Damage Computation for Concrete Towers

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Nov 29, 2013 (3 years and 8 months ago)

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Damage Computation for Concrete Towers
Under Multi
-
Stage and Multiaxial Loading


Prof. Dr.
-
Ing. Jürgen Grünberg

Dipl.
-
Ing. Joachim Göhlmann


Institute of Concrete Construction

University of Hannover, Germany

www.ifma.uni
-
hannover.de



Table of Contents

1.
Introduction

2.
Fatigue Verification

3.
Energetical Damage Model for Multi
-

Stage
Fatigue Loading

4.
Multiaxial Fatigue

5.
Summary and Further Work


Hybrid Tower Bremerhaven

Nearshore Foundation Emden

1. Introduction


Fatigue Design for …

Reinforcement

Tendons

Junctions

Concrete


0

0,2

0,4

0,6

0,8

1

0

7

14

21

28

l
og

N

S
cd,max

S
cd,m
in

= 0,8

S
cd,m
in

= 0

2. Fatigue Verification by DIBt
-

Richtlinie

N
i

= Number of load cycles for current load spectrum

N
Fi

= Corresponding total number of cycles to failure

Linear Accumulation Law

by Palmgren and Miner:


j
i
lim
i 1
Fi
N
D D 1
N

  

Design Stresses for
compression loading:

S
cd,min

=

sd

σ
c,min



c

/ f
cd,fat

S
cd,max

=

sd

σ
c,max



c

/ f
cd,fat

log N

S


N curves by Model Code 90


Strain evolution under constant fatigue loading

-0,004
-0,003
-0,002
-0,001
0
0,00
0,20
0,40
0,60
0,80
1,00
N / N
F
Fatigue Strain Evolution, ε
fat
micro-cracking
stable crack propagation
unstable crack
propagation


σ
u
=
0.05 ∙ f
c
σ
o
=
0.675 ∙ f
c

3. Energetical Fatigue Damage Model for Constant


Amplitude Loading by [Pfanner 2002]

Assumption:




The mechanical work, which have to be applied to obtain a certain
damage state during the fatigue process, is equal to the mechanical
work under monotonic loading to obtain the same damage state.

!

W
da
(D) = W
fat

(D,

fat
, N)

Monotonic Loading

Fatigue Process



   
da fat
c c c
fat
E E 1 D E

c

= (1
-

D
fat
)
∙ E
c

∙ (

c

-


c
pl
)

Elastic
-
Plastic Material Model for Monotonic Loading:


c


晡f


Damage evolution under constant fatigue loading

0
0,2
0,4
0,6
0,8
1
0
0,2
0,4
0,6
0,8
1
N
/ N
F
Fatigue Damage D
fat
Failure at "Wöhler" Test

σ
u
=
0.05 ∙ f
c
σ
o
=
0.2 ∙ f
c
σ
o
= 0.9 ∙ f
c

Extended Approach for Multi
-
Stage Fatigue Loading




S
cd,max,1

S
cd,max,2

S
cd,max,3

S
cd,min,i





Number of load cycles until failure:

N
F

=



N
i

+ N
r

Life Cycle:

L
fat

=
D
fat

(
σ
i
fat
, N
i
) /
D
fat

(
σ
F
fat
, N
F
)

≤ 1





Three
-
Stage Fatigue Process in Ascending Order

0
0,1
0,2
0,3
0,4
0,5
0,6
0
4.000
8.000
12.000
16.000
20.000
24.000
N
i
Ermüdungsschädigung D
fat
S
max,3
= 0,7
S
max,2
= 0,65
S
max,1
= 0,6
D(N
equ,2
) = D(N
1
)
D(N
equ,3
)

D(N
2
)
D
fat
= 0,58

0,6
0,65
0,7
S
max
N
1
N
2
N
3
S
min
= 0,05
N
i

Fatigue Damage D
fat


Numerical Fatigue Damage Simulation



Computed Stress and Damage Distribution



(N = 1)



(N = 10
9
)

D
fat
(N = 10
9
)

