#### Abstract

The introduction of an eccentricity in this system results in a geometric nonlinearity behavior. The midpoint of the diagonal member is connected to the corner joint using a brace member with a relatively low stiffness, thus forming a three-member bracing system in each braced panel. An iterative method of analysis has been developed to study the nonlinear load-deflection behavior of ODBS. The results indicate that the load-deflection behavior of this system follows a nonlinear stiffness-hardening pattern with two yielding points, which reflect the tensile failure of different bracings; the present study aims to investigate the efficiency of applying off-diagonal steel braces to reinforced concrete frames. To achieve this, three types of 2-story, 6-story, and 15-story structures without and with X-bracing and off-center bracing systems were modeled using SAP2000 software, and for micromodeling ANSYS software was used to achieve finite element results for an exact comparison between various retrofitting systems. The results showed that the structures strengthened by toggle bracing system revealed better behavior for low oscillation periods. Moreover, this type of bracing system is quite suitable for 10-story structures but not for higher ones. Its main problem, which requires special contrivances to solve, is the existence of a soft ground floor.

#### 1. Introduction

The earthquake catastrophe is one of the primary reasons for destruction of buildings, engineering infrastructures, and social systems [1]. Earthquake is known as the most common natural disaster since a large number of earthquakes of various magnitudes, which have caused serious damages, have been recorded in historical documents. According to these recorded earthquakes, the death toll is estimated to be around 800,000 people thus far [2]. Iran is located on the largest fold of earth surface which spreads throughout Saudi Arabia-Eurasia region with a surface area of 3,000,000 km^{2}. Therefore, located on Alp-Himalaya chain, Iran is one of the most seismically active regions for which a lot of destructive earthquakes have been recorded thus far [3]. For example, a magnitude 6.6 earthquake hit southeast Iran on December 26, 2003, resulting in thousands of casualties and total destruction of Bam city. The total number of killed people was estimated to be 80,000. Such catastrophes happen not only because of the large magnitude of earthquakes but also as a result of nonstandard structures and weak buildings. However, reconstruction cost more than 10 billion dollars [4]. Appropriate design of structures is known as a method to prevent such damages. Technological developments have made it possible to familiarize with structures behavior towards earthquake through laboratory experiments. Nonetheless, experimental studies are very difficult and time-consuming [5]. This makes the application of computer modeling methods a priority.

Nowadays, the use of reinforced concrete structures increases in Iran. Factors such as availability of required materials, simpler construction procedure, and possibility for creation of larger spaces persuade designers and engineers to employ this system [6, 7]. In concrete frames, shear wall is commonly used for lateral bracing. In recent researches, application of steel braces has also been studied as an alternative because of the earned structure ductility. On the other hand, it is worth understanding if application of off-diagonal bracing system (ODBS), which earns steel structures more ductility, makes reinforced concrete structures more ductile or not.

#### 2. Aim of the Study

The present study mainly aims to investigate the effect of high ductility of the ODBS braced frame as well as the way of damping the oscillations transmitting to the upper levels which have been strengthened with steel X-bracing system. To achieve this, different frames should be modeled and the seismic behavior of each one should separately be investigated and compared to one another. On this basis, we will firstly model several-story frames with only its first story frame strengthened by ODBS system and the upper stories braced with X-brace system. Then, another frame strengthened by X-bracing system in all stories will be modeled and compared to the former frame.

Another goal of this study is to research the reduction in the impact loads originated when changing the oscillation mode of structure. As it is known to us, a considerable impact load is exerted on bracing components if the direction of forces changes and components stretch under the influence of components’ buckling caused by pressure [8]. This study expects that the amount of lateral forces being transmitted from earth to upper levels, which have been strengthened with steel X-bracing system, and subsequently the effect of impact will decrease as a result of high energy absorption capacity of ODBS system in lower floors of the structure.

#### 3. Study Literature

The idea of steel bracing system application to reinforced concrete buildings was first suggested for seismic strengthening of concrete buildings. From the viewpoint of both research and application, this idea has been very prevalent during past two decades because of the simplicity of its implementation and its relatively lower cost compared to shear wall. For example, Sugano and Fujimura performed a series of experiments on a model of one-story frame which had been strengthened through various methods. They examined the frame samples with X- and K-Shape bracing systems and compared them to the samples strengthened by concrete- and masonry-infilled walls. They aimed to determine the effect of each of these systems on enhancement of in-plane strength and ductility of the samples [9]. Furthermore, Kawamata and Ohnuma demonstrated the possibility of the effective use of steel bracing systems in concrete buildings [10].

