Согласно (4.2.16), математическое ожидание ресурса

и согласно (4.2.17), среднеквадратическое отклонение

Следовательно, среднеквадратическое отклонение относительно наиболее вероятного ресурса

Мы не учли возможность того, что исходный размер трещины может быть с самого начала больше критического значения xf.

Характеристическая величина xf/x0, соотношения между конечным размером трещины и начальными поверхностными дефектами, имеет порядок 100. Когда исходные глубины трещин распределены экспоненциально, т.е. g=1, это дает погрешность в оценке ресурса, т.е. несоответствие действительной скорости распространения, 28%.

Скорость роста пропорциональная xs. Модель для определения скорости роста трещин, которую можно увидеть во многих работах, имеет вид

Соотношение такого рода дает теоретическая формула (4.7.81). При m=3, получим классическое значение s=1,5. В этом случае, мы можем найти промежуточную постоянную движения

которая удовлетворяет уравнению (4.7.106). Объединенная с начальным распределением, интегральная функция распределения усталостных ресурсов станет

Это трехпараметрическое распределение Вейбулла, которое преобразовывается в (4.7.108), если s=0. Характеристическая для ресурса величина tc является вероятностью разрушения 1/e, т.е. это время, при котором экспонента в (4.7.120) равна 1. Эта величина будет

Среднеквадратическое отклонение найденного ресурса относительно этой характеристической величины будет

Следует отметить, что среднеквадратическое отклонение существует, только если g больше, чем указанное выше значение, т.е. если s меньше, чем определенная в (4.7.122) величина. В противном случае, среднеквадратическое отклонение становится бесконечно большим. Однако, в качестве меры погрешности в определении ресурса, можно использовать, например, межквартильный размах.      

Список литературы для части 4.7  

1.       American Society for Metals, "Metals Handbook" Vol. 10: "Failure Analysis and Prevention. Fatigue Failures." Metals Park, Ohio 44073, 8th Edition, 1975.

2.       A.Almar-Naess, editor, "Fatigue Handbook", Tapir, Trondheim, 1985.

3.       Det norske Veritas, "Fatigue Strength Analysis for Mobile Offshore Units", Classification Note No.30.2. August 1984.

4.       British Standards Institution BS5400, "Steel, Concrete and Composite Bridges. Part 10. Code of Practice for Fatigue." 1980.

5.       Department of Energy, "Offshore Installations. Guidance on Design and Construction. New Fatigue Design Guidance for Steel Welded Joints in Offshore Structures." DoE, Issue N. August 1983.

6.       Norges Standardiseringsforbund, "Prosjektering av staalkonstruksjoner. Beregning og dimensjonering." Norsk Standard NS 3472, 1.utg. 1975, 2.utg. 1984.

7.       F.Matanzo, "Fatigue Testing of Wire Rope." MTB-Journal Vol.6 No.6.

8.       S.Gran, Evaluation of High Cycle Fatigue in Welded Steel Connections. Det norske Veritas, Report No.76-339.

9.       S.Gran, "Fatigue in Offshore Cranes". Norwegian Maritime Research, No.4 1983, 2-12.

10.    Y.K.Lin, Probabilistic Theory of Structural Dynamics. Robert E.Krieger Publishing Company. Huntington, New York, 1976 p.99.

11.    H.E.Boyer, editor, "Atlas of Fatigue Curves," American Society for Metals, Metals Park, Ohio 44073, 1986.

Postscript Equations to Article 4.7.

Section 4.7.1 - Fatigue Loading.

Equation (4.7.1):

f sub 1 (S) = g(a, h, X; S) = |h| over { GAMMA (a) X} ( S over X ) sup ah-1 e sup{-(S/X) sup h}

Equation (4.7.2):

a = 1     h = 2     X = 2 sqrt 2 sigma sub s

Equation (4.7.3):

a = 1     h = 1     X = S bar = sigma sub S

Equation (4.7.4):

f sub 2 (X) = g(b, j, B; X) = |j| over { GAMMA (b) B} ( X over B ) sup bj-1 e sup{-(X/B) sup j}

Equation (4.7.5):

f(S) = int f sub 1 (S) f sub 2 (X) dX

Equation (4.7.6):

M sub m = B sup m {GAMMA (a + m over h ) GAMMA (b + m over j )} over{GAMMA (a) GAMMA (b)}

Equation (4.7.7):

f (S) = g(d, k, D; S) = |k| over { GAMMA (d) D} ( S over D ) sup dk-1 e sup{-(S/D) sup k}

Equation (4.7.8):

a = b = d = 1

Section 4.7.2 - Fatigue Data.

