Effect of RC wall on the development of plastic rotation in the beams of RC frame structures

Abstract

The objective of this study is, to interpret the influence of reinforced concrete walls addition in reinforced concrete frame structures considering behavior laws that reflects the actual behavior of such structures, by means of Castem2000computer code (pushover analysis). A finite element model is proposed in this study, using the TAKEDA modified behavior model with Timoshenko beams elements. This model is validated initially on experimental model. Then the work has focused on the behavior of a RC frame with 3 levels and three bays to better visualize the behavior of plastic hinges. Once the plastic hinge control parameters are identified (plastic rotation, ultimate curvature), a strengthening by introduction of reinforced concrete walls (RC/wall) at the ends of the reinforced concrete frame (RC/frame) has been performed. The results show that these RC walls significantly improve the behavior, by a relocation of efforts towards the central part of the beams.

Keywords

RC/frame / RC/wall / moment curvature / plastic rotation / plastic hinge / pushover analysis / global models

Introduction

Recent earthquakes (Japan, 2011, Pakistan, 2013 and Nepal, 2015) have shown that existing structures, in particular those consisting of reinforced concrete frames, exhibit a lack of strength, compromising the safety of people during earthquakes [¹]. The most sensitive parts in structures, which are most likely to alter the performance of these, under a seismic load, are generally the nodal zones (Areas in the nodes vicinity) [²,³]. It has been found that these nodal zones are the places where load transfer occurs, located at the intersection of the beams and the columns [⁴]. The progressive plasticization of the zones in the nodes vicinity in the beams leads to the appearance of plastic hinges, thereby turning the structure into a mechanism that causes the whole collapse [¹,⁵].

The favored plasticization mode is the one caused by flexural behavior rather than the one caused by shear, as it leads to good energy dissipation [⁶]. The rational approach used to determine the seismic strength of a reinforced concrete (RC) frame structure consists of selecting the most appropriate post-elastic deformation mechanism and using an appropriate modelling approach, based on the finite element method [⁷].

The application of the finite element method on reinforced concrete structures presents some difficulties due to the complex nature of the concrete such as crack propagation, sliding between the lips of the cracks, the deformation localization and the concrete-reinforcement interface. The brittle behavior of concrete in tension leads to fast cracking, and the homogenization assumption in the classical finite element theory is no more valid as such. Furthermore, to study the seismic behavior of reinforced concrete structures, it is essential that the models can be able to consider the effect of cyclic loading on the degrading concrete (unilateral effect), which is still a challenge for most models and analysis procedures.

In order to take into account concrete cracking in modelling, two concepts have been proposed in the literature. The first is the concept of discrete cracking, which several works have been devoted to, such as the works of Refs. [⁸–¹⁰]. However, because of the difficulty of updating the mesh with the progression of the cracks and with the existing computer means, the application and the development of the discrete crack approach remain very limited. The second is the concept of diffuse cracking introduced by Rots [¹¹], which remains compatible with the finite element method since the topology of the mesh does not need to be updated during simulation, as shown in the work [¹²–¹⁴] for materials behavior laws based on local or non-local approaches.

In the two- and three-dimensional modelling of reinforced concrete structures, which the cracks are considered explicitly, much effort has been devoted to the development of the material “concrete” constitutive laws in the plasticity, fracture and damage mechanics [⁸–¹⁰,¹²,¹⁵,¹⁶,¹⁷]. However, 2D or 3D modelling at the scale of real structures such as buildings is not conceivable because the computer resources we have in hand are not powerful enough for this kind of calculations.

Besides, global models based on moment-curvature phenomenological analyses [¹⁸–²¹], or relatively complex beam-column elements [²²–²⁴], can predict very well the behavior of structures at real scale. They do this by capturing macro-scale behavior through experimentally calibrated models. These models have become increasingly advantageous because of their ability to reliably predict the global response in a short time. Several studies were conducted using these global models under SAP 2000 software on the RC frame structures such as the works of Refs. [⁷,²⁵–²⁷].