A model of two-span second stories reinforced concrete frame with a scale of 1 : 3 was chosen to represent the seismic weaknesses of the structures of this type. The strengthened frame was exposed to lateral and gravity loadings and its displacements were allowed to increase by one fiftieth of the frames’ original height (allowable drift). The strengthened inside frame by a ductile steel bracing demonstrated considerably better behavior than the preliminary reinforced concrete frame [11] or applied from outside the frame [12].

In 1999, the direct internal use of steel bracing system in concrete frame was studied in laboratory. Experiments were carried out on five one-span one-story frame samples with a scale of 1 : 2.5. Two of them had no bracing system but the other three samples were strengthened by X-bracing systems with different component connectors including bolt and nut, cover of RC column, and plates placed in concrete. The prepared frames were exposed to constant gravity and lateral cyclic loadings. Results showed, depending upon various component connectors, the bracing system considerably increases the equivalent stiffness of the frame and notably changes its behavior. When the bracing connector is implanted inside concrete, the performance of frame gets even better and further energy is absorbed. Generally, experiments demonstrated that bracing tolerates a major part of lateral load in reinforced concrete frame [13]. A ten-story reinforced concrete structure in four cases. Results revealed that the existence of shear walls extensively enhances structure’s stiffness and strength but decreases its ductility. In comparison with shear wall, the application of steel bracing system increases the structure’s ductility, although it has negligible effect on its strength [14].

Dynamic behavior of the concrete buildings strengthened with concentric bracing systems has been investigated by Abou-Elfath and Ghobarah. A three-story building was dynamically analyzed with various earthquake records and the effect of steel bracing system on building as well as the effect of bracing system distribution throughout frames’ height was studied. This study on the placement of braces investigated the seismic performance, relative and total drift of floor, and destruction index to show the effect of this type of bracing systems [15].

Maheri and Akbari first reviewed previous studies on strengthening by steel bracing systems and then investigated three models including a simple frame, a frame strengthened with X-bracing, and a frame strengthened with knee bracing system under lateral load until failure stage. They found that ductility of RC frame considerably increases when using knee bracing system [16].

In 1995, Moghaddam and Estekanchi modeled and tested off-centre bracing systems in steel frames for the first time. Later in 1999, they analyzed the seismic behavior of off-diagonal bracing system. They confirmed that this system’s behavior resembles that of seismic isolators and plays a considerable role in reduction of seismic forces [17, 18]. All previous studies confirm the effectiveness of steel braces in rehabilitating and retrofitting of any RC system.

#### 4. Study Methodology

Macromodeling method was used to analyze the nonlinear behavior of reinforced concrete frame strengthened by steel bracing system (macroelement by lowest accuracy related to micromodeling elements). The model was calibrated using existing laboratory work results and then a larger number of floors and openings were analyzed. The SAP2000 software was employed to model the reinforced concrete frame braced with ODBS system. Dynamic time history analysis was done on the concrete frames with more floors and openings. Dynamic time history analysis using earthquake accelerograms is one of the methods suggested by most regulations to investigate the seismic behavior of structures. This study used three accelerograms of Naghan, Tabas, and El Centro. Their general characteristics are listed in Table 1.

The various maximum ground acceleration after scaling is set to 0.3 g. Three groups of records are selected based on two parameters: the closest distance to a fault rupture surface [greater than 50 km (far field) and nearer than 10 km (near fault)] and the moment magnitude in every scales [19, 20]. The other characteristics of the real accelerograms such as directivity and fault mechanism are the same. The peak ground acceleration of all accelerograms is greater than 0.1 g. These accelerograms are selected from strong ground motion records. The specifications and classification of each group before doing matching procedure are tabulated in Table 1.

Since the characteristics of these earthquakes are different from other places, they have to be scaled to one scale before using them for nonlinear dynamic analyses of the studied models. To scale accelerograms using UBC-97 method, the values of natural oscillation period were firstly calculated for our three models. These three models included the models with three spans and 2, 6, and 15 floors. Models were divided into three groups of short, medium, and tall buildings and their natural period was considered to be in three categories of short, medium, and long periods [21]. Models dimensions have considered 3 m for height of stories and 4 m for uniform spans length. Suitable structural periods were selected to be smaller than 1.5T (the major oscillation period of models) (Figure 1). In the next stage, the acceleration spectra were determined for all three accelerograms and knowing the spectrum suggested by regulations, the ratios of regulation spectrum to acceleration spectrum of recent accelerograms were found to vary from 0.2T to 1.5T. The arithmetic averages of the ratios were calculated for this range. In addition, scale factors were also investigated for taller buildings. Figures 1, 2, and 3 show models.

