(behaviour) factor, R, for reinforced

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Analytical investigation of response modification (behaviour) factor, R, for reinforced concrete frames rehabilitated by steel chevron bracing Reza Akbaria; Mahmoud R. Maherib a Department of Civil Engineering, Isfahan Science and Research Branch, Islamic Azad University, Isfahan, Iran b Department of Civil Engineering, Shiraz University, Shiraz, Iran First published on: 18 May 2011

To cite this Article Akbari, Reza and Maheri, Mahmoud R.(2011) 'Analytical investigation of response modification

(behaviour) factor, R, for reinforced concrete frames rehabilitated by steel chevron bracing', Structure and Infrastructure Engineering,, First published on: 18 May 2011 (iFirst) To link to this Article: DOI: 10.1080/15732479.2011.575166 URL: http://dx.doi.org/10.1080/15732479.2011.575166

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Structure and Infrastructure Engineering 2011, 1–9, iFirst article

Analytical investigation of response modification (behaviour) factor, R, for reinforced concrete frames rehabilitated by steel chevron bracing Reza Akbaria* and Mahmoud R. Maherib a

Department of Civil Engineering, Isfahan Science and Research Branch, Islamic Azad University, Isfahan, Iran; bDepartment of Civil Engineering, Shiraz University, Shiraz, Iran

Downloaded By: [Akbari, Reza] At: 12:25 18 May 2011

(Received 29 March 2010; final version received 4 November 2010; accepted 20 March 2011) Steel bracing of reinforced concrete (RC) frames has received noticeable attention in recent years as a retrofitting measure to increase the shear capacity of the existing RC buildings. In order to evaluate the seismic behaviour of steelbraced RC frames, some key response parameters, including the ductility and the overstrength factors, should first be determined. These two parameters are incorporated in structural design through a force reduction or a response modification factor. In this paper, the ductility and the overstrength factors as well as the response modification factor (or seismic behaviour factor) for steel chevron-braced RC frames have been evaluated by performing inelastic pushover analyses of brace-frame systems of different heights and configurations. The effects of some parameters influencing the value of behaviour factor, including the height of the frame and share of bracing system from the applied lateral load have been investigated. It is found that the latter parameter has a more localised effect on the R values and its influence does not warrant generalisation at this stage. However, the height of this type of lateral load-resisting system has a profound effect on the R factor, as it directly affects the ductility capacity of the dual system. Finally, based on the findings presented in the article, tentative R values have been proposed for steel chevron-braced moment-resisting RC frame dual systems for different ductility demands and compared with different type of bracing systems. Keywords: behaviour factor; steel bracing; chevron brace; reinforced concrete frame; seismic design

1. Introduction Steel bracing of reinforced concrete (RC) frames has received some attention in recent years both as a retrofitting measure to increase the shear capacity of existing RC buildings and as a shear resisting element in the seismic design of new buildings. Earlier researchers focussed on the retrofitting aspect of bracing and studied external bracing of buildings (Sekiguchi 1988, Badoux and Jirsa 1990, Bush et al. 1991). In the external bracing system, existing buildings are retrofitted by attaching a local or global steel bracing system to the exterior frames. Architectural concerns and difficulties in providing appropriate connections between the steel bracing and RC frames are two of the shortcomings of this method. The other alternative is the internal bracing of individual bays of the RC frames (Sugano and Fujimura 1980, Higashi et al. 1981, Usami et al. 1988, Tagawa et al. 1992). In this method, the building is retrofitted by incorporating a bracing system inside the individual bays of the RC frames. The bracing may be attached to the RC frame either indirectly or directly. In the indirect internal bracing, a braced steel frame is positioned inside the RC frame. As a result, the load transfer

*Corresponding author. Email: [email protected] ISSN 1573-2479 print/ISSN 1744-8980 online Ó 2011 Taylor & Francis DOI: 10.1080/15732479.2011.575166 http://www.informaworld.com

between the steel bracing and the concrete frame is achieved indirectly through the steel frame. This method can be costly and technical difficulties in fixing the steel frame to the RC frame can be inhibiting. Another shortcoming of this method is that the retrofitted frame is susceptible to the diverse effects of dynamic interaction between the adjoining steel frame and the concrete frame during earthquake loading. In the direct bracing method, similar to the bracing of steel frames, a direct connection is provided between the steel bracing and RC frame without the need for an intermediary steel frame. Lately, the direct bracing of RC frames has attracted more attention since it is less costly and can be adopted not only for retrofitting purposes but also as a viable alternative to RC shear walls at pre-construction design level. Experimental works (e.g. Maheri and Sahebi 1997, Maheri and Hadjipour 2003, Maheri et al. 2003, Ghaffarzadeh and Maheri 2006a, 2006b, Said and Nehdi 2008), as well as analytical investigations (e.g. Abou-Elfath and Ghobarah 2000, Ghobarah and Abou-Elfath 2001, Maheri and Akbari 2003, Maheri and Ghaffarzadeh 2008) have studied the capabilities of the direct bracing system of RC frames with encouraging results.


