Evaluation of Different Methods for Considering Bar-Concrete Interaction in Nonlinear Dynamic Analysis of RC Frames by Using Layer Section Theory

: In this paper, the bond-slip effect has been applied to the numerical equations in the process of nonlinear dynamic analysis of reinforced concrete frames. The formulation is similar to that of the layer section theory, but the perfect bond assumption has been removed. The precision of the proposed method in considering the real nonlinear behavior of reinforced concrete frames has been compared to the precision of two other suggested methods for considering bond-slip effect in layer model. Among the capabilities of this method for seismic analysis are its ability of modeling the embedded lengths of bars within joints and nonlinear modeling of bond-slip. The precision of the analytical results were compared with the experimental ones achieved from a one bay two storey frame under seismic loading on the shaking table. According to the numerical results, the presence or absence of bond effect in numerical modeling and analysis will bring about considerable different results, including results for deformation and forces. All the studied methods for inserting the bond-slip effect into the layer model can relatively improve the accuracy of analytical results compared to experimental ones. The proposed method of this study has proved to enjoy the highest accuracy with regard to time-history seismic analysis of reinforced concrete frames. Among the capabilities of the proposed method, we may refer to its ability to model beam-column and joint element’s nonlinear behavior separately.

One of the most commonly used methods for nonlinear analysis of RC frames is the layer model that represents the cross section of each frame member as a set of small filaments with finite length and in series along the element. In some of researches this model named fiber method or fiber theory. In this theory the constitutive relation of the section is carried out by integration of the response of the filament, based on uni-axial constitutive law that represents the behavior of concrete or steel. This method assumes perfect bond between concrete and bar (Spacone et al., 1996;Mazars et al, 2006), but this assumption is not very appropriate and realistic and causes a considerable difference between analytical and experimental results (Kwak and Kim, 2006). Belarbi and Hsu (1994) as well as Kwak and Kim (2006) have made use of the fiber method but, in order to modify it and reduce the error of analysis resulted from the perfect bond assumption, they have drawn on an equivalent method. Limkatanyu and Spacone (2002), have used the layer model, but they have removed the perfect bond assumption.
This modified method has been used for beamcolumn elements in this study. But, for modeling reinforced concrete frames, a joint element is also needed. What matters is the compatibility and assimilability of joint elements with beam-columns elements. In initial methods of nonlinear analysis of reinforced concrete frames, the nonlinear effect of beam-column joints is considered using calibration of plastic hinges within adjacent beam-column elements. In such a situation, the joint element is not modeled separately. Based on another approach, the behavior of each of the elements of joint, beam and column is separated. The zero-length rotational spring is one such joint element which need strong calibration process (Alath and Kunnath, 1995). In some newer methods, joint elements are modeled as two-dimensional planes. These elements, however, like finite elements methods, increase the modeling time and the amount of calculations. Another type of joint elements is created by assembling a series of one-dimensional components whose calibration is carried out through experimental results (Lowes et al., 2004). Because force-deformation relations are calculated approximately, such modeling will not be completely precise and will need strong calibration.
In the present study, the beam-column element introduced by Limkatanyu and Spacone (2002) has been used for modeling beam and column elements. Also, a joint element has been defined and used which, in addition to its flexibility in modeling different type of joint elements such as interior, exterior, corner and footing, is capable of being assembled with the above beam-column element (Hashemi et al., 2009). For simplicity's sake, RCF, RCMRF, BCE, and JE will be used instead of reinforced concrete frame, reinforced concrete moment resisting frame, beam-column element and joint element, respectively.

NONLINEAR DYNAMIC ANALYSIS OF RCF:
For the purpose of nonlinear analyzing of RCFs and evaluating the proposed method, four kinds of analyses have been examined, as shown in Table 1. In order to carry our investigations, a computer program created in MATLAB software was used by the authors.

274
Evaluation of Different Methods for Considering….

SEYED SHAKER HASHEMI; MOHAMMAD VAGHEFI
Description of analyze 1: In this type of analysis, which is a nonlinear analysis using the layer model, the JE is not modeled. Formulation of each RC beam or column element is done based on the Euler-Bernoulli beam theory. The cross section of the BCE is divided into a suitable number of concrete and steel fibers (bars). In this analysis, the possible slip effect of the longitudinal bar is ignored. Since if the bar slips, the value of the longitudinal strain of the bar will not be the same as the value obtained through the above method; this is the main assumption in the layer model and is referred to as the perfect bond assumption.