Among the different analysis methods suitable for seismic analysis using global models, we can find the pushover analysis. The authors [²⁸,²⁹] have shown that this analysis can give an appropriate response of the RC-frame structures under seismic load [³⁰]. In this analysis, the post-elastic deformation mechanism is governed by the appearance of plastic hinges [³¹], and their distribution depends on several factors such as the characteristics of the ground motion and the nature of the applied load [³²].

During this study, the experimental RC frame of Vecchio [³³] is modeled through a CASTEM2000 finite element code, using Timoshenko beam elements for columns and beams with the modified Takeda model [³⁴]. The obtained results have been compared to the experiment in order to validate the finite element model.

After validation of the finite element model, a 03-storey/03-span RC frame has been modelled with the same sections as the Vecchio RC frame, in the aim of assessing of the plastic hinges damage level (Plastic rotation) in the beams, in one hand. In other hand, RC walls have been introduced at the ends of the RC frames, in order to relocate the plastic hinges (to push them to the central part of the beams) and to reduce their sizes. Those walls have been modelled with quadratic elements and an elastic behavior [²³].

Presentation of the experimental model

The experimental model stems from Vecchio F works [²,³,³⁵], where a two story’s RC frame structure is considered and tested experimentally. The size and the reinforcement of the structure are presented in Fig. 1. In order to represent the constant live loads, two vertical constant forces of 700 kN each are applied on top of the RC frame structure. A lateral displacement is imposed and the corresponding load is measured until the failure of the structure.

Finite element model

As mentioned in the introduction, the global approach is used in the current study, where the columns and beams of Vecchio RC frame (Fig. 1) are modeled by the Timoshenko beam element and the sectional behavior is modeled with the Modified Takeda model. It found that this Takeda model has enough ability to reproduce the response of the RC frame structures under lateral and seismic loads [²⁷,³⁶].

Tekeda Model

This model is assumed to be a bending model, which is characterized by a tri-linear moment curvature law of RC cross section, for more details, see Refs. [²⁷,³⁶]. The moment curvature characteristics of a given cross section can represent the deformation properties of an RC section as shown in Fig. 2.

Typically, point A (F_cr,M_cr), indicate the cracking point where the concrete starts cracking in the initial stages (M=M_cr), and the response is elastic and linear. With an increasing of the applied moment, the cracking of the concrete reduces the flexural rigidity of the section [³⁷,³⁸]. At the high load level, corresponding to point B (F_y,M_y), the tension of the reinforcement reaches the yield, followed by the crushing of the concrete at point C(F_u,M_u) [³⁸,³⁹].

The laws behavior introduced into the software (CASTEM2000) consist of a moment-curvature (Fig. 2) relationship designed to take into account the real behavior of the RC/cross section at the macro level. This law has been calculated using sectional analysis software (Response2000) based on the modified compression theory [⁴⁰]. This program uses a combination of axial, shear and moment load to find he full nonlinear response of the section, especially the moment curvature, this law taken into account the tension stiffening.

As the Tekeda model hasn’t a failure criterion in the monotone load, we proposed in this study a failure criterion in order to be as possible as close to the experiment.

Proposed failure criterion

This criterion is based on the failure of the plastic hinge, which can be evaluated by the curvature or rotation developed at each section along the length of the considered element. The developed curvatures or rotations are compared to those defined by the FEMA 273. When the plastic rotations found by the finite element model equal or exceed those given by FEMA 273, we consider that the section is failing, and when a sufficient number of sections fail, it can be considered that the entire RC frame collapses (appearance of failure mechanism).

Plastic rotation calculation

The strain energy in the structure is dissipated by the formation of plastic hinges in the end zones of an element without affecting the rest of the structure. Several analytical models [³⁷,⁴¹–⁵³] have developed semi-empirical formulae (analytical models) in order to estimate the plastic rotation q_p [³²].

The rotation of an element can be determined from the curvature distribution along the length of the element [³⁹,⁵⁴]. Therefore, the rotation between two points, A and B (Fig. 3(c)), is equal to the area under the curve between these two points, analytically it is given by Eq. (2).

