2

- -ft* IE# e m s-sip lEfi S£ * n z*

Structural Analyses of Very Large Semi-submersibles in Waves

by Kazuhiro Iijima, Student Member Koichiro Yoshida, Member Hideyuki Suzuki, Member

Summary Very large floating structures have been proposed for various applications, e. g. floating airports, cities and plants. Among them are semi-submersible type structures. This paper presents a computational method for analyzing responses of very large semi-submersibles in waves. When designing usual semi-submersibles, we need to model and analyze them as three-dimensional frame structure. The output data from this analysis is used as input at the next stage. This will be also true even in very large semi-submersible case. Because of their enormous size, some techniques are needed for such frame analyses. We have extended the formerly developed method by the authors applying sub-structure method. This is rational because the structure is expected to be composed of repetitions of simular structures. We newly introduce the concept of group body by which several columns are treated as one body in terms of hydrodynamics. The unknowns concerning both hydrodynamic and structural equations can be reduced greatly in number. In this process, both hydro-elasticity and hydrodynamic interactions among floating bodies are considered. The results obtained by the present method are compared with the results of the formerly developed method for validity check. We show the 3,000(m) structure case. The results are shown to be applicable to zoom-up FEM analyses of local structure as input data.

1. ft 16 i tmAemwaf & ###& act & -c ^a. a#, u-ctf/h-fbu-cmifi-a

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HESS 9 ¥ 1 E 10 B ## 6 k t & ¥|&9¥5H15B 282 #181#

Goo's Method Present Method c cr, A b v zx»,

Response Analysis in Waves Response Analysis in Waves ## a k %R.(D d "5 btl%> o ([A^]-[&.][Au]-'[Au]){%} STRUCTURE FLOATING BODIES STRUCTURE GROUP BODIES = {Fb}-[Z21][2„]-1{F/„} (2)

PANELS SUB-STRUCTURES FLOATING BODIES {W=[Aii]-'{FU-[A.i]-'[&,](%) (3) (SOURCE DISTRIBUTION METHOD) C C T, ^ (2) k ###—U a H -CKD Fig. 1 Philosophy of the present method ([A22] - [FaiKA,,]-‘[-K,,]) = [Kb ] ( 4 ) -CSSn-BlItt^it-o —o@ESk»»LT C #mA# ma^-caFEMta^aki^torta. ^(3)tC%?T#mftAi-$CkT#f,fi3. o#0^»rm##^.6Cktcj:-3r, a &#e t m u#*#am faf&Tta#, rga.

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A<0@m#Kkv^Ck«#L ^.Akfak, c®k#&a0-7b';zxi±a^-#gA# 5*fLT, CT#o##timuB#-Bam#&kr, $c*:i¥i$ t0kfa. #koo^#micmBLr, C0@i^#m

% k, sB##m®m8 G®a&a k » a r A^&mmrmv^^aiRe^K^a. t a. C k ^0 j; ^ K » a . #m# - #m#-c, mwo'mmmm&x’, Kn Kiz {?ii7i\_rFf.'i !RK:jE#^Tv^akfa. *wa —)ST, #*@a#kA (1) .K21 K22. ~Qe I *- Fb I kf a. A#0#*K:A*K#A 283

#L, il3„ 3® diffraction d\f f ;!/&, diffraction b g* i C±5 diffraction i&T's 'y ^ A/©*®#TgLAk #®#»tmif kf 3? b !) ?X& *®j:3K:g3fL3. [BiY X^to [Si]T Kg* z ® diffraction ##& 7 ii>?=ffs OiGidS ( 6 ) bVXXkk^3k di*T# 3. 3 3TK [g,]^ 3 K {R‘J t$»3EKt^@ 3 3 T S/f ;!/**-# if. 3iL3<,S( 7 ) R ;>?, {?&&. AAL, 3®aK(6im*;», zKfgE^-cmf 5 {A}=[g,]({®}+2[T«]1A} g*« t \*i me*, 't6fBmm@0*K:#0g**t#<K»6» + 2 (- z'

KiFfitilliTjgljf" 3 E® radiation df T > '> -y ;VK&® <1 £"C0rA C group body ©#±&#Ai"3 o 3 ©@# j=^K*i"3kdit-r#3. K j: 3f#*rm®#4 > b Kdieraction#%%#f fz^l 4>1i={Ru¥ {?} (8) [g] tmv^3 3k?t3.3©3kEj:cT,Dko©g* 33T {/?«} 0i*Kg* z A5^SC#SLT, / TjIiWS #@±©## < ®/f^-;P±® source KfSSfc^X b ;V {A} 3 K® radiation bbtAo %o^®g*tlb k * k * !) k LA# @® diffractionj: D & < ®/, f ©df-f ;f/K f®m*®#f3me#m®##m^? ^$ft3, kW^-o®(RS%T3. C®k@(12)^4;® 4>j= 4>o + 2{A,} r{ 4*f} 9« »^a-kf3g*®$&-(X b;i/(9}K^®j:i5K# +gg(-«W«W(^) {z?}=[L,-]{>?b} (13) =({a,}" + Z{A,nT^] beam SS k hull WM (Fig. 2) K X -o XX r zP{b1“ 3o + 22(- imii{Ru)T[Tij\)\{(p?i (10) node H£E«£ beam $*KfiAf fctEtt* ffltt i*j . »#M^3. hullS%Km**^@K3gA, T^gft3.AAL, A##dff ;F *, K T radiation node KHlfkSo'S iLTVi^0 § iS$tVTV>3„ Vli Kg* i ® / S|n]'\®$ti®E*$ll 3 3T, ^(10) K, A#&^Xb^l3#»^^b EK^f&%®#mTfF@-f3®**j:DlELV)kmk,di, 3 ;F^^KAi-)»B^K:»^TV)3 3kAt6A^3. A*% 3 3kKLA:. b^®^^t—d©A*dff ;f/kT3 k #©g iESdff #BfbLA */ ® diffraction d'>tzK) SH KSS hull #**tg* : cmf 3 t ® k f flK, (12) ®M# & 284 M 181 % flJfflLT 3K#'M@#»tf/F&m:'Ta#Ki-&. 4#©MR&& {Fj= -ipa>JJs (^0+ gSf+ 2 - icvviidli n y A 4 o (2X2) &—3© group body k LTSlt>, A#©#)@Sr 4 0© sub-structure + 2^f+ 2 - icovuVuj ^n dS 3. R@#f-*aKKK*^local wave A3, 1 ItliilSb, propagating term © 9*1 ©-&■£: #S = (u2[p}{v} + ia>\.v]{v] — ipa>{[cit) T f 3. +z( w[ 7),]+g - ?:,])) mKZcTM%$MA#l-#3-FE j:%R*k©ltR& iSiboEti: k©#^- fclnl^zSk L A .Fig. 4 K Fig. 3 #t-tiffl37ACi< surge ^1=1©®$®!*6^1". # o ,#*km*ax A#*#s/ = fltz source # ;Ffk©imw»5©T, b;FT,%K: [B,] L % ^ev^0 Z. ©^T/PgS© group body ti$S"T$> 0, sub-structure t fi 5kkA3k»^3. group body &&&9±e

^(14)&^(5)CRAL, REFERENCE POINT FOR DISPLACEMENT / AND BENDING STRAIN K, ^1(12) k@l%$-e-%::5@5S&#< k kT*##A©^ REFERENCE COLUMN @k, " FOR WAVE LOADS

3. & m t # m

Incident Wave 3.1 fME 4'EI#AL'Agroup body k 5 A t SUBSTRUCTURE beam element""' l sub-structure iSOtSfi^iSSTS -E> ip k V* -5 A k $ Fig. |/EI=1.05E12(k|fm2) X p =73.5E2 (kgf sec/m4) I A^.=l.0(m2)

floating body * 8 boundary nodes (Diameter. 20 (m) Height: 25 (m) i^Draft: 20 (in) Consisting one group body

Fig. 3 Schematic sketch of the model on 16 columns

hull element

S 35 ■PRESENT CALI

Sr 30

beam element g 10

0.2 0.-1 0.6 0.8 1 1.2 1.4 WAVE CIRCULAR FREQUENCY (RAD/SEC) Fig. 2 Subdivision of models by hull elements and beam elements Fig. 4 Surge forces on one column 285

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-----PRESENT CAL O 100 • o ffTHOUT INTERACTION

column

center “ 20 1,p00 (m)

.495.(13), 0-2 0.4 0.6 0.8 1 1.2 1.4 WAVE CIRCULAR FREQUENCY (RAD/SEC) node255 node135 node5 Fig. 5 Bending strains at a reference point Fig. 7 Reference points in V. L. F. S. model V.L.F.S. 2nd 1st substructure 3,000 (m)

300 (m) beam element f f'EI=1.05E12 (kgfm 2) tot) / j p =73.5E2 (kgf sec/m4) -^yvY lsAsec=1.0(m2) B 100 -----col485 —col495 ...... col505 — single body group body Substructure

Diameter: 20 (m) Height: 25 (m) Draft: 20 (m)

'•boundary node 0.2 0.4 0. 6 0.8 1 1.2 1.4 WAVE CIRCULAR FREQUENCY (RAD/SEC)

Fig. 6 Schematic sketch of V. L. F. S. model Fig. 8 Surge forces on one column in V. L. F. S. case 286 a mm#

Newman101 trapping k IrIS® S,S* si3 M&©/N# #K&oTU &c:kr&&. iT©##?*) suige*l%©^@BRUK»3Bffg^6, C^LB#)S© ©37A©B#©2#K»oT^3. C©jgg&#ia&# heave ^[=|©^6B E, < f < 1"5^|6lK® < fllnJSr^fiSJl TB/JnS Vi, (2)-"=0.67(rad!sec)fifiS© t° — y 7 tt heave ©g|

3.3 &o fy*@3©0Bg^fG:# (Fig. 12) Fig. 10 k Fig. 11 KB Fig. 7 K#$#,&3&&0 surge ft (Fig. 13) 4-Svt'o © aft * fc*B Fig. 7 TEEUfigl^ |6] ©$(5 jo J: If heave ^[6]ffl$fiIAs7S surge ^l%©g!AT3(A;# < © B, ##©M%©^&©^: # $ k kttWS"CBIi:©6i, ©f futMW)©!# k#^_ 6iL5o #itB, a)=0.67(rad/sec) it Ji, a©i^Kheave©@Wm%Tt5U, ---- C01475 o 6 -----co!485 TM6fL&e-?BBk,Lk*s#ig©@#m(Brt5% —.col495 ...... co!505 ##3*±©%©#e, heave © — single body @##88 jzD^#^kC5K:^l:%ku^i:k B###© —SISK B-sTfS StvtTt^^)11^ $7i, a>=0.8(rad/sec) B#

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0.16

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0.12

0.10

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0.00 0.2 0.4 0.6 0.8 1 1.2 1.4 0.2 0.4 0.6 0.8 1 1.2 1.4 WAVE CIRCULAR FRREQUENCY (RAD/SEC) WAVE CIRCULAR FREQUENCY (RAD/SEC) Fig. 10 Displacements in surge direction Fig. 12 R. A. O. of vertical bending strains

de5

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|

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0.2 0.4 0.6 0.8 1 1.2 1.4 0.2 0.4 0.6 0.8 1 1.2 1.4 WAVE CIRCULAR FREQUENCY (RAD/SEC) WAVE CIRCULAR FREQUENCY (RAD/SEC)

Fig. 11 Vertical deflections Fig. 13 R. A. O. of axial strains 287

Table 1 Forces and moments at each point upper: T=10.0(sec) w =0.63 (rad/sec) lower: T=21.0(sec) w =0.30 (rad/sec) 6th Substructure Fx Fv Fz My tonf rad tonf rad tonf rad tonfm rad 1 502 -0.30 85 1.48 133 -0.81 1312 -2.44

2 88 1.52 79 2.10 65 2.66 392 1.66

3 88 -0.63 75 -0.58 60 -0.33 747 2.00

4 600 -2.46 258 -1.07 150 2.44 10400 1.85

Fig. 14 Zoom-up sketch of the structure 5 500 1.75 192 2.51 27 -0.52 823 1.77 a^KC5Ck(±r@»^c L&L, < »a ^ ka^kA^c *7:, Fig. Fx Fy Fz My 13 j: f tonf rad tonf rad tonf rad tonfm rad /J\$<»7TV^5Cka%a^5c 1 171 1.39 13.3 2.34 26.4 -0.46 5170 -2.41

6%k»c-Cv^5CkkM#ai*,5k#t6ki, 2 3.6 2.74 11.6 0.24 2.9 3.10 54 2.76 ka^o < z k fuf " ©#* t 3 29 -0.83 8.8 -2.84 8.3 -2.10 309 2.19

4 247 -1.57 17.0 -0.60 43.4 -2.76 5580 1.13 %vi% $-o y 72^t;v ti^tfS^&TIsz 9 $ -5 ^ k a*T#5» 5 109 2.00 7.7 -3.12 23.2 0.08 894 2.03

(Table. 1T#S!) rt©@aM'$<»5©B, ml #K36J##K# < w7k»K@l #K#<#*$5wB-&A,#r*KBA#»K#&#^_»w T(±yv —xo j: ^»#3@@5#KAoTi)»v^#*»60 72*T&5c »©T, #lkLT37A#om:)%*bOK«TM:,F'y 1* Fly ©1/4 KmLTBO, C:©#g#© a:zfLKt7k5c 3 7A©±%kT%Ti±#^j2.5m^a* #e K &#m u t @f #5 kfi@LA:A:*)T*,5c #@fi-5 kA# %Rm#AC#5 ^ k &#%LTW5c Table. 1 ±gCjgg|I 1.0 (m) fflifSB 10(sec)©A*tE C©j:^K^#Aa*$*&fL^,B, X—A7y7"L& efL?fL©##©ow FEM#PfK"3»lf5Ck^T@, g^»k©##©AA 72l$B"C®ii6i5 J: t/lufgSA^SiLTi^o tHiikT k LTi&Bk»5ajA*#67i,5o Fig. 14 4>K^Six-5ffi®*T*SSti, Fx ii X#^1%©#= 4. m m mt, Mr B KiSfc!) ©t—> y 1 %AU©|i%# &1E) Table. lT0:KB&@mBI^LX 1.0(m)T)@^ A©emt@:5gL72@±m^g*a#im%©Km4:is# & 21 (sec) kf &A##&3S/i,72k g©±*#&#AS:gs;SfT, TviSo 37ASt * > h ###TB@rAK group body ©%&##*.T#Af' 5 C k%tk7j&#»T j: v##T#-a-d3#fLTV:'5 C k %@K kT, *KM»aK%ji,5A*]R©K^ABKW(5fC:k L7i„ #Te%o c©imT, C:fL5—o©^&%r©fB#<Rf 5, F^ k # KM #l±#K#l*gfL-nr^5. LT, Fig. 8 $ 5 B Fig. 9 r#4 ©3 7 A K# < &b© *jhHma)tftBk*$

for Huge Semisubmersible Responses in Waves, SNAME Transactions, 1990, vol. 98 6) Kagemoto, H and Dick K, P. Yue: Interactions *o among multiple three-dimensional bodies in water waves, J. Fluid Mech., 1986, vol. 166, pp. 189-209 7) #a.K amam#: 6%, p. 62-p. 70 8) Yoshida, K. and Ozaki, M: A Dynamic Response Analysis Method of Tension Leg Platforms Sub­ # # 3t E jected to Waves, 1984, Journal of the Faculty of 1) #BA%: Eng., the University of Tokyo, Vol. XXXVII, No. 4 #,#1785,1995 2) ##asm#: ?#am (e®2), #1755, KOV)T, #178 5, 1995 1994 3) %##%, 10) J. N. Newman, et. al: Analysis of Wave Effects fS^Kev^T, B#:imim#^*3t#, #180 5, 1996 for Very Large Floating Structures, Proc. of V. L. 4) ##%#gaa#m&#e: F. S.' 96, 1996 #m#wa( 4), 6?? 5, eo 11) M a##, # 5) Goo, J. S. and Yoshida, K: A Numerical Method 178 5, 1995 289 3-3

1EM M E 1S K*

An Investigation on the Dynamic Response and Strength of Very Long Floating Structures by Beam Modeling on an Elastic Foundation

by Takashi Tsubogo, Member Hiroo Okada, Member

Summary

A number of studies on the dynamic structural analysis of large scale floating structures with length or breadth to the order of several thousands of meters have been done by several authors, as reviewed by MIYAJIMA et a!.*)^). Most of the studies deal with the basic characteristics of the vertical vibrations of the structure modeled as a beam or plate on an elastic foundation^) 4) 5) G) 7) 8) Especially SUZUKI and YOSHIDA®) have given relatively precise prediction for the deflection and stress of the structure at the lower frequency zone by deriving the analytical solution of such models. This paper firstly deals with the dynamic response and strength of a very large floating struc- ture(about 4560m X 1000m) using the simple beam modeling by the analytical method. Effects of boundary conditions and structural parameters on characteristics of dynamic behavior of the struc­ ture are discussed including the characteristic wave number derived by SUZUKI and YOSHIDA. Finally, the dynamic behavior for the same structure replaced with a plane framework is also investigated by applying FEM. The modes and natural circular frequencies obtained are compared with those for the structure modeled as a beam on an elastic foundation. It is also pointed out that not only vertical bending vibrations but horizontal bending vibrations and longitudinal vibrations should be considered at the lower frequency zone.

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©$$c mz, r r?14ch*S (Ap < X < (1 - Ap)) 144314-6SS1C73 k = y/2kp,0 #*ma*e©**mi=^i43%@mmi4 (28) az d, Z&D, snZD Jbp©Si$fe©-4>Zl-T, iSttSiScttSEti k-> 0 ?#±« kl/ld = 1 £ Z a (Fig.10). f5^m#14 (31) Etz D fc = &p?#±m, 3CZ#T^a. «K&mmiezm«zAaczi4Awz#* |cc|maa: kc y/kcEI . 64ta*t. WJWffettlC45»a3|Vxl|i!|*)&fr© lotsttfe T - ^ D •5-5. $®@16^:«9tu;Upper, fciou)cr *?EE S tia Z, StS (Fig.'ll). ZoTfcpl4??^^*g|51C43l4S|BlW!S &?&%. £ttl4S5i*6©$Sm#S6)Z1t>£$S(T£3. too Z toupper mm%#et)fG©m#i4t= tp?ima®#&z%z# fcp < fci x6tl(31)3$ZD. ra*u>. s gm*?©&A#*©iB#e* |f Irani _ kc______1______, . (kl/KI = i/o,«> i)*m&saaz. a(40) ZD. 1(1 2&2Z / cop y ^ ) , ^ kiOVJer X V2too / kP < / — ^J=r- =- (41) yof 2 + Va4 4- c* 2 — 1 t.l3.Z>Z.£.tfit>ftmZ Dtop « too?14, to 0(ab-5H 14PA) #4»**ti53 © S® ©ffilft l:l4&Z/uZW#

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— dynamic cal. ------dynamic cal. --- static cal...... static cal. at 160m from near supported end supported end - co=0.40/s

co =1.00/s 500 co =1.50/S 1000 Angular Frequency [rad/sec] o> Horizontal Distance [m]

Fig.6 Frequency response of deflection am­ Fig.9 Bending moment amplitude distribu­ plitude at 160m from supported tion near supported end(long beam) end(long beam)

------dynamic cal. — dynamic cal...... static cal. -- static cal. at 70m from at center supported end

2 2.0

Angular Frequency [rad/sec] co Angular Frequency [rad/sec] to

Fig.7 Frequency response of bending mo ­ Fig. 10 Frequency response of deflection am­ ment amplitude at 70m from sup­ plitude at 1000m from free end(long ported end(long beam) beam)

------dynamic cal. — dynamic cal. ------static cal. --- static cal. near supported end at center

£ 2.0 j

0.05-

500 co =1.50/s 1000 U4 Horizontal Distance [m] Angular Frequency [rad/sec] co

Fig.8 Deflection amplitude distribution Fig. 11 Frequency response of bending mo ­ near supported end(long beam) ment amplitude at 1000m from free end (long beam) 296

$ii o ^ t tt® M & ® -£r t£H8( 98 ® Mv2 = ak

CV = wt# 4wg/ W4=0.4465/s |wj K1

t&. ST, w0mfCicpSff,t;i^x5i, %m *% C t CA&. l£(32) L kp&mmmicT

8

C C £TS&E* In] ^-rit, k c = o tt£K>®mm.wimtm.m%iits.z>. ®\z®w.n&\tn& tA%. &±m6KT#Tm*m®aic«kc& lce@T5^.A &#%.%tw < wo(D(K*tC%tT.57kf a #*Rm

Table 2 Numerical data and results for the horizontal vibration

EIh 1.088xl0 1KNm2 Eh 2.901 xl09 N/m2 &Hp 0/m WHO 0/s WH 0.2570/s VH1 0.443(0.5826)/s “H2 0.922(1.606) /s ( ): When only bending rigidity is considered

Table 3 Numerical data and results for the longitudinal vibration

EA 1.108X 1013N Ei. 2.462xl09 N/m2 WL0 0/s w* (= wLi) 1.190/s

(Miss A) Table 21:^1-. (mm g Eb) ©@Wi81lSC& Table 3CST. Tfritvtcmk (iaag 6) i:ovvrFEM##&fTt\ b'ictet^T, Fig.i2ic^f J:t>fi£*ymroESb t— w < w0roWitiilC43HTtt, $1'fc)9»Sci Table 2,31: Fig. 12 Eigen vibration modes for the floating structure on an elastic foundation >Tt-f * f• ifii(Xitii®jro IM Aj Si ®j Sc t: ft: fi -T 3 & 2SS rt* * &. 297

9 #Ss 3) Toki, N. : A Study on the Behavior of Huge Float ­ ing Structure in Regular Waves, Journal of Soci ­ @*s#*sie&#t4$E±ro$tce€ ety of Naval Architects of Japan, Vol. 146(1979), mx.zztiz£v, mmf# pp. 185-194. *6©fl,v^tii75: VfcS*^S#:

SSEiSWA 178 # (1995), pp.399-403. 6) WBS-6R : j@:*S!#*<0#m$m43«k

Mfcftf), mR*3ER##kmiBfK*6A5. 178 # (1995), pp.473 —483. 3. 3#=#m©%#!:T5*&»*m2:LT. &R^E* 7) Wffl$-ei5 : E*SU#«:C0®S^1,43 j; «T(oss#tt*ia^±tf5c^ttssTss. f ©2 : E**TH**0#aKK^E/'(7/-^6AQ. «4r @LA«7c#&cx#mis#©mf— , AW14«;!6tt14j£SSc3:ffl ViTSStl-S. 163CS, * 179 # (1996), pp. 339 —348. s) m&mm, xmwK : -^tewes*^ 4. &K3EK«* S4PI=J;eE±@##:©m&6#mi6###r, B* m 179 # (me), pp. 349 -358 . fFttiEtoa-T, **SR©S##14 (#m) a##AfN=j: 9 ) «sm, #«sb, WBistt: 6T££oTV^. Proc. 5. &%3ER@(*T Techno-Ocean ’96 International Symposium, Vol. II (1996), pp.651 - 656. o la it tt m n a w- n « a e, & v >. Appendix 6. $eSiBTHS6i8©:b©^*feEL< 86 sse@intcoviT$ij#q^*-9-A. as (41) $#^A. A.l pAut 2 < kc(0 7. tttfJEASEC-oviTtt, *«6l^©6C6T#-m AS (31) £«OA. Xc = Ci cos -^=27 cosh -7=27 + C% cos —px sinh -px 5E©Eft©B#CSA®i|>atttf-i7e$i:D, t©l8©J6

i /-< • ky ky . ky . ky Am#A$&*©i#i2#© 1/2#T$5 (S (42)(43)#E). +C3 sin —=x cosh —=27 + Ci sin —-rsinh —=27 8. #A£#14Sii£A#x5ASB»;WS»i©BWiBl» ■s/2 s/2 V2 V2 F cos kx i^T*5pffl8JSft©0t (44) EIki + (&-)4

m## A L%a< L AT^TA-mM* V C ky ky ky . ky As = Si cos —=27 cosh —=27 + 02 cos —=27 smh —px \/2 \/2 \/2 tt>J £ b T5R U & H fe'£43 5 £ ^-©®f$e$l@jWIW&>4-T F sin kx (45) frtonA^i, 43j:^B*mm#6*#%#@m#«im + EI fc4 + (ky )4 SK#©gM6e}ca#^A^#m, CTSjl.&OAA'S'SL ,-_( kc- pAw2 ^ 4 AC iSELT, fclF < 43*L 6 L ± If $ f. ^— (46)

# m * m 9 2*c r, —x2 r/n' i ^u> — =(ky) {C4COS^CO«h^ 1) sst§, *$: StCOOT — #i5©ef5E®Jt6l — , |g 13 @ ##% —, ky , ky . ky fcy f'»4gmmm — „ . ky . ky F k2 cos kx — 01 sin —=27 smh -7=*} • (47) B *786# (1994), pp. 912 —915. V2 V2 FI k4 + (ky )4 298 mm#

lgi = (AJ2{S4COS^CO.h^ d2Xc : (A+ )2{Ei cosh k*x + C2 sinh A4x dx2 — C% cosAp - Ci sinAp} - ~ ^ co&kx ^52j , n ^U) • , t-i . kw , k^j +i>3 cos —pix sinh ——x — S2 sin —=x cosh -r=x s/2 s/2 V2 V2 ei k* - (ki)4 d2Xs „ . ku . kw F k2 sin fca; =■ (A+)2{Si coshAp + S2 sinh A+x — Si sin -=xsinh —7=®} — (48) V2 V2 El fe4 + (fcp dx2 „ 7+ C - 7+ 1 E k2 sin kx . cA = cosh -p, sA = sinh ^=Z - S3 cos k+x-S4 sin * J®} - ^fc+ 4 (53) s/2 s/2 cA = cosh A4 Z, sA = sinh fej Z _ kijj , _ . ku C — COS —=1, S =r sin —7=i V2 V2 c = cos Ap s = sin A4 Z (iiSiS6ftro»-er)

C4 - I p El k4 + (k~)4

Ei = {s2 + sA2 — 2(ssA) cos kl Ei = {ccA + ssA — 1 + (c — cA) cos kl sA2 — s2 2(ccA — 1) +V2 ^"p ) (c«A — scA) sin AZ} + ( ^jT ) (s ~ sh) sin kl}

Ci 2 {—(cs + cAsA) 4- (csA 4- scA) cos AZ C13 E = E2 = {—(csh + sch) 4- ($ 4- sh) cos kl sh2 — s2 2(ccA — 1)

+V2 j ^ ) (ssA) sin AZ}, C3 = C2 - ( -j-qr ] (c - ch) sin kl}

Eli! Si = - -- {(cs — chsh 4- (csA — scA) cos kl) E3 = {—ccA 4- ssh 4- 1 + (c — ch) cos kl 2(ccA — 1)

X ( —^ ) + V2(ssA) sin AZ} 4- I — I (s — sh) sin kl}, Ci = C2

S2 = , \ { (sA2 + (ssA) cos AZ) ^ — E13 Si {(—csA 4- sch 4- (— s sA2 — s2 2(ccA — 1)

csA + scA . ------smAZ} 4-sA) cos AZ) 1 4- (c — cA) sin kl}

5,3 = 3^"{($2+(ssh)c°ski) fp E13 S2 = {(ccA — ssA — 1 4- (c 2(cch — 1)

csA + scA sinAZ}, S4 = 0 -cA) cos kl) ( — J 4- (s 4- sA) sinAZ}, S3 — Si VT”

At2 pAcu 2 > AC 6 # E13 S4 = {(—ccA — ssA 4- 1 4- (c 2(ccA — 1)

%c = Ei cosh Ap 4- C2 sinh Ap 4- C’3 cos A+ a -ch) cos kl) ( ) 4- (s 4- sA) sin kl} + C4sinfc+*+ —fc4_(fcJ)4 (49)

Xs = Si cosh Ap 4- S2 sinh A4 x + S3 cos A+ x F sin kx + S4 sin k + x + (50) El k4 - (&+)"

fc+ = f ~ ^ (51) " - \ El 299 3-4 b

ie* jf s TEm m m m tm*

A Basic Investigation on Deflection Wave Propagation and Strength of Very Large Floating Structures

by Takashi Tsubogo, Member Hiroo Okada, Member

Summary

Recently many studies on the elastic response behavior of very large floating structures have been done. Such a very large structure is relatively flexible compared with other existing floating structures like large ships. For estimating the dynamic response behavior of structure, it is important to consider the deflection wave propagation based on fluid-structure interaction analysis. This paper deals with the dynamic response and strength of VLFS except peripheral part considering deflection wave propagation for the simple beam or plate model in regular waves. From analytical results considering fluid-structure interaction, effects of wave length and direction of incident waves on the response and strength are examined. Moreover, the dispersion relation of the deflection wave is also calculated.

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Fig.l Infinite long beam on the sea

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l£. -+ o £TS. *© 4 js^is/s^tt^-erows t9*At@SSnT«-a-l* (32)Sj;0 fepA^SO, % £CTHAAMI: Table lCiPrTA3 AA#A#viJSB ©£*£R#t = tpT k,,| g 6¥®©##^ 6+a-8in&e»T©ll$SUiES©®^, PmglCl k*' (39) #lcElRl#©%lFSI8^a. fcfcbw > WQ©#%«##© ##tA$ wo©#{®abSV* t#B, mm^A#vi|6isti^,i Dx, H = y/DxDy

7k;6©SfT*l6] = $¥#;&©» Jr A A Table 2 Characteristic values

#W'a. **&©iRi$A<*ma6K#©e*mjH%© (W Incident, Floating body s&tnsi;bTf*5ct/iST*5. i #a#© ft(b wo ~1.531, 1.625/s ^pi ^p(O) 3.513, 2.031 Xl0_2/m Kp,kp { tt/6) 3.502, 2.024X10-2/m -Kp> 3.480, 2.009X10- 2/m Kp,kp ( tt/2) 3.469, 2.002 X10-2/m Hp, u/p(0) 0.5868, 0.6085/s iZp , CUp^TT j 0.5858, 0.6074/s E$ = -p^aKI2A iZp, U>p(7r /yj 0.5840, 0.6054/s iZp, Wp(?r/2) 0.5831, 0.6043/s g( : *&©G#x*;bf

E^ = £$ &^^lFl(CA6e*mm©mfl:©#f& Table 2fcSt-r.

i /x E*- dzdx' 9UG#W«©«?t#*TS. Fig^li#*K©^%M*©#f $miAl^A%* =oi = ^Pu.g|u)i 2XD tt/2 0«§l;3bTilfcfc®T«5. E'xt E’yifiiBl C ^-—y-c$a & ©&im$©K#A*a Eto : 9*T%#©@#X^lbf Fig.4it2T)©&Kmc^itaMmK9c©%*©«f & 1 p™gM 2X &1%$A% = 0 £tt/2 ©e-S-iTXiViTSVfcfetoTfea. rbf^TK&S Aiw l: tt-tn i)mn «T85^i KffiS & £p : ##: (#»&#) (Oiax^lW &5£'Ey)i$a. Ep = -p/iw2|to| 2X Fig.511 2 T3 ©StSfcfW IT 43 It £ .

E£ : ## (#»&#) Fig.6H2o©Kmmi=Awa&%©gK©«f &&im# A*g* = 0 £71-/2 ©*^l;3tiTSb&bBTib?>. **

e“L Lr-'i dzdx1 &e t> a ® ®j © isi®MS is ii“F isfita. Fig.7H%»&m©a&&wA-A©%@*«©«f t. 1 Fig.8km*&m©RK&w/<-x©mft-7( > b@# Dekk4\w\2X ©#A&, Fig.9KAW&©mRRn^-A ©%&##© tibd, (19 )a?&s*ta. uixao #?&. Fig.ioKAW%®mmmn^-A©mift —/> «®©fcp^w0ic*ftbraASfE©A"p, Oo&^*act bm$g©#?&mio]&A% = 0 ©S^rix-xiUTSLfct) ©?*s. ja&ESStofii^HSfjfcEssnTSfcfc© 305

100.0

a 10.0 r

20.0-

10.0 20.0 30.0 40.0 50.0 Angular Frequency [rad/sec] cu Wawe Velocity [m/s] V

Fig.3 Dispersion relation Fig.6 Transformation of wave velocity

------dynamic cal. $ Q ------static cal. 9 k=0, at center

c 1.0-

® q„:

0.5O). 1.0 Angular Frequency [rad/sec] cu Angular Frequency [rad/sec] cu

Fig.4 Transformation of circular frequency Fig. 7 Frequency response of deflection am­ plitude for deflection wave

X, 30.0 8 = 71/2 ------dynamic cal...... static cal. 25.0: 6 k=o, at center 20.0 : I 0 Imnx' l £ I- p w9/(2kp 2Z) 15.0;

10.0 15.0 20.0 I ! I I I

Wawe Number [/m] K Angular Frequency [rad/sec] cu

Fig.5 Transformation of wave number Fig.8 Frequency response of bending mo ­ ment amplitude for deflection wave 306 #181^

4. sas^#;omm* e.t-#ih n ammTvmmm ------dynamic cal. 14(16) EtT$S4l5. *©#6!4&lnl$AiPW#©/J\aw ------static cal. if?) fr ie-&■* s t & a # < a s. 8 k=o, at center 5. aAS@#©##A'G+#l*f!a#BrT©fSA@IQ 14 (18)(38) ST*Stl5. «©#-&l4S!Ht©A*Vi|ia# let= &p©K*t*at$l6A&#A**A6as. e. &©t>©i^AAJ$££®3£±£<|W|C-e;&-5. BleSS A^AlG@©#f ©mt44. t %m*A4masctieAs. 7. *©($#&* (32) oa. Angular Frequency [rad/sec] Q a*. * es#e©-sbi4, ^A Fig.9 Frequency response of deflection am­ (») plitude for incident wave fT^naeii, sis^©smstoiei4S*ae:MB, erenstifcA^Au fcCtSrlEUT, MS#meJP<:fe*l* UAtfSA. # # 3k m ------dynamic cal...... - static cal. 1) %*6, @!FIE|S£, AfEil,: @AS**le# < S A • 8 k=0, at center SE*A©#^®leMASE^, BAiSttF^iMS, % 179 (1996), pp. 173—182. 2) *##*, ?BJ»A^ : iAf?*; Hff ##m#lef#mAsm#A t&mie*TSlG# ©Stfiififfr, Proc. Techno-Ocean ’96 International Symposium, Vol. II (1996), pp.703-708. 3) Takaki, M.and Gu, X. : Motions of a Floating Elastic Plate in Waves, Journal of Society of Naval 0.00 O50p To 1452.0 Architects of Japan, Vol.180(1996), pp.331-339. Angular Frequency [rad/sec] Q 4) xmm, SSSA3I, T@u8fe, MBa»%, AMI : *@g*##a#m#©*#lG#lee)t^T, Proc. Fig.10 Frequency response of bending mo­ Techno-Ocean ’96 International Symposium, Vol. ment amplitude for incident wave II (1996), pp.657-662. 5) WfflS-615 : @Ag##©#im#mJ3A eX#iS#m-leMA6##, B*35*6*6*:%#, m 178 # (1995), pp.473-483. 6) mem, ##**, heiss ; #*asitf?isn> a%C6leA%. t@Am###m©mm###mieMA%#%, Proc. 5 Techno-Ocean '96 International Symposium, Vol. II (1996), pp.651-656. 7) I&A3SA, Wffl$-fi|S : @A^##©#m##4iA &-*";: ©&#©###*. seiex tfMi&mmzm-rzitm — ^t ’V->SA30?¥#s:©$t ®j©»fSM#tt(i2) iKTSsns. mB#m©it#t, #13im b 2. @XS?’?*©^#^6+4)-6HmfctiiTrSe$l-rsi6 *i&##6, (1995), pp.231-238. ©t*6©ja^©A»s&t©M«tt(i9) ST^sns. 10) AEffiln : nJISttffi>?J#A©#ilitj®JmA-%kt£WIT: A5Mi, jfsl30 UAifilM'i* m 14*$ Bii& A © AM® le It^M 1,-d &, ti&ti, titiiTtito A, (1995), pp.313-320. t5. e.le 11 FR«A

Appendix

a = -CT A.l $*£&■

w = |ui|sin(fcxE + kyy — wt) sin fcjZ = 0, sin fcjB = 0

|iu| sin{fc($cos 8 k + ysin$k) — tei} i&ont'x©®,

= |w| sin(fca;' — wt) = Cj sin(kix) sin(kBy) A.2 7)1# 6 ns. d2w 32u "d^2 dy 2 9 2to 7*9 = -2* dxdy

A.3

Txy = G'yxy

<7y = (E” cos2 0* + S' sin2 $k)zk2w

Txy = 2Gcos9 k sin 8kzk2w

A .4

G = —Dxy, E“ = Fd’

H = D' + 2DXy a.5 mmawmmmomas#)

(37)a©@##mm&*As. bw^e-hs

^ . _ e*(* cos e+9 sine)

(37) t.

6 - ±6,, 7t ± 8j

k — ±fcj, ± ifcj

cn

j — cf cosh (kix + kBy) + c£ sinh(kie + kBy)

+C3 cos(kix + kBy) + C+ sin{kix + key)

+Gj“ cosh(k® — kBy) + sinh(fc(X — key)

+G“ cos(k® — kBy) + C7 sin(fcj^ — kBy)

309 3-5

2E# ^ "&* IEfl C U IEM Z^/l/ '»z<^yx*** * e A ^*** Buckling/Plastic Collapse Strength of Ship Bottom Plating

by Tetsuya Yao, Member Osamu Niho, Member Masahiko Fujikubo, Member Balu Vargese Keisuke Mizutani

Summary To clarify the buckling/plastic collapse behaviour and strength of ship bottom plating, a series of elastoplastic large deflection FEM analyses is performed on stiffened plates subjected to combined bi­ axial thrust and lateral pressure. A part of continuous stiffened plate is considered for analysis taking into account of symmetry conditions. Series of elastic large deflection FEM analyses are also performed on continuous plating without stiffeners to examine the influence of lateral pressure on the buckling behaviour of plating. It has been found that: (1) When initial deflection is in a hungry horse mode, bifurcation buckling takes place. The buckling strength increases with the increase in applied lateral pressure. ( 2 ) The buckling strength is increased also by the stiffeners, which constrain the rotation of panel along its edge. An analytical formula is derived to evaluate the local buckling strength of stiffened plate considering the influence of stiffeners. ( 3) With the increase in applied lateral pressure, the boundary condition of the panel between stiffeners changes from simply supported condition to clamped condition. This change increases the buckling/plastic collapse strength of stiffened plate. ( 4 ) With larger lateral pressure, yielding starts earlier. This reduces the buckling/plastic collapse strength of stiffened plate. ( 5) Owing to the opposite effects described above, the buckling/plastic collapse strength of stiffened plate takes its maximum value at a certain magnitude of lateral pressure, especially when transverse compression is dominant. (6) The formulae by classification societies give conservative buckling strength under bi-axial compression, and the bottom plating has much reserve up to the ultimate strength even when lateral pressure is acting.

UWbtWPi-UE4>a0 —13, to#-OEttn:fT±©$iJ l. ti C to c

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T0-60m©*mTgHk@-&ik. — -57I 2 {tv(zr, yr)—(tVr—Wor)} 2 = 0 (6) MS • E14jHSPS) £■ !K £> fc to © sPPEttETa t> #-S#r"C

##©-A##a:*#r;g%L:t#ef 3 - x ULSAS & ZlkH, #Rl0)T#9B3fi/Ci,)a<, 3.2 a 5#-^- #muA. aMctoR^EQjc-SK, ^&emm©#^-#f- 7X/ #-&©#

©*#t, Fig. 2 . *kk»i±, a y y#@#& Rb 2 o©/<#vi/[p*©EE»&#Eo *E3Sfitt, ?i* 312 181 -if

a= 2,400 mm b = 800 mm t = 13.5mm

h = 60 m

h = 45 m

------: INITIAL DEFLECTION IN h = 30 m HUNGRY HORSE MODE

------: INITIAL h = 15 m DEFLECTION IN BUCKLING MODE h = 0 m h: WATER HEAD (a) Average stress-deflection coefficient relationships Fig. 2 Buckling behaviour of continuous panel subject­ ed to lateral pressure (longitudinal thrust)

0 m, 15 m, 30 m, 45 m 13 i. If 60 m k Ltze

TC-fhVTVi-Eic

(b) Deflection mode

Fig. 3 Deflection components and deflection mode under longitudinal thrust (h = 30 m)

g)/<^;l/Tl±^:( 1) K^2 & j: ^5 xn±, #@-f&4o©,<^P0#]%8%:ba.a&i

^TATUVSicotcSfe/vT^se-e, E@##©%ok#

®Sf5» C0M, F KjltJS-r -E-lSSCAti)

V)b#a#R%RAVE6fL&,, X b StMMtiT±iattf§£ b * o tzo Ak»A!*,6#eK:l±a4R%*k L"C©&mA!%<i,^ *#30m©#^KLO^"C, %. i/ioo »©M^$iFig.3(a)K:, ^)?ScS 6^*01^ 1/4 T, 25f5T*&. 0, #K k C«, k /b)f

Fig.3(a)c3:6 k, E#fSSA*-tfn©##±Tl±DM 313

: INITIAL DEFLECTION IN HUNGRY HORSE MODE

-: INITIAL DEFLECTION IN BUCKLING MODE

%z?L, ^-cmm^micmf^k, m h: WATER HIGHT a#*^#0mmt- p & /Ui 5. !##%:, mo#%3^#t-F®A^6%±L-nr)%. mm^mtigA^t, *EK4D±i:4:mm@st-F a = 2,400 mm

0^k»^aa9, t-F0Air/%b9B±U&v:',, C b = 800 mm tut, WftEffijjkmmz*&b, UtmmtoX^Ztzib t = 13.5 mm rt&. Fig.3(b)K^gflT^)6^^»KB*!a>;;# »gEA,T]Egil^l:A#gT»v^0B, @S0Ab»^A* %-3TV^SCk&#LTw%. ##, R0©m%BWR@

#&T# 6 titz tz to »##, O® B$/h—BiSTf# 6 titz h = 60 m

h = 45 h = 15m a&^L-cjgD, *&&<%&%# h = 30 m a-®###:#*) 6 ii/C ^ & C k 5 . h = 0 m *E#m©mnk*K, Eiwat4±#Lrff<0 ;n 14. *EK:j:3ma@@t-F07kt)»*iA#<»9, # Fig. 4 Buckling behaviour of continuous panel subject­ ed to lateral pressure (transverse thrust) A&T&&. **:, *E^2^Ag<»6k, m#%A# 4 3 E&3. CfiB, ^0ma#*w#t-F©mm#m$:A# <@^%mm ^#±L7k7kAE, 5AAk#>l64l%. ##, *E^r2A*A#<-CA#»A:k»A%^L-n,^

^mTakA/^-fbUTW^^. C0C: kB, Fig. 3(a)KiiR$4LTV^4:k) A-^T, @@© 1 BU&#aq&4f L2&0#a%@anLT^6::kK#l<^&,, Z4hB, HtTt4maA#^*»%kLTtmm%'- Ftt liftxi jg-ca* < , W*©Sj)nk * THaBS ©mm%-F^, ^k'^BA^bL-Ct^tOkat'^L 6. mwa, ±a0 4^Kmaj#*3#t-F©mm*% ^LTW%. 3.3 h 7 (a ) Average stress-deflection coefficient og'K:, RU<7X^^Hta/6=3.O0^BKK:, F9 relationships yx^[cg©El*lE®lr*» sfFfl!t"6S-a©, tztoh-b ¥iSE®lSA(DmiMi, Fig. 4 C^ET’51E= H* ©A*B, mmt-F©%#t:b*&-#&%#&©*gm& *LTV^&. $4:, *#30m©##-K:Ot)-C, A k%&a.©fmc##& Fig. 5( a ) K, g,/%cr=1.72 0

Sf, ^■?^^cf3*-p©fct)^ttn ysJ^ftnzWM&i'FfS f&#ek#»o,#2©#aik#K:#^-cfT<. efua, m3a%t-F©mmt i&xi%0mmk»&a:»r$ %. 4:4fL, m#K&C3©B#iK3E#t-F©mm'r& (b) Deflection mode

—13, mm## B a > ^lAl K#a*sfM-y % #B- k R Fig. 5 Deflection components and deflection mode under transverse thrust (h=30 m) 314 m i8i kLT®3a*fg£ Lrma 3©B Ann s£fi-"C, Flil7 yxft n yy## mT#$63%0t-FK»-3TI,^c nyy^|n] K#@a*fFm35#^kB#»0, Fig.5(a)Kt^$^L An^kR#K: A,,, A,. LONGITUDINAL THRUST Sr^nib, F B Fig. 5(b) CTKStltViS J; 9 K,

a = 2,400 mm

C0#et, *E#*©ii5nk41 cSJEWBB±#t-r b = 800 mm fr<. Fig.4Th=60m©»&, ^@#0A:k)A.® t = 13.5 mm mn$^*T%-(kL-cv)t. mB#m©±#T ^5|5©^ffiJSSr/E^SSS* 6> ^f-iLT Vi tzipSia*, C ©##y kK*jSLT^%. 3.4 ?KE £glt 5i£E/U©b©SJS5iS Fig.6K, F7yX^|#]gZTfoy^^|q]K:#m^m TRANSVERSE THRUST f &#a-f:h,?fuDA#a;@%, *mK^LTO@-ryo yFL-C^f. *E#m®#jmk#KA : CONTINUOUS PANEL jm#m*i±#LTW5. &7KE© : SINGLE PANEL #^#B, 2cK^BT#^g,^3. (1) ma©@#%%r±W5. (2) tKEKi D, EH^e— F kF©* Fig. 6 Influence of lateral pressure on buckling strength

