damage computation for concrete towers under multi-stage and multiaxial loading prof. dr.-ing....
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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 ConstructionUniversity 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
log N
Scd
,max
Scd,min = 0,8
Scd,min = 0
2. Fatigue Verification by DIBt - Richtlinie
Ni = Number of load cycles for current load spectrum
NFi = Corresponding total number of cycles to failure
Linear Accumulation Lawby Palmgren and Miner:
ji
limi 1 Fi
ND D 1
N
Design Stresses for compression loading:
Scd,min = sd ∙ σc,min ∙ c / fcd,fat
Scd,max = sd ∙ σc,max ∙ c / fcd,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 / NF
Fat
igu
e S
trai
n E
volu
tio
n, ε
fat
micro-cracking
stable crack propagation unstable crack propagation
σu = 0.05 · fc
σo = 0.675 · fc
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.
!Wda(D) = Wfat (D, fat, N)
Monotonic Loading Fatigue Process
da fat
c c cfatE E 1 D E c = (1 - Dfat) ∙ Ec ∙ (c - c
pl)
Elastic-Plastic Material Model for Monotonic Loading:
c fat
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 / NF
Fat
igu
e D
am
ag
e
Dfa
t
Failure at "Wöhler" Test
σu = 0.05 · fc
σo = 0.2 · fc
σo = 0.9 · fc
Extended Approach for Multi-Stage Fatigue Loading
Scd,max,1
Scd,max,2
Scd,max,3
Scd,min,i
Number of load cycles until failure:
NF = Ni + Nr
Life Cycle:
Lfat = Dfat ( σifat, Ni) / Dfat ( σF
fat, NF) ≤ 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 24.000Ni
Erm
üd
un
gss
chäd
igu
ng
Dfa
t
Smax,3 = 0,7
Smax,2 = 0,65
Smax,1 = 0,6
D(Nequ,2) = D(N1)
D(Nequ,3) D(N2)
Dfat = 0,58
0,6
0,65
0,7Smax
N1 N2 N3
Smin = 0,05
Ni
Fa
tig
ue
Dam
ag
e D
fat
Numerical Fatigue Damage Simulation
Computed Stress and Damage Distribution
(N = 1) (N = 109) Dfat (N = 109)
Dfat = 0,221
Dfat = 0,12
Dfat = 0,08
1st 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
fc
fcc
ft
ftt
Compression Meridian
TensionMeridian
fc = unaxial compression strengthfcc = biaxial compression strengthft = uniaxial tension strengthftt = biaxial tension strenth
/ fc
/ fc
fcc
fc
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 NF
Scd
,max
Scd,min = 0
Scd,min = 0,6
Scd,min = 0,4
Scd,min = 0,2
Scd,min = 0,8
tension loading
compression loading
cfa
t ; 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
/ fc
/ fc
log N = 0log N = 3
log N = 6log N = 7 fc
fcc
ft
ftt
TensionMeridian
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
σ11 = a · σ22
σ22
Failure Curves for Biaxial Fatigue Loading
11,max / fc
22,
max
/ f c
N = 1
Log N = 3
Log N = 7
Log N = 6
min = 0
a = 1,0 a = - 0,15
fc
fcc
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 = σ11 / σ22
ccc
0 < Scd,min < 0,8 log N ≤ 6-0,15 ≤ a ≤ 1
Scd,min = sd ∙ cc ∙ σc,min ∙ c / fcd,fat
Scd,max = sd ∙ cc ∙ σc,max ∙ c / fcd,fat
0
0,2
0,4
0,6
0,8
1
0 7 14 21 28
log N
Scd
,max
Scd,min = 0,8
Scd,min = 0
Modified Fatigue Verification
Design Stresses:
Scd,min = sd ∙ cc ∙ σc,min ∙ c / fcd,fat
Scd,max = sd ∙ cc ∙ σc,max ∙ c / fcd,fat
S – N curves by Model Code 90
a cc
1,00 0,860,96 0,850,92 0,830,89 0,820,85 0,800,82 0,790,78 0,780,75 0,770,72 0,760,68 0,760,65 0,750,62 0,740,53 0,730,35 0,740,28 0,750,25 0,760,22 0,770,18 0,780,15 0,800,11 0,820,08 0,860,04 0,910,00 1,00-0,04 1,22-0,08 1,52-0,13 1,89
Life Cycle:
Lfat = Dfat ( σifat, Ni) / Dfat ( σF
fat, NF) ≤ 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.