Effect of Maximum Size of Aggregate on the Behavior of Reinforced Concrete Beams Analyzed using Meso Scale Modeling

In this study, simply supported reinforced concrete (RC) beams were analyzed using the Extended Finite Element Method (XFEM). This is a powerful method that is used for the treatment of discontinuities resulting from the fracture process and crack propagation in concrete. The mesoscale is used in modeling concrete as a two-phasic material of coarse aggregate and cement mortar. Air voids in the cement paste will also be modeled. The coarse aggregate used in the casting of these beams is a rounded aggregate consisting of different maximum sizes. The maximum size is 25 mm in the first model, and in the second model, the maximum size is 20 mm. The compressive strength used in these beams is equal to 26 MPa. The subjects of this study are two RC beams subjected to a two-point loading designed to fail due to flexure. The RC beams under loading were studied in the laboratory as well as numerically. ABAQUS program was used for modeling and analyzing the RC beams. The mesoscale modeling that was used to model the concrete required used a special program using different programs but has not used the ABAQUS program directly. The result of the comparison between the numerical and experimental showed that the mesoscale numerical model gave results that were more approximate to the experimental ones, and the mesoscale modeling of reinforced concrete is most convenient when the maximum size of aggregate is decreased.


INTRODUCTION
Concrete is a non-homogenous material (heterogeneous material), wherein it consists of coarse aggregate, cement mortar, and air voids in the cement paste. This property makes the behavior of concrete depend upon the behavior of the material (components). In many studies, concrete is assumed to be a homogenous material for simplicity and is modeled on a macro scale, but to understand the behavior of concrete, many analytical scales were developed, such as meso, micro, and atomic scales. The mesoscale analysis model gives a good indicator of the behavior of the concrete. In the mesoscale structures, concrete is considered as a multiphase material (Al-Zuhairi and Taj (a), 2018), (Al-Zuhairi and Taj (b), 2018). In this paper, two reinforced concrete beams with dimensions of 350 × 200 × 2200 mm will be analyzed numerically by using the Extended Finite Element Method (XFEM). The results of the two models will be compared with the experimental results of the same reinforced concrete beams with approximately the same compositions. Effect of the maximum size of aggregate on the behavior of RC beams under bending analysis using mesoscale will be studied.

THE MESO SCALE MODELING
The mesoscale is a scale that falls between the macro and micro scales (Murayama, 2001), as in Fig.1. The continuum and the lattice models are subdivisions of the mesoscale modeling. In the continuum models, the material is divided into components; for example, concrete consisting of aggregate, mortar, and the interface zone between the two materials (Liao et al., 2004). In this paper, this model will be used for modeling concrete as aggregate, mortar, and air voids that result from the pouring process of concrete. The interface between the aggregate and cement mortar is modeled as fully bound and tied. This assumption is not really correct because the micro-cracks start from these areas (transition zone or the interface between the aggregate and cement paste) and form weak points in concrete. Still, this study is a start to study mesoscale in the RC beams. The lattice models of concrete are modeled as a discrete system consisting of a lattice element (Nitka and Tejchman, 2015). This model needs a massive numerical effort to model concrete. The mesoscale modeling has two approaches; image-based and parameterization modeling. The first one approaches the building on a set of 2D pictures that are assembled to get a 3D model, where the numerical model will be conducted based on this 3D model. The second approach (parameterization) is divided into two methods, direct and indirect. In the direct method, the major parameters are those such as shape, size, gradation, and distribution of the aggregate particles, the interface between aggregate particles and cement mortar, and their effect on the mechanical behavior of the concrete of the multi-phase material computation . In the indirect method, the heterogeneity of the concrete is modeled separately with a regular FE mesh (Yang, 2009), or by using lattice modeling for the aggregate and mortar phases (Leiti, 2003) and (Schlangen, 1997). The direct method is more appropriate for the mesoscale modeling process and will be used in this study.

