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.

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