FATIGUE LIFE PREDICTION FOR A VESSEL SAILING THE NORTH ATLANTIC ROUTE Anastassia Baxevani Centre for Mathematical Sciences Lund University, Sweden FATIGUE LIFE PREDICTION FOR A VESSEL SAILING THE NORTH ATLANTIC ROUTE – p.1/18
Introduction Reliability of a vessel depends on the fatigue strentgh of the material, whose properties are determined by experiments. Based on a damage accumulation rule, we derive the asymptotic distribution of the damage accumulated by the material and use it to derive the probability distribution of the fatigue life prediction FATIGUE LIFE PREDICTION FOR A VESSEL SAILING THE NORTH ATLANTIC ROUTE – p.2/18
b s b b t ) , 0 < t < t } be a random load Definitions-Assumptions Let { X ( • Damage accumulation rule - Palmgren-Miner 1 � D ( t ) : = N A i t i ≤ t FATIGUE LIFE PREDICTION FOR A VESSEL SAILING THE NORTH ATLANTIC ROUTE – p.3/18
b t ) , 0 < t < t } be a random load Definitions-Assumptions Let { X ( • Damage accumulation rule - Palmgren-Miner 1 • N A = K − 1 A − b , log( K ) ∈ N ( m K , � s 2 b ≥ 1 D ( t ) : = N A i t i ≤ t K ) , m K < 0 , FATIGUE LIFE PREDICTION FOR A VESSEL SAILING THE NORTH ATLANTIC ROUTE – p.3/18
t ) , 0 < t < t } be a random load Definitions-Assumptions Let { X ( • Damage accumulation rule - Palmgren-Miner 1 • N A = K − 1 A − b , log( K ) ∈ N ( m K , � s 2 b ≥ 1 D ( t ) : = N A i t i ≤ t b K ) , m K < 0 , • � D ( t ) : = K A i = K · D X ( t ) t i ≤ t FATIGUE LIFE PREDICTION FOR A VESSEL SAILING THE NORTH ATLANTIC ROUTE – p.3/18
t ) , 0 < t < t } be a random load Definitions-Assumptions Let { X ( • Damage accumulation rule - Palmgren-Miner 1 • N A = K − 1 A − b , log( K ) ∈ N ( m K , � s 2 b ≥ 1 D ( t ) : = N A i t i ≤ t b K ) , m K < 0 , • � D ( t ) : = K A i = K · D X ( t ) t i ≤ t • Amplitude def - Rainflow cycle count (RFC) FATIGUE LIFE PREDICTION FOR A VESSEL SAILING THE NORTH ATLANTIC ROUTE – p.3/18
m b b b m Expected nominal damage b − 2 N ( u , v ; t ) dv du b ( b − 1)( u − v ) • For N ( u , v ; t ) : nb of RFC-cycles with max > u and min < v , bdd fun. of u and N ( u , v ; 0) = 0 � ∞ � u D X ( t ) = −∞ −∞ FATIGUE LIFE PREDICTION FOR A VESSEL SAILING THE NORTH ATLANTIC ROUTE – p.4/18
Expected nominal damage b − 2 N ( u , v ; t ) dv du b ( b − 1)( u − v ) • For N ( u , v ; t ) : nb of RFC-cycles with max > u and min < v , bdd fun. of u and N ( u , v ; 0) = 0 � ∞ � u m ( u , v ; t ) = E [ N ( u , v ; t )] D X ( t ) = −∞ −∞ b − 2 b ( b − 1)( u − v ) m ( u , v ; t ) dv du • For � ∞ � u E [ D X ( t )] = −∞ −∞ FATIGUE LIFE PREDICTION FOR A VESSEL SAILING THE NORTH ATLANTIC ROUTE – p.4/18
m ( u , v ; t ) is decreas. for u , increas. for v , hence for u ≥ v Upper bound for the damage intensity m ( u , v ; t ) ≤ min { m ( u , u ; t ) , m ( v , v ; t ) } m ( u , v ; t ) m + m + After some lengthy derivations we can show that m ( u , u ; t ) m + ∂ ≤ min { t ( u ) , t ( v ) } ∂ t t ( u ) = ∂ where - upcrossing intensity of u at t ∂ t FATIGUE LIFE PREDICTION FOR A VESSEL SAILING THE NORTH ATLANTIC ROUTE – p.5/18