Fig. 6TS5>it5SB3SS©±#B, bfih 29©i?HK Z6 k#^.(,fi5. L%#e©ma%% % a y ^^leiE#-? B 212 ##/<^;i,©#^-kItR3&. aa^lAlK## MPa, FyyxTJIalEgreB 65 MPa k»5„ '&-?T, n &#e, naas^e- f©b@5Ssb, *g#& y^^i%©E*K:*LTBmma@a©±#BMw%@Tm v)#^Taa3t#t-F©*;2.iek»&. z:^B, frtk&o, cfbcafL-c, F?y 60 m 0#& k ItR L T t ^28^0$#?^#^X^[%0E#K*LTB, amagBzkEK:fSU?±#f S%#%f3k, &*%}&&. # &:ka#W3K3. #^03: *E#m0 M3#0$#&iB%T 3 k, Fig. 6 0#@a# %%,&„;:% B, ^2#$6 ;te^5„ ilM©±#tt, r©S6rEWSr5it"o 3^©#eKB, 3.5 Rmtmarn j- %. z: 0#*T@mEE##mai# mi- % k, An aa- ZLC$T0*@TTB, I0R/<^/l/®f4k)A.aiK*#A# mmt-F©A:t,»^T$3Aai r##$fLTv^kb^. ^KB, ©j&R&3&A:0'a". C:^K:*LTm*/<^;i/©#-eK: @mKBm#a%m$jt.-cv^. ter, gm#a*3.2m # i tf 3.3 @rr^A#giK: tf 0 Z ^ 6 a) &3M %tf6A*. Fig.3 *,&^BFig.5©(a)K^$fL%H&Z fi„ z: ZLTB, 30 m r F 7 y X^t=]#ma#I#33# a@t-F^a-f©6©B$*o^L»i,). f© ^0*gm»Fig.7caT3. 0f®ia*B#m'r7RLa:/<^

^vr, lScS<*otv>5,, bfeas-^T, fl23S®S: ©SUIT*, ET © fc to * B -tz n fc LTt>5<, —13, m=T#tT6, *E©^#&#a-f&k, mmtsrm%#3 —A**BRuy-xT, R##&m0a:k»%gakL A@i cfffi-r & r. k c * 5 o fiS'a©S#l$S'#"o *f, imkmR0ikRj:D, K##kL-C#9#B% *EK:j:6A:k'»t-Fkmmt-Fa^&i-&^», * m#7yi/;k#mrm(D#om#0%*T, mmamaam Ea^#^k%m»maBmfi»v). z:®#g-B, m#k 20%#LTV^Cka%a^. ——Am#BB&##@ %aa*@%K:@< k k #0A:t'»$gaKLfk#eT^IK®mEB@K/'($;i/K:f8 . i^±©zkZD, K@^K©mat#A6#a-KBmiR a33o c©*^-, v<^;i/k L-c®%D&v^a%BT*,&z: ka^a^s. *E#mKj:0^C57kk)» (##±0A:k»0m%a »#, #3@-r#Lat©B##mT©K*T&3. #$63: a#^6c ^itB, K##a*##x yf;i/#T#)N#M# 315

ELASTIC ANALYSIS 9, wf jT.t##©*WE7rr#%jG-fkg^-n,)&o a# ------: NO STIFFENER (DEFLECTION AT STIFFENER PART a, 2 33 9, IS FIXED) ------: WITH ANGLE-BAR STIFFENER (DEFLECTION ALONG #±@om©aT#9w#;*&. ^©om&mk^-am STIFFENER IS FIXED) ------: WITH ANGLE-BAR STIFFENER #**, &%, a#©'S%# ------: WITH FLAT-BAR STIFFENER kLT#^4L-C^6^#K, ELASTOPLASTIC ANALYSIS ((L-313.6 MPa) —5, 1JSB3 ------: WITH ANGLE-BAR STIFFENER

SluSo kiiC^LT, —a##HR##©#9 H!|#© 4.0 i-

^9, KlffZ+ KzGxOy + Kzdl— KifSx— KsOu + Aft=0 (8) 3.0 -

tt = 7i75-73

K2 =7179+75 77 —2 73 78 %=777,-T» Kt= 72 75+ 7176-27374 /ft = 72 79 + 7« 77 — 2 74 7s 1.0 - b = 800 mm /ft— 7276— 74 t = 13.5 mm ^(8)©#E^j:TF^(9)0##&, Appendix I C7jK~to HcF © x Ep tt, L tz BWfiSfrTS A ftliiS &aeLT#9, c^.F,©aa—am#k^v^(B&7FLT Z©CkA^, a>3;#©m#^#li#LA2*Em Fig. 7 Loading paths and strength interaction relation ­ ships (Stiffened plate ; h=0m) mmgkt, ^(8)Kj:9i@#mT^A(,fL5CkA%^ 60

(-am#) 15—20%±#L-CV^Ck^^a. &A:, a#T&$fL *EF:

x: ELASTIC BUCKLING STRENGTH BY EIGEN VALUE ANALYSIS ------: ELASTIC BUCKLING STRENGTH OF SYMPLY 0.8 SUPPORTED PLATE ------: ELASTIC BUCKLING STRENGTH CONSIDER Oy/Oy ING INFLUENCE OF STIFFENER —8------: INITIAL YIELDING STRENGTH 0.6 ------: ULTIMATE STRENGTH a, j: 9 t k r©@a» ------. LOADING PATH

iZ-DV^XWMtio PLATE: 2,400 x 800 x 13.5mm h = 0m

STIFFENER: 250 x 90 x 9/15 mm 4. ffiM'mjwmm • 0.4 CL = 313.6 MPa 4.1 h: WATER HEAD

©MffSS^ Fig. 8 0©iSlfitt Y 9 > XjjftWW1 Fig. 8 Deflection mode and spread of yielding t&EEmiBA, ##« n > s;##® u % ^ (Stiffened plate ; h = 0 m) 316 #181#

&, Fig. 10 CS^-to tfz, nyy'^l6]©l|6EII, eje*= 0.5 © 2 $6E$6:lb* J:t»yyx^|tj©l #E#©#a-©# k s e-mu-n,) a. Fig. 11(a), (b)j3i^ o^K, ay^iA] (C) KL#i-„ c©#-a-ti:mm#m©i#Ank*K, i &t-K0AE*^L, f cc#m#&oT#$ma¥SXl4iE^e-F©|W|C5l=I©/£to**5|$fi:LTfT<0 5. Fig.9(a)K E*#a^oy^^|q]K:©»f^mf5#eK:K, <:##&, #m0^(3)-r«yki®^A%^LT y^#©li-@r 3#^%t- F ©%&»*%#&%&-c## &0, Fk—acLTt^. LA^L» HJKK^-r^o b 7 yxseE"c^°*/vtt@scffi05;#

-F^©2^*j@^#^f6. f©%E©^#0kKW^ m*©^(3)©Df,T*$fL&aa@%t- #&, Fig.9(b)K^f. CfUC^LT, h9>X^|o|A6f^af5#^-, d©A +^E^L»V) * C k *!&&;*;# 3MJLK:»&k, oy^^lAll^&0@j@A!% So AL, f©$m©t-Fr^#f&. —#kLT, e,/Ez= £ilCSfLT, ej£x=0.5 © 2#E#©#ec LOO#e0##!@aR0^BjsZT/^##Kt, Fig. 9 /

(a) At ultimate strength (c, being free)

(b) In post-ultimate strength range (e» being free)

( c ) At ultimate strength (eJ,/£Jr=1.0)

Fig. 9 Loading paths and strength interaction relation ­ ships (Stiffened plate ; h = 15 m) 317

ELASTIC BUCKLING STRENGTH 4.3 TKEA^s^HKoss-sitsassstcstmge INITIAL YIELDING STRENGTH

ULTIMATE STRENGTH

LOADING PATH Fig.l2(a)j@j:tf(b)Atf. 0j:9#^

PLATE: 2,400x 800x 13.5mm h: WATER HEAD STIFFENER: 250 x 90 x 9/15 mm doma#, j= f akt,

U„ = 313.6 MPa < as k,

h = 15 m 0.2 -

fa&s, c> ^^Ai0inrmA^EKa#e0*#m@«*EAiaw#^ i9 t@<, *E 0#Alk*K:###&K@Tf%. kC Fig. 10 Deflection mode and spread of yielding at 6h 9 >x^imi0#mi(%aga(7a#^-K:«, *Eas 25 ultimate strength (Stiffened plate; h=15m) rn*mig< mT0*g-rH 2KEa:aw#^j: 9 aa%5» =K£oT, r©«-6-Kt,S$69Ktt, n y^#0##To y> ?ti'iri16o/£2ZTti:|ll

( a ) sy being free

(b) £y/£z = 0.5

( c ) ex being free Fig. 11 Strength interaction relationships for stiffened plate subjected to combined bi-axial thrust and lateral pressure 318 181#

*E*sfW#f sma-CB, R##TE%5^A4o®/<^/P ft##*'" kft*LTFig. 13(a)~(d) Iffl-S^ftfcHfctSo cO^EEflKItito^'fFfli L^#, A#®# @#KTBA(,K^®A:k'»B%^$^6. *E##W\ *f, 0(a)Rasf*E*;f#mL-Cv^w#e®R#/< 3'iPkLT®#K^&B, l@a#iHK#®#gi/<3';i/®# *E^± § vti§n scB, *ESc6tt" 5ifrlnj^® A* ivs;#k®#^-#KiecT/<$/FBime3eB^# -^, < && k#i@® 3: ^ K^-KAlfgg sa*/<3';F®#ii^#a[B#a''<3';F®#e ^®ClkB, #*&#a&#T3#5@Bt»j: 9 &bfd'K:i8< ■So 2 ,o®tSS't'2)MH®li&Sc, :iKEW$> ?> ±#2 ®#E##@gt±#^#&^-fo Bm#k®@ie-#K#^ige#ima, *E®K#?$& 4.4 /<#-mimmimizmstm # ms-rsk, 0(b)-(d)^6ga6^»j:^R. *E?gm ogK, ma

—: ELASTIC BUCKLING STRENGTH CONSIDERING STIFFENER #<&5K#cT, #W»#aB@TL-C3K3. f®##, PLATE: 2 ,400 x 800 x 13.5 mm (Tv = 313.6 MPa c:®##*ft:®tkT®#mmB, *EA*3:@<»6R: STIFFENER: 250 x 90 x 9/15 mm %?TK#/<^iPkLT®#g#gKacf^T^5. h = 5 m h: WATER HEAD L , 25 m 3#®*E#mA:f^m L T K#T 6,

h = 0 m

h = 25 m h = 15 m 4.5 Jh-;Hi5£®ifffi Sfttc, JSn@#:E®S@5®StcH# 5 iP-iPgi£® ttlcov^T^ESiDlSteAK, ip—ip#55e X 5EJE3S ( a ) Initial yielding strength #6fL^#^akjtRU^. 3#^:T%kkLA:R#v'( 0.8 r ------: ELASTIC BUCKLING STRENGTH ^;pB/\>f-f- - ^-f y®/f;p^*+ 97®#lg^KR: CONSIDERING STIFFENER fB^LTla9, R#l-*mBmK»#%#mL-C1)15m# PLATE: 2,400 x 800 x 13.5 mm k»6.f CT,*E»L43j:lf 5m k ISrnzKgmg®* STIFFENER: 250 x 90 x 9/15 mm Fig. 14 0y= 313.6 MPa h = 15 m lcEELfcE$4-aT-to 0E®,tSEB, 3#3tT^*Af h: WATER HEAD ft,f lt®#&®*##%^m@#&*To &f:, —A## h = 5 m tt5l;( 8 )rf#6Efc5PttSH5$SfflMHflSr##o — If, h = 25 m -Am#,### Ztfm&Bf ftffL, DNV", LR") ^i h = 0 m l/ABS14)®/P-;PSSrff-S$fifeE@5SSfflm6E^ NK15) IP—IP 1C 1: LR ®IP—iP^^lCi&V^ ®T, LR ®mmgL%B, ^a#%3m®$m=® & k 6 ^6®r$&.—%DNV ®ma#aTBh9y^^|q] (b) Ultimate strength ®E#1C*LT, m##®#it#&®B#tc3: &##$# Fig. 12 Comparison of ultimate strength between #K(c=1.21)t^#L-CV^. ABSTB, o>^^l=l® isolated and stiffened plates E#c*L-c6, mi#®#a38efRme#aLT^& (a Up)S^MKClEJg • M&imWMScmtE 319

STIFFENED PLATE CLAMPED PLATE SIMPLY SUPPORTED PLATE

CL = 313.6 MPa

h = dm

PLATE: 2,400x 800 x 13.5mm STIFFENER: 250 x 90 x 9/15mm

(a) h=0 m (c.) h=15 m

STIFFENED PLATE STIFFENED PLATE CLAMPED PLATE CLAMPED PLATE SIMPLY SUPPORTED PLATE SIMPLY SUPPORTED PLATE

PLATE: 2,400 x 800 x 13.5mm STIFFENER: 250 x 90 x 9/15 mm h = 5 m (L = 313.6 MPa

0.2 • PLATE: .. h = 25 m 2,400 x 800 x 13.5mm

STIFFENER: 250 x 90 x 9/15 mm

b) h=5 m (d) h=25 m

Fig. 13 Comparison between actual buckling/ultimate strength and buckling strength specified by- rules y^Z?|6]Ki® : Cl=1.1, h 9 yx5i=iE$6 : C2=l.2)0 § v^3. 3 k, * ------: BUCKLING STRENGTH BY PROPOSED FORMULA —: ULTIMATE STRENGTH BY FEM —: BUCKLING STRENGTH BY DNV —: BUCKLING STRENGTH BY ABS —: BUCKLING STRENGTH BY LR

PLATE: 2,400 x 800 x 13.5mm STIFFENER: 250 x 90 x 9/15 mm

K##0%mt#ALA:DNVj3j:{fABSi7) h = 15 m

h = 5 m

—Srt3„ —15, o y h = 0 m W 3 0T, ^( 8 ) 0A a(8)c vi/-/k#^Kj:3ie#klgK—&f3. &#, fiyyx^ |6|%rm^K*ia#-^K:«, %#±$rK*#DNVkABS Fig. 14 Comparison between actual buckling/ultimate strength and buckling strength specified by v^f^L0;k-/i/K:j:3@S%gZD rules 320 181# feS< □ y^^i=g (ay^|% ®E*A^mK»»eK:ama#iE^fT^ & ®? KB, #*mawmst#»o-c*& to, mm^mkBn^t ##e#A*t5. ZORmKMLTB, ##, #5®##% oo#*KB, ##3&aK:iawk#a.&. # EKMy 6#E&^*T#mLf:V^ ^3SS©IE.,S^i3 i/i^ii©;!/—;v-^iS4'fflv:iT fe, -tEHc 5. £ t $> $S:fl!l©5SSWtt»?titi*-E-kHi-5o —*, F7>xS|r1 ©E*A^E@9»#eca, maB#amjmT&o, Em ^mRTB, ^rnorn#:^ h %^6#emE^roa*^m&Agv^ %^T, -@p #3.0) ®f&%#®#G@T DNV K j: If ABS *B#r&SI;ffiU *Ek 2^|6]©mi%Em#m&R#K^ #%&@oEm^a^-#tTw&kB#t, zKEcaaE m*i@±#&#]#T5k, #KM@B»wk@:b:h,5. ^ZT!imLA3o©g8&K^;i/-;>©mi±, K##© O—m»%omu*:yy;yx/X^|n|©BBE*LTKMRK^eAmA#^ 5. zoma#mB, *E#m f»OT, R%#O#KKj:7)?BtoK0m#OK#B 0#mk#K±#y&. Ztitt, *E?1CJ:5 @6K, hyVA^ lnlOBE)@^Miek%&##B, 3„ T, *EKj:3Em3@gO±#tm#rT#%. (2) h9y%^AioE*g^3^E%»#e-om)miN# SB3SS K *E©#1F £ #St" 5o B@@#WK*E0B#&##f ttz, ## y5#El±fo%#'fkKov)T, ^&a3@;eo#!K''<4.^o**#aK)a^<. #^Lf=±r, iiem&'Ty^gkmkfL, #mo#EkL (5) h9 yx^iq]OE*A^Km;»#^, zkEBSrmas 4Mmy 3K%cT#i^aB±#L, tom,*E#mo#mk #&%:, $#3tTBmf*E#m*-9rSoA#gmTfP #K*#%EB@TLTfz0 #EKf5Ak*E^a»KK^UTtfUf, # # 3t E C&. 3AK$>(.ak, mmEB^ma 1) %mms, *A##K : "*E%sB&# BgoEamBKMj-smg", * gK&&9##3fL»t:'k#a.6fi/&o C^K*LT, 92# (1996), pp. 249-262 tB*E#m^A@v^#e-KB, 2) HE1SS, A##-, : “*E^$B5fflS gT#*Eca&^T;&R:mw0K:#Mrf&*&, Em 321

m 149^- (1981), pp. 144-155 x 7*: 3) 3=m#6, Nie Chun, ±WM% : "^© %=K{sin-^+Ksin^a E«#m M# k f ©fBm", ha a 2A, m i?8 ^ (1995), pp. 439-449 (A 2) 4) Smith, C. S.: “Compressive Strength of Welded 3) T, k ^ y ©@^%K: Steel Ship Grillage ”, Trans. RINA, Vol. 117 (1975), pp. 325-359 s) gma-en :(ms# ^#^5k,±^©43©^&©*©2oA#&r#%. k, *5:© m 152 ^ (1983), pp. 297-306 a(8)*:#WT,a. ^(9)©fKg[&, WTKK-f. 6) Bonello, M. A., Chryssanthopoulos, M. K. and 7i = l3i + Kifii, 72=a1 + Kia, Dowling, P. J.: “Ultimate Strength Design of Stiffened Plates Under Axial Compression and 73=0?2 + 2KK2,% + Ai/CA) ,/2 Bending ”, Marine Structures, Vol. 6 (1993), pp. 74=(ch + 2 Ki Kzou + KiKias)/2 533-552 75=03 + Ki fit + KzKafis + KiySe 7) : m 207 r*8##m©3@ 76=a3+Ki ou + Ki K% cfe + Ki ck X t ,it_ 41 (1993), pp. 323-327 77=- 78= 86 ' xb ’ ^="36 8) Yao, T., Nikolov, P. I. and Miyagawa, Y.: “Influences of Welding Imperfections on Stiffness of Rectangular Plate Under Thrust”, in “Mechan­ ical Effects of Welding ”, Karlsson, L., Lindgren, a3=%nbDP{^r+-^j L.-E. and Jonsson, M. (Eds.), IUTAM Symp., Ruled Swelden, Springer-Verlag (1992) , pp. 261- -"“•(w+w+i) 268 m27i2 , 2(1 — v) \ , mi7iiEIf 9) HcX'fSHJf, $E@grtil, Balu Varghese : hPw m27i2 4 a2 3a2 A2 4a* , m27i2GJf (1997), (mmiE) 4a2* 2 10) £jt@-a, mxvkmm, n : “spheeimsz . hDw ( m2x2 # 89 # (1995), pp. 179-190 , 4 — i7(7r + 2)) | mlxtEIf m2x2G]f 11) xmmtl: “DSB^-KOEB • EttslSSaS”, """ A2 E 2a2 ^ 4a"A2 226 SR 226-WG 2- 14-14 (1996) . hD„. o 8 \ min* , { x -2iA m 7i . x 12) DNV : Rules for Classification of Ships, Pt. 3, Ch. {(: a2* 2 16A2 1, Sec. 14 (1995), p. 104 i mijiiEIf i m2xiGJf 13) LR: Direct Calculation Procedural Document, 4a2 16a2* 2 m2;r26f &m2bt 2 m27i2bt Ch. 4 (1978) A = & = 14) ABS: Rules for Building and Classing Steel 8a 2 to* ' 15a Vessels, Pt. 5, Sec. 2 (1996), p. 46 15) NK : c m (1996), p. 183 16) : "ztgk

226 SR 226-WG 2-13-3 (1996) 4a2 ttS: xh K- X" \ b) E? -E# Dp =-12(1-^)' ^"""12(1-^)Pw flat-bar ." //=//=0 M sin sin IK sin angle-bar : If=bft//3, J/=bft}l3 a -M) tee-bar : 7/ = 6/f//12, lf = bftjl2 (A 1) 323 3-6 An Investigation of the Initial Resisting Mechanisms of Circular Hyperbolic Thin Shells

by Thomas George*, Member Hiroo Okada*, Member Nobuyoshi Fukuchi**, Member

Summary Many parts of marine structures are constituted of thin shell components, which by way of structural design or due to functional requirements may possess a negative Gaussian curvature. The load resisting mechanism of such shells are expectedly different from shells with positive Gaussian curva­ tures, and this aspect is studied in this research using the circular hyperbolic shell as a typical example. The Thin shell is defined here in the monoclinically convected coordinate system and considered to be subjected to a uniform follower load. The numerical accuracy and practical significance of all the theoretical formulations have already been substantiated through various simulations. In the simultaneous equations governing the equilibrium of shells, the presence of terms related to curvatures and their consequential factors create the distinctive aspect of ‘Form Effect', which would obviously be rendered more complicated by a negative Gaussian curvature. The resisting mechanisms of zero and positive Gaussian curvature shells have been examined earlier in detail through a similar study using the partial cylindrical and spherical shells. The present paper elaborates on the initial resisting mechanisms of circular hyperbolic shells by studying the share of different component terms associated with the extensional and bending stiffness factors, separately and in various details, to draw out an overall picture of the total load resisting equilibrium picture. All the possible combinations of circular hyperbolic curvatures are analyzed in a schematic division using a simultaneous measure based on the curvature invariants of shells. Compatible curvature ranges of hyperbolic shells are compared with the partial cylindrical and spherical shells to bring out some clear distinctions between the resisting mechanisms of each. These results are believed to be helpful in understanding the resisting mechanisms of hyperbolic shells in a clear perspective of both qualitative and quantitative terms, which would serve as a guideline for design and analysis.