Equation (4.7.9):

N sub f = N(S) = ( {S sub 1}over S ) sup m = A over{S sup m} roman   where   A = S sub 1 sup m

Section 4.7.3 - Closed-form Fatigue Life Formulae.

Equation (4.7.10):

eta = sum{n(S)}over{N(S)}

Equation (4.7.11):

eta = n int 1 over{N(S)} f(S) dS

Equation (4.7.12):

eta = n over{S sub 1 sup m} int from 0 to inf S sup m f(S) dS = n over{S sub 1 sup m} M sub m

Equation (4.7.13):

DELTA eta = n ( X over{S sub 1}) sup m {GAMMA (a + m/h)}over{GAMMA (a)}

Equation (4.7.14):

eta = n ( D over{S sub 1}) sup m {GAMMA (d + m/k)}over{GAMMA (d)}

Equation (4.7.15):

eta = n ( B over{S sub 1}) sup m {GAMMA (a + m/h)}over{GAMMA (a)} {GAMMA (b + m/j)}over{GAMMA (b)}

Equation (4.7.16):

GAMMA (1 + x) = x!

Equation (4.7.17):

N sub f =  N(S) =

left {  lpile{( {S sub 1}over S ) sup m       S > S sub 0 above  inf  S < S sub 0}

Equation (4.7.18):

DELTA eta = n ( X over{S sub 1}) sup m {GAMMA (a + m over h ; ({S sub 0}over X ) sup h )} over{GAMMA (a)}

Equation (4.7.19):

eta = n ( D over{S sub 1}) sup m {GAMMA (d + m over k ; ({S sub 0}over D ) sup j )} over{GAMMA (d)}

Equation (4.7.20):

eta = sum n(C) over N(C)

Equation (4.7.21):

N sub f = N(S) = left {  lpile{({S sub 1} over S ) sup m   S > S sub 0 above   ({S' sub 1}over S ) sup m'       S < S sub 0}

Equation (4.7.22):

m' mark = m + 2

Equation (4.7.23):

N(S sub 0 ) lineup = 1 cdot 10 sup 7

Equation (4.7.24):

S sub 0 lineup = 10 sup{- 7 over m} S sub 1 = S' sub 1 10 sup{- 7 over m+2}

Equation (4.7.25):

S' sub 1 lineup = S sub 1 ( {S sub 1}over{S sub 0}) sup{- 2 over m+2}  = S sub 0 ({S sub 1}over{S sub 0} ) sup{m over m+2} = S sub 1 10 sup{- 14 over m(m+2)}

Equation (4.7.26):

eta = n "{" ( D over{S sub 1}) sup m {GAMMA (d + m over k ; ({S sub 0}over D ) sup k )} over{GAMMA (d)} +

( D over{S' sub 1}) sup m+2 {gamma (d + m+2 over k ; ({S sub 0}over D ) sup k )} over{GAMMA (d)} "}"

Equation (4.7.27):

N sub f = N(S) = left { lpile{N sub 0 e sup{- S over B} above   inf } for   lpile{S \(>= S sub 0 above S \(<= S sub 0}

Equation (4.7.28):

eta = n over{N sub 0} int e sup tS f(S) dS = n over{N sub 0} PHI (-t)       roman where       t = -1/B

Equation (4.7.29):

eta = n over{N sub 0} d over{GAMMA (d) D sup dk} int from{S sub 0}to inf S sup dk-1 e sup{-( S over D ) sup k + S over B} dS

Equation (4.7.30):

eta = n over{N sub 0} B over{B - D} 1 over{GAMMA (d)} GAMMA (d; {B - D}over BD S sub 0 )

Equation (4.7.31):

eta = n over{N sub 0} B over{B - D} e sup{-{B - D}over BD S sub 0}

Equation (4.7.32):

eta = n over{N sub 0} 1 over sqrt pi e sup{{D sup 2}over{4B sup 2}} GAMMA \s(12(\s0 1 over 2 ; ( {S sub 0}over D - D over 2B ) sup 2 \s(12)\s0

