1. MODELLING THE PULLOUT OF HOOKED STEEL FIBERS FROM CEMENTITIOUS MATRIX Edmunds Zīle, Olga Zīle Institute of Polymer Mechanics Riga, Latvia

2. Introduction Concrete is brittle material with low fracture toughness in tension Addition of short, randomly distributed fibers greatly improve the fracture toughness. The fibers bridge discrete cracks and thereby provide increased control of the fracture process 2

3. What affects performance of FRC?  Fiber material 3  Fiber volume fraction  Fiber shape  Fiber aspect ratio  Fiber strength

4. Objectives 1.Testing of some commercially available hooked-end steel fibers. 2.Development of simple analytical model for the effect of fiber geometry on the pullout behavior suitable for practical use. 4

5. Single fiber pullout specimens HE+ 1/60 and HE 75/50 hooked steel fibers produced by ArcelorMittal 5

6. Properties of the fibers Fiber typeσ Y (MPa)r (mm)l e (mm)l (mm)ρ (mm)θ (rad) HE 75/5011000.352.02.11.70.62 HE+ 1/6014500.451.91.42.20.66 6

7. Experiment: pullout of straight fibers 7

8. Experiment: pullout of hooked fibers 8

9. Modeling of the fiber pullout process Pullout load P of mechanically deformed fiber can be split into two components: Component due to the plastic bending of the fiber in the curved matrix ducts Component due to the frictional sliding of fiber through straight matrix ducts where L s is total lenght of straight matrix ducts and τ is frictional shear stress Frictional shear stress can be obtained from straight fiber pullout tests 9

10. Bending of fiber under tension The following assumptions are made: 1.The material is isotropic and strain-rate independent. 2.The elastic strains are small in comparison with the plastic strains and can be neglected. Hence, the material is assumed to be rigid, perfectly plastic. 3.The damage of cementitious matrix around the mechanically deformed fiber during the pullout is neglected. If fiber is subjected to a tension force less than the yield tension 10

11. Increase of the tension 1.As the fiber bends at A and unbends at B there will be an increase in tension. 2.The tension will increase as the fiber slides against friction between A and B. 11

12. Increase of the tension The plastic work done on the fiber element by deforming it at A: The external work: Increase of the tension at A:The tension in the fiber after bending at A: 12

13. Friction in the curved matrix duct 13 Due to friction the tension in the fiber before unbending at B: where μ is coefficient of friction

14. Tension in the fiber after curved matrix duct Tension in the fiber after unbending at B: orTension in the fiber after ith curved duct: 14

15. Pullout of hooked-end fibers: stage 1 Length of embedded part of the fiber without hook before pullout process 15 Fiber segments in curved ducts C 1 and C 2 subjected to plastic bending. Fiber segments in straight ducts S 1, S 2 and S 3 subjected to frictional sliding.

16. Pullout of hooked-end fibers: stage 2 16 Length of the fiber segment in the curved duct C 1 decreases, which causes gradual reduction of pullout force component due to plastic bending. Fiber segment in curved duct C 2 subjected to plastic bending. Fiber segments in straight ducts S 2 and S 3 subjected to frictional sliding.

17. Pullout of hooked-end fibers: stage 3 17 Fiber segment in curved duct C 2 subjected to plastic bending. Fiber segments in straight ducts S 2 and S 3 subjected to frictional sliding.

18. Pullout of hooked-end fibers: stage 4 18 Length of the fiber segment in the curved duct C 2 decreases, which causes gradual reduction to zero of pullout force component due to plastic bending. Fiber segment in straight duct S 3 subjected to frictional sliding.

19. Pullout of hooked-end fibers: stage 5 19 Pullout force is only due to frictional sliding of fiber segment in straight duct S 3.

20. Comparison with experiment Model proposed by Alwan et al.* 20 Proposed model * J.M. Alwan, A.E. Naaman, P. Guerrero, Concrete Science and Engineering, 1 (1999) 15-25.

21. Conclusions 1.A simple model is developed to simulate the mechanical contribution of fiber geometry to the pullout response. It is assumed that fiber geometry is composed of straight and curved segments. The mechanical contribution depends on the amount of plastic work required to straighten the fiber during pullout and friction in the curved ducts. The plastic work is a function of geometrical parameters and yield stress of the fiber. The damage of cementitious matrix during pullout is neglected. 2.The model provides a reasonably good description of experimental pullout data of hooked-end steel fibers. 21

22. 22 This work was supported by ERAF via project Nr. 2010/0293/2DP/2.1.1.1.0/10/APIA/VIAA/073