Abstract
The paper presents a novel family of strain-based beam finite elements (FE) for analysis of tensile failure of a reinforced concrete bar (RC bar), with results of the analysis being independent of the applied FE mesh. The composite bar consists of a continuous longitudinal ductile reinforcing bar(s) surrounded by brittle concrete cover, which are considered separately in the model. Longitudinal slip at the contact between the concrete cover and reinforcing bars is allowed, while their relative displacements perpendicular to the axis of the RC bar are prevented. Cracks in concrete cover occur when tensile stress in concrete exceeds its tensile strength. Propagation of partially connected crack, that is, softening of the material at the crack, is described through constitutive law in form of nonlinear relationship between stresses in concrete at the crack and the width of the crack. Each separate crack is considered discretely as a discontinuity in geometry of the element. In the analysis of cracking of concrete, it is commonly assumed that the discrete crack can occur at the nodes of FE only. However, this assumption leads to dependence of the analysis results on the employed FE mesh. The presented family of FE enables occurrence of the crack anywhere along the FE. In order to achieve this, the discrete crack is excluded from equations of FE and additional boundary conditions are introduced at the discontinuity. This approach ensures that the location of the cracks, their number and their propagation are independent of the applied FE mesh. Advantages of the novel family of FE are thoroughly presented in a parametric study which investigates influence of number of FE as well as influence of degrees of interpolation and integration on the cracking of RC bar under tensile loading. Experimental results of tensile tests on the analysed bar are available in literature. It can be concluded that the results obtained with the minimal possible number of novel FE and sufficiently high degree of numerical integration scheme, applied for solving integrals in equations of FE, are considerably more accurate than the results of previous analyses with model of discrete crack at the nodes of FE only.
Highlights
Concrete is a heterogeneous and brittle material, where even very small tensile loading can result in occurrence of cracks
That the entire considered structure is modelled with finite elements, whose lengths are consistent with characteristic length Lc, it can be assumed, that the results of the analysis are completely independent of finite element mesh (Bažant and Planas, 1997; Federation internationale du beton, 2013)
The paper presented a novel family of strain-based beam finite elements for analysis of tensile failure of a composite bar
Summary
Concrete is a heterogeneous and brittle material, where even very small tensile loading can result in occurrence of cracks. When the tensile strength of concrete is exceeded, localisation of extensional strains in concrete, εc, occurs on a band of the element with limited length, Lc, as shown in Figure 1(b).The length of this band of concrete, over which the cracks are smeared, is not arbitrary Instead, it must be defined as a material parameter and depends on fracture energy of concrete, Gf, and constitutive law of crack propagation. Notional extensional strain of concrete εrc and width of actual crack r is related through well-known equation εrc 1⁄4 εct þ r If it is ensured, that the entire considered structure (or at least parts of the structure, where cracks are most likely to occur) is modelled with finite elements, whose lengths are consistent with characteristic length Lc, it can be assumed, that the results of the analysis are completely independent of finite element mesh (Bažant and Planas, 1997; Federation internationale du beton, 2013).
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