**(a) Two-story models of flexural frame (A), X-Steel (B), and off-diagonal bracing system (C)**

**(b) Six-story models of flexural frame without bracing system (A), with X-Steel bracing system (B), and with off-diagonal steel bracing system (C)**

**(c) Fifteen-story models of flexural frame without bracing system (A), with X-Steel bracing system (B), and with off-diagonal steel bracing system (C)**

**(a)**

**(b)**

**(a)**

**(b)**

Time history analysis should be performed according to previous accelerogram multiplied by modified scaled factor. Simulated accelerograms should be done according to peak ground acceleration of the region. Seismic hazard of the region could clear the quantity of PGA according to related seismic manuals. Details of scale factors are in Table 2.

A rectangular building with dimensions of 15 m × 25 m was chosen for frames. Dead loads which equaled 60 kN·m^{−2} and live loads which were 20 kN·m^{−2} for floors and 15 kN·m^{−2} for ceiling were considered to uniform linearly be 300 kN·m^{−1} for dead loads and 100 and 75 kN·m^{−1} for live loads of floors and ceiling, respectively. The effect of wind and other similar effects were neglected. Tables 3, 4, and 5 show characteristics of the sections for various 2nd, 6th, and 15th story frames. It is worth mentioning that the strain hardening strength of steel materials in postplastic region was considered 3 percent positive during the modelings. After scaling accelerograms, to obtain exact analysis results for verifying the recent responses of the modeling analysis and for using them in subsequent conclusions, there is a need to calibrate the software results with experimental responses. In both of software analyses, SAP2000 and ANSYS, this calibration process is done.

Calibration of obtained results in SAP2000 is performed for flexural frame and X-bracing systems according to previous experimental investigation by Maheri and Ghaffarzadeh in laboratory of Shiraz university [22].

According to Figure 3, the calibration process is iterated till the differences between two experimental and numerical curves are decreased. These differences should be lower than 10 percent without any curve-fitting. According to these curves, software results are acceptable.

To perform nonlinear static analysis for various models, reinforced concrete beams and columns action are in Life-Safety performance level according to FEMA-356 and ATC-40. Also components of steel bracing system (all components of X-bracing and 1st and 2nd components of ODBS bracing system) were analyzed through Life-Safety performance level. For third member of ODBS, the Collapse Prevention performance level is used.

#### 5. Case Study Results

Table 6 gives the obtained values of relative drift of floors. This table illustrates the type of second, sixth, and fifteenth floors under the effect of mentioned earthquakes for flexural frame and the frames strengthened with X-bracing and off-diagonal bracing systems.

After performing dynamic nonlinear analyses by scaled earthquakes, the drifts of different nodes in numerous time steps were investigated for each of the described models. The results obtained for each of the columns listed in Table 6 along with the structural responses for three earthquakes of Naghan, Tabas, and El Centro and the Hysteresis curves obtained for each model using time history nonlinear analysis were evaluated. Regarding the general shape of the modeled fifteen-story building, in two-story structure the node number 2, in six-story building the node number 7, and in fifteen-story building the node number 76 were monitored for their drifts. These curves show that as the number of floors rises the difference between static excess load and time history curves (dynamic excess load) increases. In addition, for two-story model, the variation in dynamical characteristics of structure oscillation results in the variation of the position of Hysteresis curve’s zero point. Such a phenomenon cannot be determined and evaluated by pushover analysis at all.

According to Table 6, in two-story samples the largest drift for Tabas earthquake occurs for the first floor and the second floor’s drift is quite less than that of first floor. In addition, application of X-bracing system results in a decrease of drift to 70 percent of that of the flexural frame. This amount is even smaller for ODBS system since its capability to absorb deformation is quite greater compared to that of flexural and X-braced frames.