(1){{\displaystyle \theta }}_{\mathrm{A}\mathrm{B}}={\displaystyle {\int }_{\mathrm{A}}^{\mathrm{B}}\varnothing (x)}\mathrm{d}x,

where q is the rotation of an element, x is the distance of the elementary element dx from B, and ∅ is the curvature between points A and B (see Fig. 3(c)).

The condition at the ultimate load stage of an RC member is shown in Fig. 3, for loads values under the yield moment M_y, the curvature increases gradually from the free end of the member (point A) to the point B supported by the column.

There is a large increase in curvature at first yield of the tension steel. At the ultimate load stage, the value of the curvature at the support increases suddenly, causing a large plastic deformation. Since the concrete around the cracks can carry some tension (tension-stiffening), a fluctuation of the curvature along the member length can be noted. Each of the peaks of curvature corresponds to a crack. The actual curvature distribution at the ultimate load can be idealized (simplified) into elastic and plastic regions (Fig. 3(c)), thus the total rotation, q_t, over the member length can be divided into elastic and plastic rotations. The elastic rotation, q_e, (before reaching the yield stress of the reinforcements) can be obtained using the curvature at yield. The plastic rotation can be determined, on each side of a section by Eq. (3) [³⁷,³⁹].

(2){\theta }_{\mathrm{p}}={\displaystyle {\int }_{\mathrm{0}}^{l_{\mathrm{y}}}\left|\mathrm{\varnothing }(x)-{\mathrm{\varnothing }}_{\mathrm{y}}\right|}\mathrm{d}x

Where \varnothing (x) is the curvature at distance x from the section at the ultimate load, {{\displaystyle \varnothing }}_{\mathrm{y}} is the curvature at yield, and l_y is the length of the member segment over which the maximum moment exceeds the yield moment (yielding length). The hatched area in Fig. 3(c) is the plastic rotation (q_p), which occurs after the elastic rotation of the plastic hinge at the ultimate load [³⁷,³⁹].

Provisions for plastic hinge rotation capacities of RC members by FEMA 273

The nonlinear procedures of FEMA require definition of the nonlinear load deformation relation. Such a curve is given in Fig. 4.

Point A corresponds to the unloaded condition. Point B corresponds to the nominal steel yield strength. The slope of line BC is usually taken between 0% and 10% of the initial slope (line AB). Point C has resistance equal to the ultimate strength. Line CD corresponds to initial failure of the member. It may be associated with phenomena such as fracture of the bending reinforcement, concrete spalling or shear failure. Line DE represents the residual strength of the member. It may be non-zero in some cases, or practically zero in other cases. Point E corresponds to the deformation limit [⁵⁵].

However, we usually consider the initial failure at point C, as well as the deformation limit, and therefore points E, D and C have the same deformation. In this study, we consider that the failure is reported to the points: Immediate Occupancy (IO), Life Safety (LS) and Collapse Prevention (CP), are used to define the damage level for the plastic hinge [⁵⁵]. Table 1 represents the damage level (IO, LS, and CP) of the plastic hinge according to the developed rotation on the element. In this study, we consider that the failure is occurred when the CP damage is reached.

Table 2 represents the plastic rotations calculated for each damage level of the RC frame studied in this paper for the considered section presented in Fig. 1.

Simulation results of the Vecchio RC frame

Figure 5 represents the comparison between the numerical and the experimental results. In global manner, the numerical model can predict in acceptable way the elastic and the plastic response of the RC frame.

This figure shows also that, the numerical model predict very well the load and the displacement corresponding to the failure of the beams, as it is shown in Fig. 5.

Table 2 presents the local obtained results of the simulated RC frame in terms of damage level with its corresponding load and displacement for each beam.

The beam of the first level undergoes a damage level of IO (immediate occupancy) at the left and right zones, near the two nodes of the RC frame, for loads of 271.194 kN and 278.969 kN, respectively. With load increasing, the damage level of IO passes to damage level of LS (life safety) for loads of 300.284 kN and 301.981 kN at the left and right zones of the beam, respectively. After reaching the two mentioned damage level, the failure of this first beam appears in the left and right zones for loads of 324.017 kN and 325.569 kN, respectively. At those loads, the plastic rotation developed in the finite element model in the left and right zones of the beam are greater than 0.02 rad (those calculated by FEMA 273), so the beam is considered as fail.