Equation (4.7.33):

eta = n over{N sub 0} e sup{{D sup 2}over{4B sup 2}} \s(12"{"\s0 e sup{- 1 over 2 ( {sqrt 2 S sub 0}over D - D over{sqrt 2 B}) sup 2} + sqrt pi D over B [ 1 - PHI ({sqrt 2 S sub 0}over D - D over{sqrt 2 B} ) ] \s(12"}"\s0

Equation (4.7.34):

DELTA eta = DELTA eta sub 0 = ( Z over{S sub 1}) sup m

Equation (4.7.35):

DELTA eta mark = 1 over{S sub 1 sup m} "{" psi sup m Z sup m + (1 - psi ) sup m Z sup m (e sup{- alpha T/2} + e sup{- alpha T}) sup m [ 1 + e sup{- alpha Tm} + e sup {-2 alpha T m} + cdot cdot cdot ] "}"  lineup = ( Z over{S sub 1} ) sup m "{" psi sup m + (1 - psi ) sup m {(1 + e sup{- pi lambda}) sup m}over{2 sinh pi lambda m} "}"

Equation (4.7.36):

DELTA eta = ( Z over{S sub 1} ) sup m "{" psi sup 3 + 15 (1 - psi ) sup 3 "}"

Section 4.7.4 - Natural Dispersion.

Equation (4.7.37):

DELTA eta sub 1 , DELTA eta sub 2 , DELTA eta sub 3 , cdot cdot cdot  DELTA eta sub j cdot cdot cdot

Equation (4.7.38):

eta (t) = eta sub n = DELTA eta sub 1 + DELTA eta sub 2 + DELTA eta sub 3 + cdot cdot cdot + DELTA eta sub n

Equation (4.7.39):

xi = 1 over{N(S)} = ( S over{S sub 1}) sup m = r S sup m        roman with      r = S sub 1 sup -m

Equation (4.7.40):

f( xi ) = g(d, k over m , rD sup m ; xi )

Equation (4.7.41):

xi bar = M sub 1 ( xi ) = int from 0 to inf xi f( xi ) d xi = r D sup m {GAMMA (d + m over k )}over{GAMMA (d)} = TU

Equation (4.7.42):

M sub 2 ( xi ) = int from 0 to inf xi sup 2 f( xi ) d xi = (r D sup m ) sup 2 {GAMMA (d + 2m over k}over{GAMMA (d)} = TV

Equation (4.7.43):

M sub 3 ( xi ) = int from 0 to inf xi sup 3 f( xi ) d xi = (r D sup m ) sup 3 {GAMMA (d + 3m over k}over{GAMMA (d)} = TW

Equation (4.7.44):

U = {xi bar}over T = {M sub 1 ( xi )}over T V = {M sub 2 ( xi )}over T W = {M sub 3 ( xi )}over T

Equation (4.7.45):

mu sub 2 ( xi ) = sigma sub xi sup 2 = M sub 2 ( xi ) - M sub 1 sup 2 ( xi ) =  nu sup 2 xi bar sup 2  roman where

nu sup 2 = ( {sigma sub xi}over{xi bar} ) sup 2 = {GAMMA (d + 2m over k ) GAMMA (d) - GAMMA (d + m over k ) sup 2}over {GAMMA (d + m over k ) sup 2}

Equation (4.7.46):

mu sub 3 ( xi ) = M sub 3 ( xi ) - 3M sub 2 ( xi ) M sub 1 ( xi ) + 2M sub 1 ( xi ) sup 3 = lambda sigma sub xi sup 3 = lambda nu sup 3 xi bar sup 3 roman where     lambda = {GAMMA (d + 3m over k ) GAMMA (d) sup 2 -

3 GAMMA (d + 2m over k ) GAMMA (d) GAMMA (d + m over k ) + 2 GAMMA (d + m over k ) sup 3}over

{[ GAMMA (d + 2m over k ) GAMMA (d) - GAMMA (d + m over k ) sup 2 ] sup 3/2}

Equation (4.7.47):

phi (s) = int from 0 to inf e sup{s xi} f( xi ) d xi  Re "{" s "}" < 0

Equation (4.7.48):

phi (s) = int from 0 to inf [ 1 + s xi + 1 over 2 s sup 2 xi sup 2 + 1 over 6 s sup 3 xi sup 3 + cdot cdot ] f( xi ) d xi