The tolerance towards drift is limited for El Centro earthquake and both floors have almost the same amount of drift, while the tolerance for Naghan earthquake is something between those for Tabas and El Centro earthquakes; that is, the amount of drift for two-story sample increases in the sequence of El Centro earthquake < Naghan earthquake < Tabas earthquake. According to the results obtained for six-story building, it is obvious that the greatest amount of drift occurs in lower floors for intense earthquakes like Tabas earthquake and in higher floors for the earthquakes of medium intensity but longer duration like El Centro earthquake. For six-story building, the amount of drifts in different floors of flexural frame is almost the same and does not vary considerably. Nonetheless, X-bracing system decreases the drift of lower floors and ODBS decreases that of second, third, and fourth floors. The result is that the off-diagonal braces act similar to seismic isolators. The criterion for performance of ODBS as seismic isolators is the entrance of concrete and steel construction materials into nonlinear or postelastic region. Therefore, a considerable drift is absorbed by first floor and the drifts of upper floors are balanced to a great degree. In fifteen-story flexural frame samples, the maximum amount of drift belongs to third and fourth floors, while the minimum drift in ODBS system was observed along first, eighth, and fifteenth floors. The reason is the greater number of oscillation modes in taller buildings. Therefore, the high-period earthquakes such as Tabas earthquake have greater effect on fifteen-story tall buildings and result in the maximum drifts. For the earthquakes of higher oscillation period and PGA, like El Centro earthquake, the maximum relative deformations occur for fourth and eleventh floors. The earthquakes like Naghan earthquake, which have lower oscillation period, have smaller effect on fifteen-story building which possesses a great natural period and hence the amount of relative deformation for its floors is smaller. Considering the reducing effect of X-bracing system on structure period, this system intensifies the effect of low-oscillation-period earthquakes, such as Naghan earthquake, on floors’ relative drift. Nevertheless, the relative drift of a fifteen-story concrete building braced with X-bracing system is generally smaller than that of a flexural frame. In the case of ODBS, the effect of bracing system on drift of a fifteen-story concrete structure is a considerably reducing effect for the floors from second to eighth but a negligible effect for the floors from ninth to fifteenth and the ratio obtained for flexural frame and X-braced frame is also obtained here. Eventually, the effect of various steel bracing systems on concrete flexural structure is related to the formation of plastic hinges in flexural frame and steel bracing system components. Adding the X-bracing system to concrete flexural frame, the formation of hinges in steel components is restricted and the possibility of formation of plastic hinges in beam components, especially in concrete frame column, will be weak.

Adding the ODBS to concrete flexural frame, not only is the formation of more plastic hinges in components of beam and column, but also the systems ductility increases specially for three- to ten-story structures, as a result of producing axial plastic hinges of ODBS components. Considering the design of frame sections based on linear static analysis, a limited number of plastic hinges are formed in flexural frame and all of the members will not be able to produce plastic hinges, but when the steel off-diagonal bracing system is added to flexural frame, increasing the rotational capacity of RC members and also increasing the number of composed plastic hinge are some indices of increased ductility of ODBS braced RC flexural frame. Many results are generated along this analysis, in ODBS braced RC frame before performing any damages, the third member of this system has been rotated and deflected near the plastic limit. In this hand, the initial sever vibrations have been damped through the flexibility of ODBS system and also its members elongation and energy absorption. Figure 4 illustrates the formation of plastic hinges and their rotational capacity.

**(a) Flexural frame**

**(b) X-braced frame**

**(c) ODBS braced frame**

Contour shape of plastic hinges is indicated in Figure 4. As shown in this figure, the off-diagonal system has the highest level of deformation not only in RC frame members but also in steel members, especially in third member of steel ODBS. The rotational capacity is increased in ODBS, the most ductile system. Performance levels of flexural and X-braced frames are limited to Life Safety (LS) level, but for ODBS, level of performance has been extended to higher ductility about related design criteria. Six-story flexural frame has plastic hinges more than X-braced frame. On the other hand in ODBS, more members are contributed in absorption of defined existing energy by nonlinear ductile behavior of plastic rotation and deformation proportional to flexural frame. Nonlinear static analyses as well as dynamic step by step seismic analyses are performed and special purpose elements are employed for the needs of this study. Results show the influence of the maximum rotation of the multistory frame members in terms of ductility requirements and rotational requirements of the frame members [24].

Acceptance criterion for flexural frame of LS level is 0.02 for primary components and also the acceptance criteria for ODBS braced frame of CP level are 0.025 and 0.05 for primary and secondary components, respectively. A more detailed scrutinizing of the results reveals that the hinges formed in ODBS system endure the maximum deformation and earn the structure a very high performance level along ductile behavior. In addition, the more performance levels of plastic hinges are gathered in the structure, so by this level of ductility, the structure will absorb more quantity of energy. These results are deduced based on nonlinear dynamic step by step analyses. The time steps for this analysis is considered less than s. Future research will be dedicated to the full time history analysis and investigate the proportional hysteresis curves. Assessing the stiffness and/or the strength degrading is the most important to diagnosing the exact behavior of this system.