s s b b s s b G p b b s t s t b G t p Upper bound for expected nominal damage b − 2 min { b ( b − 1)( u − v ) m + m + • Damage intens. d X ( t ) = d ( E [ D X ( t )]) bdd from above dt � ∞ � u d X ( t ) ≤ t ( u ) , t ( v ) } dv du −∞ −∞ FATIGUE LIFE PREDICTION FOR A VESSEL SAILING THE NORTH ATLANTIC ROUTE – p.6/18
b b s t s t b G t p Upper bound for expected nominal damage b − 2 min { b ( b − 1)( u − v ) m + m + • Damage intens. d X ( t ) = d ( E [ D X ( t )]) bdd from above dt � ∞ � u d X ( t ) ≤ t ( u ) , t ( v ) } dv du s X ( t ) , s ˙ −∞ −∞ b − 1 b s ˙ s b • For X ( t ) zero-mean, loc. stat., Gaussian load, G 2 p C [ X ( t ) , ˙ X ( t ) } → C [ X ( t ) , ˙ X ( t )] ≪ min { X ( t )] = 0 � � X ( t ) ( t ) 3 X d X ( t ) ≤ 2 2 + 1 2 FATIGUE LIFE PREDICTION FOR A VESSEL SAILING THE NORTH ATLANTIC ROUTE – p.6/18
Upper bound for expected nominal damage b − 2 min { b ( b − 1)( u − v ) m + m + • Damage intens. d X ( t ) = d ( E [ D X ( t )]) bdd from above dt � ∞ � u d X ( t ) ≤ t ( u ) , t ( v ) } dv du s X ( t ) , s ˙ −∞ −∞ b − 1 b s ˙ s b • For X ( t ) zero-mean, loc. stat., Gaussian load, G 2 p C [ X ( t ) , ˙ X ( t ) } → C [ X ( t ) , ˙ X ( t )] ≪ min { X ( t )] = 0 � � X ( t ) ( t ) 3 X d X ( t ) ≤ 2 2 + 1 2 b − 1 b s ˙ t ) s t ) b G t 2 p • Hence � � t X ( ( � 3 ˜ X E [ D X ( t )] = 2 2 + 1 d 2 0 FATIGUE LIFE PREDICTION FOR A VESSEL SAILING THE NORTH ATLANTIC ROUTE – p.6/18
s X ( t ) and s ˙ Upper bound for expected damage Suppose X ( t ) known for [0 , t ] . Then, for t large enough the difference b − 1 s ˙ t ) s t ) D X ( t ) − ˜ t E [ D X ( t )] ≈ 0 2 p hence s 2 � t X ( ( X D ( t ) ≈ K d 0 with log( K ) ∈ N ( m K , K ) FATIGUE LIFE PREDICTION FOR A VESSEL SAILING THE NORTH ATLANTIC ROUTE – p.7/18
Fatigue life distribution Failure time: P ( T f ≤ t ) = P ( D ( t ) ≥ d crt ) If D X ( t ) ∈ N ( m ( t ) , s 2 ( t )) F f ( z ) d s K � ∞ m K + log m ( t ) + log(1 + s ( t ) � m ( t ) z ) − log d crt � P [ T f ≤ t ] = −∞ FATIGUE LIFE PREDICTION FOR A VESSEL SAILING THE NORTH ATLANTIC ROUTE – p.8/18
Fatigue damage accumulated by a vessel a d t ) t Let X ( t ) be a variable load applied at a vessel. Then, after certain considerations / simplifications � t D ( t ) = K H s ( 0 where H s ( t ) - significant wave height process encountered by the vessel FATIGUE LIFE PREDICTION FOR A VESSEL SAILING THE NORTH ATLANTIC ROUTE – p.9/18
Notation • X ( t ) - random load, e.g. stress at some point on the vessel • t 0 - departure time • T - duration of trip • V ( t ) = ( V x , V y ) - velocity of vessel, known in advance • s ( t ) = ( x ( t ) , y ( t )) - deterministic position of vessel at time t FATIGUE LIFE PREDICTION FOR A VESSEL SAILING THE NORTH ATLANTIC ROUTE – p.10/18