different theoretical and practical investigations 1'2'. In 1. Introduction a practical situation, the present state of theoretical as Structural design is performed to meet the strength well as numerical developments in shell theory envelop a and functional requirements based on the respective sphere of criteria which are derived from not so clear codes and standards, which are updated continuously visualizations of the characteristics of each particular according to the latest experiences and needs posed by type of shell. new challenges. This makes it all the more important In the earlier stages of this research, a very detailed to upgrade the theoretical knowledge of the practitioner general governing equation has been formulated from through new analytical methods and interpretations the fundamental principles of continuum mechanics, for adaptable to his needs. the finite deformations of thin shells subjected to fol ­ Shell components are advantageously placed to play lower loads 3'. The shell continuum was defined in the an increasing role in the future demands for deep sea monoclinically convected coordinate system and the constructions, since their inherent strength and stability deflected states were expressed using the metric and due to ‘Form, Effect’ would make it possible to create curvature tensors of the deformed state written in terms lighter and safer structures in the ultra-high pressure of the original metric and curvature tensors and the environments of the ocean depth. The existing level of deflection values. Several numerical investigations knowledge about shells go into intricate details in have followed to substantiate the feasibility and significance of that formulation. At one stage of the research, the dynamic stability of shells under disturbed * College of Engineering, Osaka Prefecture equilibrium conditions were determined4' through an University ** Faculty of Engineering, Kyushu University investigation of the frequencies of natural vibrations Received 10 th Jan. 1997 and the dynamic responses to small disturbances of Read at the Spring meeting 15 th May 1997 shells at their equilibrium positions. Also, several new 324 Journal of The Society of Naval Architects of Japan, Vol. 181 aspects have been brought into light and many impor ­ uniformly distributed follower type loading is assumed tant conclusions have been derived on the dynamic for numerical calculations. The resisting mechanism is stability mechanisms of different shells, using the calculated by considering the entire shell as an integral method of disturbed small motions 5’. unit, and not for some selected points only, which In an immediately preceeding study 6’, the equilibrat ­ makes these results equally true for all points on the ing mechanisms of some particular shell types were shell for both qualitative and quantitative applicability. studied by analyzing the variations in the relative con ­ 2. Theoretical Formulations tributions from the groups of extensional and bending stiffness terms to the total load-resisting mechanism The process of deriving the general governing equa ­ from the initial and small deformation to the large tions for a thin shell undergoing finite deformations deformation stages. Further, detailed analyses of the from the fundamentals of tensor continuum mechanics contributions to the resisting mechanism from different is not given here1,3’. The treatment in this section is terms in the extensional and bending parts containing strictly limited to the statement of the final form of the the principal strains and their cross components, the governing equations and thereby proceed to the details inplane deflections, the Christoffel symbols and the necessary for the purpose of this paper. However, curvature tensors of mixed variance were also carried background materials and the terminology are ex­ out. plained to the minimum necessary levels, wherever In this paper, investigations similar to the above are found essential. conducted for the initial resisting mechanisms of circu­ The shell continuum is defined here in a system of lar hyperbolic (HP) shells. It is understood that an monoclinically converted coordinate axes defined over infinite number of curvature combinations exist for the the middle surface of a thin shell of uniform thickness general hyperbolic shell types. A particular method for t, as shown in Fig. 1. Also, the Kirchhoff-Love hypoth ­ identifying the curvature combinations of the circular eses are assumed to be true during the entire finite rotational shell types is devised here using the curvature deformation process. invariants field of such shells for a joint measure based The range convention for all the Greek indices on the Gaussian curvatures and mean curvatures, from appearing here and elsewhere in this paper are to be which the circular hyperbolic shell range is analyzed in taken as a, /5, y, ••• = !, 2. this paper. 2.1 Equilibrium Equations The subsequent analyses of the resisting mechanisms The general governing equations for the finite defor ­ of different hyperbolic shells reveal the significance of mations of thin shells are stated below. negative Gaussian curvatures, as a consequence of the IV" (1) joint action of ‘cable effects’ and ‘arch effects’ in the (2) two principal directions. As can be expected, the exten­ sional stiffness part solely dictates the entire resisting The symbol ‘||’ represents the 2-dimensional (surface) mechanism at large Gaussian curvatures for all values covariant differentiation of the preceding middle sur­ of mean curvatures, which transforms to the plate face quantities with respect to the subscripts. The mechanism at very small Gaussian curvatures for very symbol (a) denotes that these differentiations are to be small mean curvatures, and the cylindrical shell mecha­ done over the quantities after deformation. nism for large mean curvatures. All the quantities in these equations represent the The curvature ranges at which the extensional and state of the middle surface after deformation. IV", M" bending parts share the strength scenario in varying and Be,? are the membrane force tensor, the moment proportions is clearly brought out here. Also, the tensor and the curvature tensor of the middle surface significance of inplane deflection terms and the after the deformation, respectively. The tangential and Christoffel symbols for deep shells are illustrated. It is the normal component in the positive z direction of the shown that the hyperbolic shell as a whole is not pos ­ sibly a candidate for classification or consideration as a single structural type, but differs in various details and each of the differing types of curvatures has to be considered as a separate entity with particular charac­ teristic governing equations. It becomes possible to formulate the governing equation for the normal deflection of hyperbolic shells on a case by case basis depending on the curvature under consideration, there­ Upper surface by obtaining sufficient accuracy and saving unnecessary Middle surface numerical calculations. Lower surface A simply supported boundary condition with the in­ Fig. 1 The Definition of monoclinically convected plane deflections arrested along all the edges, and the coordinates An Investigation of the Initial Resisting Mechanisms of Circular Hyperbolic Thin Shells 325 applied load, which is of the uniformly distributed (10) follower type, are denoted respectively by if and p 3. Also, m“ is the sum of the surface tractions and body Here, Uk represent the deflection coefficient and forces contributed moment load. 8 Z) may be considered as a double trigonometric Now, the membrane force tensor, the moment tensor function, where 8 1 and 8 2 are the angular coordinates and the curvature tensor can be expressed in the follow ­ of the middle surface in the principal directions. ing forms. 2. 2 The Mechanism of Normal Deformations The initial resisting mechanism during the normal ###)%, (3) deformations of a thin shell is analyzed in this paper. (4) As a result, Eq. ( 2 ) which is the governing equation for the normal direction is to be considered for bringing out BaP= bctpP Kap (5) In the above equations, D and K are respectively the the characteristics of different load resisting mecha­ Bxtensional and the Bending stiffness parameters. aam nisms at the initial state of a shell. Here, Eq.( 2 ) is to is the Elasticity tensor of the middle surface before be represented in the following convenient form, where deformation. The initial curvature tensor is given by the £ term denotes the group of terms that associate bap whereas bl represents the curvature tensor of mixed with the Extensional stiffness parameter D and the B variance. is the strain tensor and £# is the change of term denotes the group of terms that associate with the curvature tensor of the middle surface. The above Bending stiffness parameter K. quantities are defined as given below, where E is the £ + B — ~P3 (11) Young ’s Modulus of Elasticity and v is the Poisson ’s The terms £ and B are to be further subdivided into ratio. their components as shown below, where the E terms Ef E f and the B terms represent grouping of terms having D= K = (6 ) (1-^) identical characteristics. £ =E1+E2 + E3 + E41 . , (7) B =B1 + B2 + B3 + B4J U

Sap — -^{Aap — aap) This subdivision is purely arbitrary, although its significance would become more visible when each

—2~{Ua,p+Up,c 2( tapUz + F Is til) group of terms is studied from the following detailed representation of their component terms. For the ease + 6i6^(%)'+ of visualization, these equations are given in terms of + bi[{bpuf + us.p) Ui + (T ipUfl — u>,p) uz] the strain, curvature and the change of curvature. + bp( UlUz.a — Uz.aUz)} (8 ) Kap = Uz,aP — babpMs ~ T apUz.A El = D(

3. Numerical Analysis of the Resisting Shell Thickness : 11 a-0.01 Mechanism The hyperbolic shells to be analyzed for their resist­ ing mechanisms in this paper form a distinctive class of shells by themselves due to the presence of a negative curvature in one principal direction, which renders at least one or both of their curvature invariants to be negative. An attempt is made here to clearly demon ­ strate the meaning and significance of this aspect through a systematic numerical analysis. 1l\ / R22 The particular class of circular hyperbolic shells are selected here as the most prominent representative of the negative Gaussian curvature shells. The boundaries The General The Circular are considered simply supported, with all the inplane Shell Dimensions Hyperbolic Shell deflections arrested along the edges, and the loading is considered to be the uniformly distributed follower type Fig. 2 The coordinate definition of toroidal shells and acting on the entire shell surface in the anti-radial the shell dimensions for numerical calculations direction. The differential geometry of a toroidal shell definition is used to express the basic values of the distance from the origin measured on the X-Y plane can tensor geometry of the middle surface. The coordinate be expressed as follows. definition of a rotational surface formed by a circular Ro=B + R cos B2 (15) generatrix is shown at the top of Fig. 2, where (part of) The distance Ro and the radius R are to be denoted a circle of radius R at a distance B from the origin of hereafter as the ‘primary ’ and the ‘secondary ’ radii coordinates O is assumed to rotate about the origin. respectively. The principal radius in the primary direc­ The semi-subtended angles fli and &2, as shown at the tion, Ru is calculated from Eq. (15) by substituting the bottom of the figure, determine the extend of the sur­ value of d2 as zero for shells generated by the right half face generated by rotation in the ‘primary ’ (51) and the of the generatrix circle and n for the left half, and the ‘secondary ’ (8 2) principal directions. Now, the ‘primary secondary principal radius R22 is R. When the absolute direction ’ (see sec. 2.2) of partial cylindrical shells is value of the principal radii are equal or when one of chosen in its longitudinal direction. them is infinity, as in the case of plates or cylindrical Rotations by the right half of the generatrix circle shells, the radius will be denoted by R in the presenta­ produces positive Gaussian curvature surfaces, and the tion of numerical results. left half gives negative Gaussian curvature (hyperbolic) Here, the shell geometry is considered to be that surfaces. When the distance B is zero and the radius R which produces a projected square base of unit area, by take finite values, a spherical surface is generated. A which the principal chord lengths [a = /i=&] become cylindrical surface may be generated by taking the unity, as shown in Fig. 2 for a general shell. The semi- distance B as infinite for finite values of the radius R, subtended angles and the principal radii are also which transforms to a plate surface for infinite values of shown in the figure, where the opposite curvatures in the R. The left half of the generatrix circle, which is shown primary directions of a hyperbolic shell are also shown. as a thick line curve in the figure, produces the hyper ­ In order to satisfy the thin shell assumptions, the shell bolic surface that transforms into the cylindrical or thickness t (f/a = 0.01) is considered to be uniform and plate surfaces according as to whether the distance B is sufficiently small enough in comparison with the infinite or both B and R are infinite. principal radii. For any point P(X, Y, Z) on the generatrix circle, the Now, the orthogonal Cartesian coordinates of point P An Investigation of the Initial Resisting Mechanisms of Circular Hyperbolic Thin Shells 327

can be expressed in terms of the curvilinear coordinates In this paper, the region ACD representing the HP (d\ 9 2, R) from which the differential geometry of the shells become the area of interest. It must be noted middle surface before deformation may be derived immediately that the two halves of this region lying on using the fundamental relations of tensor continuum both sides of line OA are identical in every way other mechanics. Thus, equations for the metric tensors, the than that they represent the same HP shells with inter­ curvature tensors and the Christoffel symbols of the changed principal radii values. Thus, it becomes surface may be written as follows. sufficient to investigate the resisting mechanism within the region AGO only and the mechanisms for the other ""-ow ' (a=t=!S) half can be found by interchanging values for the 6n=-Ro cos 62, 622=-R , 6#=0 {a+$) principal directions appropriately. The equilibrium equations [Eqs. (1), ( 2 ) ] under 61=—, 6,5=0 • static considerations are solved using the Galerkin ri Rsin 8 2 p 2 R« sin B2 method. Here, 16 modal combinations [m,n=4 in Eq. Tn R Flz R„ ’ (10)] are used. In the numerical analysis of Eq. (11) to More details on the derivations of these equations determine the resisting mechanism of shells, instead of may be obtained from the published literature1’3’ on this considering only a single monitoring point, for example subject. the center of the shell, the average value of several As stated earlier, the possible combinations of curva­ symmetric mesh points is calculated. However, the ture values exist in infinite number of permutations, allowable error in satisfying Eq. (11) is restricted well which makes it difficult to conduct a complete and within 1.0% by adjusting the number of mesh points. systematic analysis of the full range of circular shells. The value of E=1.96xlO n N/m2 for the Young ’s To overcome this, a particular scheme is devised here modulus and the Poisson ’s ratio of v=0.3 are used for based on the curvature invariants of Gaussian curvatures numerical calculations. (0=6162 — 6261) and mean curvatures (M = 61 + 62) as 3.1 The Initial Resisting Mechanism shown in Fig. 3. This figure covers a range of subtended The shares of extensional and bending stiffness parts angles from zero to tz, which gives a principal radii of the equilibrium equation for deformations in the (R/a) range of °° to 0.5 by virtue of the particular normal direction have been determined in an earlier choice of the shell geometry. study 31 using the corresponding principal diagonal Thus, for circular shells of revolution, the figure gives coefficient of the stiffness matrix for small deforma ­ a Gaussian curvature range of —4.0 to 4.0 and a mean tions. This qualitative result was verified in a further curvature range of —2.0 to 2.0 within which the plate, study 6’ in a more quantitative way by considering the cylindrical shells, spherical shells, elliptic (EP) shells whole equilibrium equation, Eq.(ll) in the present and the hyperbolic (HP) shells occupy their respective study, for partial cylindrical and spherical shells which positions. The plate is represented just by the origin O have zero and positive Gaussian curvatures, respective­ and the cylindrical shells are distributed along the line ly. The variations in the shares of extensional and OC. The curve OS represent the range of spherical bending stiffness parts were clearly depicted in that shells, and the region OCS gives the different variations study for the small deformation states of those shells, of EP shells. from the shallow to the deep curvature ranges. The present study forms a continuation of this research extended to the negative values of Gaussian curvatures. First of all, it becomes necessary to pin point the t hyperbolic shell type that can directly be compared o 3.0 with the previous results for partial cylindrical and spherical shells. It was made clear from this research ,-^x Partial spherical shells that the negative Gaussian curvature shells with zero (curve OS) mean curvature values fit exactly into this category, EP shells (area OCS) and this result is shown in Fig. 4 where the resisting Partial cylindrical shells {line OC) mechanism of zero mean curvature HP shells can be seen to occupy a place right between the partial cylin ­ 1.0 2.0 3.0 4.0 Gaussian curvature drical (zero Gaussian curvature) and partial spherical (positive Gaussian curvature) shells. Here the term, Influence parameter indicates the fraction of influence -2.9' v shared by a part for equilibrating the total load resisting mechanism. Fig. 3 A method of identifying different circular For all the three types of shells, shallow shells shown partial shells based on the ranges of their towards the right have their resisting mechanism Gaussian curvatures and Mean curvatures dominated by the bending part (8 ) and deep shells at 328 Journal of The Society of Naval Architects of Japan. Vol. 181

nisms, but for the interchanging of principal curvature directions. The aggregate influences of £ and B shown in Fig. 4 can be analyzed in more details for the shares of contri ­ butions from each of their component terms. This is 2 o.7 • Extensions! part Bending part illustrated in Fig. 5 where the earlier results for the partial cylindrical shells in Fig. 5(a) and partial spheri­ cal shells in Fig. 5(b) are shown along with the present cylindrical shells spherical shells - results for zero mean curvature HP shells in Fig. 5(c). A direct comparison is made posssible between the 5 0.3 Wstiefc HPshels - three shell types, where the natural similarities at the (zero mean (zero mean shallow shell ends towards the right of the figures are curvature) curvature) - clear and the striking dissimilarities towards the left end for deep shells are logically explainable. The influence of any component term omitted from these figures means that the corresponding share due to Radius (R/a) that term is negligibly small. Reasons for the Deep shells Shallow shells differences between the cylindrical and spherical shells have already been explained previously 19 . Now, the HP Fig. 4 The initial resisting mechanism of zero mean shell mechanism shows a very large influence from one curvature HP shells in comparison with those of of the principal component terms at the deep shell end partial cylindrical and spherical shells and a negative influence from the component E2. These can be explained easily when each of the mutually opposed curvatures in the principal directions are con ­ the left extreme have the extensional part (£) sidered to be developing the arch effect and the cable dominating the scenario. In between these two effect, respectively giving rise to compressive and tensile extremes, the shares of both S and £ are present in forces. As a result, El in the first principal direction different proportions. For deep HP shells, the resisting develops compressive stresses (and strains) from the mechanism approaches that of deep cylindrical shells, arch effect for very deep curvatures, and tensile stresses and for shallow HP shells it approaches the spherical (and strains) are developed for E4 in the other principal shell and eventually the plate mechanism. This result is direction due to cable effects. The same effects are the not any more intriguing when the geometrical reason for the negative E2 due to Poisson effects, which significance of each is considered. More interesting is shows that the arch effects are considerably large at the the fact that out of all the possible combinations of deep shell end. The total effect of these negative negative Gaussian curvature shells, the zero (or very influences are poised by the cable effect component small) mean curvature HP shells occupy a place which takes an influence value of more than unity. between zero and positive Gaussian curvature shells. It As the curvatures gradually diminish towards the should be remembered here and elsewhere in this paper right of the figure, the arch effect slowly decreases that both positive and negative mean curvatures of along with the cable effect and the components El and equal abolute values have identical resisting mecha­ E4 eventually become almost equal. The influences due

S 0.8-1 E 0.6 - E 0.6 - ® 0.4 ro 0.4 ffl 0.4

® 0.2 ® 0.2 ® 0.2 B1.B4 ® 0.U c -0.2-:

Radius (R/a) Radius (R/a) Radius (R/a) (a) Cylindrical shells (b) Spherical shells (c) Hyperbolic shells

Fig. 5 The shares of different components of extensional and bending stiffness parts in the initial resisting mechanism of zero mean curvature HP shells in comparison with those of partial cylindrical and spherical shells An Investigation of the Initial Resisting Mechanisms of Circular Hyperbolic Thin Shells 329 to all the bending components are seen to be identical ed by the resisting mechanism of the cylindrical shell. with those of the cylindrical and spherical shells for the The Gaussian curvature for which the contributions whole curvature range. from extensional and bending stiffness factors sharing The above results have demonstrated the resisting an equal influence was found to be —1/300 for the zero mechanisms of the particular class of HP shells with mean curvature end at which the principal radii are zero mean curvatures and their comparison with two (RnAz) = (Raa/s) ~ 17.32 for equal subtended angles of other types of shells. It could be understood from about 3.31° and a center rise of about 0.72# for each Fig. 3 that the zero mean curvature HP shells occupy only principal direction. Similarly, the cylindrical shell for the line OA and the rest of the possible combinations of which equal contributions from both the factors was HP shells occupy a large area. In order to conduct found to be having a radius of (RId) ~ 15.0 for a subtend­ a systematic analysis on this region, the present research ed angle of about 3.82° and a center rise of about 0.83# focusses on the positive vertical direction of increasing in the secondary principal direction. mean curvatures starting from different values of Gaus­ Thus, even for very small values of the subtended sian curvatures on the line OA. Thus, the following angle or the center rise, values which are sometimes results subdivide the HP shells based on the Gaussian considered as shallow curvatures in practical analysis, curvatures. the effect of the extensional part is already in par with Figures 6(a) and (b) show the results of this analy ­ that of the bending part. This shows the importance of sis for HP shells with Gaussian curvatures ranging from using more complete equations for numerical analyses very shallow to very deep for the corresponding ranges of shell mechanisms. of mean curvature values. It should be understood that Similar to Fig. 5, the detailed analyses of the resisting large values of either the Gaussian curvature or the mechanisms due to different component parts are shown mean curvature represent deep shells, and shallow shells in Fig. 7(a), (b) and ( c) for three different HP shells. are obtained when both the invariants become simulta­ The Gaussian curvature values of (a) and (c) are neously small. chosen arbitrarily on both sides of the Gaussian curva­ Now, both the figures show the one and the same ture value of —1/300 in (b) which is the case of equal thing, (a) for the aggregate extensional influence and aggregate influence from both extensional and bending (b) for its complementary effect of bending influence. stiffness terms. In all the cases, the large mean curva­ At very small values of the Gaussian curvature, the HP ture ends give the influence pattern of the deep cylindri ­ shell effect identifies with the cylindrical shells for all cal shells, where El and E4 are interchangeable, as values of the mean curvature, as shown by dotted lines mentioned earlier. in the figure, deep cylindrical shells being represented At the small mean curvature end, smaller Gaussian towards the right. This is the same for large Gaussian curvature shells show the shallow double curvature curvature values also at very large values of the mean effect, as could be expected, and larger Gaussian curva­ curvature. Thus, the resisting mechanisms of HP shells ture shells give the deep double curvature effect, except of all curvature combinations can be found to be bound ­ that in these cases the component E2 is negative for

(Very deep HP shells) 1.0-_ G=-1.0 G=-0.000001 (Partial cylindrical shells) G=-0.1 •0.0002

G=-0.02 G=-0.0005 & [G: Gaussian curvature ] G=-0.001

G=-0.02 "t [ G : Gaussian curvature ] G=-0.05

(Partial cylindrical shells) _ G=-1.0 (Very deep HP shells) 0.001 0.010 0 1.000 0.001 0.010 0 1.000 Shallow Mean curvature Mean curvature (a) Extensional stiffness part (b) Bending stiffness part

Fig. 6 The charts of aggregate influences of extensional and bending stiffness parts in the initial resisting mechanisms of HP shells of different Gaussian curvatures 330 Journal of The Society of Naval Architects of Japan, Vol. 181

1.0--

CL 0.4" 0.4-

» 0.2 - O 0.2 ;: B1,B3,B4

2 0.0--'

-0.2 —-0.2-;

0.001 0.010 0.100 1.000 0.001 0.010 0.100 1.000 Mean curvature Mean curvature Mean curvature (a) Gaussian curvature =-1/10000 (b) Gaussian curvature =-1/300 (c) Gaussian curvature =-1/100

Fig. 7 The shares of different components of extensional and bending stiffness parts in the initial resisting mechanisms of HP shells of different Gaussian curvatures reasons already explained. Generally, the larger the terms61 for the partial cylindrical and spherical shells, Gaussian curvature or the mean curvature, the larger here also the inplane deflection components ua (or u“) becomes the extensional influence. However, the strik­ or their derivatives, the Christoffel symbols Fie and the ing differences between these three figures numerically curvature of mixed variance 6#are selected to study the illustrate the variety of resisting mechanisms that exist influences of terms containing them in the extensional within the range of HP shell curvatures. and the bending parts, as shown in Fig. 8. Here, only the 3. 2 Initial Influence of Small Factors zero mean curvature HP shells are shown to facilitate The aggregate influences due to the extensional and a direct comparison between them and the results for the bending parts and their components were examined the cylindrical and spherical shells obtained earlier. already. What might be remaining is some more Initial influence of the inplane deflection terms is detailed investigation of the influences due to some of illustrated in Fig. 8(a) where the shallow shell end at the much smaller, but obviously interesting components the right of the figure has similar influence values for all in the shell governing equations, which are composed of the three shells. However, the deep shell end has a very several small terms. It might be worthwhile to study characteristic curve for the HP shell which could be some of those terms and determine their shares in the easily explained using Fig. 5(c). Initially for very deep total resisting mechanism. This process might further shells, the cable effect induces a very large net value of the aim of understanding the overall resisting mecha­ extensional stress and reduces slowly along with arch nism in much finer details. effect as the curvatures become smaller, where the net In line with a previous investigation on similar value of membrane stresses from both the principal

0.14; - 0.12 -

0.10- / HP shells " 0.08 (zero mean curvature) 0.06 - 0.04- , Spherical shells 0.02 V\ 0.00- 1.0 10.0 100.0 0.5 1.0 1.5 2.0 2.5 Radius (R/a) Radius (R/a) (a) Inplane deflection (b) Christoffel symbols