Equation (4.7.49):

phi (s) = 1 + M sub 1 ( xi ) s mark + 1 over 2 M sub 2 ( xi ) s sup 2 + 1 over 6 M sub 3 ( xi ) s sup 3 + cdot cdot

lineup = 1 + T U s + 1 over 2 T V s sup 2 + 1 over 6 T W s sup 3 + cdot cdot

Equation (4.7.50):

PHI (s, t) = int from 0 to inf e sup{s eta } rho ( eta , t) d eta  Re "{" s "}" < 0

Equation (4.7.51):

eta (t + T) = eta sub n+1 = eta sub n + xi

Equation (4.7.52):

PHI (s, t+ T ) = PHI (s, t) phi (s)

Equation (4.7.53):

{partial PHI (s, t)}over{partial t} = 1 over T [ PHI (s, t + T ) - PHI (s, t) ]

Equation (4.7.54):

int from 0 to inf e sup{s eta} {partial rho ( eta , t)}over{partial t} d eta mark = 1 over T PHI (s, t) [ phi (s) - 1 ]

lineup = U s PHI (s, t) + 1 over 2 V s sup 2 PHI (s, t) + 1 over 6 W s sup 3 PHI (s, t)

Equation (4.7.55):

int from 0 to inf e sup{s eta} [ {partial rho}over{partial t} + U{partial rho}over{partial eta} - 1 over 2 V{partial sup 2 rho}over{partial eta sup 2} + 1 over 6 W{partial sup 3 rho}over{partial eta sup 3} ] d eta  - [ e sup{s eta} "{"

rho (U + 1 over 2 sV + 1 over 6 s sup 2 W) - {partial rho}over{partial eta}( 1 over 2 V + 1 over 6 sW) + {partial sup 2 rho}over{partial eta sup 2}1 over 6 W "}" ] from{eta = 0} to {eta = inf} = 0

Equation (4.7.56):

{partial rho}over{partial t} + U{partial rho}over{partial eta} - 1 over 2 V {partial sup 2 rho}over{partial eta sup 2} + 1 over 6 W {partial sup 3 rho}over{partial eta sup 3} = 0

Equation (4.7.57):

{eta sub n}bar mark = sum{DELTA eta}bar = n xi bar

Equation (4.7.58):

mu sub 2 ( eta sub n ) lineup = sum mu sub 2 ( DELTA eta sub i ) = n cdot sigma sub xi sup 2 = n nu sup 2 xi bar sup 2

Equation (4.7.59):

mu sub 3 ( eta sub n ) lineup = sum mu sub 3 ( DELTA eta sub i ) = n lambda sub 3 sigma sub xi sup 3 = n lambda nu sup 3 xi bar sup 3

Equation (4.7.60):

{sigma sub {eta sub n}}over{{eta sub n}bar} = {sqrt{mu sub 2 ( eta sub n )}}over{{eta sub n}bar} = nu over sqrt n

Equation (4.7.61):

lambda sub 3 = {mu sub 3 ( eta )}over{mu sub 2 ( eta ) sup 3/2} = lambda over sqrt n

Equation (4.7.62):

rho ( eta , t) = |h| over{GAMMA (a)} e sup{ah( eta - u )} e sup{-e sup{h( eta - u )}}

Equation (4.7.63):

psi '' (a) over{psi ' (a) sup 3/2} = {lambda sub 3}over sqrt n

Equation (4.7.64):

h = \(+- {sqrt{psi '(a)}}over {sqrt n sigma sub xi} + for lambda sub 3 < 0   and   - for lambda sub 3 > 0

Equation (4.7.65):

u = n{DELTA eta}bar - 1 over h psi (a) = n xi bar + sqrt n sigma sub xi {psi (a)}over{sqrt{psi ' (a)}}

