1. 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

2. 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

3. Hybrid Tower Bremerhaven Nearshore Foundation Emden 1. Introduction

4. Fatigue Design for … Reinforcement Tendons Junctions Concrete

5. 0 0,2 0,4 0,6 0,8 1 07142128 log N S cd,max S cd,min = 0,8 S cd,min = 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: 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

6. Strain evolution under constant fatigue loading

7. 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  c = (1 - D fat ) ∙ E c ∙ (  c -  c pl ) Elastic-Plastic Material Model for Monotonic Loading: cc  fat

8. Damage evolution under constant fatigue loading

9. 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

10. Three-Stage Fatigue Process in Ascending Order NiNi Fatigue Damage D fat

11. Numerical Fatigue Damage Simulation

12. 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

13. 4. Multiaxial Fatigue Loading Junction of Hybrid Tower Floatable Gravity Foundation Joint of Concrete Offshore Framework

14. Fracture Envelope for Monotonic Loading Main Meridian Section fcfc f cc ftft 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 fcfc

15. Fatigue Damage Parameters for Main Meridians  c fat ;  t fat

16. Main Meridians under Multiaxial Fatigue Loading fcfc f cc ftft f tt Tension Meridian Compression Meridian

17. Failure Curves for Biaxial Fatigue Loading  11,max / f c  22,max / f c N = 1 Log N = 3 Log N = 7 Log N = 6  min = 0  = 1,0  = - 0,15 fcfc f cc

18. Modification of Uniaxial Fatigue Strength S cd,min =  sd ∙ cc ∙ σ c,min ∙  c / f cd,fat S cd,max =  sd ∙ cc ∙ σ c,max ∙  c / f cd,fat

19. 0 0,2 0,4 0,6 0,8 1 07142128 log N S cd,max S cd,min = 0,8 S cd,min = 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 Life Cycle: L fat = D fat ( σ i fat, N i ) / D fat ( σ F fat, N F ) ≤ 1

20. 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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