THE EXENDED FINITE ELEMENT ETHOD (XFEM)
The extended finite element method (XFEM) is a numerical method based on the finite element method and on the partition of unity methods (PoUM or PUM). It is the use of partition of unity functions which are functions whose values sum up to unity at each point in the domain (Ahmed, 2009). The XFEM is used to solve the discontinuity problems that occur in brittle materials such as concrete (Belytschko and Black, 1999).
Localized enrichment functions are used in the XFEM, which are an enrichment of nodes and developed near the discontinuity (Khoei, 2015). There are two types of discontinuities in concrete structures. The weak discontinuity and strong discontinuity. Weak discontinuity in concrete results from bi-material problems (aggregate particles and cement mortar). The strong discontinuity is described by the crack interface in the domain (Khoei, 2015). The enrichment is done mathematically by adding the enrichment part to the standard part of regular interpolation, as shown in Eq. (1) (1) (2) regular interpolation enrichment interpolation Where: ( ) : The standard or regular shape functions ̅ ( ) : The enhanced shape functions ̅ : standard DOF ̅ : enrichment DOF : the set of all nodal points : the number of enrichment node ( ) : the enrichment function : the number of enrichment function

ENRICHMENT FUNCTION
The enrichment is an act of improving the approximation displacement field based on the properties of the problem. The choosing of the enrichment functions is related to the type of Journal of Engineering Volume 26 May 2020 Number 5 146 discontinuity, and its influences on the kinds of solutions. These functions are such as the signed distance function, level set function, branch function, Heaviside jump function, and so on.
The level set function is used for weak discontinuity, where it is the signed distance function ( ).
N ⃗⃗⃗ : is the local unit normal at ⃗ taking the value (+,−) that referred to the outside and inside regions.
For modeling the strong discontinuity (cracks), two enrichment functions are used: Heaviside or step function (jump function), and asymptotic near-tip enrichment function (Belytschko and Black, 1999), and (Khoei (2015). Where: : The set of enriched nodes whose support is bisected by the crack : The set of nodes which contains the crack tip in the support of their shape functions enriched by the asymptotic functions ̅ : The unknown standard nodal DOF at ℎ node ̅ : The unknown enriched nodal DOF associated with the Heaviside enrichment function at node J ̅ : The additional enriched nodal DOF associated with the asymptotic functions at node .

EXPERIMENTAL WORK
The experimental work consisted of casting and then testing two simply supported reinforced concrete beams with dimensions of 2200 × 350 × 200 mm using minimum reinforcement, as in Fig.2. The first beam used rounded coarse aggregate with a maximum size of 20 mm. The second beam used rounded coarse aggregate with a maximum size of 25 mm. The amounts of the material used to casting these beams are explained in Table (1) and were designed and executed according to (ACI 211-02). These beams are tested under two-point loads, and these beams are designed to fail in flexure.  The load was applied incrementally. The amounts of strains and deflection at the beam mid-span corresponding to each load increment were recorded. Two electric strain gauges were attached to each RC beam specimen. One of these gauges was attached in the middle of the front face of the specimens in the tension zone, the other gauge was attached at the middle of the bottom face of the specimens. These two strain gages were used to measure the tensile strain at midspan of RC beam specimens. Dial gauge was used for deflection measurements. It was located at midspan of the bottom face of the RC beam specimens The first beam, which was constructed using rounded coarse aggregate with a maximum size equal to 25mm, was failed at a load of 106 kN. The first cracking was observed at the midspan at a load of 27 24 kN. This crack is very fine then when the applied load is increased, and other cracks had appeared, Fig. 3. The second beam, which was constructed using rounded coarse aggregate with a maximum size equal to 20mm, was failed at a load of 99.8 kN. The first crack was observed at the mid-span at a load of 24 kN, Fig. 4.

NUMERICAL WORK
Two-dimensional mesoscale finite element models were used in modeling the two RC beams. Concrete was modeled as a bi-phasic material consisting of coarse aggregate particles and cement mortar. Air voids were assumed as spaces (voids) in the concrete model without any material properties (Al-Zuhairi and Taj (a), and (b), 2018).
Where: is the coarse aggregate weight in kg/m 3 used in the mix design, equals to 1249 kg/ 3 when using coarse aggregate with a maximum size of 25 mm, and is equal to 1180 kg/ 3 for coarse aggregate with a maximum size of 20 mm The absolute volume of aggregate used in the model is calculated as: Where represents the specific gravity of the coarse aggregate Calculate the area of aggregate used in the model in 2D This area is the total area calculated. However, it is needed to find the area for each gradient. The area of aggregate for each gradient of maximum size 25mm and 20mm in the 2D model is shown in Tables 2 and 3, respectively.   Air voids content was assumed to equal to the theoretically assessed percentages given in (ACI 211-02, 2002) for the design mix properties. Air voids content percentage equals 1.5% and 2% for the beams have a maximum size of coarse aggregate particles of 25 and 20 mm, respectively. EXCEL sheets were used to calculate the area required for drawing the rounded aggregates. To model the rounded aggregate and air voids, they were assumed of elliptical shapes, as this representation is more accurate because the rounded aggregate is not necessarily a circle. The ABAQUS software program was used in the modeling and analysis of the RC specimens. Cement mortar and aggregate materials were represented as a linear elastic material. Materials' properties fed to the ABAQUS program are explained in Table 4. (Wang et al., 2015).