The beam of the second level follows the same damage path as the first beam and the failure is occurred in the same way as the first beam. The failure load on the left and right zones are 327.004 kN and 325.569 kN, respectively.

As indicated in Table 2, the load and the displacement corresponding to the failure of the beam of the first and the second level in the experiment are 327.716 kN and 0.08617 m, respectively, and in the numerical is around 327 kN and 0.08618 m, respectively. From theses loads and displacements values corresponding to the failure, it can be concluded that the numerical model can predict very well the experimental response of the RC frame structures.

Regarding the columns, no failure is observed in the experiment as well as in the numerical results. However, the columns underwent only damage at the base for the failure load of beams (see Fig. 5). It is also interesting to mention that only the failure of the beams leads the failure of the whole RC frame.

Study of the effect of RC wall in the RC frame

The studied RC frame structure (Fig. 6) is made of columns and beams of the same dimensions as the experimental RC frame, in which it added one level and two frames (Fig. 6) in order to view the positioning and the size of the plastic hinges in the beams. In order to reproduce the effect of an earthquake on the RC frame, a pushover analysis has been used [²⁸,²⁹]. The RC frame has been subjected to a triangular lateral load with intensity increasing gradually (incrementally pushed) [³²,⁵⁵]. The increasing lateral load, at which the various structural components reach failure, is recorded function of the roof displacement (load-displacement). This incremental process continues until the ultimate structure displacement is attained (occurrence of plastic hinges).

The RC walls used have a length of 80 cm (Fig. 7), modeled in CASTEM2000 by quadratic shell elements (4 nodes, each having 6 degrees of freedom, 3 translations and 3 rotations) with linear elastic behavior law. The common nodes between the concrete walls and the RCF structure are merged and are subjected to the same distortions, the same lateral loading as the reference RCF structure is applied (Fig. 7).

The behavior of the RC wall is considered linear elastic; this assumption can be justified in one hand, by the important stiffness in plane of the wall in respect to the beams and columns, in other hand. Odile Morand [²³], tried to represent the behavior of a concrete wall using plastic concrete model (available in CASTEM2000), but after the beginning of the cracking at displacement of 0.7 mm, the nonlinear procedure of CASTEM2000 diverged [²³]. For those reasons, we chose the use of a simplified elastic linear model for the study of the wall behavior.

Presentation of the results

Figures 8–10 show the results in terms of plastic rotation developed in the beams before and after the introduction of the RC wall, and the maximum values of plastic rotation are given in the Tables 3, 4 and 5.

Results interpretation

Figures 8‒10 show the resulting curvature field of the studied RCF structure beams. These results clearly show damaged areas in the beams (Particularly, the beams of first level with f_u=0.4137 m⁻¹); consequently the plastic hinges are formed at the ends of these beams. The comparison between each field provides information about the penetration of hinges in plastic infield. The beams of the first level (Beam 1, Beam 2 and Beam 3) are most damage in respect to the other levels by comparison of the maximum values of curvatures (Figs. 8(a), 9(a) and 10(a)); however, the last level has a better behavior in terms of propagation of curvature (see Table 6).

Table 6 shows the results in terms of plastic rotation, ultimate curvatures. The beams of the first level have larger values in terms of plastic rotations. Indeed, all values are between 0.095 and 0.10573 rad, which mean this level is the most damage specifically the zones 1 Beam 1) and zone 4 (Beam 3) with plastic rotation as 0.10573 and 0.10411 rad, respectively. These latter have reached advanced plastics damage compared to the other level, in the other hand, the beams on the last level sustained minor damage, on which almost elastic behavior is observed with plastic rotation (q_p) values ranging from 0.00882 to 0.06568 rad for the beams 8 and 9 (Fig. 11).

The results obtained concerning the appearance of plastics hinges in the vicinity of the nodes, are in good agreement with the experimental results obtained by Negro and Verzeletti [⁵⁶]. Although a slight difference is noticed which is due to the taking into account of the fillings which prevents the dissipation of energy in the RC frame structure, but as shown by Ref. [⁵⁷] these fillings do not affect the appearance of the hinges in a significant manner.