Fig. 8 The shares of inplane deflection terms and Christoffel symbols in the initial resisting mechanism of zero mean curvature HP shells in comparison with those of partial cylindrical and spherical shells An Investigation of the Initial Resisting Mechanisms of Circular Hyperbolic Thin Shells 331

directions become additive. Thereafter, the net effects of hyperbolic shells are only a variation of the zero slowly become smaller for shallower shells. Gaussian curvature shell mechanisms with an added For HP shells, a maximum influence of upto 8.0% negative principal curvature to the longitudinal direc­ exists at the deep curvature end, as different from the tion. This research has clearly shown that the hyper ­ other two shell types where the maximum influence was bolic shell mechanisms exist in infinite combinations, obtained within the medium curvature ranges. This limiting cases of which are the plate and the zero result shows that the inplane deflection terms can not be Gaussian curvature shell mechanisms. neglected altogether, even for the so called shallow The classification of shells based on the curvature shells. However, detailed results have shown that the invariants field is found to be a very convenient tool for same may be neglected in the bending part S since its differentiating between the combinations of curvature significance lies only in the extensional part G for all types. A very clear comparison between the resisting types of shells. mechanisms of zero mean curvature hyperbolic shells, The share of initial influence of the Christoffel sym ­ zero Gaussian curvature shells and spherical shells are bols are shown in Fig. 8(b) for the HP shell along with obtained in this study. Also, the charts which depict the the earlier result for partial spherical shells. Although variations of resisting mechanisms in the curvature the shallow curvature end shows negligible influence invariants field, and the results for the numerical due to the Christoffel symbols, the large curvature significance of inplane deflections and the Christoffel range shows an influence exceeding 14.0% for HP symbols would serve as a useful guideline for design by shells, which is more than double that of the case of analysis. The quantitative nature of present results spherical shells. Here again the significance lies only could serve as a useful guideline for other types of with the extensional part G for both types of shells. hyperbolic shell problems too. Thus, the Christoffel symbols can not be neglected References altogether from the equilibrium equations of deep HP shells. 1) Fltigge. W.: Tensor Analysis and Continuum Mechanics, Springer-Verlag, New York (1972). The curvature tensors of mixed variance were found 2) Pietraszkiewicz. W.: Geometrically Nonlinear to be exhibiting negligible initial influence for all curva­ Theories of Thin Elastic Shells, Advances in ture ranges. These investigations have shown that the Mechanics, 12 (1989), pp. 51~130. influences of inplane deflection terms and Christoffel 3) Shinoda. T., George. T., Fukuchi. N.: A Detailed symbols exist only in the extensional part, and their Analysis of the Theory of Thin Shells and some effects are considerably large for deep shells. Also, the Particular Applications, Trans, of the West case of the zero mean curvature HP shells shown in the Japan Society of Naval Architects, No. 80 (1990), pp. 171—193. above figure was found to be fairly representative of 4) Fukuchi. N., George. T., Shinoda. T.: Dynamic other HP shells also for their qualitative nature, and Instability Analysis of Thin Shell Structures quantitatively the maximum influences were generally subjected to Follower Forces, (2 nd Report) obtained in the case of zero mean curvature HP shells. Numerical Solutions and some Theoretical Con ­ In the shell governing equations, corresponding varia­ cepts, Journal of the Society of Naval Architects tions are possible to include or exclude a particular of Japan, No. 171 (1992), pp. 597— 609. 5) George. T., Fukuchi. N.: Dynamic Instability term or group of terms, according to the shell type and Analysis of Thin Shell Structures subjected to the curvature range. Such a process of evaluation Follower Forces, (6 th Report) Dynamic Thresh­ conducted before making the detailed numerical ana­ old Characteristics and Post-critical Stability, lyses would be beneficial from the view point of sim­ Journal of the Society of Naval Architects of plifying the theoretical process for a better understand­ Japan, No. 176 (1994), pp. 319 —330. ing of many underlying phenomena. 6) George. T., Okada. H., Fukuchi. N.: The Resist­ ing Mechanism during the Large Normal 4. Conclusions Deflection of Thin Shells subjected to Follower Forces, Journal of the Society of Naval Archi­ It is generally assumed that the resisting mechanisms tects of Japan, No. 179 (1996), pp. 281— 292.

333 3-7

Eg

lEit iH m m m* em m eb 1# m* EM ± m # m** On the Shear Buckling Collapse of Continuous Rectangular Panels

by Koji Masaoka, Member Hiroo Okada, Member Yukio Ueda, Member Summary

It is important that ships are prevented from fatal failures such as the hull girder collapse under severe sea conditions. On the other hand it is also important to know the reduction of the strength of a ship with lightening caused by the constant demand for lighter ship hulls. Severe shear force acts in the side shell near the bow of the ship during slamming conditions. Thus it becomes important to investigate not only compressive behavior of the deck panels but also shear collapse behavior of the side shell panels to determine the collapse behavior of the ship's hull girder. Simply supported and uniform shear load boundary conditions are suitable to examine the plate behaviour until the shear collapse load. But strength evaluation considering the continuity between side shell panels is needed to analyze the load carrying capacity accurately during the post shear collapse. In this paper it is intended to solve only one panel considering the inplane and rotational continuity between panels as the boundary condition instead of solving many panels to investigate the panel behavior of the post ultimate strength region, such as the load carrying capacity under shear.

i ©jftigi-i4©fcib&v\ iot.

It» K ii iff IS < a. jovrfctfrtlTfc^© J; 5

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2 5Eg^/U';u©«#^ftlcs|-rs^W RB Model Lt ^ 6 % fotPit'CySt'#; £ & 6#m&##L±a?z)m@©z©##^?#6. u^

S§ f£ il ffl-f S © tt B H T'& 5. tZT, L, *#"t a/t-tA-ta 9"/ £ ©*%*&#* L

UF Model

Fig.2 Displacement control model with rigid edge bars (RB Model)

2.2 (RB Model)

Fig. 2ttiE'(ilSJffliS©^TyVT-&a.

61^ Ty>-C& i>. l©X?y©Sr RB Model (Rigid Bar Model) £^-5. Fig. 2 (b) it Fig. 2 (a) SrESlcg tfcVSoAfcEltfe 5. Fig. l (b) ©^T'yvfc tthiiot, ttri4 < /<^yi- & L#o*a C £^T#-CV'6. tfc^-j-r, AAybM© Bm©mmaa#<©aK m*i±, ^.tn^ if comm Fig.l Load control model subjected to uni­ form shear force (UF Model) 2.3 yUoizto^ESroBA^eoaEtt

2.1 (UF Model) ccr, Fig. mftmmfflK x^xmmnMwmfflMZMVt AlCFig. 3©R*K*r#A6. 5 1-&#a-©0T&a. JiiBfc-SSE* 5 -tirt© S L, 3; ^[p]^©3E(&& u, y^lRl^toEili^ V, -*|S]^© r, jaE©Bi*3*T6]tt, Mfriaig^ihfeafcfe©^^^ E(£4- wit5. ux v' * v \ mm mm a-r a s -e «js

Simply Supported PanelB

Stiffner

SS Model

Panel A Clamped

Fig.3 Displacement of the panels and the CL Model stiffener Rotationally Continuous

Fig. 5 Rotational boundary conditions of the panel edges

Table 1 Inplane and rotational boundary con ­ ditions

Inpiane Boundary Condition LTF Uniform Forced Weak RB Rigid Bar Strong 1C inplane continuous Moderate Rotational Boundary Condition SS Simply Supported Weak CL Clamped Strong KC Rotationally Continuous Moderate

l±/<;&yF©#&&l #©&#&## LIT (D&ID. Fig.4 Displacement control model with inplane-continuous edges (IC Model) DUz©##U: 9 z© Fig. 4©$#&/<^yD|W©gl^© tm&D, ^©IrxyFST IC Model (In-plane Continuous Model) ct X7N7d-©^:/tey;S-|6]tctetofBtoV;Mtfc&iD i"5. 2.4 5 il&tc. fcfcD, A Lr^*/v b ttm©@iEt4 /<^y>©immgii©iEiRA©a##Kc©ir Fig. £ fc D'i-VMifi-5 ©T-, A ©J9®©E)5: i-SrSx. 5&##lc#A5. *yv © 3S$ £ $£flj # x 5 & 6 HJ9ESr*-E5:# t ©^#a© r *"|p]^©#Ei^M Supported). 2%, ^Td7f^##l:#

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#&a. #*3E*T&a ss Model i-a#erm@^jai'TA#i-^aw@i:*a. cl Model fix tV y-rtm < ^^©©ISSr^: <»£ vvefc 9 XT-f 7T©iiH4S:iS:kFF-ffiLT LStcf&E&flJ i^otlSi x tM 7-Vto^ES©@|SP|iJtt£«l® V fc rc Model ttnie^ ©m*% sr # x a t tsa/^f yvtsgffei-av tds-C't a. a_B©#AlcS 9, #fTa|gfCi±RC-IC Model (Rotationally and In-plane Continuous Model) Sr# 9 © g 3 "C fo a. m«<£f*ic:bfcoT^Tvv©iWSMto-fXtW =ev\ v^v. #a©3E^ist^/ta^±# &% 9 m vc%K#e#j* vtmaw* MVfSr RC-IC Model  ztKS9fr5 zt a. 3 G®m£& Fig .6 Fundamental in-plane displacements &.±: fteM*it^yL©«#^#©S-attVov'-c5£^fc (a) ~ (e) and the combined displace­ d:, fji^©##Aft=»#tV'kd:6^ri5g*&-C##& ment (f) of the panel (9 nodal points model). i-ae&^ov'-c*#Tis)e^a. T0, #mmmca7d'y/<7/HJ y7$*© —3-c& a 4 @P A MITC SEX (Mixed Interpolation of Tensorial 3.2 Components Element) Srffiofc®) . ##pi©###^)'iC ffi ft 0## mm&ft s- # tS)!f9x/yt;t7K/v#^Srmv\ -##©#^A©Rlisi- a fc »ic *»%-e it, te fc-it a Ett s 6 s © Brtii'2 X 2 A. Eff^[Bli-6 AiroTt'a. t 9*tcx*SrLfc. ^©*&%@Slc|fc^i-afci61C9 3.i imi&ffiMtDtiim' is a © m &s- ® f*} m fs i c o v' r © ^ % x a. Fig. elcfiS-IES^e-K irTtv^-e-j&LfcES^TT. mvTV'a. *#&#ici3s--gx©8HR&#a&amv Fig. 6 (a) W:.>:;9-[B]'--©#ITJ!X— K^rVi-C'fe bt> V, -Cj39,*#a#lSMc$#^A%Kjac%'pT^a. (b) tt y^lfil^tottV^e-K Sr t/2T', (c) teBEESx- v-c##f&fro-cv^a. KSr fLT-feVbi-. £ Vic, (d) tt^^iaiH-ES^e-K 9,*f@^XTyy@K ^ % V %T, (e) tt±Tffl|Wl-ES^-F Sr Uek U?X Newton-Raphson ftSriSffl V, THIRSTZ> S "XflljlSr &6kVCV'a. L, 9 &5 BrtSEt4 as = Vs, as = U2, U4 — fV . U4 — Ur, ^0wm#&&#&Tat&tc%,

^ ^#a*Aa. Us = U5, a.5 — V5 , u6 = U3 + Uc, V(, = U2 + Ur, 337

ut = Ui, vj = 0, us = Ui + U4, vs = Us, 2 2?, a i4/^yV©fl:5, 6 iiitl, t ttEJ¥, Ei±X>?

«g = !7j + l/3, vy = U2. $micAii/-cv\%v\ $fc, s^/i'CofflMMz-cofrm&az 5t(l)£"7l' !/ fcit(2) fc*a. (5)©jL?K#^.5^).

wq sin(7T.ir/i) sin(7ry/6), {«} = mm (2) (0 < x < 6/2), wo sin(wy/b), K\ {•u} = {ui VI ■■■ u 9 yg} T w(z,y) = < (6/2 < x < a — 6/2), ’ wo sin(7r(a — -r)/6) sin(?ry/ 6), {[/} = {u5 Vs Ui ■■■ Uj}T (a — 6/2 < x < a).

[T\ »s;(i) ^fex^na. a: (2) {/},

m = m^{/} (3)

{/} = {Zrl fyl "• 1x9 Sys}T (a) Stress-strain curves m} = {A: Fi ... f?}^ -.. 1C mti, we/: /: t

[A"] = [T]t[A][T] (4) 0.6 LT^© 3:5 ^3E*-7 h y -CV'6 /: A < $/,© n-smffl fc H-SS»as£.$ bt£oT L £ 5 ©T, SSffl'Jtt - ©iaaSr-f-5. El-htt 2 x 2 **##©#&&#/: Lt#MT&a#, z©#&ic^6-3-c#©metmf&fT5 /L/r^-et a. 3.3 aemm&oit&xjvi'&wffatAwpr eeit

(b) Stress-deflection curves ---■ - B DR

#g#t#*;%dia-&-r5#e©i%iatcami-ac/:ai -et a. t> v, 2 m±mo%-v?m hxmmtim'f® tm

$#©*#*&##LTt, x^yv©)lEtt@rtirtta

4 fiSfirSB*

::n i$Ett%#SVfc^*/v&a®$SiS-e/f©J:

5 $7t, se©&h* ©8efl!#iSKdoV^-Ot/f©3: 5 &mvv©$S©i$iEt£ &#Aa#-a-ai%-?&a;Nccv'T%/

Fig.7 Average stress-strain . curves and */v©#Ett*8iA#Nff£ff 5. stress-deflection curves of the square plates subjected to shear, 0 — 4, ##i-a±-ceg^/<7/-^^&#i-a. wo l i = 0.1 « = a/6 (iMtfcfc), 0 = -^/-^- (tofttti) 338 181#

Sflt) 1$fc A/US: v;; k5. -4i, S (a) 8=3.0 rt©EEtt^?SS U*V' UF ^e^/VT-ttA^aSflirS Z i: B*©$Eti%*S V fc ic ^T?/H4^ ©*raw*fi:^h5't-- i z©

©HH5#©^#:Sf*©t)^v'icJ:-oT5SSir^7y*^ ±c6m*#mic#v'-c#@mmm, tue^iSE© IC-RC Model *fcoTFFff4-5 c t##wrc&a.

tw!, #m& &m-t6Wm^te'?X','6 z Fig. 814 (a) tfS/3 = 3 ©##©#mm6#m-C& 9 , (b) &0 = 2 ©##-©mm#l##-e&6. (a) ©##14, Fig. 7© it Bt'14»t^T-14*V^s b LTtthbIS

(b) 8=2.0 C©##l4Fig. 7©t#R „....RB ■---- IC / UF" ^14*#mA%©%a@TI4» R U 4 9 St@ e lpl&^i-Ct(4kA^. ^ = 3©#-8-l4###Al4f© 0.8 - ^-^lUot^-Cih#irBEtt#14r-iijaL-Cjb'9Sl4 -e©#©##tc##&#ai*6fi/a. (b) ©,a = 2 ©##lci4#tEK#lK©§g©.Rm#%^#©t7vHc 0.6 - JSV'T b

■ / C©-y-^©##, 0.4 - F*s$#©^K j: "D-cEit-fbaima L-cv'6.

0.2

/ Y\ °0 1 2 3 4 5

Fig .8 Average stress-strain curves and stress-deflection curves of the square plates subjected to shear, wo/t = 0.1, (a) ,6 = 3, (b) ,6 = 2

4.1 lE^tetDjgtir

Fig $ it & t it,6 = 4

6. #*@©3E)»&^6t, ©A, B, C, D ©###4.3 maicMTS##

CftE,©**©### Fig. loa#*

(b) (c) ttJi B@tei$E®;eTyv-e&5). iiifttt/? = 4 ©itvxtglco^ -c, 5 (SK&ofc i # ©0

SrSL-O'5. #@fc A, B©8P&o(ft, #0© A, Btt (a)l:j6t'-CI4, B©04EA^#* (b) Kkt'Ttt, 19E^BS-efo5©T-, A, B

(c)14A, BT©

A©#ai4L'f^CA,Tt'6C6^kA^.

x//. *. ’It 'sl 'k. •*. Xx x % % >n v \ \ X X H X \) |X x x % % % X X X x X k X /X k * H x X X X x x X k X /X * * x x x X X X X XXX x k X / X k X X X X X X x x X x X X / X k X X X X x k X x X * X V J X x X X X x k x XXX X X i X X X X x x X x X X x X X 1 X XX* x X x X X X X k X / X X x k X X X X X X x k X / X XXX X X X X XXX 4 /X x x X X X x x X X A x X X x XI k\ X X * a3 >CL: •%. ■*.

Fig. 11 Principal sectional forces and nodal forces of the rotationally continuous square plates. 0 = 4. 7 = 5"yy . u’o/t = 0.1, IC-RC Model

4.4

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3. A, B, C,

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CC&-CI4, jE*S6C-ot'riStff^fTV'iiEAM«% L

#m©^-&/<^/be%& L-c#*r^ & ^ e Fig. 10 Deflection shapes and their contours of the inplane-continuous square plate, p = 4, -y — 5-yy , Rota ­ C Z 4N4, ilK#ltdi 2, 3 ©/t, ^A@i©aa#M3tR (RC Model) #«rKRLTI4%%#^©*#$l4 Wo = (/10&r#tTV''6. )K#jt^2©e7'/Wt32x 16 ##], Wit^S 3 ©^7VW4 48 X 16 ^#11 V7t. 340 mm# ■ft±x-, aamot'-c#w Lt. t©# (a) 0=3.0 #, m«/<^/wcjot^ii/<^/FR©mmigm, sis Z i SrIMUfc.

it it * m # * # © sii « % bj m i z -t % % m * m v' fc.

3. l t. t©**, K*#*iz#z);m5zt^w^t. #m©* 1C Model tc£EflMz ^)©Stv^B1tMe»Dx:fc. @Ste© s-BiiiEjjstsj; y t#m@T©m^a^#i/'z t

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# # * SR

1) Y. Ueda, 3.M.H. Bashed. Y. Abdel-Nasser: An improved ISUM rectangular plate element: taking account of post-ultimate strength behavior. J. of SNAJ, Vol. 171, (1992). 2) T. Yao and P.I. Nikolov: Progressive collapse anal­ ysis of a ship ’s hull under longitudinal bending, J. of SNAJ, (1st report) Vol. 170, (1991), pp. 449-461, 1C Model (2nd report) Vol 172, (1992), pp. 437-446. 3) K. Easier: Strength of Plate Girders in Shear, J. of the Structure Div. Proc, of the ASCE, (1961), pp.151.

°0 1 2 3 4 5 4) #B, ##: EJE£Ett£#SLfcft/hSeEH- -%

Fig.12 Stress-strain curves of the rectangular #119 (1966), pp. 200-208. plate subjected to shear, (a) ,6 = 3. 5) P.C. Davidson, J.C. Chapman, C.S. Smith, P.J. (b) f = 2 Dowling: The design of plate panels subject to in­ plane shear and biaxial compression. Trans. RINA, (1989), pp. 267-286. ###*& Fig. 121 CTpf. (a) 11,6 = 3 ©S-S-Ol+S 6) #m, R*,

i. »m-r?>^^mk^r^7i-k

iem it # iem m m m m ^ ^ B***

A Study on A Posteriori Error Estimation of Finite Element Analysis for Eigenvalue Problem of Frame Structures

by Mitsuru Kitamura, Member YuHua Chen, Student Member Hisashi Nobukawa, Member Hideomi Ohtsubo, Member

Summary A posteriori error estimation method for finite element solutions of natural frequencies and natural mode shapes of structures is proposed in this paper. The governing equations of a posteriori error for free vibration problems are introduced. A method used for solving the error equations is presented and an efficient solution strategy is developed since it is difficult to solve the error equations directly. A technique of eliminating the degrees of freedom at the mid-node of element is developed in order to reduce the size of matrix in error equations. The finite element solutions of eigenvalues and eigenvectors are improved by adding the estimated error onto the original solutions.

i. m w 5>tLSo . ^ f) _.. a'wfi. f) ^ az' in Q (1) u = u on f, (2) I'D 'D II s (3) Zienkiewicz^#fe. : :T, «(%, y) A *, <0 5. @@ O K##r. mAliEr C j: 9 k c&L, a 3 *165: BgfiSttglW®R®, #E7U % =0 (4) ■So C i: T « 6. ^ (4) j: 9 2. «Eig^FFfiE£<7)$lB b/!/&#&& C: «(z, f) k#Bg*# «(z, f) KDIToi ^

u(x, t)=U(x)sincot (5) ** u(x, t)= U(x)sma>t (6) c ^-r f/w k & mess 9 * i a 9 a 9, i7(i) k a K *Gg9^5H15a 342 #181^

3 7 {#(z)} k K ^ ^ 3 = — oj2U(x ) + co2U(x ) (19)

RSr^73^^ Wi/(i/}, KiD$#$ft3. gfg i Efetf 3 '&&WM fc MSEE 3. C0R#MjR&^t^WKg%&Sr^ d7) cam? 3 k^o#R#*^e^. [4f](e.}+[^]{e«}=(G)-W (22) ^73k%^. k kT{G} k M H, f jtf ft, E k#A%:R@#a A(, 0#^E 3: D#6ft3 W} EjSB73##m!jUrT&3, (9) mmm&MmK: #v^-c, g#, Em-3#&% #,(#) a WT0j:^K#eAKM*f 3#E^G) k*%±EM^ 1-3* #,'(#)*:##?# 3. (10) E,(G)=E,'(G)+E,'(G) (23) ±^Sf 03(,Hg#, k;M®g*##^K^^3%# ^(23)@^#&m^3k, mgKjeai-3#m*^A^^ *0:R*Rar&3. ^(5), (6)»^(4), (9)CftA HM± 73k^*l#^fL3. (G},= -j[ W(j?KG)+7??(G))^0< (-^w4pw=[mi+^AE^-^n ={G'},+{G'}, (24) E»3. C03:^»{G}0^-ietm^3^kKj:D^(22) i±%^E»3. (-^r/l^WO+^AE-gr^O-AE^^]^ [M]{e.}+[A"]{g,}={G'}-(A}+{G'} (25) ^(25) @ A##E^^3##K%^T$ — 2 vAa—J^vRdQ^)sindl = 0 (12) 3* 1, i%%»mM(±7^-Ag*0f®mo#aaegra-E ZCT, ^LTtgRa-4-3. ##m^0^^i±^c0WRg**$f E = -Ap&'G-AE-^ in O (13) cmw*:#Rg#0m(B(MR0%&i o & ima&r&a TsA, 2ms (12@AA) y-YV/^^l-Vyryg^Kj: ^■(^f).-(^f: ^ on Fa (14) o 3 %& (is sincvf =#=0, sinS#*0 T*$>3?i, 5$ (11), (12) 7/ 7^9 7 h V ^7g*%m^7j#f73. #Tt3. 2.2 1ggSJUA A##k#g%K:j3ij-3##m : j: D^iK'lt3.

