This paper presents numerical modelling of braided fibre, to be used as concrete reinforcement. Ultimately, a corrosive and fire resistant concrete will be produced. A Cubit script (geometry and mesh software) was created to mesh braided yarns under dierent geometric parameters. Fibres were represented using elastic transversely isotropic materials, for which the fibre directions for every yarn were precisely determined from the gradients of the resultant stream functions of potential flow problems. Applying an elastic interfaces between yarns to preventing penetration and having free sliding, convergence studies were conducted on a coarse and fine mesh, using hierarchical higher order approximation [1] for uniform p- and hp-refinement

Cortis, M.

Hussain, Mohammed S.

Javadi, Akbar

Kaczmarczyk, L.

Pearce, C.

English

Durham Research Online

Proceedings of the 22nd UK Conference of theAssociation for Computational Mechanics in Engineering2 - 4 April 2014, University of Exeter, ExeterComputational Modelling of Braided Fibre for Concrete Reinforcement*M. Cortis, L. Kaczmarczyk and C.J. PearceThe University of Glasgow, Rankine Building, School of Engineering, Oakfield Avenue, Glasgow, G128LT, United Kingdom*m.cortis.1@research.gla.ac.ukABSTRACTThis paper presents numerical modelling of braided fibre, to be used as concrete reinforcement. Ulti-mately, a corrosive and fire resistant concrete will be produced.A Cubit script (geometry and mesh software) was created to mesh braided yarns under di↵erent geomet-ric parameters. Fibres were represented using elastic transversely isotropic materials, for which the fibredirections for every yarn were precisely determined from the gradients of the resultant stream functionsof potential flow problems. Applying an elastic interfaces between yarns to preventing penetration andhaving free sliding, convergence studies were conducted on a coarse and fine mesh, using hierarchicalhigher order approximation [1] for uniform p- and hp-refinement.Key Words: Fibre Reinforced Concrete; Composite; Bond Strength; Interface Cohesion Elements;Hierarchical Refinement1. IntroductionThis paper presents a numerical investigation into the modeling of braided carbon fibre ropes. Theseropes can provide an alternative to steel reinforcement in concrete, whereby the braiding can providegood bond adhesion between the fibre and concrete [2] and mitigate the need for weak resin binders[3]. In order to realize this ambition and to develop an appropriate modeling framework, a number ofmodeling challenges need to be overcome. First, an accurate geometrical representation of the com-plex geometry of the braided rope needs to be achieved. Second, the fibre direction during deformationneeds to be accurately determined. Third, the interface between fibre yarns needs to be modeled. Novelhierarchic approximation functions [1] are also adopted and the paper demonstrates the convergence per-formance of h, p and hp-refinement. The fibres are modeled as transversely isotropic, although restrictedto linear elasticity in this paper.2. Numerical Modelling Approach2.1. Mesh ModellingDue to the lack of tools for modelling braided geometries, a flexible Python/APERPRO script was writtenin Cubit to generate 12-strand plaited sinnet geometry as shown in Figure 1. The algorithm of this scriptis as follows: a) sets of vertices representing the centre of axis for every yarn were created followingthe pattern of braiding, b) axis splines were formed along every set of vertices, c) circular surfaces wereextrude along the same splines to form the braiding geometry. The input parameters were: i) the diameterof the yarn, ii) permissible tolerance between the yarns, iii) pitch and iv) the number of turns required.The interfaces between the yarns were created by subtracting one yarn from the overlapping adjacent one.Two square clamps were modelled at both ends of the braided model. Tetrahedral mesh was generatedusing Cubit mesh generator. The respective boundary conditions and material parameters were assignedin Cubit and directly read by MoFEM (Mesh Oriented Finite Element Method, our group’s open-sourceFEM software).Figure 1: Type of Braided FibreModelledFigure 2: 5 Material Parametersfor Transversely Isotropic Mate-rial (with reference to fibres)Figure 3: Flow Direction Vectorsin Rope Model2.2. Transversely Isotropic MaterialA suitable material to represent fibre yarns was chosen to be transversely isotropic material. This materialis described by 5 independent material parameters, which are the principal sti↵ness Ez, and poisson’sratio ⌫z along the transverse direction (z-axis) of the fibre and the sti↵ness Ep, poisson’s ratio ⌫p andshear modulus Gzp in the plan of orthotropy (xy plane) as shown in Figure 2.The fibre directions can be determined from the spline axes of the yarns. However, this is inaccuratewhen non-uniform cross-section are considered and/or rapid change of direction exists. An alternativeway of obtaining the fibre directions is to solve a steady laminar incompressible potential flow problem,where the gradients of the stream function  are the resultant flow velocities defining the fibre directionsF at every Gauss point as shown in Figure 3. By using a higher order approximation for the potentialflow solution very accurate stream functions are computed for coarse meshes.To transform the material response between the local fibre direction and global axes, the axis of rotationis expressed as A = F⇥Z, where Z is the unit vector (0, 0, 1) representing the global z-axis and the angleof rotation is expressed by ✓ = cos 1 F.ZkFk kZk . This rotation matrix can be computed from R = I+ sin ✓N+(1  cos ✓)N2, where I is the identity matrix, N = ⌦! , ! = +qA21 + A22 + A23 and ⌦ =26666640  A3 A2A3 0  A1 A2 0 03777775 .2.3. Interface ElementsInterface elements represented by prism elements in the FE mesh, were inserted between yarns, andan elastic cohesive model was used for its formulation [2]. Orthogonal