From the results shown in Table 6, it can be seen that significant damage occurred; they must be reduced in order to improve the performance of this RC frame structure. Since that, the most damaged areas were precisely located; a strengthening technique using the introduction of RC walls has been adopted in order to investigate the possible behavior improvement.

These walls have been therefore placed in a manner to prevent the development of the plastic hinges (Fig. 12) in different zones (ex. Area 1 in beam1 and area 4 in beam 3). The mean idea in this technique is to eliminate the appearance of the plastic hinges in these areas (1 and 4), this in order to understand if the introduction of the RC walls will lead to the disappearance or at least the relocation of the plastic hinges.

Concerning the associated beams with the RC wall, a relocation of the plastic hinge is observed on the reinforced areas. The addition of walls leads to the transfer of effort (load) from nodal zones to the beams center. The initial hinges appeared in the ends of the beams in the initial RCF structure are reduced and relocated.

The beams of the second level (Beams 4, 5 and 6) sustained the most damage in this model with minimum and maximum curvatures of 2.87×10⁻³ m⁻¹and 5.1×10⁻³ m⁻¹ (see Table 7), respectively. Comparatively the beams of the 1st level that have achieved the most significant damage before strengthening, have shown curvatures between 1.42×10⁻³ m⁻¹ and 4.09×10⁻³ m⁻¹, and exhibited improved behavior after strengthening where damage was less significant, particularly in zones 1 and 4.

The comparative analysis of the results (Fig. 13) revealed that plastic rotations exceeds the level 3 of FEMA273 (collapse prevention) and plastic hinges training sequences were almost similar for all three levels (before introduction of the RC/Wall). However analyzing the results after introduction of the RC/Wall, it is clear that all damaged areas have undergone a relief were plastic rotations are all less than the 0.005 rad (Immediate Occupancy) or the rang of a performance level increased from collapse prevention (CP) to immediate occupancy (IO). Moreover, this can be explained by the high rigidity of the RC/Wall by comparing it to that of RC/frame, because the objective of the introduction of the RC/Wall is to make them take maximum seismic efforts to minimize the plasticizing of the elements constituting the RC/frame.

The introduction of the RC wall of 80 cm (L_v=0.80 m), in the RCF structure leads to the relocation of the plastic hinges, from the vicinity of the nodes (in the initial model) to the central area of the beams, are delocalized to a distance X = L_v. The rigidity of the wall had also a reducing effect on the size of these plastic hinges (minimization of plasticized sections and the plastic rotation).

These results (relocation of the plastic hinges), are consistent with those found by Fahjan et al. [⁵⁸], where he used two models (Multi-Layer Shell Model and Mid-Pier Frame Model) for modeling the wall under SAP2000, and with experimental results of Kato et al. [³¹]. Whose main objective was to study the overall behavior of an RC frame structure reinforced by a wall, (or 03 specimens were prepared and analyzed). And these results show that the major damage (appearance of the plastic hinges), followed in the ends of the connecting beams with the wall and the progression of damage occurs beams the first level to the beams of the upper levels.

Conclusions

This paper presented a finite element model under CASTEM2000 code that capable of reproducing satisfactorily the force-displacement global response (Fig. 5). After validation of the approach (using global modified Takeda model with Timoshenko elements) with an experimental result, the study has focused on the study of the development of plastic hinges (local behavior) in a RCF structure.

After the appearance of plastic hinges in the vicinity of the nodes in the RCF structure, and in order to minimize damage to the lamination sections, we introduced bracing RC walls at both ends of the RCF.

The introduction of these walls, enabled us to prevent the formation of plastic hinges in the vicinity of nodes, and then push them to away to the junction between the wall and beam (at distance X = L_v), minimizing the plastic curvatures developed on the one hand, and reducing their length (hinge size) on the other hand.

This resulted in a significant percentage in the majority of areas that form the hinges. The walls have consequently greatly relieved the beams.

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