II e||=-/|l e“\\ 2 + ||e6!2 (26) -^cApa'wo+^yLE^-^o \\e\\i=A\eu ;+k*M (27) lle“ll22lk“| |2=2j[ elE~le5^(16) SlOK k%^o ±^*|e"|| k ||e*| K,

/)VK~Ch V, ea k ev n.iB* km@®aa#T&3„ (I?) (30) e*#, ^%@@kAma@@KjsW3m%T& ev=coU(x) — aiU(x) (31) 5o 3. mmmM& eu(x) = U(x) — U(x) (18) ea(x) = {- a?U {x))—{—&2U{x)) Ipfi? h 1> y 7 7, (consistent mass matrix) Sr 31 71'-AmmstpmmunmKmhm 343

LAA, e%K:-^^6^Aa$-m^-C#(,fL7t#{G}k #»###*# w j: 0 6A# < »37kA, a tt5$(44)£7@JS"C§6o0 -0*9, w = (o+Aco (32) {[K}~

Ai ([an-w'[M])W={G'}-U} + #G'} (34) A {A} = [ir]{m (47) 0 = 1-2 -^--4^-4^ (35) -An- v^S STxB^e.iifcsUiijSVi'T {ej Bfiv b 9 7 X X KmL-nEafkgfL-cv^. o$9, ({G)+{g,})lM]({G}+(g,})=1.0 (36) [M}{U} (48)

iBj 7T3&@rt37kA, ###*#%: ,9 &9W&f&;:kW:;# ccr%gwPitFsft/hK-rsr jK%»)9&m^»0 (0=aw (0< or K » 3 $ ?, step-b) ([A"]-afa'[M]){e:)={G'}-W+j8°{G'} (39) —step-d) §•§§ 9 M~fo ±5$Kj3V>T o'Ofg^SibSrktkJ: 9, f) {G}'-» k a"-" C i 3#JE$ fuTk iS(39)k&B#5%K:»9, {e£} ttz, @ HW-^7 h k ISWBligte 6 3 „ b/i/H*®j:i)E#Z^3Z:k*;T#3. {G9}={G}+{e:} (40) 4. Vh^y^^Xroil {G°} kK 3: 9#]EgfLA@#^ %®W®gS$F#tEffl^?igSJ: 9 fe 1 ^b;pT*»9, ^Oj:^K:IE#'(kr@3o $C&##m^*pTE#^i-3g'm^37kA, ##K## {U}=-^—{U°}=-^—{{U} + {e'£i) (41) #07 b 9 7 7X1MX'ti:A§ < *9, fiSsfciSAaLr Ul/ UiK L&5. CCT1±, ^(34)&#o [K\i={[K]-ct2w2[M])i (50) {uYvmm (43) 3-t^ {gi}, k L, (G}^[M]{G} g***6BAK:is)j- 3 t O %r W, k f 3 k, ^(49) ##3% a SrS*3 kan 3. (\K}-w2[M}){U}=Y>} (44) 344 181 -t

"[&i] lKl2] 1 r{e,h = 7-f V/<9X h V y _[K2i] [-K22]. A{ea}J ■ ###'ref kK f{#h rMi}) f(Gf}i 3: 0, 3: 0 #1E3ihfk# 'esf ©*#%&# 1{W' l(vWJ' (51) (Gf), (Gf), (Gf) {Gf} ###?&&#, WWRg an#, ##©m# LA^L, 7l/-A|R@K:jSV)-Cg @*R$* Table 1 #, 3: 0 *%K X^S/SttftfflgXk^LT ##*©%# maZ < @W@*Rt#lE3 fi)k C k Table2 A 0$ 0, W K-tfo mi,m4,m6@#Bm#K:B#aBmM#F k/<9/- ±^K*LTC©#:m=&F8lt^k ^ a©M#t^TA*, *#K:3:D#lE3fi^B|W#m»a

[Ku\i{ei}i + [ KvA i{e 2} • aE%#L»v^k^3fLTt^o Tkki —{Giit — {yli} i + /?{Gf} i (52) If, m6 0##KKK:3Sl)T, aA*0.4-0.9(3H3»aK

[ &i] i{ ei] i+[KzA,•{62} ,• 0.64)©#BT^-fkLTt#%K:Av^*A#6flTH3.

={G2"},+^{Gf}, (53) /

[KbHK^-lKuUKzAAlKzAi (55)

{61} i—{Gf} i — [ K\o\ [[^22] i 1{Gi} i (55) ©53*1)7 Fig. 2,3 CzF3 iKTV:> 1fi, T}~f tb-x-^/ls^ — E {6*}i={Gf} i-[Ki2]i[K:22]71{GI}i (57) Mi-6#^*#%Kit3 6* Tv^6 C kA%5. t— F ©^EAS'SSat-5 KW-3 X 1E*% —EMi-6 Ex^ffi^ssseBscj: vm'&mmm ^:gAM»«^©*Kg*#$r©b©k^L<»e. 3 6 ;w^'-y ;fa©53* i)iffirnltt#^— &. Ak)Uf, #35XM:#llt-FKj3^-CK^V;yA

A*s#M\3w##K:#^-rv)&. ?#l#^/;PAk##K#//FA©jtA;Fig.4 E^g 5. ft fi m ffi m ^gXKjaw-C^eir/leHr^lK^-oT

3: t #Ef a. Table 1 Comparison of eigenvalues for the plane frame problem 5.1 ¥®7U-AF^M Fig. 1-a) CS?il55FH7 v —A©^®^ Fig.l-b) © Mode # U) GJest uref F a 1 15.87 15.35 15.35 0.003 0.967 2 24.45 23.28 23.28 4.689 0.952 3^7-f y/<9/ h V &%, 3#^ 3 56.97 52.77 52.77 5.444 0.927 4 131.87 112.58 112.58 0.587 0.850 1 5 260.72 196.80 196.80 2.132 0.755 6 475.82 304.88 304.88 0.159 0.640 / = 4.19xl0" 5m4 /=6.49xlO"V A = 2.4xl0"3m2 / = 6.49xl0" 5m4 A = 6.8xl0* 3m2 A = 6.8xl0 ‘3m2 •_ - l = 20m E-6.67x10 N/m2 - Table 2 Relationship between the modified eignvalue © h = 3 m P = 2620 Kg/m 3 ™ and papameter a for the plane frame problem

a) Strucutre model of plane frame First mode Fourth mode Sixth mode a w"* a w"* a w"* i®«®»®i®i®»®»®«®i®.®i 0.50 15.35 0.50 112.58 0.40 304.90 © @ ® Ci; — @ Element Number ® 0.70 15.35 0.70 112.58 0.55 304.90 © © 0.90 15.35 0.85 112.58 0.64 304.88 0.967 15.35 112.62 b) Finite element sub-division 0.95 0.80 305.45 1.00 15.35 1.00 114.02 0.95 309.72 Fig. 1 Plane Frame Problem ^ 345

^—1 "7 0.0006 3^ z= -y 0.0005 -y 77, 77 / w sc -VT 0.0004 77. U; . b b b b 0.0003

0.0002 a-1.0m t w || c = 1.0 m «2=7mm Ik 1: tif = 0.5m h = 130mm . ~iF . 0.0001 e~ 1.8m w= 150 mm |t ,| a) Structural model of space frame

Element Number a S^%> ® Fig. 2 Strain and kinetic energy norm of estimated ■ m t ® - error for the first mode • ® - # e © @ . ©f <3 ■03 |2) ” © ~ © ~ © ■ lie"II lle*ll ■ (T)~® Element Number

b) Finite element sub-division Fig. 5 Space Frame Problem

Table 3 Comparison of eigenvalues for the space frame problem

Mode # W U)est u>ref F a 1 88.16 75.90 75.90 0.313 0.86 2 119.07 110.45 110.45 2.796 0.95 1 2 3 4 5 6 7 8 Element Number 3 240.53 209.01 209.01 5.616 0.85

Fig. 3 Strain and kinetic energy norm of estimated error for the sixth mode Table 4 Relationship between the modified eigenvalue and papameter a for the space frame problem Heir/llellref First mode Second mode Third mode a w"* a w"* a U)est 0.60 75.98 0.60 110.56 0.60 209.30 0.70 75.91 0.70 110.47 0.70 209.05 0.80 75.90 0.80 110.45 0.80 209.01 0.86 75.90 0.85 110.45 0.85 209.01 0.90 75.90 0.90 110.45 0.90 209.01 0.95 75.90 0.95 110.45 0.95 209.01 1.00 75.90 1.00 110.45 1.00 209.02

1 2 3 4.5 6 7 8 Element Number

Fig. 4 Ratio of estimated energy norm of error to Fig.5-b) referred energy norm of error

Table 3 R: ^ $ fi T o «t < # C kd*a-6„Table4 5.2

Fig. 5-a) d?, rn'^mStfsSm^, EWSerEli/'fy/-^' s@|gf 346 mm#

# # 3t iff g K 0.86 T $ & K 6 M b (, f, #lf-FTKa^ 0.60-l.omr^ikL-C6|%-oem ^ 1) Zienkiewicz OC and Zhu JZ, “A Simple Error Estimation and Adaptive Procedure for Practical %. *(2t-F, #3t-FKjsv^-C6|%#»il6#*#e, Engineering Analysis ”, Int. J. Numer. Methods iff k^&o a#., Vol 24, (1988), 337-357. 2) Ohtsubo H and Kitamura M, “Element by Ele­ 6. US E ment A Posteriori Error Estimation and Improve ­ j: 6 @ ment of Stress Solutions for Two-Dimensional Elastic Problems ”, Int. J. Numer. Methods Eng., ###?#&!?»©, y;vA©y$ij k m^smwct Vol 29, (1990), 233-244. @#-<7 1$K, 7V-A@%K#g 3) Ohtsubo H and Kitamura M, “Element by Ele­ ment A Posteriori Error Estimation of the Finite Rayleigh Element Analysis for Three-Dimensional Elastic emcav'T, ###fe&Kj: 9#& Problems ”, Int. ]. Numer. Methods Eng., Vol 33, (1992), 1755-1769. 4) Ohtsubo H, Suzuki K and Sato K, “A Posteriori nam#a&#Lfk>s. Error Estimation in Finite Element Analysis for »6ii&#^R^7;i/Ai±/<9 7-7 aK:%#D»v^# Elastodynamic Problem ”, JSME, Vol 57-541 (A), lS*arf C ^R0##Km#Pf (1991), 1979-1985. Ka^-ca/<9/-7 a»*MBK:S*3Ch»<, ##K 5) Wiberg NE and Li XD, “A Postprocessed Error Estimation and an Adaptive Procedure for the Smf&fr»^CkA*T^5. CitZO, 7C0#mB%# Semidiscrere Finite Element Method in Dynamic Analysis ”, Int. J. Numer. Methods Eng., Vol 37, (1994), 3585-3603. $#% 1:

IE* Jl| # # B* _

Development of the Automatic Quadrilateral Mesh Generator based on the Paving Method for the automatic crack propagation analysis

by Yasumi Kawamura*, Member Yoichi Sumi*, Member

Summary In order to investigate the structural integrity of large scale structures, the prediction of the crack path is very important. For this reason, a step-by-step finite element approach has been proposed by one of the authors, Sumi, in which a cracked domain must be remeshed by an automatic mesh generator as the crack propagates in the domain 1121. In this approach, the modified quadtree algorithm is used for automatic mesh generation. One of the problem of this method is that it is difficult to generate mesh near the boundary and therefore reliable mesh generation is sometimes not possible. In this paper, the automatic quadrilateral mesh generation based on the paving method 3’4’ is developed for the crack propagation analysis. By the developed mesh generator, completely automatic simulation and reliable analysis are possible. Also, a method for automatic choice of the increment size of the crack propagation in each step is proposed in this paper. In this method, proper increment size is first selected based on the numerical data from the computational experiences by authors. Furthermore, if the computed error of Kn/Ki exceeds the allowable limit specified by the user, re-analysis of the previous step is automaticaly carried out by decreasing the increment size so that the crack path prediction can be performed accurately.

1. §StiSSSfTr©fc£>CSlE05>*i£CSoA >tz/ y S'iS

(2)

wm&m ¥s8 9 ^ i b io b Y1&94P5H15B 348

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C Element Row Generation ) 2. Paving iitcSoT'fcy -y>;i.£fijfc Smoothing >k Boundary * node - smoothing 2.1 Paving ;£ £ 12 Interior - node • smoothing \~) E#BXyi/zL^gmkL-Clj:, H^BX-y^ou*^© 'Seaming N TRIQUAMESH 5 V)f ^ t Interior - node - seams Fixed - node - seams. - k a# LW. Paving *""(2, Intersection Checl 6 12##i

C Closure Check Seaming ^feteit^X ^#-hf S"CE*»5®fl069 K15E $ foT T t1 k Interior - node - w ^ A# Fig. 1 12, Paving *K 2 5g%± ^—/Element"^—> rows are generated ^s^one round 1// 9 A^6 13 **#&#, C:W#DiSgtt.51:kE20# (Wedge Insertion and Tuck Formation) # (Paving## ^mm^k»%LT^?T, #KAK:B Intersection Checl W>L&asf» Paving&©7;p=ry XA(2, ##^121? (Closure Check Seaming Interior - node -

'""'Element*' Generation is -+-\t8\17mi5\!4 13 •Nsfinished/ jj_

( Cleanup of Mesh ) 33333331 ;HJHJH7 s 3333332 £. (a) (b) (c) (d)

Fig. 1 Overview of the Paving Method Fig. 2 Flow chart of the mesh generation algorithm Paving

*f, Paving##±Kt&g;atA#®mmK:j:^-C tuT, &. 0^g<-^-a»6,lfEndNode, 4J^T©6©©^^T#6B#®]5i'^@*'#K^&. 6L *^-©»f ^%# a/4 HIT© t ©#» 8<-jx %b SideNode, ^n<6<-^Ti %b ComerNode, »#, 3%#©#mKj:D» a-^eeW:^K(Fig.5(b)), @rL,, xa—'Xyx'&k V 2.2.6 IISOfi')1? (Closure Check) TK, Paving##E#f 6B#B#&%#T6A:A©r# Paving###6(ai^T©mA&-g-tf#^KK, #mA©xA-$;yyjk, mm©B*#R&%#?&rm f©Paving##E&)j-5 ^®ja©xA-s;>yj ©2%#tmv^Tv^. cfLEj: &. 6(B©m^^tf##tmC&#e -oT, A#kLT©;(yS/a.©XA—X$&##f&j:5 B, Paving##©#%#% Fig. 6© 4: Ef-5. Tg#t#Afa. 2.2.4 (Seaming) 2.2.7 •? x y ^©#A • 9 y X©£$c (Wedge Inser­ Paving ##±E X 9 y 9 ^©@5^- (R A a/6 mT©# tion and Tuck Formation) S*#m»5#^E(±, Paving##^* (S/-S>-y)BBf;&fT% Fig.4(a)©j:^E, Tfz< k Fig. 7 (a-&) ©4: ^ K##1hT X#A# < » D a#a*&ki-3@!i^©m#©#A$-ie^LT i o©*&a -f^6#A-#$&. f©Z-)»#a-Ki4:. 9^.y^©#A KftckTy-a>ytfri). *AC:©#a-, (Fig.7(a)). S%, S**-# a >y#%& & k##Eiav^#r $ % $ >y# ##*#±©%ATt±, a#©y-&vy» &#&B?y?©&&#fr:bfl,& (Fig.7(b)). fr) (Fig.4(b) 2.2.8 y(yya.©9D->7yy (Cleanup) a. c©j=i)»#eEK%E%©g*^j$-±^L, f©# Wk#, A fit ^ y y ^ © Ci/-:=>:y&fT3(Fig.4(b)&). y-svy&fr 9#c, i*a-f'<#2c©g%©a©^@$-)t'<, mwa Fig.4(c)© i 5 KjEX©f$A$rfj'? => 2.2.5 rSiltn(Intersection Check) rTT^

“I

(a) SideNode (b) ComerNode (c) ReversalNode (d) EndNode

Fig. 3 Element row generation for each angle 350 mm#

(a) (b) (c) Fig. 5 Intersection check

Rectangle Triangle Semicircle Circle Special LU A A e fA h rl A A fA ;>-) A A 9 A Fig. 6 Closure check

Fig. 9 An example of the case in which element row generation must be stopped

2.3 (a) Wedge insertion ±B0 Paving m07/l/=fUXAK 3:0,

ecir, *#%'ri±c:0M 6**K,±K0T/k=f OXA0%^^froA:. 2.3.1 gamma® 4# k LT#L < e, fLik#^-K±K07;>d''; XAtamftk, Fig. 9 0 3: Fig. 7 Wedge insertion and tuck formation 5 C/F$ Paving ## (Fig. 9 -hi® KAS 6^5 Paving## (Fig. 9 T*) %0B*?!l& ^0% y y.3.0? V-^Tyyfr #&*&# & k, /|\$ ^g%0 Paving# 3. Fig.8(a)03:3K, &&mdW2o0mRkW4@L #MKA0i&tfZ:kK»&o f0A&, C0#K3^0% Ttr'&V) k # K(±, f 2 O0B3S&—oK -fto Fig. 8(b)03:^K, #Kg*Amc-cLm-5. fhr, mRfmi&agKai,)

y-s >y^^0@a0yotx^k#a&B*)5. Pavingsc .t smaiEgiM y y 351

2.3.2 XA-yyX&©%# AA&ykL, KE^fb/t-j, &J,= y-ykT PavingmKl3W-Ca#*F!l^m%K r###A©XA a. -^yxj k r^#A©^A-yyxj *;fTti3ia. L (Iq+Id)/2 if (lg> Id) ##mA©%A-yyx&fr$BSK:. *&k Id otherwise f amAcjBRTaB%T-xc 3= amA y ©#ma j »af-^X&#o2o©#^-$-ieA/A#gK, *SA,A#© %k»a3:i)K:TaA»©#lE# Jx k,m^*A y&S k, < E3W3&

Vi^omibLIjfaZi&lE-t & Jb * m^X A={A A #eAi&a (Fig. 11(a)). d^&a^aAAK, *6/CA# +^,)/2k*kT. ]GR3)4)R:#^XW:, @#gia/J\2V)##©0:#K^:b-&a 3: ^ KTa (Fig. Aamc^^^v^xv^a. 11(b)). |^#cy—ayXKKnxt, y —ay/#©# Aa= VI- V (1) #&, /J\gwE*@lK-ak-g:a3:^K:i-a. AAL, y=-^^(yv+y,-y^) (2) 2.3.4 E#©m©%#T& Fig. 12(a) (b) i±, XA^# zzT, «HmAy*qeLxwa##*, -t^h/Fy,, < m»a*g-K±E©7;>x v XA tarn L Agu^^ ux y., yu(3w#g©s*©#c#A©mm-<^h;FT*, v^a. Fig.l2(a)©3: 5 C:A$»B#k/h2#E3Ra:&l& a. c©a8 , mA y $fiaK:o^x, ^©MmTy-ayXaifrt)^, a# L V iSn-K it ESStt & ¥^to CTfTE3a#ClgV)#% K Fig. 12(b)©3: i) y y^ k»a. C©#g-,^:#» f aAv^&T&a**, ES&tM X*^# < K, T2V)E*##L<3/uTL&3 Ck#&a. f CX, f a aa-cmgwBcm'aaw^ < ^ $ 3ix u * ^. ^W9ST*tt±^S-^^© J: -5 KSIELTfll©TV^5o tcx, T@f%xi±, ^yyzL&ig#K:g9Am0im#& 2-Ky,+ yu-y*) %#T a AT© 3:^»7;i/XVXA L A. y, = J!=lU----- (3) *f, @#©^i:*,aT^X©@iAK:oV)X, *#T - ^ aga%**5mA±©()©K:owx, #RTaB#%©$ AAc, ^|y.-y |+|y-y |

$A, j, ktmawA y ^6 y^^mc^ig^aiE fa^^h^X&D, @Ay*^ 2#©gak@LXwa #g-©*AT©3: 3 iC^ALTf^a. Fig. 10(b)© 3: ^ K:,

##±X-< 9 1 )V Pai{= Vii- Vj) k i> b /V Paz( = Vu -y)#%f#&2##LA;5igAk, y*^ 6y^© Before connection After connection 3emL-t^ b^ A(=y-y) k&?%LA^ig» A k Fig. 11 Connection of two segments when the size of fa. f ©^igK(5)^Tm-#$^a^# the segments are different

Fig. 10 Boundary node smoothing 352 IB 181 -g-

(a) On the way of mesh (b) Generated mesh without (c) Generated mesh with generation conversion procedure conversion procedure Fig. 12 Mesh generation for the problem with gradation of elements size

3 Z^KA, A, Fig.l3(a)K^?2gm)#*Ba'6, Fig. 13(b)

I nA~41 +1 nB~4| + |«e — 4|

Fig. 13 Improvement of the shape of quadrilateral >| n'A~i I + | yiB~ 41 + | n'c~ 41 (6) elements Wx, B, »» B, CK*

e#L»m3K:, Fig.l4©j:3K:eBL%maA0B* 0 K: t B, C K: jgf 5 ftf ft 4 #. 3 #T

-So eo>ei (7) tztiU I A—?-| + | i—w| + 1 A—f-| + | 2—w| Fig. 14 The case in which the improvement procedure eo= 4 18 > is not applied | A + 4>i—f- +| t~K l+“7fX3 ------^------5------(9) Vi/eiCJitT©®S4rfT5o £i = (i) Fig.l3(a)@j:3K, Paving a 353

(b)

(d) Fig. 15 Examples of generated mesh

(2 ) 6 W±®*A a^ta, tzr, #@m7°oy9AKiswT, ggww@a (l)C##a*4=2®*&#%f@&Ab-&a'&, g# -cttTyTV- h SfflvifiEiijto»E^SSS®EE$-6 i-a 2B#egt@i-a. KWcfr5 J: -5 CU. Fig. 12(c)a^ejac j: 0 ElA y y y a. Tt a 3.2 s a y-y a y i±^%#h#c j: -? Tfzk^a 2.4 > -y '> a. % V 9 fOB# Ka#ME»a. ^msTmwT^a#sm##M7°ny 9ATK#gte%m%®#^K, A^C^cT^'fkgib m^^AAf-yEL-c-^^-cv^. ##a /z3o@B^/<9y-ft#tf#— ai^#m#^0m##k?fT-r»T C0f 9 7^gC, EG//EG KRf a#f#m#^RSLA# ^A»E^BcH»6»v^», #Bfmaa%<»a#AAc, %a^gg#@i@M#^^s-fa^ka*Tga. fg 354 181#

Fig. 16 3: 9 70,/TO 3 %k LTz 0.23X7&m #^EK, Ja/7?mn=0.23@akT6"<#T$6Zk^k #7oy9Af#j%±(Dj:-5»m&K:j:9, f@#e, #^TyyK^^3#a!m#*Jaa, f $©g®K®S-fTrjT©6*3', Wrg ElotE ©XTyXE#wCiH-#$7i,:&7&an&@t:'T, Ja= TO,/TO 6 c k6. f C r, mcE, Tr«/To Efi, 1 %tv ^BU©# SUM* 2/a & CY.Aa{C = 0.7 k LTW6)E®@i-6j:i)Eg#K#L, ##(-##$

3.3 mmm Fig. 17 ©3: ^ »M@E^:Xn X? A . Fig. is a, 70,/To 5 %T©##imaE# 5 6 titz J y'> z. Ti> 50 llfflipE, §S5fe

