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

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, 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

Strain evolution under constant fatigue loading

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

Damage evolution under constant fatigue loading

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 NiNi 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

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

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

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

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

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

0 0,2 0,4 0,6 0, 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

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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