2

R. Akbari and M.R. Maheri

To be able to carry out seismic design of steelbraced RC frames, some key response parameters, including ductility and behaviour factor, should first be established. Comparative experimental and numerical works reported on model X-braced (Maheri et al. 2003, Ghaffarzadeh and Maheri 2006b, Maheri and Ghaffarzadeh 2008) and knee-braced unit frames (Maheri et al. 2003) and for full-scale steel X-braced and knee-braced moment-resisting RC frames (Maheri and Akbari 2003, Maheri and Ghaffarzadeh 2008) have shed some light on these parameters. However, for steel chevron-braced RC frame dual systems, which is a commonly used bracing system, these parameters are still unknown. In this paper, in continuation of the earlier work by the authors regarding the seismic behaviour factor parameters for steel X and knee-braced RC frames (Maheri and Akbari 2003), these parameters are evaluated for the steel chevron-braced moment-resisting RC frames of different heights and brace-frame configurations. The parameters including ductility, ductility reduction and overstrength factors are extracted from response curves obtained from inelastic static pushover analyses of brace-frame system. 2.

Behaviour factor parameters

In force-based seismic design procedures, behaviour factor, R (EC8), or Rw, also referred to by other terms, including response modification factor (FEMA 1997, UBC 1997), is a force reduction factor used to reduce the linear elastic response spectra to the inelastic response spectra. In other words, behaviour factor is the ratio of the strength required to maintain the structure elastic to the inelastic design strength of the structure. The behaviour factor, R, therefore accounts for the inherent ductility and overstrength of a structure and the difference in the level of stresses considered in its design. It is generally expressed in the following form taking into account the above three components, R ¼ Rm Rs Y

ð1Þ

where, Rm is the ductility dependent component also known as the ductility reduction factor, RS is the overstrength factor and Y is termed the allowable stress factor. With reference to Figure 1, in which the actual force–displacement response curve is idealised by a bilinear elastic–perfectly plastic response curve, the behaviour factor parameters may be defined as: Rm ¼

Ve Vy Vs ; Rs ¼ ; Y¼ Vy Vs Vw

and the behaviour factor, R is redefined as:

ð2Þ

Figure 1. Typical pushover response curve for evaluation of behaviour factor, R (Maheri and Akbari 2003).

RðRw Þ ¼

Ve Vy

Vy Vs Ve ¼ Vs Vw Vw

ð3Þ

where, Ve, Vy, Vs and Vw correspond to the structure’s elastic response strength, the idealised yield strength, the first significant yield strength and the allowable stress design strength, respectively. For structures designed using an ultimate strength method, the allowable stress factor, Y, becomes unity and the behaviour factor is reduced to: R ¼ Rm Rs ¼

Ve Vs

ð4Þ

The structure ductility, m, is defined in terms of maximum structural drift (Dmax) and the displacement corresponding to the idealised yield strength (Dy) as: m¼

Dmax Dy

ð5Þ

Many investigators have discussed the two main components of R factor presented in Equation (4), in particular, the ductility dependent component, Rm, has received considerable attention. Reviews of these discussions can be seen in works by Uang (1991), Miranda and Bertero (1994), Kappos (1999), Elnashai (2001) and Mwafy and Elnashai (2002). Ductility reduction factor Rm is a function of the characteristics of the structure including ductility, damping and fundamental period of vibration (T), as well as the characteristics of earthquake ground motion. Newmark and Hall (1982) arrived at a set of equations expressing Rm in terms of the above characteristics. They concluded that for T 4 0.5 s, Rm is effectively equal to the ductility factor of the structure, m. Later

Structure and Infrastructure Engineering works by others including those of Nassar and Krawinkler (1991) and Miranda and Bertero (1994) showed the T-dependence of Rm for T higher than 0.5 s. They also showed the influence of underlying soil type on the values of ductility reduction factor, Rm. Nassar and Krawinkler presented a relation for Rm in the following form: Rm ¼ ½cðm 1Þ þ 11=c

ð6Þ

cðT; aÞ ¼ Ta =ð1 þ Ta Þ þ b=T

ð7Þ

where,

In Equation (7), a is the post-yield stiffness given as a percentage of the initial stiffness of the system and b is parameter given as function of a.


