Fracture energy-based optimisation of steel fibre reinforced concretes
Introduction
The addition of steel fibres into concrete at a certain volume fraction improves the ductility of concrete [1], [2], [3], [4], [5]. Steel fibres, randomly distributed in matrix, show its effect after matrix cracking by delaying the crack formation and limiting the crack propagation by reducing the crack tip opening displacement [6], [7]. Steel fibres also contribute the steel fibre–matrix bond strength by increasing fracture toughness of the steel fibre reinforced concrete (SFRC). Steel fibres in the matrix act as crack arresters by bridging mechanism, undergo a pull-out process, delay crack formation and limit crack propagation [6], [7]. The use of steel fibres greatly increases its energy absorption and ductility [8]. The performance of steel fibre depends on fibre type and orientation of fibres in matrix, aspect ratio (length/diameter), volume fraction and tensile strength of fibre as well as matrix strength influence the performance of SFRC [9], [10], [11]. SFRC has a wide-range of applications such as; pavements and overlays, industrial floors, precast products, hydraulic and marine structures, repairing and retrofitting of reinforced concrete structures, tunnel linings and slope stabilization works [12].
Fracture mechanics has been developed and applied for many decades. Its scope of application has been extended into numerous fields of materials, such as metal, ceramic and concrete [13].
Since Kaplan firstly introduced fracture mechanics into concrete beam to measure fracture toughness in 1961 [14], [15], [16]. Although concrete material has longer history than other construction materials, the modelling of its mechanical behaviour under complex loading paths still represents a challenging task, especially when the fracture of concrete members and structures are of interest [17]. In order to model the response of concrete structures to external action it is vital to include the effects of cracks in concrete. Since concrete is composed of different sized aggregates, the material is very heterogeneous. Therefore the cracking in concrete may be modelled at different scales as presented in Ref. [18]. Damage in concrete during its initial loading phase appears in the form of distributed micro-cracks. These micro-cracks are primarily due to the shrinkage of the cement paste around the aggregates. In due course of time, under sustained loading, the diffused damage expands and forms a distinct large crack due to coalescence of microcracks which would propagate and cause final failure of the structural member [19]. So fracture toughness of concrete, which represents crack resistance capability of concrete, is one of fundamental fracture parameters in the fracture analysis of concrete.
Fracture energy is very important parameter in understanding the properties of concrete and determining the design criteria of large concrete structures. The fracture energy (GF) is defined as the area under the load–deflection curve per unit fractured surface area. The most widely used fracture mechanics models for analysing concrete structures is the fictitious crak model (FCM) proposed by Hilleborg et al. [20], [21], [22], RILEM [23], [24] and Petersson [25] recommended a method for the determination of GF using simple three-point bending test.
In this study, a multiobjective simultaneous optimisation technique is used to optimise concrete with special emphasis on optimum solutions for fracture energy, in which the Response Surface Method (RSM) is incorporated. The RSM uses statistical techniques for empirical model building; it comprises regression surface fitting to obtain approximate responses, design of experiments to obtain minimum variances of the responses and optimisations using the approximated responses. The RSM also aims to reduce the cost and numerical complexity of other expensive analysis methods such as finite element and finite difference methods. The RSM has been widely used to optimise products and processes in manufacturing, chemical and other industries, but it has had very limited use in the concrete technology [26].
The main objective in this work is to optimise the fracture energy of SFRCs to obtain a significantly more ductile behaviour than that of plain concrete. The test method used in the experimental part is based on EN14651 [27] in which toughness is calculated from the load–deflection curves obtained by performing a third-point test on notched beam. Experimental design of test results is made by using RSM, and a multiobjective simultaneous optimisation technique is used for optimisation and obtaining desirability functions. Optimum solutions are carried out by maximising fracture energy and splitting tensile strength while minimising the cost (i.e. volume fraction of fibres).
Section snippets
Specimen characteristics and production
CEM I 42.5R Portland cement and silica fume (SF) with 81.35% SiO2 were used for the program. Their properties are presented in Table 1. Crushed limestone sand (CLS) up to 4 mm, and course crushed limestone (CCL-I) up to 12 mm and course crushed limestone (CCL-II) up to 19 mm were used as fine size and coarse size aggregate, respectively. Gradations of them are presented in Fig. 1. Also physical properties of aggregates are given in Table 2. Fibres used are cold drawn steel fibres with hooked-ends
Test results
Test results of fresh concrete and mechanical properties are given in Table 5. The addition of steel fibre into concrete has negative influence on the workability. Steel fibres decrease workability of fresh concrete. Vebe time increases with the increasing of steel fibre volume fraction for each water/cement ratio. On the other hand, the unit weight of the fresh concrete increases with the increasing of steel fibre content. This can be attributed to the specific gravity of steel fibres (7.82 g/cm
Optimisation
A multi-objective simultaneous optimisation technique incorporating Response Surface Method (RSM) as the basis for finding the best solution was used in this study. A commercially available package program (Design-Expert) was used [38]. A common response surface experimental plan which can be used to find optimal settings is a two variable (i.e. W/C and Vf), three-level (i.e. W/C = 0.35, 0.45 and 0.55; Vf = 0.26%, 0.51%, and 0.76%) full factorial experimental design for each steel fibre tensile
Conclusions
Regression analysis by using Response Surface Method (RSM) for predicting of fracture energy of SFRCs, incorporating low and high steel fibre tensile strengths, presented good correlation with the experimental ones in the range of factors investigated. Fitting quadratic models, usually assumed to represent each concrete property of interest, can be done in identifying optimal mixes to obtain an adequate representation of the responses. In this study, the results show that the predictiveness of
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