Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2018 / Article

Research Article | Open Access

Volume 2018 |Article ID 5364564 | https://doi.org/10.1155/2018/5364564

Massimiliano Ferraioli, Angelo Lavino, "A Displacement-Based Design Method for Seismic Retrofit of RC Buildings Using Dissipative Braces", Mathematical Problems in Engineering, vol. 2018, Article ID 5364564, 28 pages, 2018. https://doi.org/10.1155/2018/5364564

A Displacement-Based Design Method for Seismic Retrofit of RC Buildings Using Dissipative Braces

Academic Editor: Junwon Seo
Received03 Aug 2018
Revised05 Nov 2018
Accepted30 Nov 2018
Published27 Dec 2018

Abstract

The paper proposes a displacement-based design method for seismic retrofit of RC buildings using hysteretic dissipative braces. At first, a fully multimodal procedure based on an adaptive version of the capacity spectrum method is applied to the 3D model of the damped braced structure. Then, the properties of an idealized bilinear model are defined using the seismic characteristics of the compound system thus accounting for the frame-damped brace interaction. Finally, an iterative procedure is developed to provide an optimal distribution of dampers. The proposed method overcomes the limitations of the design procedures in the literature that generally neglect the frame-damped braces interactions. Moreover, it addresses the main issues of seismic design of damped braces: effect of force demands applied to the frame due to the damper yielding and strain hardening, higher modes contribution, effect of soft-storey irregularities, and torsion effect in asymmetric buildings. The proposed design procedure is first validated using nonlinear static and dynamic analyses of a numerical example. Then, it is implemented to a real case study of a RC school building to assess its applicability in current practice.

1. Introduction

Experience from recent severe earthquakes has confirmed that RC buildings designed without earthquake-resistance requirements (precode structures) or following outdated structural codes are extremely vulnerable to seismic excitation. From literature review, considerable effort has been made to evaluate the seismic vulnerability of structures on both global and local levels [15]. This is especially true for public buildings such as hospitals or schools, whose seismic resistance is very important in view of the consequences associated with failure [69]. Low-rise RC school buildings often have large windows with a long rectangular floor plan thus producing a seismically weak direction along the internal corridor. Furthermore, the buildings designed without consideration on seismic loading often are irregular in plan and/or elevation. The consequent lateral-torsional coupling leads inevitably to considerable increase in the seismic vulnerability [1012]. Finally, the seismic gap between adjacent structures often is not adequate to accommodate their relative motions, thus resulting in a significant seismic hazard of pounding during earthquake excitation. Therefore, the existing RC building structures often need retrofit to resist the design earthquake demand. Many techniques are available for seismic upgrading of existing buildings and have proved to be effective in increasing the capacity of the structure and/or reducing the seismic demand. Among these techniques, the hysteretic dissipative braces (buckling-restrained braces (BRBs) and steel hysteretic dampers) have been extensively applied for the seismic retrofitting of RC framed buildings and proved to give additional energy dissipation capacity, control the interstorey drifts and lateral displacements, and, finally, encourage dissipative collapse mechanisms [1319]. Steel bracings have also other advantages such as their relatively low weight, their suitability for prefabrication, and the possibility of allowing inner and outer openings. Moreover, the braces may be directly connected to concrete members without using steel frames fixed to the concrete structure. Finally, these devices can be used as structural fuses since they are easy to replace, concentrate damage while the rest of the structure remains undamaged, and fix a limitation of the brace force that is transmitted to the highly stressed anchorage.

