UNIVERISTY OF SPLIT FACULTY OF ELECTRICAL ENGINEERING, MECHANICAL ENGINEERING AND NAVAL ARCHITECTURE Mdeling ship structural systems by events Brank Blagjević
Mtivatin Inclusin the cncept entrpy rm inrmatin thery as the nly ratinal measure system uncertainty, r the estimatin structural prperties. Rbustness and redundancy systems events may be cnsidered as additinal characteristics engineering systems (ship structures). The prblem distinctin cmplex engineering systems that have the same reliabilities and prbabilities ailure, but dierent number pssible events.
Event riented system analysis System events S E E... E 1... p E p E p E 1 n n Events E i, i = 1,,,n are STATES the system S. Entrpy Uncertainty individual randm event, E, with prbability ccurrence p(e) is measured by entrpy: H lg p E Entrpy a system events (Shannn): H( S ) H ( S ) H ( p, p,..., p ) p lg p n n 1 n i i i1 n
Types events particular interest r engineering purpses: Operatinal, E Failure, E Transitive, E t Cllapse, E c System events, S, can be subdivided int subsystems: peratinal and ailure O E E E 1 p E p E p E 1 N N F E E E N1 N NN p E p E p E N1 N NN
Then system, S, can be viewed cnditinally: S / O Entrpy a subsystem is: E / O E / O E / O E / O 1 N p E p E p E p E 1 N po po po p O E / F E / F E / F E / F N1 N N NN S / F p E p E p E p E N1 N N NN p( ) p( ) p( ) p( ) F F F F H m i mi p Eij S / Si lg p S j1 i i p E DOES NOT DEPEND n the prbability the system p(s) DOES NOT DEPEND whether the system is cmplete r incmplete p ij S
Rbustness Rbustness is regarded as the system s capability t respnd t all pssible randm ailures unirmly. The system rbustness is related nly t the ailure mdes the system: H N NN p Ei S / F lg p F in 1 p E p i F S / F S S / F ROB ROB H N ROB (S) = 0, i there is n ailure, r ne ailure event has a prbability p=1 ROB (S) = max when all the ailure mdes are equally prbable. Rbustness increases with increasing number events in the system
Redundancy capability a system t cntinue peratins by perrming dierent randm peratinal mdes given prbabilities in case randm ailures cmpnents The system redundancy is related nly t the peratinal mdes the system: H N N p Ei S / O lg p O i1 p E p i O S / O S S / O RED RED H N RED = 0 i there are n peratinal states r i nly ne peratinal state is pssible RED = max. i all peratinal states are equally prbable
Rbustness a beam Example 1 F y Mean COV Distributin Side a 6 mm 0,01 Nrmal Side b 5 mm 0,01 Nrmal b a x Length L 500 mm 0,01 Nrmal Mdulus elasticity E 06000 N/mm 0,01 Nrmal A = a b Yield stress F 5 N/mm 0,06 Lg- Nrmal L Lad (Frce) F t 150 kn 0, Nrmal Crss-sectin Area A 900 mm 0,1 Nrmal Critical buckling stress ( x ) Cx 0,4 N/mm 0,07 Lg-nrmal b a F Critical buckling stress ( y ) Cy 19, N/mm 0,07 Lg-nrmal Limit state unctins: Cmpressin: g 1 = A F F t Buckling (x): g = A Cx F t Buckling (y): g = A Cy F t A1 =,1511 A =,140569 A =,15947
Example 1 Series system, basic events: A 1 A A Cmpund events E i = n =8 System events: E E E E E E E S 0,9897 8,74 10 1,510 8,08 10 1,44 10 7,18 10 8,0810 1 1,1,, 1, 1,, -4 - -4-5 -7-4 Reliability series system: R S pe 1 0, 9897 p S 0,01708 Uncnditinal entrpy: H(S) = 0,14191 (,80755; 0,047800) System rbustness: ROB S H S F i 7 p Ei p Ei N / lg 0,574081 p F p F Maksimum rbustness: ROB max (S) = lg (N ) =,58496
Example 1 Event riented system analysis a beam Requirements: A = 900mm = cnst. L = 500 mm = cnst. Results: Max. ROB r a = b = 0 mm ROB max = 1,780
Fr a=b=0 mm cnsider the change L t rbustness.,5 Example 1 Cnditin: R (S) > 0,999 Result: ROB(S ), R(S ),0 1,5 1,0 0,5 R(S ) ROB(S ) ROB max r L = 4 mm 0,0 0 100 00 00 400 500 600 Beam length L, mm Cnclusins r example 1 Rbustness changes r dierent dimensins. It is pssible t calculate max. Rbustness Fr the same reliability level the rbustness varies signiicantly.