According to Fema-356, the plastic rotation of mentioned beams and columns of flexural frame is 0.025 rad and 0.02 rad, the plastic rotation of mentioned beams and columns of flexural frame is 0.025 rad and 0.02 rad, respectively. These quantities are 0.05 rad 0.03 rad in the system like ODBS by high ductility and therefore the structural damage can be prevented to a great extent. This is why the structure’s ductility and its capacity of energy absorption decrease considerably when the structural performance is limited to the formation of first crack [25]. As it is obvious, when the hinges occur in the beam and columns’ concrete elements, the drift and rotation of the frame attain to C and D points in the ODBS pushover curve, which will satisfy the safety requirements versus any deterioration related to its hazard and risk level. Thus, it is concluded that the application of steel ODBS in concrete flexural frame is quite suitable and earns the structural outstanding characteristics from the viewpoint of the economy and ductility of reinforced concrete frame.

Also in micromodeling phase the flexural RC frame was designed according to ACI concrete manual. In all modeling specimens main RC frame is flexural and these bracing systems are added to the main flexural frame. ANSYS scale modeling is 2/5 because that in experimental models used the same scale factor for geometry characteristics (Figure 9). The finite element frame model had 1.76 m length and 1.36 m height and 0.16 × 0.14 m of rectangular section dimensions. The top and bottom longitudinal reinforcements of the beam section were 2 M10, by means of 4 M10 totally (Rebar’s diameter of 11.3 mm). The column longitudinal reinforcement was 4 M10 (Rebar’s diameter of 11.3 mm). In the plastic hinge regions (350 mm of each end), the transverse reinforcement of beams and columns consisted of 6 mm steel wires spaced at 35 mm and for other places far from plastic regions, the 6 mm steel wires spaced at 70 mm are used. The beam-column joint was transversely reinforced with two 6 mm wires. Reinforcement properties and other specimens’ details are shown in Figure 5.

**(a)**

**(b)**

For the braced RC frames, two, mm, steel plates were placed at each of four inner corners of the RC frames prior to casting the specimens [26]. Each plate was anchored to the RC frame using four-5/8 inch headed studs as shown in Figure 5. Self-consolidated concrete with 28-day compressive strength of 40 MPa was used to cast the frames. The yield strength of the steel reinforcement was also measured as 400 MPa. Bracing members were then installed by welding their gusset plates to the previously anchored steel plates. A double-angle brace cross section, consisting of two mm angles, giving a cross-sectional area of 300 mm^{2}, was chosen for the flexural frame and a 30 × 3.5 mm channel with a cross-sectional area of around 500 mm^{2} selected for the frame ; see Figure 5. The brace members had yield capacities of 300 MPa. For frame braced by off-diagonal bracing system (), all RC cross sections and first and second steel braces are same as details and properties of model (x-braced frame) and their difference is only in third member of bracing system. The difference in the brace member cross section, therefore, made the frame somewhat stronger than the frame.

The beam-column joint of the moment frame was transversely reinforced with two 6 mm steel wires in accordance with the special seismic provisions of the ACI code (ACI Committee Manual, 318-02). For the braced frame, the stirrups of the column were continued in the joint resulting in one 6 mm wire in the joint area. It can be observed that the strains in the one 6 mm wire of the braced frame were about 40% those of the two 6 mm wires of the ductile moment frame. Thus, the use of braced frames is expected to eliminate the undesirable shear failure of beam-column joints without the need for any special joint detailing [25]. However, this needs further testing to reach final recommendations [26]. Comparison results of pushover analysis indicate that off-diagonal bracing system can increase lateral flexibility by velocity control or on the other hand by increasing damping characteristics. In X-bracing systems after yielding steel members, the strength of RC frame was decreased suddenly.

According to Figure 6 and Table 7 and by considering pushover curves, to calculate the response modification factor, some basic formula is needed. As shown in Figure 7, usually real nonlinear behavior is idealized by a bilinear elastic perfectly plastic relationship. The yield base shear coefficient of structure is shown by and the yield displacement is . In this figure, corresponds to the elastic response strength of the structure. The maximum base shear ratio in an elastic perfect behavior is [27]. The ratio of maximum base shear coefficient considering elastic behavior to maximum base shear coefficient in elastic perfect behavior is called force reduction factor:
The overstrength factor is defined as the ratio of maximum base shear coefficient in actual behavior to first significant yield strength in structure :
The concept of overstrength, redundancy, and ductility, which are used to scale down the earthquake forces, needs to be clearly defined and expressed in quantifiable terms. To design for allowable stress method, the design codes decrease design loads from to . This decrease is done by allowable stress factor [27]:
The range of this factor is about 1.4 to 1.5. In this paper allowable stress factor was considered as 1.4 [28]:
Equation (4) shows the seismic response modification factor () in ultimate strength design method. Also, (5) indicates seismic response modification factor in allowable stress design method. Structural ductility, *μ*, is defined in terms of maximum structural drift () and the displacement corresponding to the idealized yield strength (), as given in
The response modification factor is included in the inherent ductility and ductility and overstrength effects of a structure and the difference in the design methods and limitations about related manual. Also Ductility reduction factor is a function of both of the characteristics of the structure including ductility, damping, and fundamental period of vibration and the characteristics of earthquake ground motion, where is the overstrength factor and is termed the allowable stress factor [16].