D
fat

= 0,221

D
fat

= 0,12

D
fat

= 0,08

1
st

Principal Stress

Fatigue Damage


4. Multiaxial Fatigue Loading

Junction of

Hybrid Tower

Floatable Gravity Foundation



Joint of

Concrete Offshore Framework



-2,00
-1,50
-1,00
-0,50
0,00
0,50
1,00
1,50
-1,5
-1
-0,5
0
0,5
Fracture Envelope for Monotonic Loading

Main Meridian Section

f
c

f
cc

f
t

f
tt

Compression
Meridian

Tension

Meridian

f
c

= unaxial compression strength

f
cc

= biaxial compression strength

f
t

= uniaxial tension strength

f
tt

= biaxial tension strenth



/ f
c



/ f
c

f
cc

f
c


Fatigue Damage Parameters for Main Meridians

0
0,2
0,4
0,6
0,8
1
0
5
10
15
20
25
30
35
log N
F
S
cd,max
S
cd,min
= 0
S
cd,min
= 0,6
S
cd,min
= 0,4
S
cd,min
= 0,2
S
cd,min
= 0,8
tension loading
compression loading

c
fat
;


t
fat


Main Meridians under Multiaxial Fatigue Loading

-2
-1,5
-1
-0,5
0
0,5
1
1,5
-1,5
-1
-0,5
0
0,5

/ f
c

/ f
c
log N = 0
log N = 3
log N = 6
log N = 7
f
c

f
cc

f
t

f
tt

Tension

Meridian

Compression
Meridian


-1,4
-1,2
-1
-0,8
-0,6
-0,4
-0,2
0
0,2
-1,4
-1,2
-1
-0,8
-0,6
-0,4
-0,2
0
0,2

σ
11
=
a
· σ
22
σ
22
Failure Curves for Biaxial Fatigue Loading


11,max

/ f
c


㈲Ɑ,x

/ f
c

N = 1

Log N = 3

Log N = 7

Log N = 6


min

= 0

a

= 1,0

a


-

〬ㄵ

f
c

f
cc


Modification of Uniaxial Fatigue Strength

0
0,5
1
1,5
2
2,5
-0,2
0
0,2
0,4
0,6
0,8
1
a
=
σ
11
/
σ
22
c
cc
0 < Scd,min < 0,8
log N

6
-0,15


a


1
S
cd,min

=

sd


cc


σ
c,min



c

/ f
cd,fat

S
cd,max

=

sd


cc


σ
c,max



c

/ f
cd,fat


0

0,2

0,4

0,6

0,8

1

0

7

14

21

28

l
og

N

S
cd,max

S
cd,m
in

= 0,8

S
cd,m
in

= 0

Modified Fatigue Verification

Design Stresses:

S
cd,min

=

sd


cc


σ
c,min



c

/ f
cd,fat

S
cd,max

=

sd


cc



σ
c,max



c

/ f
cd,fat

S


N curves by Model Code 90

a

cc
1,00
0,86
0,96
0,85
0,92
0,83
0,89
0,82
0,85
0,80
0,82
0,79
0,78
0,78
0,75
0,77
0,72
0,76
0,68
0,76
0,65
0,75
0,62
0,74
0,53
0,73
0,35
0,74
0,28
0,75
0,25
0,76
0,22
0,77
0,18
0,78
0,15
0,80
0,11
0,82
0,08
0,86
0,04
0,91
0,00
1,00
-0,04
1,22
-0,08
1,52
-0,13
1,89
Life Cycle:

L
fat

=
D
fat

(
σ
i
fat
, N
i
) /
D
fat

(
σ
F
fat
, N
F
)

≤ 1


5. Summary and Further Work


The linear cumulative damage law by Palmgren und
Miner could lead to unsafe or uneconomical concrete
constructions for Wind Turbines.


A new fatigue damage approach, based on a fracture
energy regard, calculates realistically damage evolution
in concrete subjected to multi
-
stage fatigue loading.


The influences of multiaxial loading to the fatigue
verification could be considered by modificated

uniaxial Wöhler
-
Curves.


Further Work:


Experimental testings are necessary for validating the
multiaxial fatigue approach.


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