3.

Determination of R factors

3.1. Variable parameters In the braced frames under study, behaviour factor, R, is dependent on a number of variables including the number of storeys, the location of bracing system and the share of bracing system from the base shear. In this study, 4-storey, 8-storey and 12-storey frames are considered. These are typical numbers of storeys used by author in a similar research (Maheri and Akbari 2003) and some other investigators to cover low- to medium-rise framed buildings (e.g. Kim and Choi 2005). All frames are three-bay wide with the central

Figure 2.

Geometry of the dual systems under investigation.

3

bay braced in the braced dual systems. Akbari (2002) investigated the pushover response of both the central and the side braced bay RC frames. He showed that the position of the braced bay has little effect on the pushover response. A study carried out on braced steel frames by Assaf (1989) showed that the number of bays in a frame has little effect on the R values. Geometry of the frames under investigation is shown in Figure 2. Chevron bracing system is considered. In designing the dual systems, the share of bracing system from the lateral load is a parameter to be decided at design stage. In other types of dual systems, very little work is reported on the effect of assigned share of brace from base shear on the R factor. Considering the dual purpose of bracing and RC frame, both as a combined load-resisting element at design level and as a retrofitting measure, it was found necessary to investigate this effect. Three different shares from the base shear of 0, 50 and 100% for bracing are investigated. The steel bracing systems are thus designed to resist the above load shares and the RC frames are designed to resist the remaining codespecified base shears. 3.2.

Design, modelling and analysis

To study the effects of the three variable parameters described above and to determine the R factors for the dual systems under consideration, design base shears were determined for a peak ground acceleration (PGA)


4


of 0.3g. The weights of the systems were assumed to consist of total dead load plus 20% of live load. Since the R factors for steel-braced moment-resisting RC frames were unknown, base shears were calculated using R ¼ 6 for all systems [with comparison to tentative R values of steel X-braced RC frames proposed by author (Maheri and Akbari 2003)]. This, in effect, resulted in all systems of the same height being designed for the same base shear. The storey loads were determined using an inverted triangular distribution over the height. In a study by Mwafy and Elnashai (2002), it is shown that the inverted triangular distribution of load produces better estimates of the maximum drift and R factor compared with uniform and multimodal distributions. The design of frames and bracing systems were carried out using the above loading. The moment-resisting RC frames were designed on the basis of weak beam-strong column principle with intermediate (medium) ductility using the ACI-95 code provisions (ACI Committee 318, 1995) and the steel bracing systems were designed using AISC-LRFD code of practice (1994). Similar dimensions were assumed for all beams and columns in each set of 4-, 8- and 12-storey frames and the desired strength and ductility for each member were achieved through changing the reinforcement steel ratios within the code-specified limits. To evaluate the structural response curve for use in determination of the R factor parameters, two different methods of analysis are generally used. These include: the inelastic pushover (static) analysis and the inelastic dynamic analysis. Owing to the simplicity of the former method compared to the latter, the inelastic pushover analysis is more widely used. In a comparative study, Mwafy and Elnashai (2002) examined the applicability and limitations of the two methods. They concluded that the inelastic pushover analysis is more suitable for the short period (low- to medium-rise) regular frame structures, but for long period (high-rise) and special buildings, the inelastic dynamic analysis is preferable as it is better suited to account for the effects of higher modes. In line with the above conclusion, the inelastic pushover analysis is used in the present study to calculate the R factor parameters. DRAIN-2DX program (Prakash et al. 1993) was utilised to carry out nonlinear pushover analysis of each system. The RC beam and column elements in the program are capable of modelling reinforcements and axial-bending capacity interaction in columns. Takeda’s (1970) bilinear curve was used to model the concrete behaviour with a recommended strain hardening of 0.2%. To model the desired ductility in the RC beams, three beam elements were used to represent each beam, the two outer elements, representing the critical regions for formation of plastic hinges, were designated lengths and properties to achieve the