**(a) Force-displacement curve of flexural frame and deformed shape and sum of elastic and plastic strain contour**

**(b) Force-displacement curve of X-braced frame and deformed shape and sum of elastic and plastic strain contour**

**(c) Force-displacement curve of ODBS steel braced frame and deformed shape and sum of elastic and plastic strain contour**

#### 6. Material Nonlinearity

Concrete and steel are the two constituents of RC braced frame. Among them, concrete is much stronger in compression than in tension (tensile strength is of the order of one-tenth of compressive strength). While its tensile stress-strain relationship is almost linear, the stress-strain relationship in compression is nonlinear from the beginning. But for specification of nonlinear geometry of ODBS, concrete nonlinearity is added to material nonlinearity in this paper. Only steel nonlinearity for third member of ODBS is considered in this paper’s analysis. Steel, on the other hand, is linearly elastic up to a certain stress (called the proportional limit) after which it reaches yield point () where the stress remains almost constant despite changes in strain. Beyond the yield point, the stress increases again with strain (strain hardening) up to the maximum stress (ultimate strength, ) when it decreases until failure at about a stress quite close to the yield strength. The elastic-perfectly-plastic (EPP) model for steel, which is used in this work, assumes the stress to vary linearly with strain up to yield point and remain constant beyond that [29].

In this research the Willam-Warnke, the yield and failure criteria, is considered for concrete model behavior. Also, since the SAP2000 assumption applies the Drucker-Prager criteria for concrete material modeling and its behavior, both of mentioned criteria’s are considered in analysis of models. By this method, the analytical comparison of applied criteria’s is done.

In steel material modeling, the bilinear curve of behavior is used. This model is included in two parts, linear and elastoplastic behavior. The elasticity modulus is kg/cm^{2} for linear part and kg/cm^{2} for the nonlinear part of the behavior. These specifications are indicated in Figure 8.

**(a)**

**(b)**

**(a) ODBS frame (deformed shape)**

**(b) X-braced frame (deformed shape)**

Nonlinear static procedures use equivalent SDOF structural models and represent seismic ground motion with response spectra. Story drifts and component actions are related subsequently to the global demand parameter by the pushover or capacity curves that are the basis of the nonlinear static procedures. Nonlinear dynamic analysis utilizes the combination of ground motion records with a detailed structural model and therefore is capable of producing results with relatively low uncertainty. In nonlinear dynamic analyses, the detailed structural model subjected to a ground-motion record produces estimates of component deformations for each degree of freedom in the model and the modal responses are combined using schemes such as the sum of squares square root.

In nonlinear dynamic analysis, the nonlinear properties of the structure are considered part of a time domain analysis. This approach is the most rigorous and is required by some building codes for buildings of unusual configuration or of special importance. However, the calculated response can be very sensitive to the characteristics of the individual ground motion used as seismic input; therefore, several analyses are required using different ground motion records to achieve a reliable estimation of the probabilistic distribution of structural response.

Since the properties of the seismic response depend on the intensity, or severity, of the seismic shaking, a comprehensive assessment calls for numerous nonlinear dynamic analyses at various levels of intensity to represent different possible earthquake scenarios. This has led to the emergence of methods like the incremental dynamic analysis [30].

Complete comparisons of the studied Retrofitted Frames in ANSYS (version10) software with the micromodeling structural element indicate that ODBS steel bracing RC frame has two yielding points that were related to main RC flexural frame and third steel member of ODBS. It is so useful for structures that are under impact loads and loads by high velocity specifically according to Figure 2 in last pages.

The main flexural RC frame is calibrated by results of experimental modeling of the same flexural frame and X-braced frame that were constructed in laboratory [31].

According to Figure 10, compared results of experimental and numerical modeling have been shown by force-displacement diagram control. Numerical modeling is based on exact modeling and real parameters [32]. This diagram shows that errors in modeling were minimized and convergence condition was satisfied.

