Q(x,t)
x .
,
. , Q(x,t),
t x,
,
, x.
((x,t), ,
Q(x,t) x
, x
[t,t+dt] ((x,t)dt. (4.7.94) ,
((x,t)
((x,t)
(4.7.95) ((x,t)
, U=U(x) .
, ,
xf, .
, ,
Pf(t), ,
xf. (4.7.91)
.
4.7.6 .
(4.7.94).
t=0 g(x)
U(x) (4.7.89). ,
.
.
(4.7.100),
-
. , ,
E[x]
, (=1. ,
, ,
.
, ,
, .. ( . ,
x0=0,1
.
, x t
xi=xi(x,t) , xi t=0
. , Q(x,t)
xi
Pf(t) xi
x (4.7.101). xi,
U , (4.7.89).
, x,
g(x).
. ,
.
x
x .
(4.7.101),
.
-
(, .. ,
. , ,
- .
. ,
, ..
, g((x)
(4.7.84) m da/dN 2.
, xi
(4.7.106).
(4.7.101),
(4.2.6) ,
.
, .. , tc
(4.2.16),
(4.2.17),
,
,
xf.
xf/x0,
, 100.
, .. (=1,
, ..
, 28%.
xs.
, ,
(4.7.81). m=3,
s=1,5. ,
(4.7.106).
,
,
(4.7.108), s=0.
tc 1/e, .. ,
(4.7.120) 1.
, ,
( , , .. s ,
(4.7.122) . ,
. ,
, ,
, .
4.7
American Society for Metals, "Metals Handbook" Vol. 10: "Failure
Analysis and Prevention. Fatigue Failures." Metals Park, Ohio 44073, 8th
Edition, 1975.
Det norske Veritas, "Fatigue Strength Analysis for Mobile Offshore
Units", Classification Note No.30.2. August 1984.
British Standards Institution BS5400, "Steel, Concrete and Composite
Bridges. Part 10. Code of Practice for Fatigue." 1980.
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.
Norges Standardiseringsforbund, "Prosjektering av staalkonstruksjoner.
Beregning og dimensjonering." Norsk Standard NS 3472, 1.utg. 1975,
2.utg. 1984.
F.Matanzo, "Fatigue Testing of Wire Rope." MTB-Journal Vol.6 No.6.
S.Gran, Evaluation of High Cycle Fatigue in Welded Steel Connections.
Det norske Veritas, Report No.76-339.
Y.K.Lin, Probabilistic Theory of Structural Dynamics. Robert E.Krieger
Publishing Company. Huntington, New York, 1976 p.99.
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)
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)