There are many studies available in the literature focusing on experiment tests and analytical results that have proved their excellent hysteretic behaviour without any strength and stiffness degradation [2025]. However, although conceptually clear as general principle, the application of hysteretic dissipative braces require more complex procedures if compared with other retrofit strategies. For example, the seismic retrofit with base isolation may be studied using a simple linear elastic analysis [26, 27] while the traditional techniques to increase strength and/or ductility require the plastic section analysis. The design of hysteretic dissipative braces is certainly more complex. Application of damped bracing systems in Europe is seriously limited by the lack of a standardized European design procedure. In fact, despite this technique proved its effectiveness for seismic upgrading of RC structures, Eurocode 8 [28] does not give any rules for design of hysteretic dissipative braces. The traditional methods appropriate for conventional structures (such as the use of a q-factor in Eurocode 8 [28] or the R-factors as prescribed by ACI) become worthless. The nonlinear response history analysis is complex and computationally demanding and, thus, its implementation is generally avoided in current practice for design of hysteretic dissipative braces. Thus, many procedures have been proposed in the literature for designing the hysteretic dissipative braces. Some of them are based on the Direct Displacement-Based Design (DDBD) method [2932]. Practically, the hysteretic damping provided by inelastic deformation of the damped braces is replaced by an equivalent viscous damping to convert the nonlinear system into an equivalent linear system. Bergami et al. [33] developed a displacement-based procedure based on two performance objectives: protect the structure against structural damage or collapse and avoid nonstructural damage as well as excessive base shear. Mazza et al. [34] developed a design procedure that combines a proportional stiffness criterion (which assumes the elastic lateral storey-stiffness due to the braces proportional to that of the unbraced frame) to the displacement-based design. Choi et al. [18] proposed an energy-based design method using hysteretic energy spectra and accumulated ductility spectra. Bosco et al. [35] proposed a design procedure for steel frames equipped with BRBs and proper values of the behaviour factor were developed through numerical investigation. Bai et al. [36] proposed a performance-based plastic design (PBPD) method for dual system of buckling-restrained braced reinforced concrete moment-resisting frames. Guerrero et al. [37] developed a method for seismic design of buildings equipped with BRBs that uses seismic records to solve the dynamic equation of motion for dual oscillators. Barbagallo et al. [38] proposed a design method for seismic upgrading of existing RC frames by BRBs that allows a direct control of storey drift demands. The equivalent viscous damping ratio to account for the energy dissipated by the damped braced frame is a dominant parameter in procedures based on displacement-based design. Ghaffarzadeh et al. [39] presented results of experimental and numerical investigations performed for estimating the equivalent viscous damping in DDBD procedure of two lateral resistance systems, moment frames and braced moment frames. Mazza et al. [40] and Dwairi et al. [41] developed analytic expressions of the equivalent-damping considering the energy dissipated by the hysteretic dampers and the framed structure.

The main drawbacks of the design methods based on DDBD [2934] are that they neglect the frame-damped brace interaction and are based on the proportional stiffness criterion. In fact, the equivalent single-degree-of-freedom (SDOF) system of the RC bare frame is defined from the pushover analysis of the existing RC structure. The effective period and secant stiffness of the structure are then calculated combining the damping characteristics of the structural components. Finally, the same value of the stiffness ratio between damped braces and bare frame is assumed for each storey. Thus, the damper stiffness is distributed along the height of the building according to the profile corresponding to the fundamental mode of the structure. This approach can produce a nonuniform distribution of peak storey drift under earthquake ground motion and, therefore, it is not able to prevent soft-storey mechanisms.

As an alternative, Kasai et al. [4244] developed a method based on the SDOF idealization of the multistorey building structure and proposed a rule to arrange the damper stiffness over the height of the building so to produce the uniform distribution of drift angle and ductility demand under the design shear force, although those of the frame without dampers may be nonuniform. The damper design method giving optimal dampers distributions was initially proposed for elastic steel structures as well as elastoplastic structures including steel and timber structures [45] and then extended for retrofitting of existing RC structures with elastoplastic dampers [31, 46]. This design approach is very useful since it gives a closed form expression for the required damper to RC frame stiffness and the optimal damper stiffness for each storey. However, it suffers some main limitations for applying in current practice. First, all these damper design methods giving optimal dampers distributions are generally based on a shear beam model that consists of a mass and two springs (one for damper and the other for frame) for each storey. This model represents the hysteretic characteristics of the dampers but neglects some considerable interaction effects between the damped braces and the RC frame that may influence the seismic characteristics of compound system. In fact, despite the increase of strength and stiffness, the introduction of damped braces leads to an increase of the axial forces in the columns. This effect reduces the deformation capacity of the RC columns and may lead to their premature failure. Thus, the design method should include the effects of frame-dampers interaction so to relate the damped brace capacity with the deformation capacity of the RC columns. Moreover, the damper quantity obtained in the equivalent SDOF system is distributed to each storey of MDOF system by using the constraint condition that the equivalent stiffness of passive control MDOF system at the target drift angle is proportional to the storey shear force acting on each storey. This design approach is very sensitive to the design shear force and frame stiffness distributions along the height. The existing RC buildings typically show significant reduction in cross-section of the column along the height. Thus, the frame stiffness may decrease rapidly along the height and the storey drift at upper stories increases (upper-deformed type structure). In this case, the required damper to RC frame stiffness becomes high value at the storey expected to have large drifts of frame without dampers. On the contrary, the closed form expression of the optimal damper stiffness gives negative values in the first stories of the structure which means that no damper should be inserted in these stories. This optimal design solution obtained for an ideal shear beam model cannot be easily implemented in current practice due to the effects of frame-damper interactions. In fact, in the retrofitted building all the lateral load resisting systems (including the damped braces) should run without interruption from their foundations to the top of the building. If no damper is inserted in the first storey, the uniformity in the development of the structure along the height of the building is lessened, and this gives a concentration of shear forces in the first storey columns below the damped braces that might prematurely cause collapse.