Example EOSA: Rbustness ship structural elements L = 17,15 m B = 1,4 m D = 11 m = 47400 tdw b e =80mm hw= 0mm t w tp Lads: DnV 4 ailure mdes Trsinal buckling Lcal buckling Plastic ailure Fatigue ailure
Series system (4 basic events) Example A 1 (trsinal buckling) A (lcal buckling) A (plastic dermatin) A 4 (atigue) System: E E 1 1,1 E, E, E4,4 E1, 0,90987 8,95 10,90 10 4,06 10 5,8 10,90 10 S E1, E1,4 E, E,4 E,4-4 -9-5 -11-11,90 10 5,77 10 9,0 10 1,85 10,60 10 - -4-4 -8-4 Reliability: R (S) = 0,909 p (S) = 0,09014 Uncnditinal entrpy: H(S) = 0,44471 Rbustness: ROB (S) = 0,804 Max. Rbustness =,1
Analysis: Rbustness change r dierent lngitudinal s height and under cnditins: A = cnst. 10 mm < h w < 0 mm R > 0,909 Example 1,00 0,95 0,90 R(S ) ROB(S ), R(S ) 0,85 0,80 0,75 0,70 ROB(S ) 0,65 Cnclusins: 0,60 05 10 15 0 5 0 5 Lngitudinal's height h w, mm Rbustness a system events can be related t design variables ship structural elements max. ROB can be achieved within acceptable gemetry limits it is pssible t use ROB t cmpare dierent design slutins with varius cnstraints applied
Redundancy ship structures Example Deck panel (tanker) b b b tp 1 y' t w hw ht t t HP HP1 HP Functinal levels: b t tb 1. Level intact structure z'. Level ailure ne lngitudinal: n. r n.. Level ailure tw lngitudinals (nn-redundant structure) b b b tp 1 y' t w hw ht t t HP HP1 tb HP b t z'
Lads: DnV Failure mdes: buckling plating between stieners () plastic ailure a deck girder (1) plastic ailure lngitudinal () trsinal buckling a deck girder (1) trsinal buckling lngitudinal () Example 1st peratinal level: n = 11 basic events, nd peratinal level: S S S S S 1 1 1 i 1 t 1 c 1 1 1 1 1 N = 11 = 048 cmpund events S S, S E,..., S E,..., S E, S 1 i 1 t 1 t 1 t 1 c 1 1 j j 1 t 1 t N N 15 peratinal states, N = 971 cmpund events rd peratinal level 18 peratinal states, N = 1017 1 i 1 c 1 t 1 t 1 t S S 1S E1 S E 15S E15 t 1 t t 1 t t 1 t 1S E1 E1 S E E1 S E E1 t 1 t t 1 t t 1 t S 4S E4 E 5S E5 E 6S E6 E S E E S E E S E E t 1 t t 1 t t 1 t 18 18 9 18 18 9 18 18 9
Operatinal levels, states and transitive events a system 1 S = 1 1 S i 1 1S c 1 1S t 1 E t 1 1 E t 1 E t 1 E t 4 1 E t 5 1 E t 6 1 E t 7 1 E t 8 1 E t 9 1 E t 10 1 E t 11 1 E t 1 1 E t 1 1 E t 14 1 E t 15 1 S S S 4 S 5 S 6 S 7 S 8 S 9 S 10 S 11 S 1 S 1 S 14 S 15 S S = 1 1 S i 1 1S c 1 S S 5 S 6 S 7 S 1 E t S 1 E t 1 1 E t 4 S 1 E t 4 1 E t 5 1 E t 6 1 E t 7 1 E t 9 S 1 E t 9 8 1 E t 10 1 E t 11 1 E t 1 1 E t 1 1 E t 14 8 S 10 S 11 S 1 S 1 S 14 S 15 S 1 E t 15 E t 1 E t E t E t 7 E t 8 E t 9 E t 1 E t 14 E t 15 E t 16 E t 17 E t 18 E t 4 E t 5 E t 6 E t 10 E t 11 E t 1 1 S S S 7 S 8 S 9 S 1 S 14 S 15 S 16 S 17 S 18 S 4 S 5 S 6 S 10 S 11 S 1 S S = 1 1 S i 1 1S i ( 1S E t 1) 1 E t 1 ( 4S E t 4) 1 E t ( 7S E t 7) 1 E t ( 10S E t 10) 1 E t 4 ( 1S E t 1) 1 E t 6 ( 16S E t 16) 1 E t 9 ( S E t ) 1 E t 1 ( 5S E t 5) 1 E t ( 8S E t 8) 1 E t ( 11S E t 11) 1 E t 4 ( 14S E t 14) 1 E t 6 ( 17S E t 17) 1 E t 9 ( S E t ) 1 E t 1 ( 6S E t 6) 1 E t ( 9S E t 9) 1 E t ( 1S E t 1) 1 E t 4 ( 15S E t 15) 1 E t 6 ( 18S E t 18) 1 E t 9
Analysis: Changes the redundancy r dierent spacing between lngitudinals. Cnstraints: Reliability must be equal r larger than that the riginal structural cniguratin. Weight the panel = cnst. Example 1,8 1,6 1,4 RED S i Result: R, RED 1, 1 0,8 R RED max = 1,77694 0,6 0,4 0, 0 6 67 7 77 8 87 9 Spacing between lngitudinals [mm]
Cnclusins Traditinal prbabilistic engineering system analysis based n physical and/r technical cmpnents a system, may be extended by an EOSA. Numerical examples cnirm the relevance EOSA and indicate ptential imprvements in system design (ROB, RED). Prblems Pssible numerical prblems when dealing with larger systems. Fr a cmplete event riented system analysis an enumeratin ALL the pssible events is needed.
General remarks 1. General relatins amng the prbability, uncertainty the system and uncertainties the subsystems are derived by using inrmatin thery.. The uncertainties in system s peratin riginate rm the unpredictability pssible events.. The system uncertainty analysis is based n entrpy. 4. The entrpy, as the nly ratinal measure system uncertainty, des nt depend n anything else ther than pssible events and in this sense is entirely bjective. 5. The entrpy a system itsel in general is nt particularly helpul in the assessment system perrmance. 6. The uncertainties imprtant subsystems events, such as the peratinal and ailure mdes, as well as their relatins t the uncertainty and reliability the entire system, can prvide a better insight int the system perrmance. 7. May be helpul in dierent ields engineering in the reinement system perrmance.