Equation (4.7.66):

a mark approx n over{lambda sup 2}

Equation (4.7.67):

h lineup approx - n lambda over{sigma sub xi}

Equation (4.7.68):

u lineup approx n "{" xi bar - {sigma sub xi}over lambda ln [ n over{lambda sup 2a} ] "}"

Equation (4.7.69): (xxx)

rho ( eta , t) = 1 over sqrt{2 pi n} 1 over{sigma sub xi} e sup{- {( eta - n xi bar ) sup 2}over{2 n sigma sub xi sup 2}}        t = n T

Equation (4.7.70):

j = eta over L     roman or     eta = j L

Equation (4.7.71):

Pr ( eta = j L ) = Pr (j; n) = ( cpile{n above j} ) p sup j (1 - p) sup n-j        n \(>= j

Equation (4.7.72):

p = (1 - p) = 1 over 2

Equation (4.7.73):

Pr(j; n) = ( cpile{n above j} ) 1 over{2 sup n}

Equation (4.7.74):

{eta sub n}bar = L n p  and  sigma sub eta sup 2 = L sup 2 n p (1 - p)

Equation (4.7.75):

{sigma sub eta}over{eta bar} = 1 over sqrt n sqrt{{1 - p}over p}

Equation (4.7.76):

L = xi bar (1 + nu sup 2 )     and     p = 1 over{1 + nu sup 2}

Equation (4.7.77):

L = {M sub 2 ( xi )}over{M sub 1 ( xi )}     and p = {M sub 1 ( xi ) sup 2}over{M sub 2 ( xi )}

Section 4.7.5 - Fracture Mechanics Approach.

Equation (4.7.78):

sigma sub ij = R(r) THETA sub ij ( theta )

Equation (4.7.79):

R(r) = r sup {n over 2 - 1}

Equation (4.7.80):

sigma sub ij = K over sqrt{2 pi r} THETA sub ij ( theta )

Equation (4.7.81):

sigma sub ij = sqrt{x over 2r} sigma sub inf  THETA sub ij ( theta )  roman {so that} K = sqrt{pi x} sigma sub inf

Equation (4.7.82):

DELTA K = K sub max - K sub min

Equation (4.7.83):

DELTA x = left { lpile{ C( DELTA K ) sup m above     above 0}  for lpile{ DELTA K > DELTA K sub 0 above     above  DELTA K < DELTA K sub 0}

Equation (4.7.84):

DELTA K = sqrt{pi x} g'(x) S = g(x) S            g(x) = g'(x) sqrt{pi x}

Equation (4.7.85):

DELTA x = left { lpile{ C g(x) sup m S sup m above   above 0}  for lpile{ S > S sub 0 (x) = {DELTA K sub  }over{g(x)} above    above S < S sub 0 (x)}

Equation (4.7.86):

DELTA x sub 1 , DELTA x sub 2 , DELTA x sub 3 , cdot cdot cdot DELTA x sub j cdot cdot cdot

Equation (4.7.87):

eta = {x - x sub 0}over{x sub f - x sub 0}     and DELTA eta = {DELTA x}over{x sub f - x sub 0}

Equation (4.7.88):

{DELTA x}bar = C g(x) sup m int from{S sub 0} to inf S sup m f(S) dS = C g(x) sup m D sup m { GAMMA (d + m over k ; ({DELTA K sub 0}over{g(x) D}) sup k )} over{GAMMA (d)}

Equation (4.7.89): (xxx)

U = dx over dt = 1 over T dx over dN = {{DELTA x}bar}over T = 1 over T C D sup m {GAMMA (d + m over k  }over{GAMMA (d)} g(x)

Equation (4.7.90):

Pr( roman{crack depth} \(<= x roman{at time} t) = F(x, t)

Equation (4.7.91):

Q(x, t) = 1 - F(x, t)

Equation (4.7.92):

rho (x, t sub 1 ) = {partial F(x, t sub 1 )}over{partial x} = - {partial Q(x, t sub 1 )}over{partial x}

Equation (4.7.93):

{partial Q}over{partial t} dt = -{partial Q}over{partial x} dx = -{partial Q}over{partial x} U(x) dt

Equation (4.7.94):

{D F(x, t)}over{D t} \(== ({partial F}over{partial t} + U {partial F}over{partial x}) = - ({partial Q}over{partial t} + U {partial Q}over{partial x}) = 0