Finally, it should be highlighted that the design methods available in literature generally include only the first mode contribution while the contributions of higher modes in evaluating the response of MDOF elastoplastic system is neglected. Moreover, the design methods based on optimal dampers distributions [31, 4247] are generally tested on two-dimensional planar shear-bar models and their efficiency for 3D asymmetric building structures should be further investigated. Thus, they are fully reliable only when applied to symmetric-in-plan buildings and this is a very strong limitation since many existing RC buildings (including schools, hospital and other public buildings) are asymmetric in plan and/or in elevation. On the other side, the application of design methods based on DDBD [2934] to asymmetric-plan buildings creates some problems of accuracy. In fact, the dynamic properties of the damped braced structure are calculated combining the damping characteristics of the RC bare frame and the damped braces. However, the response of the RC bare frame is dominated by the torsional effects, while the response of damped braced structure is poorly conditioned by these effects since they are mitigated by the damped braces. Clearly, this would affect the effectiveness of these design methods.

In this paper, a design method is proposed to address the main issues of seismic design of damped braces: effects of frame-damper interactions, higher modes contribution, effect of soft-storey irregularities, and torsion effect in asymmetric building. The proposed method is based on the Direct Displacement-Based Design (DDBD) and is an evolution and improvement of the procedure proposed by Mazza et al. [34]. The design procedure explicitly considers the frame-damper interaction (i.e., the force demands applied to the frame due to the damper yielding and strain hardening). In fact, the seismic design is carried out using the pushover curve of the dual RC-brace system thus considering the yield mechanism of compound system and the corresponding nonlinear drift demand for achieving the expected seismic performance. To solve the drawback of fixed-load pattern, the pushover analysis is carried out by using the Displacement-based Adaptive Pushover (DAP) method [48] together with an adaptive version of the capacity spectrum method [49]. Based on a 3D model and a fully multimodal procedure, the proposed method allows accounting for the higher modes contribution and torsion effect in asymmetric buildings. The equivalent viscous damping of the dual RC-brace system is characterized using a specific formulation calibrated on experimental results for steel braced RC frames. The lateral force distribution is based on the inelastic state of the structure and this gives the stiffness and strength of the hysteretic dissipative braces to prevent undesired failure modes (i.e., partial mechanisms and soft‐storey mechanisms). The effectiveness and reliability of the proposed procedure is investigated using a case study example. Its practical applicability is investigated with reference to a real case study that is an asymmetric in plan school building. Nonlinear dynamic time history analyses have been carried out to evaluate the seismic performance of the retrofitted building and validate the displacement-based design procedure.