desired ductility. Truss elements having bilinear response curves with 2% strain hardening were adopted to model the steel braces in the chevronbraced frames. In all the analyses carried out, the P 7 D effects has been included. The connections between the steel braces and between the bracing system and RC frames were modelled by specifying appropriate rigid zones as shown in Figure 3. However, no specific provisions were made for the connections between RC beams and columns. For the finite element modelling of RC frames, a same procedure as Kim and Choi (2005) for the design of steel chevron braces has been adopted. To validate the numerical models used and the assumptions made in the inelastic pushover analysis of the frames, the modelling assumptions were the same concept as another study by author (Maheri and Akbari 2003). It has been shown that good agreements between the numerical and experimental results may be noted for both the pushover curves for unit X and knee-steel braced RC frames. Inelastic pushover analysis of the multi-storey systems under investigation was carried out at horizontal load steps equal to 2% of the design capacity. A constant gravity load equal to total dead load plus 20% live load was also applied to each frame. Selective pushover curves are shown in Figure 4 for chevronbraced frames. To gain a comparative view of the pushover curves, selective curves are also normalised to their respective design capacities as shown in Figure 5 for the unbraced frames and Figure 6 for the braced frames. Figure 5 shows that the response of the 8- and 12-storey frames are somewhat closer to each other,

Figure 3. Finite element representation of the RC framesteel brace system.


Figure 4. shear.

5

Selected pushover response curves for 4-, 8- and 12-storey Chevron-braced frames with different shares from the base


with 100% brace share of load show stronger responses than the systems with 50% brace share of load (Figure 6).

Figure 5. Normalised pushover response curves for unbraced RC Frames (Maheri and Akbari 2003).

Figure 6. Normalised pushover response curves for Chevron-braced RC frames.

whereas the shorter 4-storey frame exhibits a different response to the other two. It should be noted that dual systems, in which the bracing system is designed for 100% share of the lateral load, are in fact designed to resist 125% of the base shear, as the RC frame itself is designed for the code-specified minimum 25% share of the base shear. This point can be seen in the normalised pushover response curves in which dual systems

4. Results and discussion A number of performance parameters may govern the capacity of a structure. In order to carry out an inelastic pushover analysis, one or a number of these parameters should be considered for determination of the displacement limit state (Dmax). For the type of regular, low- to medium-rise, ductile (weak beam, strong column) buildings considered in this study, the global drift (maximum roof displacement) or the interstorey drift are commonly used failure criteria. In a comparative study conducted by Mwafy and Elnashai (2002) on different classes of buildings, a number of global collapse criteria, including interstorey drift limit, column hinging mechanism, limit on drop in the overall lateral resistance and stability index limit, were considered. They concluded that the interstorey drift is the collapse parameter that controls the response of buildings designed to modern seismic codes. For evaluation of R parameters in the present study, the ultimate capacity of each frame was assumed to have been reached when the global drift equalled 1.5% of height of the system. This is in line with National Earthquake Hazards Reduction Program (NEHRP) recommendations for moment-resisting RC frames with intermediate (medium) ductility (FEMA 1997). The R factor parameters for each system were extracted from the respective pushover response curve. The ductility dependent component, Rm, was calculated using Equations (5)–(7) and overstrength component, Rs, was determined from Equation (2). The fundamental period of vibration T was also determined by elastic dynamic analysis of each frame using DRAIN-2DX program. Selective fundamental periods of vibration, T and the behaviour factor parameters, m, Rm, RS and R, evaluated from the pushover curves are listed in Table 1. As expected, with increasing height, the increase in

6


Table 1. Fundamental period of vibration and R factor parameters of the frames. Number of storeys 4 8


12

Share of brace (%)

T (s)

m

Rm

RS

R

0 (Unbraced) 50 100 0 (Unbraced) 50 100 0 (Unbraced) 50 100

0.722 0.342 0.329 1.007 0.577 0.543 1.286 0.926 0.831

2.32 4.76 4.6 2.27 3.85 3.6 2.03 2.89 2.73

2.32 3.55 3.39 2.4 3.14 3.01 2.16 2.32 1.85

2.32 3.1 2.98 1.92 3.06 2.82 1.86 2.98 2.75

5.3 11 10.1 4.6 9.6 8.5 4.0 6.9 5.1

the fundamental period of vibration, T, of the frame is well marked. The reduction in T when an RC frame is braced is also quite noticeable. However, the change in fundamental period of vibration is comparatively less when frames of the same heights are braced with bracing systems having different load capacities. A discussion on the results for R factors follows. 4.1.