**(a) Flexural frame calibration**

**(b) X-braced frame calibration**

According to Figure 10, the calibration process is iterated till the differences between two experimental and numerical curves are decreased. These differences should be lower than 10 percent without any curve-fitting. According to curves comparison, software results are accepted strongly for flexural frame and are accepted relatively for x-braced frame.

Figure 11(a) indicates differences between capacities of load-deflection curves for three types of reinforced concrete frame that are analyzed in ANSYS software. Two types of them consisted of high strength steel bracing (X-brace) and ductile steel brace (ODBS) and another one is flexural frame. This figure shows the maximum strength is through X-braced RC frame and the maximum ductility is through RC off-diagonal braced frame. The flexural frame has minimum strength and moderate ductility. Figure 11(b) is the same as Figure 11(a) but in SAP2000 software. According to Figure 11(b), the curve for mixed braced frame is about a structure that consisted of frames by ODBS in first story and X-brace for other stories but the curve of OBS braced frame is about a structure that consisted of ODBS in all stories.

**(a)**

**(b)**

Comparison between shown results in Figure 11 and performing additional calculation to obtain -factor of behavior that is in accordance with Table 7 indicate high flexibility of ODBS braced RC frame, although -quantity was obtained by linear relation between yielding displacement and maximum displacement of monitored point on top of the flexural RC frame, that maybe not exact. To retain exact solution result of -factor (response modification factor) experimental assessment is required.

Also to optimize eccentricity ratio according to Figure 12, several models by various eccentricities have been modeled (see Table 8). The effect of normalized eccentricity on the stiffness is shown in Figure 13. For the ( on shape) values lower than 0.2, the influence of normalized eccentricity is on the stiffness and the ductile behavior is decreased. It can be noticed that, for constant values of , maximum stiffness is obtained when normalized eccentricity ratio is equal to 0.5, by means that the best influence of eccentricity () will occur when the OA (length of the third member of ODBS) is parallel to concrete frames’ diagonal. Also, the maximum stiffness of the frame is obtained when point O lies on the diagonal BC (i.e., ). Maximum ductility is by considering eccentricity and related design code limitations on story drift, concurrently.

**(a)**

**(b)**

**(a) Maximum Acceleration**

**(b) Ductility Assessment**

After performing pushover analysis and results of deformations along shear force, many parameters have been recorded. Ductility and factor of behavior (response modification factor ()) are indicated in Table 7. Results indicate ductility parameter increase with retrofitted RC frame braced off-diagonal system. Recent researches on base-isolations indicated that they are so ductile and hysteretic. The off-diagonal bracing system behaves the same as base-isolations in ductility behavior (regardless of drift limitations). Table 7 components and numbers demonstrate off-diagonal bracing system has higher ductility even more than flexural frame.

Effect of lateral force will decrease when the factor of behavior is increased in a structure. Also frame resistance was optimized in off-diagonal bracing system related to flexural frame.

The dynamic behavior of the mentioned frames under three records (the earthquake records considered in this research are Tabas, Naghan, and El Centro whose peak ground accelerations are 0.93 g, 0.72 g, and 0.35 g, resp.) of Tabas, Naghan, and El Centro has been studied. Each typical 4-bay frames with different stories (2, 6, and 15) has been adopted accordingly. Structural response for each frame versus eccentricity ratio was determined and the optimum eccentricity value is obtained comparing each other. The effect of the parameters such as period time of vibration, initial stiffness, and displacement at first yield point compares various eccentricity ratios. Optimum eccentricity was around to 0.4 according to normalized eccentricity in Figure 13. Primary relationship has been proposed between several eccentricity ratios in different levels of ductility.

Also the dynamic behavior of the 6-story frames retrofitted by off-diagonal bracing system under three records of Tabas, Naghan, and El Centro has been compared for optimizing normalized eccentricity ratio. Overall result related to several eccentricities has been shown in Figure 13.

In next step, the dynamic time history analysis was applied on single off-diagonal braced frame. Acceleration excitation was exerted on joint-1 at the base level of ODBS retrofitted frame and then joint-2 response at floor level of frame was monitored (Figure 14). It can be seen that the displacement, velocity, and acceleration response of off-diagonal braced frames in all cases are less than frames without bracing system for all different time durations of earthquake. The response of El Centro earthquake (scaled time history response analysis of El Centro by 0.3 g compared with response of floor level joint-2) is compared in Figure 13. These results show that installation of ODBS in simple flexural frames causes 40% to 60% improvement in dynamic response. The structure will remain in elastic region due to this amount of reduction in displacement and internal forces of structural members when ODBS characteristics select correctly.