2. Proposed Retrofit Design Method

2.1. Preliminary Design

A preliminary design of the damped braces is carried out with the displacement-based design procedure proposed by Mazza et al. [34]. According to this procedure, the capacity curve of the framed RC structure is selected from the pushover analysis that is carried out under constant gravity loads and increasing lateral forces with different distributions over the height of the building. The lowest capacity curve (base shear V(F) versus top displacement d(F)) is used for the preliminary design. The design displacement is selected from the seismic performance of the existing RC building. In particular, may be selected as the value of the top displacement corresponding to the Limit State of Life Safety (LS). Typically, this limit state is defined from the ultimate chord rotation capacity of ductile members (beams, columns, and walls), the interstorey drift capacity of the building or, even, the width of seismic gaps to prevent the structural pounding between adjacent structures during earthquakes. The capacity curve is idealized as bilinear for design purposes. The original RC structure is then represented by an equivalent SDOF system [50] characterized by a bilinear curve with yield displacement and corresponding base shear , and stiffness hardening ratio rF (Figure 1). The Jacobsen formulation [51] is used to calculate the equivalent viscous damping due to hysteresis of the RC structure. The equivalent viscous damping of the damped braced structure () is estimated by summing the elastic viscous damping for RC structure (=5%) and the equivalent viscous damping of the system composed of the framed structure (F) and the damped braces (DB) evaluated as a weighted average as follows:The base shear in the hysteretic dissipative braces is unknown since the effective strength properties of the equivalent damped brace is one of the parameters of the design procedure. Thus, an iterative procedure is required. At first, an attempt value of the equivalent viscous damping is imposed. Then, a damping reduction factor Rξ defined as multiplies the spectral ordinates to get the design displacement values. The effective period Te of the damped braced structure is estimated as the period of the -damped displacement spectrum corresponding to the performance displacement dp (Figure 2). The equivalent stiffness Ke of the damped braced structure is calculated as follows:The equivalent stiffness of the damped braces is given byFinally, since the constitutive law of the equivalent damped brace is idealized as bilinear, the performance and yielding base shears ( and ) are calculated as follows:The base shear from (5) may be used as a new attempt value of in (1) to calculate the equivalent viscous damping of the damped braced frame. This iterative procedure progresses very quickly to a converged solution. To complete the design procedure, it is necessary to define the distribution of base shear along the height. The preliminary design is based on the proportional stiffness criterion; that is, at each storey the same value of the stiffness ratio between the lateral stiffness values for damped braces and bare frame is assumed. Moreover, the mode shapes of the structure are considered unchanged even after inserting the damped braces. Thus, the base shear is distributed along the height of the building according to the profile corresponding to the fundamental mode. This gives the yielding design shear force of damped braces at i-th storey and allows defining the strength of the damper and the stiffness of the damped brace (i.e., brace + damper). In the case of one bracing in a single bay, the strength and the stiffness are given bywhere and are, respectively, the i-th and (i-1)-th components of the fundamental mode shape vector .

2.2. Final Design

The preliminary design procedure based on the proportional stiffness criterion may not give the expected results when applied to real case studies. The main drawbacks are as follows:(1)The mode shapes of the structure are considered unchanged even after inserting the damped braces.(2)The dampers are distributed along the height according to the proportional stiffness criterion.(3)The frame-damped braces interaction is neglected.

These drawbacks can affect the accuracy of the design method when applied to buildings with soft stories, plan irregularities, or nonductile columns. Many existing RC buildings were designed without earthquake-resistance requirements and partial collapse mechanisms, such as soft-storey mechanisms, are not necessarily avoided. Thus, if the mode shapes of the structure are maintained unchanged even after inserting the damped braces, then even the retrofitted building may develop undesired partial collapse mechanisms under the design seismic action. Furthermore, it may be necessary to use the damped braces to improve the seismic behaviour of the building. In fact, the buildings designed without consideration on seismic loading often are irregular in plan and elevation. The consequent lateral-torsional coupling leads inevitably to nonuniform displacement demands among resisting planes during the pushover analysis of the existing RC building. Thus, the mode shapes and, in general, the seismic response of the building can vary greatly after inserting the damped braces. For example, the torsional effects are greatly reduced in braced frame buildings since the damped braces are positioned in plan to increase the torsional stiffness and strength and minimize the eccentricity between the centres of mass and stiffness. As a result, also the choice of the design displacement from the seismic performance of the existing RC building becomes questionable.

Moreover, it should be highlighted that the damper distribution along the height of the building according to the proportional stiffness criterion may be not able to assure very large plastic strains in all the dampers.

Some authors [31, 4247] have tried to overcome this limitation by developing damper design methods that give optimal dampers distributions, so that the distribution of ductility demand becomes uniform, although that of the frame without damper is nonuniform. However, these methods are generally based on a simplified shear beam model, while applications to 3D buildings models accounting for frame-damper interaction, higher modes effects, and plan irregularities are still lacking. Moreover, these optimal design methods provide that no damper must be inserted in the first storey of “upper-deformed type” frames that are the frames in which the storey drift at upper stories increases. This gives a concentration of shear forces in the first storey columns below the damped braces that might prematurely cause the collapse of the structure. This effect cannot be highlighted by the shear beam model that consists of two springs in parallel (one for damper and the other for frame). This is a general drawback of many design procedures that neglect the effect of frame-damped braces interactions (i.e., the force demands applied to the frame due to the damper yielding and strain hardening), which may significantly affect the failure mechanism. In fact, the internal force demands of RC beams and columns intersected by braces are underestimated if interaction is neglected. In particular, the increase of the axial forces in the RC columns decreases their deformation capacity (nonductile columns) and this may lead to their premature failure.