Figure 7. The effect of brace share of base shear on R value of chevron-braced frames.

Effects of variable parameters on R factor

Figure 7 show the changes in R factor due to changes in the brace share of base shear for chevron-braced systems. When we consider the effect on different components of R factor, higher shares of load for the chevron-braced system appear to result in lower overstrength factors (Rs). As for the ductility-based component, with the increase in brace load share, Rm decreases, irrespective of the number of storeys. To investigate the effects of the number of storeys on R factor, the two parameters are plotted against each other for different bracing load shares in Figure 8 for chevron-braced frames. It is evident from Figure 8 that the relationship between the number of storeys and R factor is somewhat influenced by the share of steel bracing from the base shear, but a general decrease in R occurs as the number of storeys increases. A similar decrease in R value due to increase in the number of storeys has been reported by Assaf (1989) and Maheri and Akbari (2003). The effects of number of storeys on R are more profound when bracing systems are included to form a dual system. Shorter dual systems (4 storey) exhibit much higher R values when compared to taller systems (12-storey). The reason for the large difference becomes apparent when we examine the different components of the behaviour factor, separately. The overstrength component, Rs, appears to be comparatively little affected by the number of storeys. However, large variations in the ductility dependent component, Rm, may be noted for frames of different heights. In shorter frames,

Figure 8. The effect of number of storeys on the R value of chevron-braced frames.

introduction of the bracing system tends to increase sharply the pre-yield stiffness of the system, hence reducing the value of Dy in Equation (5), while the specified global drift limit, Dmax, remains constant as 1.5% of the height of the system. This in turn greatly increases the ductility and consequently the R value of the braced dual system. In taller frames, however, the bracing system does not increase the pre-yield stiffness of the dual system to the same extent, hence the increase in ductility and R value are of lower magnitudes. 4.2. Tentative R factors for steel chevron-braced RC frames Before proceeding to discuss the R factor for steel chevron-braced RC frames, it is useful to draw some

7


Structure and Infrastructure Engineering comparisons between evaluated R values of the unbraced RC frames considered in this paper and those of the similar RC frames given by some codes of practice. For an intermediate ductility, moment-resisting RC frame, behaviour factors of R ¼ 5.0, R ¼ 5.5 and R(q) ¼ 3.75 are given by FEMA (1997), UBC (1997) and Eurocode 8 (1998), respectively. It should be noted that the R factors given by Uniform Building Code (UBC) and NEHRP are generally much higher than those given by Eurocode-8. Uang (1991) and Miranda and Bertero (1994) are critical of the R factors given by UBC and NEHRP for being too high. The reason presented for their argument is that these codes do not consider the T-dependence of the ductility component of the behaviour factor. On the other hand, in a study, Mwafy and Elnashai (2002) have concluded that the R values recommended by Eurocode-8 underestimate the reserve strength in regular, ductile, buildings. They included the effects of shear displacements and vertical ground motion in their study and used an incremental dynamic collapse analysis, as opposed to the conventional inelastic pushover analysis, to arrive at their conclusion. The R values evaluated for similar momentresisting RC frames in the present study range from 4.0 for the 12-storey frame to 5.3 for the 4-storey frame. These values fall within the range of values given by different codes of practice. This can be considered as a further validation of the methodology used in evaluating the R values, particularly the level of global drift limit, Dmax, adopted in this study, as this parameter highly influences the value of R. Also, bearing in mind the variation in R due to the number of storeys, it follows that NEHRP and UBC slightly overestimate the R factor for taller frames, whereas the R factor given by Eurocode-8 is somewhat conservative in estimating the R value for shorter frames. Drawing our attention to the steel-braced RC frame dual systems, it is evident that little basis for similar comparison exists between the R values evaluated for this newly proposed load-resisting dual system and the R values of the well-established dual systems presented in codes of practice. Of the two variable parameters discussed in this paper as affecting the R value, the number of storeys appears to be the predominant variable. The other variable, the share of bracing from the applied load, has more localised influences and therefore do not warrant a similar generalisation. The significant effect of the number of storeys on R factor of steel-braced RC frames, as highlighted in the previous section, stems from the fact that shorter braced frames exhibit larger ductility than taller frames, therefore possess higher ductility ‘capacity’. It is therefore prudent to recalculate the R factors for the frames under consideration using specific