RESULTS AND DISCUSSION
A 2D plane stress approach was used to analyze the flexural members (reinforced concrete beams) tested under two-point loading. The non-homogeneity of concrete is taken into account using the mesoscale model. Then, the numerical results are compared with the experimental results. Table 5. shows the experimental and numerical maximum applied load for RC beams. The serviceable load is expected to be between 60 and 70% for the ultimate load. In this study, 70% of the ultimate was taken as a serviceable load to measure the percentage of convergence between the experimental and numerical works.   . 7 and 8 show the load-deflection curve at the mid-span, and the tensile strain curve measured at the mid-span using strain gage, respectively, for rounded coarse aggregate with different maximum sizes.   Fig. 7 shows the maximum deflection in the experimental work at the ultimate load is equal to 5 mm and 8.2 mm for first and second beam, respectively. Also, it can be seen that the experimental deflection approximately has the same amount of deflection until 40 kN, while Journal of Engineering Volume 26 May 2020 Number 5 152 when increasing load from this amount, the deflection of the second beam is increased if compared with the other one. From the above, it can be noted that the amount of deflection increases with the decrease of the coarse aggregate maximum size when compared with the deflection at the same load. When comparing the experimental with the numerical mesoscale modeling that has the same maximum sizes at the serviceable load, the percentage of convergence equal to 74% and 80% for maximum sizes of 25 mm and 20 mm, respectively. Fracture mechanics is the material mechanical behavior when subjected to load in the presence of cracks (Hamed et al., 2016). The fracture energy of the material can be calculated by computing the area under the stress-deflection curve (Bažant1992).
The first beam (beam No.1) has the upper values of the fracture energy because it has the most significant values, of coarse aggregate (1249 kg/m 3 ), Table 1. This is due to the fracture energy increases with the increase in coarse aggregate content. Because the cracks need to travel around the coarse aggregate particles, in this case, area of the crack surface increases, that led to increasing the energy required for crack propagation that means the fracture toughness increase with an increase in aggregate size (Moavenzadeh and Kuguel, 1969). In addition to the fracture, energy decreases when increasing in porosity that makes the cross-sectional area to be fractured smaller. Porosity in the first beam (beam No.1) where the air void contains is equal to 1.5%, Table 1, (Ioan et al., 2015), and the fracture toughness increases with a decrease in the water-to-cement ratio (Nallathambi et al., 1984).
From Fig. 8, it can be seen that the experimental results are closer to the numerical results, where the percentage of convergence between them is equal to 92% and 98% for maximum sizes of 25 mm and 20 mm, respectively. Also, it can be noted that the amount of tensile strain in the RC beam at the mid-span increases with the decrease of the coarse aggregate maximum size in both experimental and numerical analysis. Fig. 9 explains the sections taken to measure the bending and shear stresses developed in the RC beams. Fig. 10 and Fig. 11 show the distribution of bending stress across the section A-A taken at the middle part of the beam for rounded coarse aggregate with a maximum size of 25 mm and 20 mm, respectively. While Figs. 12 and 13 show the shear stresses in MPa at section B-B for RC beams that have a maximum size of 25 mm and 20 mm, respectively.  size of 20 mm.
The zigzag shape of the bending and shear stresses may be imputed to the effect of the nonhomogeneity of the concrete material. The coarse aggregate particles and air voids near and at the section of the beam causes stress concentration problems at the place of interface position, as can be seen in the above-mentioned figures. The high values of stress due to stress concentration problems, that result from two reasons. The first reason the Modulus of Elasticity for aggregates is three times larger than mortar. The second reason is assumed fully bound (tied) between the