https://orcid.org/0000-0002-3816-6431
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ABSTRACT
An analytical modelling approach is developed in this article to simulate the pull-out behaviour of a single straight/hooked end fibre that is embedded in concrete matrix. The partial-interaction model is used to simulate the interfacial bond between the fibre and matrix along the entire fibre length throughout all stages of loading where additional axial and frictional forces due to straightening a hook is incorporated by simulating it as plastic hinges at the apex of bends. The analytical model is presented in a concise matrix form that helps to minimise the number of solution steps for the involved individual cases and provides a monolithic simplified implementation. A significant advantage of the approach is that it does not require fitting of a smoothing polynomial to show the full-range of fibre load-slip behaviour as it simulates the forces for straightening the fibre as it pulls through a bend and directly couples this to the interfacial bond forces.
Notations
| Ac | = | = effective cross-sectional area of concrete (Ac = (Ef / Ec).Af); |
| Ae | = | = area of the elastic region in fibre cross-section; |
| Af | = | = cross-sectional area of fibre; |
| Ap | = | = area of the plastic region in fibre cross-section; |
| Bm | = | = Bond force at fibre-concrete interface of infinitesimal segment m; |
| c1,2,3 | = | = initial coefficient vectors; |
| df | = | = diameter of fibre cross-section; |
| dx | = | = infinitesimal length along the embedded length; |
| Ef | = | = modulus of elasticity of fibre material; |
| Ec | = | = modulus of elasticity of concrete; |
| = | = fibre force immediately after bend 1; | |
| = | = fibre force immediately before bend 1; | |
| = | = fibre force immediately after bend 2; | |
| = | = fibre force immediately before bend 2; | |
| FPH | = | = force necessitates formation of plastic hinge; |
| FPH1 | = | = force necessitates formation of plastic hinge at bend 1; |
| FPH2 | = | = force necessitates formation of plastic hinge at bend 2; |
| Fμ1 | = | = force due to additional friction at first bend; |
| Fμ2 | = | = force due to additional friction at second bend; |
| fc | = | = concrete compressive strength; |
| fy | = | = steel fibre yield strength; |
| If | = | = second moment of area of the fibre section; |
| L | = | = embedded length of the fibre; |
| Lp | = | = fibre bonded perimeter; |
| My | = | = yielding moment when extreme fibre is just started yielding; |
| MPH | = | = moment necessary to produce plastic hinge; |
| m | = | = 1, 2, 3, …, m (infinitesimal segment number from loading end); |
| N | = | = normal force acting normal to the inclined segment of the fibre; |
| N1 | = | = normal force acting at bend 1; |
| N2 | = | = normal force acting at bend 2; |
| P | = | = pull-out load; |
| Pc | = | = reaction force on concrete from support in the pull-push experiment set up; |
| = | = concrete force at the same level of infinitesimal fibre element (m); | |
| Pf | = | = global pull-out force in fibre; |
| = | = fibre force in the infinitesimal element m; | |
| r | = | = radius of fibre cross-section; |
| S1,2,3 | = | = solution matrix for Zone I, II, and III. subscript (1,2,3) refers the zone; |
| x | = | = position from end of embedded length; |
| y | = | = vertical distance of a point from neutral axis in fibre cross-section; |
| y1 | = | = vertical distance of elastic end from neutral axis in fibre cross-section; |
| δ | = | = local slip at any point along fibre length; |
| δ(m) | = | = local slip at m along fibre length; |
| δ0 | = | = local slip at fibre free end; |
| δ1 | = | = local slip at ultimate bond-shear stress (τf); |
| δ2 | = | = local slip when τ = τr; |
| ɛf | = | = fibre strain; |
| ɛf(m) | = | = fibre strain at segment m; |
| ɛc | = | = concrete strain; |
| ɛc(m) | = | = concrete strain at segment m; |
| θ | = | = angle at the bend of hooked end fibre (rotation angle for formation of plastic hinge); |
| μ | = | = frictional coefficient; |
| μN | = | = tangential frictional force; |
| μN1 | = | = tangential frictional force acting at bend 1; |
| μN2 | = | = tangential frictional force acting at bend 2; |
| τ | = | = local bond-shear stress; |
| τ(x) | = | = local bond-shear stress at x distance from free end; |
| τ(m) | = | = local bond-shear stress at segment m; |
| τf | = | = ultimate bond-shear stress; |
| τr | = | = frictional bond-shear stress at local slip of δ2 or more; |
| χ | = | = curvature in bending (); |
| = | = curvature at yield at extreme perimeter of fibre cross-section in bending; | |
| Δ | = | = global slip (slip at loading end); |
| = | = slips strain; | |
| (dδ/dx)m | = | = slip strain at segment m. |
Acknowledgments
This material is based upon work supported by the Australian Research Council Discovery Project 190102650. First author acknowledges gratefully the Adelaide Scholarship International to facilitate this research work.
Disclosure statement
No potential conflict of interest was reported by the author(s).
Data availability Statement
All data used, and models generated in the study appear in the manuscript.
Additional information
Funding
Notes on contributors
Iftekhair Ibnul Bashar
I. I. Bashar, is a PhD candidate at the School of Civil Environmental and Mining Engineering, The University of Adelaide, Australia. His main research interests include ultra high performance fibre reinforced concrete and geopolymer concrete.
Alexander Bonaparte Sturm
A. B. Sturm, is an Assistant Professor at the Department of Civil Engineering, National Cheng Kung University, Taiwan. His main research interests include the development design approaches for fibre reinforced concretes and Ultra-high Performance Fiber Reinforced Concrete (UHPFRC).
Phillip Visintin
P. Visintin, PhD, is an Associate Professor at the School of Civil Environmental and Mining Engineering, The University of Adelaide, Australia. His main research interests include the development of generic, mechanics-based design approaches for reinforced concrete and their application to novel materials such as fibre reinforced and geopolymer concretes.
Abdul Hamid Sheikh
A. H. Sheikh, is an Associate Professor at the School of Civil Environmental and Mining Engineering, The University of Adelaide, Australia. His main research interests include Innovative finite element modelling of stiffened plates and shells, Numerical simulation of high velocity ballistic impact response of laminated composite and sandwich panels, Micromechanics based coupled damage plasticity model for progressive failure analysis of laminated composite structures.