Table2. Selected models for material behavior and their interactions Relationship Description
Concrete stress-strain Park et al. (1972) and later extended by Scott et al. (1982) for monotonic compressive envelope curve it is assumed that concrete behavior is linearly elastic in the tension region before the tensile strength and beyond that, the tensile stress decreases linearly with increasing tensile strain Yassin (1994) rules is adopted for hysteresis behavior Steel stress-strain The initially proposed model by Giuffre and Pinto (1970) and later used by Menegoto and Pinto (1973) Bond stress-bond slip Eligehausen et al. (1983) model Shear stress-shear deformation in the JEs Anderson et al. (2008)  is the tensile strength of the concrete. ρ is the ratio of the bar cross sectional area to the cross sectional area of the whole RC section, which must be more than 0.005.

Description of analyze 3:
For the purpose of reducing errors induced from the perfect bond assumption, in this type of analysis the yield stress of the bars is modified into an equivalent value as described in analysis 2. But here, not only the yielding point but also the bar elasticity modulus are modified. Based on the function of slip distribution between cracks and assuming a linear relationship between slip and bond-stress, the axial force balance of a concrete length segment and of adjacent bars is studied, and the axial force is appropriately divided between the bar and the adjacent concrete, thereby creating a balance. Finally, in this type of analysis an equivalent elasticity modulus will be used for the steel bars. For more information regarding the numerical calculation of the equivalent elasticity modulus and using this calculation in layer model analyses, please, see (Kwak and Kim, 2006).  Limkatanyu and Spacone (2002), in the layer model, the slip effect between concrete and bar is implemented without ignoring the compatibility of the strain between concrete and bar. In this method, a length segment of an RC frame element is considered as a combination of a length segment of a 2-node concrete element and a number of steel bar elements (i.e., longitudinal bars). 2-node concrete elements follow the Euler-Bernoulli theory, and 2-node bar elements are in fact trusses elements. Contact between concrete and the longitudinal bars are provided by a constant bond force around the bars. Using the internal forces balance equations, the governing equations of the length segment of the BCE are obtained. A weak form of the governing equation in a finite elements method is obtained using the shape functions based on displacement and using the principle of stationary potential energy. More information on this element can be found in (Limkatanyu and Spacone, 2002) (Figure 1-a&b). For the sake of simplicity in modeling of different types JEs, at first, a reinforced concrete sub-element, a concrete sub-element and a bar pull-out mechanism are defined which for simplicity's sake, RCSE, CSE and PM will be used instead of them respectively. The RCSE and CSE follow the Timoshenko beam theory. These sub-elements are capable of considering shear deformation and bond-slip effects in nonlinear behavior. According to the location of the JEs in a two dimensional RCF, four types of element is defined. Type 1 of JEs is basically modeled on PM and rigid links that simulate the behavior of footing connections (Fig. 1-c). Type 2 of JEs is used as the substitute of the corner connection in the frame, embracing two RCSEs, two CSEs, and two PMs ( Fig. 1-d). These parts are assembled according to Figure 2 and the effect of boundary conditions (BC) is considered on the specified side sections based on the results of internal forces in the related sections during analysis. Type 3, which can be used as an exterior connection in the frame, is the assemblage of three RCSEs, one CSE, and one PM ( Fig. 1-e). Type 4 is a representative of an interior connection in which the PM is not considered because all longitudinal bars have been passed through the element. This type is a combination of four RCSEs (Fig. 1-f). More information on about JE can be found in (Hashemi et al., 2009).
Behavior of materials: Behavior of materials selected as . This frame is modeled as the combination of BCEs, JE type 1, 2 and type 3. Some details of that are shown in Figure 3 and more details are given in (Clough and Gidwani, 1976). Newmark's method has been used for solving the equation of motion in the numerical investigation, and the equation of motion has been solved incrementally. Acceleration changes have been SEYED SHAKER HASHEMI; MOHAMMAD VAGHEFI storey level calculated in the four previously defined analyses, have been compared with the corresponding experimental results. Table 3 shows a comparison of experimental and analytical results, including the results for the levels' lateral displacement and base shear and drift. In this comparison, a relative error percentage with a plus sign (+) reveals a higher value for the analytical result compared to experimental, and a minus sign (-) reveals a lower one. Based on the results, if the bond effect is excluded, analytical and experimental results will be considerably different (analysis 1). Including the equivalent bond effect (analyses 2 and 3) leads to more accurate results, but as shown in Figure 4, there is still no good agreement between analytical and experimental figures. The use of proposed method (analysis 4) results into more accurate answers and a better agreement between analytical and experimental figures. Conclusion: According to the results, the presence or absence of bond effect in numerical modeling and analysis will bring about considerable different results, including results for deformation and forces. All the studied methods for inserting the bond-slip effect into the layer model can relatively improve the accuracy of analytical results compared to experimental. The proposed method of this study has proved to enjoy the highest accuracy with regard to time-history seismic analysis. Among the capabilities of the proposed method, we may refer to its ability to model beam-column and JEs' nonlinear behavior separately. Finally, the authors of this paper suggest this method will be useful and remarkably accurate for nonlinear dynamic analysis of RCMRF.