^(±$W%rN#L*:Pavmgm&mw-r/ ^LT^6.Fig. 19 #,##frk©30,/70 ©^-fb$#LT V^6. ^^ifSTEP2A^6 STEPS Eg%#e, mgQSg 0.23 0.0 0.1 0.2 0.3 0.4 0.5 Act / 7?mln #, f ©#©7C##T»fT^ k, 30,/30A#3##(5%) W±k»7-cv^%. 6©A*), @*K)E#gm5R$*3#0 Fig. 16 Kn!Ki vs AajRmm for various analysis #$71/2 IBI©Xf yy©^9aLAifTk»n,T^6oeLT, #*&ME70,/30 i±9f@K^K0^E»cTW3©^7p^ 0= 10 MPa 6. C©3:9E, *%3tT#ALA, #####$©@# m###E j: 9 , -@#amr© #k » L&L, ^©#h#ri3, yXE^^X

iY 15mm #^r©^9mLA:e67i,T&9, #%T'yXE%WT##] E-#^. 671/6 #amB#©R^9 m < t^TV)»v^Ck 6mm 7, !k»^6. 67W3, Fig.l6©*@§U&&fr?*#©$#a:

^#A^A7s»7gkm7)7l/6. f »7)6, 70,/TO A## ------500mm------K#WmE»6 3:^E, #XT y XEjSi76 $#)###» gtSf6#^Ei±, ###&&$##?#<##%#©# *x-

Fig. 18 Generated mesh in the crack propagation analysis Paving SC T % y V 3-ffehScS0 355

Fig. 20 K, @f#K3#3%, 5%, 10%, 2O%0#g-C^ 3T#K^K: j: D, #T##2#3%0*& C0RHg0*e, 3~5%&TFR:f S ^ Lv^k#^ajra. R=@%#&±#

STEP1 Kir/ K, 0 Paving arc#, @ < glfki" 3#^0 / 7;P=fUXA0%$k@# BK0%#^&»#AT5CkK:j:D, ##-9^X0#% CftKiD, @1E H 0.2 - E9^*#tMV)a i !1 6#*%0Kw#BmMWf*^T^ a j: ^ (2) #xfyyT0a%»#gmmR3&.w.?&0# STEP2 STEPS STEPS STEP4 STEPS @me%9 a em k # mmamo §*@!sa -0.05 STEPS STEP? HB^AfackciD, mf#m%K^0^'ft:K#fa#S0 140 x-coordinate(mm) mm@&0^'fi:$'&m*K#^A^fackA*T#^. 4: Fig. 19 Variation of Kn/Ki (Allowable limit=5%) emcjic-c, -S/g>^nT#K:»ak#^.6fta. # # X IK 1) f§#—, #S¥ : SSilSgSS^ 5 41.1/-^ 3 ytc a#iEEaa-*@K^m&0iGm, #%#, # 167^, (1990), pp. 253-260 2) ##-,*# :##e^#B*B?ai0%&0## j/ $ i u~'y 3 >#&0M% m 174 -§-, (1993), pp. 607-615 3) T. D. Blacker, M. B. Stephenson : Paving : A New Approach to Automated Quadrilateral Mesh Generation, Int. J. Num. Meth. Eng., Vol. 32, (1991), pp. 811-847 4) Hl##3, A#-, A#m# : Paving &K j: a 2 * 7Cg3m#m&&@!l8;, #ia%# 1: is Id a %@##f& #119#, (1995), pp. 285-290 5) D. F. Watson : Computing the n-dimensional Delaunay Tessellation with Application to Vor- O -150 «7.\ onoi Plytopes, The Computer Journal, Vol. 24, No2, (1981), pp. 167-172 6) T. Taniguchi, K. P. Holz and C. Ohta: Grid Generation for 2 D Flow Problems, Int. J. Num. o- •*■ Meth. Fluids, Vol. 15, (1992), pp. 985-997 7) M. S. Shephard, M. A. Yerry : Approaching the Automatic Generation of Finite Element Meshes, Computers in Mech. Eng., 1 (4), (1983), pp. 49-56 8) P. L. Baehmann, et al.: Robust, Geometrically Based, Automatic Two Dimensional Mesh Gener­ ation, Int. J. Num. Meth. Eng., Vol. 24, (1987), pp. 1043-1078 110 120 130 140 150 9) J. C. Cavedish, et al.: Automatic Triangulation of Arbitary Planar Domains for the Finite Ele­ x-coordinate(mm) ment Method, Int. J. Num. Meth. Eng., Vol. 8, (1974), pp. 679-696 Fig. 20 Simulated crack paths for each allowable limit 356 181#

' ment Method, Int. J. Num. Meth. Eng., Vol. 8, Yagawa, T. A. Cruse (Eds.)), Vol. 1, Springer, (1974), pp. 679-696 (1995), pp. 350-355 10) S. H. Lo : A New Mesh Generation Scheme for 14) B. P. Johnston, J. M. Sullivan Jr., A. Kwasnik: Arbitary Planar Domains, Int. J. Num. Meth. Automatic Conversion of Triangular Finite Ele­ Eng,, Vol. 21, (1985), pp. 1403-1426 ment Meshes to Quadrilateral! Elements, Int. J. 11) & Num. Meth. Eng., Vol. 31, (1991), pp. 67-84 15) M. L. Sluiter, D. L. Hanse: A General Purpose S 175 #, (1994), pp. 291-298 Automatic Mesh Generator for Shell and Solid 12) D. O. Potyondy, P. A. Wawrzynek, A. R. Finite Elements, Computers in Engineering, Ingraffea : An Algorithm to Generate Quadrilat­ ASMS, New York, (1982), pp. 29-34 eral or Triangular Element Surface Meshes in Arbitrary Domains with Applications to Crack m s? Propagation, Int. J. Num. Meth. Eng., (1995), Vol. 38, pp. 2677-2701 13) K. Shimada, T. Itoh : Automated Conversion of (JR*8$) 2D Triangular Mesh into Quadrilateral Mesh, Computational Mechanics’95 (S. N. Atluri, G. 357 3-10

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Bayesian Reliability Analysis for Non-Periodic Inspection with Estimation of Uncertain Parameters (2 nd Report) Inspection Schedule for Wing Lower Surface Structures

by Hiroshi Itagaki, Member Masanobu Shinozuka Hiroo Asada, Member Seiichi Ito

Summary The purpose of this study is to perform a Bayesian reliability analysis for developing optimal non ­ periodic inspection schedules and estimating values of uncertain parameters from field data collected during in-service inspections for practical aircraft structural elements. It is suitable that information obtained from inspection results through the long service term is utilized in order to examine the effectiveness of this reliability analysis. However, it is close impossible that such actual data is obtained to the actuality. Then, the previous report adopted the damage-tolerant structural element in typical fuselage structures modeled by referring to various design data and fracture processes. Furthermore, the failure rate function which uses for this reliability analysis was also examined, and a sufficient knowledge was obtained. Transport wing lower surface structures with a number of fatigue -critical elements are used as a realistic structural model for the present analysis. Each element consisting of a skin panel and stringers, is subjected to flight-by-flight loading and is designed by the damage tolerance criterion. Probabilistic factors considered in this analysis are fatigue crack initiation and propagation, crack detection capability and failure rate before and after crack initiation. Monte Carlo simulations are conducted to demonstrate the efficiency of the Bayesian reliability analysis for development of inspection schedule and for the estimation of uncertain model parameters.

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A ©*@[»#)@*%Sf g©#&%K O V) 6. - 2-bay h ©T\ i ^-Ctt^-© * External inspection &. mTT«, mimm, Fig. 1 Critical Location of Wing Lower Surface m^#a©#^kms, ##k%@%%, #@#sk#si &ma$, -Flight y 5 a y — h Lt, & 5„ #amm#SB k a . 2.1 sxk LTffli'ssiisiHi i>o tas, ##Bic^m##©y s ^y-y 3 y&©#ge //&)- byyy^m{-?%D±kf3±mTmg?S#^ Fig. 1 K^i-. C©gg#l± 3*© Z#X h 2.1.2 m^@mi©%^kmj# V yjf k^HR^6#^$tb% 2-bay 7 a-—fv • -te —7#st T$D, ^XMl>^kf^R$-ie^-L-CW5';"<7b?L yh?L&^©?L#c@E^#m^#K%±L, maftk ©^©?L#K:g^##Ai%AU, ffL&6l#eK:#ms $. ma#gao^fTlg»&K:%^i-3k#*-C, <-LT, 6 B#iS©2 #*7^ 7"/F^-# W(6 : a, A g% h!) y^KTffAL, f ©%^C©fF^m&©##t %KK<.

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Table 1 Element Model for Bayesian Reliability Analysis

Structural item Reliability model

•Fatigue crack initiation f;(t;r)=a/p*(t;/p*r 1exp{-(t;/p*n (Tl) •Fatigue crack propagation das/dt=C(as)b'2, C=10z (T2) D(a;id)=l - exp[- {(as~ami[])/(d-amiI1))e] •Probability of detection for a crack D(a;id)= 1- D(a*ld), a' = as-rh (T3) •Failure rate before crack initiation h(t) = exp(r) = h0 (T4) •Failure rate after crack initiation h(t) = af/p f(t/p r)“' '+ h0 (T5) 360 B $#####:%# #181-#

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Table 2 Values of Parameters in Numerical Example

Bayesian analysis Item Values for Range for true model Model value estimation • Service life (flights) 50,000 • Minimum level of reliability 0.8 • Total number of critical elements M 50 ■ Parameters of TTCI 2 -parameter Weibull a 4 4 B(flights) 40,000 p*(flights) Unknown 25,000 to 65,000 • Rivet head radius th (in) 0.34 0.34 • Initial half crack length for true model a c 1. Clear air turbulence(CAT) 0.105 0.995 2. Thunder storm 0.270 0.005 Truncation of o y (g) < 0.1 • Parameters of POD e 1.2 d(in) 1.2, 1.6 • Parameters of failure rate r -18 cti 1.8 B,(flights) Unknown 5,000 to 45,000 Notes: 1 ksi=6.895 MPa, 1 in=2.54 cm, 1 ksWin= 1.099 MPaVm

Amin=5,000 flights, A-max = 45,000 flights, 4.2 -<-fP7>msnz

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Number of detected cracks 31 412 121 22 Failure after crack initiation 1 1 1 10000 20000 30000 40000 50000 Service time (Number of fligfhts) 1) d=1.2 inches

Failure after crack initiation 2 1 2 11 ).7_ ” ■ _l • ■ i ' ' ■ ” ■ i r , j , . , , | “ ■ i t ' i | 0 10000 20000 30000 40000 50000 Service time (Number of flights) 2) d=l.6 inches

Fig. 4 Inspection Schedule and Structural Reliability (Uncertain Parameters: 0*, z and 0,) 363

P* ~“ 60000

1) Pm =15000

P* 6000oTioF Pf

2) zm = -3.5

-2.7

3) P*M = 45000

(1) The tenth inspection (2) The twentieth inspection

Fig. 5 Posterior Joint Probability Density Functions (d = 1.2 inches)

Table 3 Statistics on Frequency of Inspections during 50,000 flights (Number of Simulations = 30)

Distribution of frequency Frequency of inspections Case Standard cov 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 deviation

d=1.2 0 0 1 2 4 2 3 3 2 3 3 1 2 0 1 1 24.4 4.03 0.165 d-1.6 2 0 3 0 1 1 3 2 3 4 3 2 1 0 1 2 1 24.6 4.61 0.187

@KAgt) . f Table 5 1: 364 a mm#

Table 4 Mean, Standard Deviation and COV on Cumu­ lative Frequency of Detection and Failure of Structural Element WrjTWRga* f * a a tf e z © # (Number of Simulations =30) W»L, Table5©##@ Detection Failure 0 &* = 13.560—14,800 Bights &, B%6B@©m&a@S/ Case Standard Standard Mean COV Mean COV deviation deviation m 0fU = 12,900 flights ttmtit, Table 5 d=l,2 26.9 336 0.125 6.37 1.75 0.275 d=1.6 26.9 3.63 0.135 6.53 1.90 0.291 @R#©##8B@K:gSf

Table 5 Mean, Standard Deviation and COV on vi5 Z €> tiio Modal Value of Posterior Joint Probability Density of Uncertain Parameters 5. a t> v iz (Number of Simulations =30) Mean bayF$#iSg]i'f44-WD±bf, Case P* z Pf d=1.2(30) 47,850 -3.49 13,720 d=1.6 (30) 47,350 -3.52 14,800 (): Number of simulations Standard deviation y $ y$ilfflurcn'9 0#@iig$'S»fio Case P* z Pf fLT, C:h,6@j|@*&9SmR7'-?kU-cmwT/

Mean ##0#Jmu#-31t%±f & ai, ~jf, ##»0i#)ma# Pf Case p* z 5. LA:*^7^#B0#rmmmK: 4=1.2(27) 46,300 -3.43 13,560 d=1.6(25) 46,000 -3.45 14,480 (): Number of simulations Standard deviation kE»5 Case P* z Pf d=1.2(27) 2,540 0.124 2,320 . d=1.6 (25) 2,890 0.133 2,660 ^#%0agfTKi*L, (): Number of simulations G.Deodatis (Princeton ± COV #), T. Swift k S. Sampath CO MR (JR B # ^ M Case P* z Pf FAA) CIS <»Wb±lf*t. d=1.2 (27) 0.055 0.036 0.171 d=1.6 (25) 0.063 0.038 0.184 (): Number of simulations

(2) The Twentieth Inspection T, ^»X@@0%m0 Patch

S-fBv^TSSLfc^ffl© Patch © 3$ £(

5) Asada, H., Itagaki, H. and Ito, S.: Effects of /(

(Al) ability, Proceedings of the 4 th International Conference on Structural safety and Reliability, C CT, /Ei9SS 1 Jiti: Clear air turbulence (CAT: B#5c ICOSSAR’85, IASSAR Publication, Vol. 1 (5/ 5L®, M 2 /SO: Thunder storm (##) £ it, # 1985), pp. 1,87-1.96. »Ci, c2#cAT»6#cm#KaaLT^&#igc)&i6) Itagaki, H. and Yamamoto, N.: Bayesian Analy ­ FFr^mv^z sis of Inspection on Ship Structural Members, (A2) Icoosar ’85, IASSAR Publication, Vol. 1 (5/1985), pp. Ill 533 III 542. y(kAt)={Sy(a>o)Aa)}'12 7) Fujimoto, Y., Itagaki, H., Ito, S., Asada, H. and + 2 2{ S^0zf)dco=l (A3) Ill 533 III 542. 8) Deodatis, G., Fujimoto, Y., Ito, S., Spencer, J. and (A2) h/l/^K (PSD) 0 S. Kkt, Itagaki, H.: Non - Periodic Inspection by Cttfc a ft if-* - ^ y t G If-# LtzlfctKD PSD & Bayesian Method I, Probabilistic Engineering Mechanics Journal, Vol. 7, No. 4. (1992). pp. 191- LT, 204. 9) Kawano, A., Itagaki, H. and Ishizuka, T: An S» R*#iLLt PSD, P tt 0~7T OflC-fS-fttliJSL application of Bayesian Decision Theory to the Design and Inspection of Marine Structures, Jour ­ nal of Society of Naval architects of Japan, Vol. # # x it 176 (12/1994), pp. 587-595. 1) Yang, J. -N. and Trapp, W. J.: Reliability Analy ­ 10) mm, me, A* : sis of Aircraft Structural under Random Loading mm*#, s *###*# 72 and Periodic Inspection, AIAA Journal, Vol. 12 (4/1995), pp. 240-241. (12/1974), pp. 1623-1630. 11) mm, ##, 4N&, a* : 2) Shinozuka, M.: Development of Reliability- a cmsm, voi. 3. Based Aircraft Safety Criteria : An Impact Anal­ JCOSSAR'95#:%# (11/1995), pp. 687-690. ysis, AFFDL-TR-76-31, Vol. 1 (4/1976). 12) #g, ##, mm, M* : 3) Goranson, U. G.: Damage Tolerance-Facts and Fiction, Proceedings of the 17th Symposium of #, m 179 m (5/1996), pp. 359-368. the International Committee on Aeronautical 13) Swift, T.: Damage Tolerance Capability, Fatigue (ICAF), Vol. 1 (6/1993), pp. 3-105. Fatigue, Vol. 16, No. 1 (1994), pp. 75-93. 4) Shinozuka, M., Itagaki, H. and Asada, H.: Reli­ 14) j: ability Assessment of Structures with Latent ssmcMf ammamme#, % a (3/1991 ). Cracks, Proceedings of US-Japan Cooperative 15) Military Specification, “General Specification for Seminar “Fracture Tolerance Evaluation ” (12/ Aircraft Structures”, MIL-A-87221 (USAF, 1981), pp. 237-247. 1985).

367 3-11 Sensitivity Analysis on Fatigue Reliability and Inspection of Ship Structural Members

by Sung Chan Kim*, Member Yukio Fujimoto*, Member Fiji Shintaku*, Member

Summary Sensitivity analysis on fatigue reliability and inspection is carried out for a typical structural member of bulk carrier. Fatigue properties of the member and probability of crack detection at the field inspection are obtained by the questionnaire asked to the naval engineers. Changing stress level (fatigue life) of the member, crack growth curve and inspection interval as parameters around a standard condition, failure probabilities of member during 20 years ’ service are calculated by the Markov Chain Model. The result of analysis can suggest the solutions for design requirements such as improvement of reliability level, reduction of structural weight and the simplification of inspection. In the latter part of the paper, effectiveness of combined inspection method and sampling inspection method is discussed through the sensitivity analysis assuming member sets having different number of initiated cracks. It is found that combined inspection has more advantage than sampling inspection, when the number of members having initiated crack is a few in the inspecting member set.

structural member. There exist certain uncertainties on 1. Introduction stress level and crack growth curve for ship structural In-service inspection on ship structure plays an members. At the design stage, further, there are several important role for maintaining safety against fatigue requirements such as improvement of reliability, reduc­ damage. When we want to perform reasonable inspec­ tion of structural weight, and simplification of inspec­ tion, it is necessary to clarify the present reliability tion. In order to respond to these problems, failure level of structural members as well as the desirable probabilities of the member were calculated by chang ­ level. ing stress level (fatigue life) of the member, the crack From this point of view, in the previous study, we growth curve and inspection interval as parameters conducted questionnaire about “Fatigue damage and around a standard condition. Markov Chain Model was inspection of bulk carrier” to naval engineers 1),2),3). In employed in the reliability analysis. the questionnaire, six locations of bulk carrier reporting From the limitation of cost and time at real inspec­ frequent damage occurrences were selected and the tion, an inspection is applied in the actual ships through number of cracks initiated in the members, the fatigue long distance visual inspection, combined method with characteristics of members, the probability of crack long and close visual inspection, or sampling inspection. detection by field inspection, the target reliability level The effectiveness of these inspection methods are dis­ of members, etc., were asked. cussed through the sensitivity analysis in the latter part Then, fatigue reliability analysis considering of the paper. repeated inspections was carried out using the informa ­ 2. Outline of questionnaire and reliability tion obtained by the questionnaire. From the analysis, analysis how much present inspections contribute to maintain the reliability level and what adequate inspection inter­ The six locations of bulk carrier reporting frequent val is to accomplish the target reliability level were damage occurrences were the object of the question ­ made clear. naire (Ml to M 6 in Fig. 1). In the questionnaire, In this study, sensitivity analysis was carried out answer was requested to reply subjectively the average around the present inspection condition of a typical condition of fatigue damage and inspection per a ship during 20 years ’ service on the basis of the damage * Faculty of Engineering, Hiroshima University examples he experienced in the past. Namely, frequent damaged ship, very sound ship, different sea routes and Received 10th Jan. 1997 different operating methods were mixed in the average Read at the Spring meeting 15th May 1997 condition. To get more accurate answer, critical crack 368 Journal of The Society of Naval Architects of Japan, Vol. 181

(M1)Connection between lower stool (M2) Connection between side longl. (M3) Lower bracket end of of trans.bulkhead and tank top and trans. ring at top side tank trans. web frame

HATCH END

HOLD

HATCH SIDE

(M4) Kunckle connection between (MS) Slant plate of hatch corner (M6) End of connection between bottom inner bottom and hopper tank longl.and tripping bracket

Fig. 1 Structural members selected as the object of questionnaire. length was defined as 200mm or 500mm which is often experienced at ordinary inspection and damage exam­ ples. The crack exceeding the critical crack length was Crack initiation life(20 mm crack) considered as failure event. The crack initiation was defined with 20 mm length. The detailed content of Fatigue life at 200mm length questionnaire and its result are in Refs.1*' 2*' 3* (including crack initiation life) 2.1 Fatigue characteristics of standard member Under the limitation of cracks which initiate within Fatigue life at 500 mm length 20 years ’ service, their crack initiation time and crack (including crack initiation time) propagation lives from 20 to 200 mm and from 200 to 0- 0.05 500mm were asked for the six members. In the results, the fatigue characteristics for the five members except member (M 3) were similar1*. The aim of this study is to discuss the inspection reliability of a typical member of ship structure. From this point of view, member Service years (M 3) was removed from the sensitivity analysis. Fig. Fig. 2 Fatigue life distributions of standard member. 2 shows the distributions of crack initiation time, fatigue life to 200 mm in length (crack initiation time is included) and fatigue life to 500 mm obtained by out for the four kinds of patterns shown in Fig. 3. averaging the properties of five members except mem­ Pattern ® or (2) is usually obtained by the experiment ber (M 3). These distributions were used as the fatigue using fatigue specimen. However, fatigue crack growth property of a standard member in the sensitivity analy ­ of actual ships may take pattern (3) or © due to the sis. It should be noted again that these fatigue life change of stress distribution accompanied by crack distributions are for the limited cracks which initiate growth or the change of crack shape. In order to cover within 20 years ’ service. the wide possibility of crack growth, the above four In the questionnaire, crack growth pattern of each curves were chosen. Each curve is expressed by the member was asked to select from the several patterns following equation. prepared beforehand. But the trend of crack growth a={acr-a

Close visual inspection a„r= 200 or 500 mm (0.5 m distance)/^

Long distance inspection (2 to 5 m distance)

Crack length (mm)

Fig. 4 POD curves for close visual and long distance Service years visual inspections.