**(a)**

**(b)**

Another investigation of ODBS is about its dynamic behavior under various earthquakes. Many difference ratios in 15-story drift are shown in Figure 15. These drift quantities which are assessed along several earthquake accelerograms, Tabas, Naghan, and El Centro, are investigated for assessing ODBS braced frame. Results as shown in Figure 15 indicate the ODBS decreases drift for stories from 2 to 15 and increases 1st story drift. This first drift quantity is just for investigation, but for obtaining exact results, 1st story drift should be limited to permitted drift and allowed plastic rotation in accordance with related structural manuals by increasing required stiffness in ODBS bracing members and/or in flexural rigidity of reinforced concrete frame.

**(a) Flexural frame**

**(b) X-braced frame**

**(c) ODBS braced frame**

#### 7. Conclusions

Performing dynamic analyses on various models, hysteresis behavior curves were obtained for the studied models. Results showed that as the structure period increases the energy absorption capacity also rises [33]. The more the intensity of selected earthquake is, the higher the performance of ODBS in concrete flexural frame will be. The consequences of intense earthquakes are more noticeable for ODBS. The ODBS bracing system increases the basic shear endured by structure as well as structure’s drift, causing the increase of hysteresis curve’s surface area which, in turn, leads to the enhancement of modified ductility or system flexibility. The flexural frame undergoes a higher level of drift but its shear strength is quite less compared to the frames with X-bracing or off-diagonal bracing systems. Indeed, as the X-bracing system is added to frame, its ductility decreases considerably, but in the case of the ODBS system, not only the ductility does not decrease but also the system strength and finally, its energy absorption performance rises.

Application of ODBS to tall buildings weakens ground floor columns; that is, in the case of using ODBS which offers no resistance to column’s stretch and pressure forces on foundation, all the above forces are transmitted through columns. Thus, to achieve a suitable performance of ODBS in tall buildings, the ground floor columns should be strengthened through special design of the columns and beams with specific flexural connections. Modification of ground floor columns and creation of special flexural frame with special flexural connections prevent the formation of soft story on ground floor of tall buildings. The above studies on fifteen-story building, which was designed based on linear static analysis, were carried out based on Naghan and El Centro earthquakes which caused greater destruction in ground floor columns.

In addition, plastic hinges did not form in most of the beams, columns, and X-braces of upper floors and before the ODBS system could show its efficiency in making tall buildings ductile, ground floor columns transformed to mechanism and reduced the efficiency of entire system energy absorption. By neglecting P-Δ effects and M-P interactions, the model will continue its efficiency at various performance levels. As a general result, the different stiffness of floors is proportional to each floor’s shear absorption level. Considering the drift and energy absorption potentials of various floors, the stiffness of each floor should be controlled.

Eventually, nonlinear dynamic analyses of flexural frame, X-bracing system frame, and steel ODBS frame revealed a greater ductility for the frame strengthened by ODBS system. The effect of ODBS system on concrete buildings up to ten-story is desirable but not considerable for taller structures. The height of a building will be acceptable only when the structure’s basic oscillation on first mode can cause basic shear and the amount of this shear is at least 40 percent of the final shear of the structure. If the number of floors is greater, ODBS system will not be suitable for reinforced concrete frame and may result in undesirable consequences. Structure underwent three accelerations which were different in intensity, period, and oscillation range and their responses to each case were investigated. The findings demonstrate that application of ODBS is suitable for the structures which undergo low-oscillation-period earthquakes. ODBS itself does not possess a considerable lateral strength but when combining with concrete flexural frame system, the strength of entire system rises by 10 to 45 percent. This effect is because of the behavioral nature of ODBS system so that when composed to stretching, its components start to form an internal force that helps the system to tolerate the stretch. As a result, an extra strength of about 10 to 45 percent is earned when concrete frame is strengthened by steel ODBS system.

#### Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The author would like to express his deep gratitude to Professor Mahmoud Reza Maheri, Professor Abdolrasoul Ranjbaran, and Professor Seyed Ahmad Anvar, for their patient guidance, enthusiastic encouragement, and useful critiques of this research work in Shiraz University. The author would also like to thank Dr. Hamid Seyedian, for his advice and assistance in keeping his progress on schedule. He would also like to extend his thanks to the technicians of the laboratory of the structural department of Aisan Disman Consulting Engineers for their help in offering him the resources in running the program. Finally, he wishes to thank Roza for her support and encouragement throughout his study.