ductility ‘demands’. R factors are re-evaluated for all the systems under consideration using Equations (4)– (7) and ductility demand values of m ¼ 2, m ¼ 3, m ¼ 4 and m ¼ 5. The results are presented in Table 2. The variations in the R factors for a particular ductility demand, as seen in this Table, highlight mainly the effects of different variables. Considering the above discussion and with a view at the results presented in Table 2 and those presented by the authors in an earlier study considering X and knee-braced RC frames (Maheri and Akbari 2003), tentative R values for steel-braced intermediate ductility, moment-resisting RC frame dual systems are presented in Table 3. The proposed R factors are given for different ductility demands that constitute the generally accepted range of ‘intermediate ductility’ response. Further work is needed to arrive at R factors for low ductility and high ductility steel-braced RC frame dual systems. It needs to be emphasised that the main purpose of this work was not to reach at definitive values for R factor for the type of system under study. However, the results of this study in conjunction with those of the previous paper of the authors will provide a general sense regarding the limits and variation of the values of R factor for similar concrete-framed buildings having different steel bracing systems. In fact, the ever-increasing need for further knowledge by practical structural engineers on steel-braced rehabilitation of RC buildings provided a suitable justification for the authors to further their previous work on the X-braced and the knee-braced

Table 2.

R factors for different ductility demands.

Number of storeys

Share of brace (%)

4

0 (Unbraced) 50 100 0 (Unbraced) 50 100 0 (Unbraced) 50 100

8 12

Ductility demand m¼2

m¼3

m¼4

m¼5

4.8 5.3 5.1 4.1 4.8 4.5 4.0 4.4 4.2

7.1 8.3 8.0 6.0 8.1 7.8 5.8 7.0 6.8

9.4 9.8 9.5 7.9 9.5 9.3 7.7 8.3 7.9

11.7 12.1 11.0 9.8 11.3 10.2 9.6 10.1 9.8

Table 3. Tentative values of R factor for steel-braced, RC frame dual systems. Ductility demand Type of bracing system Chevron brace (this study) X and knee brace (Maheri and Akbari 2003)

m¼2

m¼3

m¼4

m¼5

5.0 5.0

6.5 7.0

8.0 9.0

10.0 12

8


RC frames and similarly investigating the R factor for the commonly used chevron bracing system. Further research works are needed in order to improve the accuracy of the proposed R factors. 5.

Conclusions


The above discussion on the results of the nonlinear pushover analysis, evaluation of the behaviour factor and the effects of certain parameters on the behaviour factor leads us to draw the following conclusions: (1) When designed for a specific base shear, steelbraced RC dual systems possess much larger ductility capacities than their equivalent unbraced moment-resisting RC frames. (2) So far as behaviour factor is concerned, in the chevron-braced dual systems, it is beneficial to apportion the base shear between the bracing system and the RC frame more evenly. (3) Ductility of a moment-resisting RC frame is to some extent affected by its height. When bracing systems are included, the height dependency of ductility is greatly magnified. Shorter steel-braced dual systems exhibit higher ductility and therefore higher R factors. (4) Considering the range of ductility capacities shown by different systems discussed in this paper, tentative R factors for different ductility demands are proposed for steel chevron-braced RC frame dual systems. References Abou-Elfath, H. and Ghobarah, A., 2000. Behaviour of reinforced concrete frames rehabilitated with concentric steel bracing. Canadian Journal of Civil Engineers, 27, 433–444. ACI Committee 318, 1995. Building code requirements for reinforced concrete (ACI 318-95) and commentary (ACI 318R-95). Detroit, Michigan: American Concrete Institute. AISC Manual Committee, 1994. Manual of steel construction, load and resistance factor design, LRFD. 2nd ed. Chicago, IL: American Institute of Steel Construction. Akbari, R., 2002. Seismic behaviour factor, R, for steel braced RC buildings. Thesis (MSc). Shiraz University. Assaf, A.F., 1989. Evaluation of structural overstrength in steel building systems. Thesis (MSc). Northeastern University. Badoux, M. and Jirsa, O., 1990. Steel bracing of RC frames for seismic retrofitting. Journal of Structural Engineering, 116 (1), 55–74. Bush, T.D., Jones, E.A., and Jirsa, O., 1991. Behaviour of RC frame strengthened using structural steel bracing. Journal of Structural Engineering, 117 (4), 1115–1126. Elnashai, A.M., 2001. Advanced inelastic static (pushover) analysis for earthquake applications. Structural Engineering and Mechanics, 12 (1), 51–69.

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