penetration/separation of theinterface was controlled using a sti↵ness about thousand times the principal material sti↵ness, while nosti↵ness was used to control shear movement, i.e. fibre were free to slide.2.4. Convergence StudyA convergence study was conducted using p, h and hp-refinement on the problem described in Section2.1. Hierarchical higher order (HO) approximation was used to perform uniform p-refinement up to 4thorder polynomial, where a minimum of 45 Gauss points were necessary for the numerical integration.Hierarchical HO shape functions for edges, faces and volumes were constructed using standard linearnodal shape functions (used in linear FEs) and legendre polynomials, as described by Ainsworth [1].This allows local p-refinement, which is computationally cheaper that global p-refinement. However, thework presented here is restricted to global p-refinement.Two meshes were considered: coarse mesh (49,624 elements) and a fine mesh (396,992 elements). Thecoarse mesh was analysed using 1st, 2nd, 3rd and 4th order approximations. The fine mesh was analysedusing 1st, 2nd and 3rd (5,823,150 DOF) order approximation.The error was computed as |uz   uhz |, where uz is the displacement of a random node, and uhz is thedisplacement at the same location on the coarse mesh with 5th order approximation. The 5th order p-refined mesh is assumed to provide the reference solution.3. ResultsCreating the geometry and mesh using a Python/APRPRO script has proven to be an e↵ective dynamicsolution to generate braided type geometries. The flexibility of this script is not only the capability ofvarying the parameters of such braiding, but could be easily adopted for other types of braiding. Loosebraiding was formed and hence pre-stressing using nonlinear geometry (large rotations and small strains)is required at a later stage.Figure 4: Linear Elastic Materialwith interface between yarns4.5 5 5.5 6 6.5 732.82 .62 .42 .221.81 .61 .4log 10(DOFS)log10(Error)Coarse Mesh p-re fF ine Mesh p-re fFigure 5: Convergence Results for Coarse and Fine Mesh with dif-ferent order of approximationIsotropic steel material was used for the clamps (not shown in fig. 4), and carbon fibre transverse isotropicmaterial properties was used for the yarns. Figure 4 shows the deformed shape of the braided rope.Negligible penetration between yarns was observed.P-refinement on the coarse and fine mesh resulted in similar rates of convergence (Figure 5). This reflectsthe theoretical rates of convergence for both the energy and displacement norms [4].P-refinement on the coarse mesh is found to be the optimal refinement, achieving accurate results withthe least number of DOFs. Hence uniform hp-refinement is not recommended in this case, although onemight investigate the use of adaptive p and/or hp-refinement, where the theory suggest an exponentialrate of convergence would be achieved.4. ConclusionsAn e↵ective method was achieved to model and mesh complex geometries such as braided ropes. Me-chanical modelling of such geometries, required the implementation of an appropriate material, i.e. trans-verse isotropy, that well represent the behaviour of the individual fibre yarns. Although such a materialcould be orientated using the centre axis of the yarns as a mean of representing the fibre directions, thiswould lead to problems when the model is subject to large deformations and yarn cross-sections do notremain uniform. An e↵ective solution for this problem, was to solve a potential flow problem, whereby the velocity vectors at every gauss point (later used as the fibre directions) were computed using thegradients of the resultant stream function ( ). Furthermore, convergence studies, shows that although alocal error was computed (a nodal displacement was used to compute the error), linear convergence holdswith the theory, achieving about 93% of accuracy for 2nd order of approximation with the coarse mesh.After this initial study, the process to investigate the bond behaviour between the fibre rope and concretewill be as follows:• Include a non-linear interface between the yarns.• Pre-stress the rope model using large rotations/small strains.• A better representation of the geometry using higher order elements.• Implement the potential flow solution for every iteration of the nonlinear mechanical analysis,hence fibre direction will be updated every time the geometry changes.• Encase the pre-stress ropes in concrete and use a suitable cohesive damage law for the interface• Investigate the pull-out bond strength for plain and ribbed braided ropes• Implement concrete fracture to investigate cracking induced by reinforcementAcknowledgementsThe research work undertaken in this publication is funded by the Strategic Educational Pathways Schol-arship (Malta). The scholarship is part-financed by the European Union European Social Fund (ESF)under Operational Programme II Cohesion Policy 2007-2013, “Empowering People for More Jobs and aBetter Quality of Life”.References[1] M. Ainsworth, J. Coyle, Hierarchic finite element bases on unstructured tetrahedral meshes, International Journal for Nu-merical Methods in Engineering 58 (14) (2003) 2103–2130.[2] M. Cortis, L. Kaczmarczyk, C. J. Pearce, Computational modelling and experimental investigation of the bond behaviourbetween concrete and braided fibre ropes, in: International Conference on Computational Mechanics (CM13), no. March,2013.[3] A. Katz, N. Berman, Modeling the efect of high temperature on the bond of FRP reinforcing bars to concrete 22 (2000)433–443.[4] L. Demkowicz, Computing with hp-adaptive finite elements: Volume 1 One and Two Dimensional Elliptic and MaxwellProblems, Chapman & Hall/CRC Applied Mathematics and Nonlinear Science Series, Taylor & Francis, 2006.

Computational Modelling of Braided Fibre for Concrete Reinforcement

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Computational Modelling of Braided Fibre for Concrete Reinforcement

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