-
,H
x
y H xsystem
input output
x y
Response of a linear system
sinx x t
siny H x t
-
sinx x t
siny H x t
sinx x t
siny H x t
( ) sin ( )x x t ( ) ( ) sin ( ) ( )y x H t
2
0
1 ( )( ) lim
2xx
xS
22
0
( ) ( )1( ) lim
2yy
x HS
2
( ) ( ) ( )yy xxS H S
Response spectrum of a linear system
-
( )F t
( )u t
k c
( )F t
( )u tm t
t
;2
e
k c
m k m
A single-mass-spring system under random loading
-
1 2
2 22
1( )
1 2e e
H H
k
2
2arctan with 0
1
e
e
(0.1)
frequency
; natural frequency
damping factor ; 2
e e k m
c k m
Dynamic single degree of freedom system
k c
( )F t
( )u tm
-
FFS
( )H
uuS
1
2k
0
0
2
u
Analysis of the single-mass-spring system
-
FFS
2H
uuS
2
1
2k
e
0
2area
4
e S
k
0S
21 k
2
0S k
Response spectrum for a single-mass-spring system under
stationary load with white-noise spectrum
-
wind
10 s
-
respons
10 s Te = 0.5 s
-
FFS
0S ( )FF eS
white-noise approximation
real spectrum
e
Approximation of the load spectrum by a white-noise
spectrum with intensity
0 ( )FF eS S
-
S Du u u Fk
2 2 22 2 2( )( )
( )21
FF eFFuu
ee
SSS
k k
Response to arbitrary load
Split the response into
a quasi static part and a white noise part:
22
2 2
( )1
4
F e FF eu
F
S
k
-
3 6 2
3 2
0
9 5
70 10 N ; 8 10 Nm ; 20,000 kg
50 10 N s ; 3 m ; 0.01
50 years 1.5 10 s ; 10 Nm
F
p
EI m
S l
T M
( )F t
T t
F
e
( )FFS
0S
( ) ( )F t m u t
l HE 200 A
Portal frame with data
-
Spring stiffness k :
6
3
247.1 10 N/m
EIk
l
Natural frequency:
3
2419 rad/se
k EI
m m l
Mean response
10.0098 m 9.8 mmu F
k
Variance of the response:
2 6 20
21.47 10 m
4
eu
S
k
Standard deviation:
0.0012 m 1.2 mm u
( ) ( )F t m u t
l HE 200 A
-
( )F t( )u t
= full-plastic hinge A
Considered limit state of the frame
-
1(0)
( ) at
u
u t t t
( )u t
0 T t
( )u t
10 t T t
The event failure in [0,T]
-
( )u t
t
1a
2a 3a
ia
na
T
( )aF
Peak values of the narrow-band process
-
Yield occurs if: 1
4A p pM F l M
Corresponding value of u: 4
0.0188 m 18.8 mmp p
p
F Mu
k kl
Reliability index: 18.8 9.8
7.51.2
p u
u
u
Exceedance probabtilities:
APT:
single peak:
50 year:
( )u t
t
1a
2a 3a
ia
na
T
example portal frame
14{ ( ) } 1 ( ) ( ) 3 10p N NP u t u
213{ } exp 6.1 10
2i pP a u
3{ ( ) in [0, ]} { } 2.7 10p i pP u t u T n P a u
-
Fatigue strength of steel
2 double amplitude [N/mm ]s
400
100
200
50
20 5 6 710 10 10 N
reinforced beam
welded beam
s
st
average
exceedance probability
-
Fatigue
105 106 Nf
S
bfS
CN
S = stress range
Nf = number of cucles till fracture
C, b = material constants
2
-
Fatigue Miner Rule
)2
b1(}22{
C
Tf)]T(D[E
ds2
sexp
s
C
)s2(Tf)]T(D[E
)s(N
ds)s(fn
)s(N
)s(n)]T(D[E
)1Diffailure(N
n...
N
n
N
nD
bs
e
2
2
2s
b
e
i
i
2
2
1
1
-
54
3
2
1
00 1 2 3 4 x
( )x
0.50 1.772
0.75 1.225
1.00 1.000
1.25 0.906
1.50 0.886
1.75 0.919
2.00 1.000
x x
The Gamma function
for integers: (x) = (x-1)!
-
/ 2 / 2k c c k
v1
2
105 tons
300 tons
m
m
Schematisation of the coal-dust mill