Equation (4.7.95):

{partial rho}over{partial t} + {partial rho U}over{partial x} = {partial rho}over{partial t} + U {partial rho}over{partial x} + rho {partial U}over{partial x} = 0

Equation (4.7.96):

int from 0 to inf rho (x, t) dx = 1

Equation (4.7.97):

chi (x, t) = {partial Q(x, t)}over{partial t} = - {partial F(x, t)}over{partial t}

Equation (4.7.98):

chi (x, t) = U rho (x, t)

Equation (4.7.99):

{partial chi}over{partial t} + U {partial chi}over{partial x} = 0

Equation (4.7.100):

P sub f (t) = Q(x sub f , t) = 1 - F(x sub f , t)

Section 4.7.6 - Life-time Probability.

Equation (4.7.101):

Q(x, 0) = e sup{- ( x over{x sub 0} ) sup gamma}       t = 0

Equation (4.7.102):

E[x] = x sub 0 GAMMA (1 + 1 over gamma )        t = 0

Equation (4.7.103):

sigma sub x = x sub 0 [ GAMMA (1 + 2 over gamma ) - GAMMA (1 + 1 over gamma ) sup 2 ] sup{1 over 2}       t = 0

Equation (4.7.104):

Q(x, t) = Q( xi )        xi = xi (x, t)       xi (x, 0) = x

Equation (4.7.105):

{partial Q}over{partial t} + U{partial Q}over{partial x} = ( {partial xi}over{partial t} + U{partial xi}over{partial x} ) {partial Q}over{partial xi} = 0

Equation (4.7.106):

U = 1 over T da over dN = dx over dt = - {partial xi / partial t} over{partial xi / partial x}

Equation (4.7.107):

xi = x - Ut     U = 1 over T da over dN = roman constant

Equation (4.7.108):

P sub f (t) = Q (x sub f , t) = e sup{-({x sub f - Ut}over{x sub 0}) sup gamma} = e sup{-({x sub f /U - t}over{x sub 0 /U}) sup gamma} t < {x sub f}over U

Equation (4.7.109):

E[t] = 1 over U [ x sub f - x sub 0 GAMMA (1 + 1 over gamma ) ]

Equation (4.7.110):

sigma sub t = {sigma sub x}over U = {x sub 0}over U  [ GAMMA (1 + 2 over gamma ) - GAMMA (1 + 1 over gamma ) sup 2 ] sup{1 over 2}

Equation (4.7.111):

da over dN = C x       roman and      U(x) = C over T x = cx

Equation (4.7.112):

xi = x e sup -ct

Equation (4.7.113):

P sub f (t) = Q(x sub f , t) = e sup{-({x sub f}over{x sub 0 e sup ct}) sup gamma} = e sup{-e sup{- gamma c ( t - 1 over c ln {x sub f}over{x sub 0})}}

Equation (4.7.114):

t sub c = 1 over c ln {x sub f}over{x sub 0}

Equation (4.7.115):

E[t] = 1 over c [ ln {x sub f}over{x sub 0} + 0.5772 over gamma ]

Equation (4.7.116):

sigma sub t = pi over sqrt 6 1 over{gamma c}

Equation (4.7.117):

{sigma sub t}over{t sub c} = pi over{sqrt 6 gamma ln {x sub f}over{x sub 0}}

Equation (4.7.118):(xxx)

da over dN = C x sup s      roman or U = C over T x sup s = cx sup s      s \(!= 1

Equation (4.7.119):

xi = [ x sup 1-s - (1 - s) ct ] sup{1 over 1-s}

Equation (4.7.120):

P sub f (t) = Q (x sub f , t) = e sup{-({x sub f sup 1-s - (1-s)ct}over {x sub 0 sup 1-s} ) sup{gamma over{(1-s)}}}

Equation (4.7.121):

t sub c = {x sub f sup 1-s - x sub 0 sup 1-s}over{(1 - s) c}

Equation (4.7.122):

{sigma sub t}over{t sub c} = { [ GAMMA (1 + 2(1-s) over gamma ) - GAMMA (1 + 1-s over gamma ) sup 2 ] sup 1/2} over{( x sub f / x sub 0 ) sup 1-s - 1}       gamma > 2(s - 1)

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