In order to overcome these drawbacks, a retrofit design method is proposed in this paper. In the first step the damped braces are defined from the preliminary design developed in Section 2.1. Then, the pushover analysis is carried out on the dual RC-damped brace system thus accounting for the effects of frame-damped braces interactions, the yield mechanism of compound system, and the corresponding nonlinear drift demands. This allows accounting for the increase in the axial forces of the columns given by the damped braces and its effects on the ultimate chord rotation capacity of the RC columns. The pushover analysis is carried out by using the Displacement-based Adaptive Pushover (DAP) method proposed by Antoniou and Pinho [48]. This method overcomes the assumption that the structure vibrates predominantly in a single mode and that the dynamic properties of the structure remain unchanged, but it provides no solution to determine the target displacement. Thus, an adaptive version of the capacity spectrum method is developed in this paper. The classical formulation proposed by Fajfar [50] is useless to define the equivalent SDOF system in adaptive pushover analysis. In fact, the capacity spectrum method assumes that the response of the multiple degree-of-freedom (MDOF) system is entirely in the fundamental mode, while no contribution is considered from other modes. This allows defining an idealized bilinear SDOF system from the transformed capacity curve ( versus , where is participation factor of the fundamental mode shape). In case of adaptive pushover, the lateral force pattern and, thus, the equivalent SDOF system changes during the analysis. At each step of the pushover analysis, a different equivalent SDOF system is defined as a function of the actual lateral displacement pattern. In particular, the equivalent mass of the SDOF system at the i-th step of the pushover analysis is expressed as a function of the j-th storey displacement as follows [49]:where N is the number of stories, mj is the mass of the j-th storey, is the lateral displacement of the j-th storey at the i-th step of pushover analysis. The corresponding participation factor is defined as follows:The capacity curve (base shear versus top displacement d) of the damped braced structure is transformed step by step into the capacity curve ( versus ) of the equivalent SDOF system, as follows:where and are the base shear and the corresponding top displacement increments at the i-th step of pushover analysis, defined as follows:Finally, according to the principle of energy equivalence, the capacity curve is idealized as bilinear.

It should be highlighted that the capacity curve ( versus ) referred to the dual RC-brace system and thus explicitly considers the frame-damped braces interaction. Moreover, (10)-(12) that control the transformation from MDOF to SDOF model account for the dynamic behaviour of the damped braced structure. This overcomes the main limitations of the preliminary design procedure that do not consider the frame-damped braces interaction and define the equivalent SDOF system of the RC bare frame from the pushover analysis of the existing RC structure. This includes many effects (such as lateral-torsional coupling, torsional eccentricity, and partial or soft‐storey mechanisms) that are present in the existing RC building but may disappear in the dynamic response of the building after retrofit. Finally, it should be observed that the pushover analysis of the damped braced structure allows calculating the capacity curve of the RC bare structure and the capacity curve of the damped braces, separately. In fact, the top floor displacement is the same in both cases while the base shear V(F) of the RC bare structure may be calculated as the difference between the total base shear and the base shear in the damped braces.

The equivalent viscous damping of the damped braced structure () is estimated using the equation proposed by Ghaffarzadeh et al. [39] for steel braced RC frames:where =5% is the elastic viscous damping for the RC structure while the coefficients a, b, and c are calibrated matching experimental and numerical data [39], which gives a=70, b=43, and c=4.7x10−5.

Equations (2)-(6) give the equivalent stiffness Ke of the damped braced structure and the performance base shear and the yielding base shear in the damped braces. The yielding base shear should be distributed along the height of the building. The conventional displacement-based design procedures [2934] are based on the proportional stiffness criterion and distribute the yielding base shear along the height according to the profile corresponding to the fundamental mode of the RC bare structure. As aforementioned this approach may fail when the bare structure shows lateral-torsional coupling effects and/or undesired failure modes. Moreover, it is not able to assure large plastic strains in all the dampers.

As an alternative, in this paper the damper quantity of the equivalent SDOF system is distributed to each storey of the MDOF system by using the following rule to distribute the yielding base shear along the height of the structure:where is the i-th storey displacement at the performance point calculated from the pushover analysis. The dampers and the braces are dimensioned from the storey shears using horizontal equilibrium.