Fig. 3 Crack growth pattern 2.3 Member reliability under present inspection condition mean life to arrive to critical crack length and k is 4.0 In the previous study, fatigue reliability analysis was for pattern (D, 2.0 for pattern (D, 1.0 for pattern (3) and carried out employing Markov Chain Model (MCM)1>,4). 0.7 for pattern @. This model was first applied to fatigue problems by Failure was defined by the crack exceeding 200 mm or Bogdanoff and Kozin 61. The MCM can describe the 500 mm in length as described above. So, the cumulative entire probabilistic feature of fatigue process (including probability of failure is defined as follows. fatigue crack initiation life, propagation life, crack P/(200) : Cumulative probability of failure that a crack, growth pattern and failure event) in the state vector which is initiated within 20 years and crack and the transition matrix. Also repeated inspections initiation time follows the distribution in Fig. and repair processes can be incorporated into the 2, propagates over 200mm in length (probabil ­ model. The cumulative failure probabilities and detect­ ity of failure per one member) ed and residual crack length distributions at respective Pf (500) : Cumulative probability of failure that a inspection time can be calculated. The details of the crack, which is initiated within 20 years, MCM was written in the Refs.l) ,4) and 5). Perfect propagates over 500mm in length (probabil ­ repair model was assumed in the model. The fatigue ity of failure per one member) life distribution and the POD curve of respective mem­ 2. 2 Probability of crack detection bers were determined from the reply of questionnaire. POD (Probability of crack detection) for visual The crack growth pattern © was assumed for all the inspection was asked in the questionnaire under good members. Five kinds of inspection interval as 3 months, condition without influence of corrosion and painting. 6 months, 1 year, 2 years and 4 years were examined. In the results of reply, it was seen that detection proba ­ Fig. 5 summarizes the calculated P/ (200) and P/ (500) bility was influenced by crack orientation, crack open ­ at 20 year ’s service for the six members, in which the ing breadth, existence of hint for crack detection and failure probabilities obtained from the questionnaire existence of adjacent structural member hindering are also plotted for the comparison. In the figure crack detection. However, average POD for close “MCM 200 analysis ” means the calculation using Mar­ visual inspection and long distance inspection was kov Chain Model with critical crack length of 200 mm, obtained as shown in Fig. 4. Close visual inspection is to “MCM 500 analysis ” means that of 500 mm. The P/ inspect the member from about 0.5m distance and long (200) of the questionnaire result is near to that of distance visual inspection is from 2m to 5m distance. calculated result with 2 years inspection interval, and The POD’s in Fig. 4 can be approximated by the follow ­ the Pf (500) of the questionnaire result is near to the ing equations. calculated result with 4 years interval. Four years and Close visual: (POD) i two years intervals correspond to regular or intermedi­ expf — 6.18 + 1.451og(ffl —20)} , „ , ate inspection interval of ship structure. l+exp{-6.18 + 1.451og(a —20)} If inspections are not carried out during the entire Long distance-. (POD)2 life, about 90% of initiated cracks would propagate _ expf — 5.18 + 0.861og( a —100)} / « , over 200 mm and about 60% propagate over 500 mm. 1 + exp{ - 5.18 + 0.861og(a- 100)} 1 d } However, P/ (200) and P/ (500) is maintained in the Where a is visible crack length. level of 0.3 and 0.08, respectively, on the average of six 370 Journal of The Society of Naval Architects of Japan, Vol. 181

—o No changing the stress level (fatigue life), inspection inter­ inspection val and crack propagation pattern around the standard 4 years condition. The analysis was carried out by using the 2 years Markov Chain Model. Details of the sensitivity parame­ ters are as follows. 1) Eight fatigue lives such as 25%, 50%, 75%, 100%, 150%, 200%, 300% and 400% of standard life. 6 months 2) Six inspection intervals such as 3 months, 6 Inspection interval months, 1 year, 2 years, 4 years and no inspection. 3 ) Four kinds of crack growth pattern in Fig. 3. ■ Questionnaire Table 1 shows eight fatigue lives and corresponded o MCM200 analysis equivalent stress levels which are calculated from S = CA-"(m=0.2). (M1) (M2) (M3) (M4) (M5) (M6) 3. 2 Results of sensitivity analysis Member Figs. 6 (a), (b) show one example of cumulative prob ­

inspection ability of failure to service years when inspection inter­ val is 2 years, crack growth pattern is© and critical 4 years crack length is 200 mm or 500mm. These calculations were carried out by the above mentioned Markov Chain Model. Figs. 7 (a), (b) show the cumulative probability of failure after 20 years ’ service to the ratio of fatigue life (or ratio of equivalent stress) for all the cases, Inspection interval calculated. It can be seen that cumulative probability of failure 6 months becomes to decrease with the decrease of stress level ■ Questionnaire and inspection interval. For crack growth pattern (D, o MCM500 analysis failure probability is quite high, but big differences were not seen for the other patterns. The Pf (200) and Pf (M1) (M2) (M3) (M4) (M5) (M6) Member (500) of crack growth pattern ® are about two times and five times of that of pattern ® respectively at the Fig. 5 Failure probabilities of six members for respec­ standard condition. tive inspection interval. Table 2 shows the classified level of P/(200) and Pf (500) after 20 years ’ service. Level A means very safe members. These probabilities mean that if there are ten state, level F means dangerous state. Tables 3(a),(b) initiated cracks about three cracks among them would summarize the 7/(200) and Pf (500) following the propagate over 200 mm, and about one crack would classification levels shown in Table 2. In the table it is propagate over 500 mm under the present inspection understood that stress level influences most to struc­ conditions of actual ships. From the comparison with tural reliability, but reliability level is also changed the case of no inspection, it is understood that the very much due to the change of inspection interval. In present inspection can reduce Pf (200) to about 1/3 and the case of standard stress level with 2 years inspection Pf (500) to about 1/7, respectively. In the sensitivity interval, Pf (200) ’s almost belong to level E and Pf (500) analysis 2 years interval of close visual inspection was ’s belong to level D. chosen as the standard inspection condition. In this sensitivity analysis, the fatigue characteristics of the member and the inspection capability were deter­ 3. Sensitivity Analysis mined based on the questionnaire asked to naval engi ­ 3.1 Contents of sensitivity analysis neers. Therefore, if the analytical condition represents As the standard condition of sensitivity analysis, the the fatigue and inspection condition of real structure, it fatigue characteristics of member shown in Fig. 2 and may be possible to suggest solutions for the several close visual inspection with 2 years interval were design requirements. The following solutions can be selected. The sensitivity analysis was performed by extracted from the Tables 3(a),(b).

Table 1 Eight fatigue lives and corresponded stress levels used in the sensitivity analysis.

Fatigue life ratio (N/N0) 0.25 0.5 0.75 1.0 1.5 2.0 3.0 4.0 Stress ratio (S/So) 1.32 1.15 1.06 1.0 0.92 0.87 0.80 0.76 Sensitivity Analysis on Fatigue Reliability and Inspection of Ship Structural Members 371

o IQ-

Close visual inspection Close visual inspection 2 years inspection interval 2 years inspection interval Type © crack growth pattern Type @ growth pattern

(a) Service years (b) Service years Fig. 6 Relationship between failure probability and service years with the change of stress level.

Table 2 Classified level of cumulative probability of (1) Present Pf (200) belongs to level E. If we want to failure after 20 years ’ service. improve the reliability level to D, it is necessary Level of Failure probability to reduce inspection interval from 2 years to about 4 months, or to reduce stress level by 15% reliability after 20 years under 2 years inspection interval. P/(200),P/(500) (2) If we want to reduce stress level by 10% with sustaining present Pf (200), it is necessary to A Less than 0.0001 Very safe reduce inspection interval from 2 years to 6 B 0.0001 to 0.001 months. C 0.001 to 0.01 (3) If we want to sustain present Rr (200) with no inspection, it is necessary to reduce stress level by D 0.01 to 0.1 20%. E 0.1 to 0.5 (4) Present P/ (500) belongs to level D. If we want to F Larger than 0.5 Dangerous improve the reliability level to C, it is necessary

Table 3(a) Level of cumulative probability of failure, Table 3(b) Level of cumulative probability of failure, Pf (200), after 20 years ’ service. Pf (500), after 20 years' service.

Crack Inspection Stress Ratio $/Sn Crack Inspection Stress Ratio S/S„ Growth Interval 1.32 1.15 1.06 1.00 0.92 0,87 0. 80 0.76 Growth Interval 1.32 1.15 1.06 1.00 0. 92 0.87 0.80 0.76 3 months F mm E D D C A A 3 months E Hi C B A A A A 6 months F F F E D D C B 6 months F E D A A A A © 1 year F F F E E P C C © 1 year F E E D C A A A 2 years F F F CF) E E D C 2 years F F E CE) D C A A 4 years F F F F E E ' 0 D 4 years F F F E D D B A NO insp. F F F F 1 F F £ D NO insp. F F F F E . C A 3 months F E D D C B A A 3 months E D b A A A k ; A 6 months F E E E D C B A 6 months F D 8 A A A 1 A 1 year F F E E D D C r 1 year F E D ' ) B A A : A 2 years F F F (E) E E D c 2 years F E CD) C B A .; A 4 years F F F K LL E D 0 4 years F F E E E> C A .■ NO insp. F F F F | F F E NO insp. F F F E E c A 3 months F D D | C B A 3 months E B A A A A A 6 months F E E D 1 0 C ; B • 6 months E D C B A ® 1 year F F D- D C B 1 year F D C A A A A 2 years F h CE) E D C . 2 years F E E CD) C A A A 4 years F F F F E E D C 4 years F F E 0 C A A NO insp. F F F F F F - D NO insp. F F F F E C A 3 months F E D D C R A A 3 months E C B A A A A A 6 months F E • E D c C A A 6 months E D C ■’ A A A A @ 1 year F F E D c B © 1 year F E 0 C A A A ! A 2 years F F F (E) E D C c 2 years F E 0 CD) C A A 4 years F F F E D c 4 years F F • E E 0 C A j A F-1 NO insp. F F F F | F F D NO insp. F F 1 F E c i c nm 372

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Sensitivity Analysis on Fatigue Reliability and Inspection of Ship Structural Members 373

Table 4 Solutions to achieve design requirements obtained by sensitivity analysis

Requirements Solutions Shorten inspection interval from Improve present reliability level 2 years to 4 months by one step (from level E to D) (or) Lower stress level by 15 % under 2 years inspection interval P/(200) Reduce weight by 10 % without Shorten inspection interval loss of present reliability level from 2 years to 6 months Remove inspections without Lower stress level by 20 % loss of present reliability level Shorten inspection interval from Improve present reliability level 2 years to 10 months by one step (from level D to C) (or) Lower stress level by 10 % under 2 years inspection interval P/(500) Reduce weight by 10 % without Shorten inspection interval loss of present reliability level from 2 years to 8 months Remove inspections without Lower stress level by 15 % loss of present reliability level

to reduce inspection interval from 2 years to following four inspection methods changing the number about 10 months, or to reduce stress level by 10% of members having initiated crack. under 2 years inspection interval. Case A : Long distance visual inspection for all the (5) If we want to reduce stress level by 10% with members is carried out (2m to 5m from sustaining present P/ (500), it is necessary to inspector ’s naked eye) . reduce inspection interval from 2 years to 8 Case B: Combined inspection is carried out. At first, months. long distance visual inspection for all the mem­ (6) If we want to sustain present P/ (500) with no bers is carried out. If at least one crack is inspection, it is necessary to reduce stress level by detected, close visual inspection for all the mem­ 15%. bers is carried out (0.5m from inspector ’s naked Table 4 shows the summary of these solutions. eye) . Otherwise, inspection is terminated. 4. Combined Inspection and Sampling Inspection Case C10: Sampling inspection with 10% sampling rate is carried out. First, close visual inspection 4.1 Method of analysis for sampled members is carried out. If at least As ship structure is composed of a lot of members, the one crack is detected among the sampled mem­ application of close visual inspection for all the mem­ bers, close visual inspection for all the remaining bers is usually very difficult on the time and cost aspect. members is carried out. Otherwise, inspection is Therefore, long distance visual inspection, combined terminated. one or sampling one is often applied in the actual field. Case C 20: Sampling rate is selected as 20% and the In this study, the effectiveness of combined and sam­ inspection procedure is same as Case C10. pling inspection was studied assuming a member set. Case D: Close visual inspection for all the members is The member set consists of two types of members ; ‘n carried out. ’ members will be cracked during 20 years ’ service and The probability of crack detection for Case B, {POD) B, the rest members will not be cracked during this period. is calculated by The cracked members Cn’ members) have the fatigue characteristics shown in Fig. 2, and have the crack +[!-(!-W]x(fQD)i (4) growth pattern (3). The critical crack length of the where n is the number of members having initiated member was defined as 500mm. The probabilities of crack during 20 years in the member set. Qi is the crack detection for close visual and long distance visual detection ratio of crack by the long distance visual inspections were evaluated by Eqs. (2) and (3), inspection and is calculated by multiplying (POD) 2 by respectively. The analysis was carried out for the the distribution of crack length,A (a) (crack state 374 Journal of The Society of Naval Architects of Japan, Vol. 181

vector of Markov chain model) , at each inspection close visual inspection for all the members. Following time. results can be extracted from the Fig. 8. fa (POD)2xfa(a)da (5) (1) The smaller the inspection interval is, the bigger the difference of cumulative probability of failure The probability of crack detection for Case C10 and between Case A and Case D is. Case C 20, (POD) c, is expressed by (2) Combined and sampling inspections become (POD)c = rx(POD),+(l-r) effective as the increase in the number of mem­ x[l-([email protected]]x(fO 0). (6) bers having initiated crack in the member set. where r is sampling rate and Qi is the detection ratio of (3) Combined inspection has more advantages than crack by the close visual inspection and is calculated by sampling: inspection when number of members the following equation. having initiated crack becomes small in the @,= / X/.(a)dz (7 ) member set. 4. 2 Results of analysis 5. Conclusion Fig. 8 shows cumulative probability of failure after 20 Hot spot stress amplitude and crack growth pattern years for the case of n = 5, 10, 20, 50 and 100 members are usually uncertain in ship structural members. with respect to different inspection methods. Failure Furthermore, there are several requirements such as the probability for long distance visual inspection is the improvement of reliability, reduction of weight and highest among the five cases. The order of failure simplification of inspection at the fatigue design. In probability is 10% sampling inspection, 20% sampling order to respond to these problems, sensitivity analysis inspection, combined inspection, and the lowest is the of fatigue reliability and inspection was carried out based on the information obtained from questionnaire. Also, the effectiveness of combined inspection method and sampling inspection method which has been often used in the actual field was discussed through the sensi­ tivity analysis. The major conclusions are drawn as follows : (1) Under present inspection condition of actual □ Case A ships, it is thought that about 70% of initiated • Case B A Case C10 cracks are detected below 200 mm and about 90% V Case C20 are detected below 500mm. This reliability level 1 year inspection interval I Case D CD 10 is maintained by the close visual inspection with 2 years interval. £ 101 (2) For the design requirements such as the improve ­ ment of reliability, reduction of weight and simplification of inspection, the solutions to respond to the requirements are extracted as shown in Table 4.

0 Case A (3) Combined inspection, which is performed by both • Case B long distance and close visual inspection, has A Case C10 more advantage than sampling inspection, when V Case C20 2 years inspection interval I Case D the number of members having initiated crack is a few in the inspecting member set.

4 years inspection interval References 1) Fujimoto, Y., Kim, S. C., Shintaku, E. and Ohta- ka, K„ “Study on Fatigue Reliability and Inspec­ □ Case A tion of Ship Structures Based on questionnaire • Case B Information ”, J. of the Society of Naval Archi­ A Case C10 tects of Japan, 180, pp. 601-609 (1996) (in V Case C20 Japanese) 1 Case D 2) Kim, S. C., Fujimoto, Y., Shintaku, E., “Study on Fatigue Reliability and Inspection of Ship Struc­ tures Based on questionnaire Information ”, Pro ­ 10 20 50 100 ceedings of PACOMS’96, pp. 65-72 No. of initiated cracks , n 3) SR226 Committee, “An Investigation Study on Rational Structural Design for Bulk Carrier”, Research Committee of the Shipbuilding Fig. 8 Effectiveness of combined and sampling inspec­ Research Association of Japan (1996. 3) (in tion. Japanese) Sensitivity Analysis on Fatigue Reliability and Inspection of Ship Structural Members 375

4) Fujimoto, Y. et al, “Reliability Assessment of tures Based on Sequential Cost Minimization Deteriorating Structure by Markov Chain Method ”, J. of the Society of Naval Architects of Model ”, J. of the Society of Naval Architects of Japan, 170, pp. 755-768 (1991) Japan, 166, pp. 303-314 (1989) (in Japanese) 6) Bogdanoff, J. L. and Kozin, F., “Probabilistic 5) Fujimoto, Y., Swilem, S. A. M. and Iwata, M., Models of Cumulative Damage ”, John Wiley & “Inspection Planning for Deteriorating Struc­ Sons, Inc., New York (1985). 377 3-12

|b[ m IE ** s ± # IEm zh # s IE 1ES m A m IE Temperature Effect on Corrosion Fatigue Strength of Coated Ship Structural Steel

by Masahiro Takanashi Akio Fuji Yuki Kobayashi, Member Masao Ojima Yasushi Kumakura, Member Masaki Kitagawa

Summary Corrosion fatigue tests were carried out in synthetic seawater, in order to clarify the temperature effect on the corrosion fatigue strength of coated ship steel. To simulate the shipbuilding construction process, KA 32 TMCP steel in 10 mm thickness was sprinkled with water in outdoor twice a day for 20 days. After the exposure, the surface preparation was conducted, and the tar epoxy resin was painted with 50,100, 200, and 300 /on in thickness. These specimens were subjected to cyclic axial stress at 0.17 Hz in 25, 40 and 60°C synthetic seawater. For the comparison, corrosion fatigue tests of uncoated specimens were also carried out. In the low cycle region, the corrosion fatigue strength of the coated specimen was almost equivalent to that of uncoated. On the contrary, it was improved in the high cycle region. The temperature effect of the corrosion fatigue strength was assessed by introducing corrosion factor Kc, which implies the reduction rate of the corrosion fatigue strength against that at 25°C. The corrosion factor Kc indicated 1.03—1.13 at 40—60°C. In addition the maximum depth of the corrosion pits was found to be 30—40 um after 20-day exposure. In the long-life corrosion fatigue, cracks were initiated from such corrosion pits under the coating film before it was destroyed. Finally, effect of corrosion pits on the fatigue crack initiation behavior was considered under the conditions of surface preparation of specimens.

IB 1.

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Table 1 Chemical compositions (wt%) c Si Mn P S 0.14 0.20 1.14 0.016 0.004

Table 2 Mechanical properties

Yield Stress Tensile Strength Elongation Chamfer (R2) (MPa) (MPa) (%) 390 496 29 A-A' Cross section G.L. = 200 mm

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SSW Synthetic seawater Fig. 10 Surface observation of coated specimen after Number of cycles to failure, N, removing coating film. (AS=270 MPa, film Fig. 8 Relationship between Nc and Nf thickness 100 /im, 60°C) 381

0^m, mg#0^##jRA&»@aH#mg#kitRL v^. uta^-c, @^A*a-n±mma#@f a#c, -c$#$fL-cv^a. 4.3 Mg#<7)S#i$^5SSiz# <£IStiMMm^mi *A:, emijKAkfyt'&'im^Z^KL-caRLTW? m#a^ur&3LBmaa%^«tfg%* 2:®3:3»#a«, m&0*#ey a«l=]?&^. mg#0MA#**^r@ m##mg#01 %, k ##» o, Bm# k»a. L^AicT, #*f0m#mmwGBmK:%3:# 4. * m fmm0»#»^»a^*, *#k»ag^m@%js 4.1 ####0 A%* k***0&***f)tkR =aW" W2#LT|*AK: 3: am^g@^S:*f »* Fig.3Kal6Ck^T#3. 2:03 Table 3 Standard curves of coated specimens in syn- thetic seawater 3K#g%A@#K, RiGAm^m&m#kmG;&#a Thickness Standard curve ^m*g^T*^:89K#»ak#A63ia. m#$%a^a (ym) (25°C-basis) 3 2—3 X10" iM f ;i/iaeT0 B a ^ # 50 A S= 1539 Nf -01440 100 A S= 1929 Nf -01640 4.2 wmtmmmum-k^m^ < te-oiir J:4f*y##mm@K#Tlf»Kl,5^2.0%T*,a.4-ia 0MKTB%A^K:3:au-f»#4ia**,a6@®, CNS k-f 60k#A6ft, m#*!m^kg^%g0@T$6d\ tr^g»u-f»0%±A#^E(. $-^^^->0@#'\0@3(3g< %a.Fig. 11BC®## £> tl&o $3: <#LTj3 9, 300frni@mg$-m^-|f, mg®K#B -^, g##M##*rB9aaa^mg@%mBmm# BBkAkSB»w. 200MPamTK»a k##g#klt^-C#^ w±02:k^(,, mg®#^#B#m (2s°o %M-am& gBI»l±fa. 2:®3:3»fE*#HrB, mKB##»A k L T&f C k a*r #, B 1/1.03—1/1.13 0 Z#*A^a#rTa%@l<t^a k»»-tc k*!T# MT@Tfa. #6fmaiC03:3K:/!\2#@&k?%m a. La^L, ^^O^^gklt^akAik^mmAi*, AkLTB, *0 2dA^#^g,3ia. -^T6, t0@E##Skl@TL-Cwa. A%# afk&^0&g±#K:#L% 6 *®@KrB^TB%t^kv^a. a0(6A#aT0m# %*B, «#ag#afr&azK^»^'f^yo##^0 ^9^##A:G0°C^#mT*/k3TA/kAk#^63ia.# *a&m&-&-rv)ac:kK»a. i@^f#@*kK%@a f-;k.z:df:fy#ggm#09-7>:^me&g@KL, *#<^.A:#e, *0 B 3 a*B a a4C^#%B 3: W'. L ^7XR»^»#SLA: k ^ a 60°Cia#T$c*: k®# ata^T, **KKR0^»3$#T@#K:amLTV^a #6$a"'. mt0asasg6c±#fMmBW:fAfB t00, Bm0#8&mmawfTV)ak#a$iT,a. f #KA9. «*#^aai3@TfaW*#ai&a. LA:** 382 mm# c -c, ^ 4= 6o°c$ r-rt 5^. Fig. 11 KT±M#C, i

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Table 4 Corrosion factor (25°C—basis)

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O Coating film thickness 50 p m A Coating film thickness 100nm sm#-r#, #%kL^0Tiimm^^c»?AK:6^ □ Coating film thickness 200 p m k , mirF0@#*mK:#± 30-40 as* 0m*y V Coating film thickness 300 p m y % C k *:#>(. ^ft&o I»^K: Fig. 10 ftA 40 /zm @&0m#e v bwAmmm* K#AKtLA60ra%<, ±kL7mg#0#*K:j: Temperature, TfC) OBlgg^ACkA^^S. C0jt3»Byb0#&6m Fig. 11 Temperature dependency of corrosion factor kA*T#%. Mg. 14 K^**#&mLA#mmf0f-? e. m% 4.4 rnmmm kjtRLAt0T

# Fig. 10 i 0 M A.

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£ 10' %. 6/1-So 3) Tested in SSW

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3) : m 201 weas# 10) m 4 rngm-fb#*# 29 ##, A#, (1993 ), p. (mo). 95. 4) wm-% ie##, #mm#, *$#1^, 11) H##?: 220 "/?? 4tme, : B$)gm#^#^#, No. iso, (1996), p.521—530. (1996), p.401—410. 5) 4-#wi, 12) = B^m## Vol. 13, No. 2, (1981), p.237—257. ##3:*A*8, Vol. 53, No. 496, (1987), p. 6) EEE# : S**B, fcffiJS, (1967), p. 552-561. 2267—2273. 7) Vol.59, 13) ^UHE#, BBZEmag : B^## No. 557, (1993), p.1-11. Vol.B, No. 940-37, 8) : 7VX 3>7'J-h, Vol. 31, (1994), p. 407—408. No. 2, Mar. (1989), p.7—15. 14) # fe- 9) &BE#, m: an : Vol. A, *, No. 157, (1985), p.498— 506. No.96-10, (1996), p.631—632.