a dt during one voyage � t 0 + T m ( s , t ) = E [log( H s ( s , t ))] The mean and variance of H s ( t ) t 0 Let log( H s ( s , t )) be a locally stat. Gaussian r.f. with • • r (( s 1 , t 1 ) , ( s 2 , t 2 )) = C [log( H s ( s 1 , t 1 )) , log( H s ( s 2 , t 2 ))] m ( t ) = E [ Y ( t )] = E [log( H s ( t ))] = m ( s ( t ) , t ) Then encountered pr. Y ( t ) = log( H s ( t )) = log( H s ( s ( t ) , t )) is locally stat. Gaussian with • • r ( t 1 , t 2 ) = C [ Y ( t 1 ) , Y ( t 2 )] = C [log( H s ( t 1 )) , log( H s ( t 2 ))] = r (( s ( t 1 ) , t 1 ) , ( s ( t 2 ) , t 2 )) FATIGUE LIFE PREDICTION FOR A VESSEL SAILING THE NORTH ATLANTIC ROUTE – p.11/18
a dt during one voyage a dt ] : = � t 0 + T The mean and variance of H s ( t ) t 0 s 2 ( t ) am ( t ) + a 2 s 2 = r ( t , t ) � t 0 + T � t 0 + T m ( T ) = E [ H s ( t ) h ( t ) dt t 0 t 0 a dt ] = a 2 r ( t 1 , t 2 ) − 1 � � with h ( t ) = exp and 2 � t 0 + T � t 0 + T � t 0 + T � � s 2 ( T ) = V [ H s ( t ) h ( t 1 ) h ( t 2 ) e d t 0 t 0 t 0 FATIGUE LIFE PREDICTION FOR A VESSEL SAILING THE NORTH ATLANTIC ROUTE – p.12/18
Distribution for D ( n ) - damage accumulated during n voyages t ijr , T ijr , j = 1 , . . . , 12 , r = 1 , 2 and i = 1 , . . . , n rj : departure, duration of i th voyage in j th month in r th dir. n 1 j = n 2 j or n 1 j = n 2 j + − 1 and T ijr indep. of a dt : indep., mean m jr and var. s 2 ∈ [ t ijr , t ijr + T ijr ] and vessel stays at port t . H s ( t ) = 0 , t / enough time for Y ( t ) to become independent. Then � t irj + T irj • Z i H s ( t ) jr = jr t irj FATIGUE LIFE PREDICTION FOR A VESSEL SAILING THE NORTH ATLANTIC ROUTE – p.13/18
Distribution for D ( n ) - damage accumulated during n voyages t ijr , T ijr , j = 1 , . . . , 12 , r = 1 , 2 and i = 1 , . . . , n rj : departure, duration of i th voyage in j th month in r th dir. n 1 j = n 2 j or n 1 j = n 2 j + − 1 and T ijr indep. of a dt : indep., mean m jr and var. s 2 ∈ [ t ijr , t ijr + T ijr ] and vessel stays at port t . H s ( t ) = 0 , t / enough time for Y ( t ) to become independent. Then � t irj + T irj • Z i H s ( t ) jr = jr t irj • CLT → D jr = � n rj i = 1 Z i jr ≈ N ( n rj m jr , n rj s 2 jr ) FATIGUE LIFE PREDICTION FOR A VESSEL SAILING THE NORTH ATLANTIC ROUTE – p.13/18
Distribution for D ( n ) - damage accumulated during n voyages a dt ≈ N ( Hence � t 0 + T 12 2 12 2 12 2 � � � � � � n rj m jr , D X ( n ) = D jr = H s ( t ) n rj s t 0 j = 1 r = 1 j = 1 r = 1 j = 1 r = 1 F f ( z ) dz with n = � 12 � 2 s K r = 1 n rj and j = 1 � ∞ m K + log( m ( n )) + log(1 + s ( n ) � � m ( n ) z ) P [ T f ≤ n ] = −∞ FATIGUE LIFE PREDICTION FOR A VESSEL SAILING THE NORTH ATLANTIC ROUTE – p.14/18
Remarks • If r ( t 1 , t 2 ) is unknown but > 0 , setting r ( t 1 , t 2 ) = 0 results to an underestimation of s 2 ( T ) which consequently leads to a fatigue failure time more concentrated about the median • Most of damage occurs during such operations as loading cargo or supplying with fuel, but these loads are not considered here. Obviously though, this damage could also be included in the analysis by allowing the function h ( t ) to take on suitable values during the times between travelling. FATIGUE LIFE PREDICTION FOR A VESSEL SAILING THE NORTH ATLANTIC ROUTE – p.15/18
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