Practically, the distribution of the lateral loads carried by the damped braces at the yielding point is based on the inelastic state of the structure calculated from the adaptive pushover analysis of the dual RC-brace system.

The distribution of the base shear according to (15) is focused on optimizing the seismic response of the building after retrofit. In fact, it increases the stiffness of the damped braces for the stories with higher interstorey drifts (thus preventing undesired soft-storey mechanisms) and tends to assure very large plastic strains in all the dampers. In the case of one bracing in a single bay, the yield strength of the dampers is given by (7), while the stiffness of the damped brace at i-th storey is calculated as follows:where and are, respectively, the i-th and (i-1)-th storey displacements at the performance point. The damped braces defined by (16) have varied with respect to the previous iteration. Thus, the procedure should be iterated until convergence is achieved, that is until the damped braces and, therefore, the equivalent viscous damping do not vary with respect to the previous iteration with the prefixed tolerance of 5%.

3. Case Study Example

The proposed procedure is applied to a regular 5-storey RC moment frame (Figure 3) that is considered one of the elements of the lateral force resisting system of a regular RC framed building. The storey height is 3.5m for all floors. The bay length is 5.00 m in both orthogonal directions. The steel material used is B450C with tensile strength value of 450 MPa. The concrete is assumed to have a nominal compressive strength =25N/mm2 (compressive strength class C25/30). The building is designed for vertical loads only, and then retrofitted to Eurocode 8 [28] requirements for soil class A, damping ratio 5% and design Peak Ground Acceleration for Life Safety Limit State is =0.35g. The SeismoStruct program [52] is used in the simulations presented in section. Distributed plasticity beam-column elements are used to account for material nonlinearity. Specifically, inelastic force-based fibre elements (infrmFB) are used for modelling components. The sectional stress-strain state is obtained by integrating the nonlinear uniaxial material response of the single fibres in which the section is subdivided, fully accounting for the spread of inelasticity along the member length and across the section depth. The reinforcing steel bars are modelled with a bilinear hysteretic model. The concrete is modelled accounting for the amount of confining with the well-known Mander model [53]. While the material nonlinearity is accounted for flexural and axial degrees of freedom, sections are assumed to behave elastically under shear and torsion. The bracing in a single bay of the framed structure is considered for retrofit. The dampers used have the following mechanical parameters: stiffness hardening ratio rD=0.020, design ductility =10, lateral stiffness ratio between damper, and brace . These parameters may correspond to different hysteretic dissipative braces (such as buckling-restrained braces (BRB) or steel hysteretic dampers (HBF)).

Figure 4(a) shows the pushover curves of the RC bare structure under three different lateral force distributions along the height: (a) uniform distribution, (b) equivalent static force distribution, and (c) first mode distribution. Figure 4(b) shows the lowest capacity curve of the RC bare structure and the corresponding performance points for three limit states: Damage Limitation (DL), Life Safety (LS), and Collapse Prevention (CP). A soft-storey mechanism occurs at the fourth level of the structure during pushover analysis. The limit states for the ductile and brittle failure modes are determined according to Annex A of EC8-3 [28]. The capacity of the ductile and brittle members is estimated in terms of chord rotation and shear strength, respectively. The value of the capacity of both the ductile and brittle components and mechanisms are then compared to the corresponding demand for the safety verification.

The damped braces are designed applying the proposed design method. Figure 5 shows the pushover curve of the structure with damped braces at each step of the iterative procedure. The points corresponding to yielding and failure of each device are evidenced on each pushover curve. The first step (Step 0) corresponds to the structure retrofitted by damped braces sized by the preliminary design. In this case, the failure of the 4th storey damper occurs before the target displacement dp is reached. This is due to the soft-storey mechanism that is formed at the fourth level of the structure. The other dampers are very far from failure. This behaviour derives from the proportional stiffness criterion that is applied for the preliminary design. In fact, the mode shapes of the structure are maintained unchanged even after inserting the damped braces and, thus, also the retrofitted building develops a soft-storey mechanism at the 4th floor level of the structure. The last step (Step 7) corresponds to the structure retrofitted by damped braces sized by the final design. In this case, the failure in the first damper occurs when the target displacement is reached and the other dampers are very near to failure. Figure 6 shows the variation of the properties of the damped brace with height at each step of the iterative procedure. The stiffness of the damped brace is plotted in Figure 6(a), while the strength of the dampers is plotted in Figure 6(b). Figure 7 shows the variation during the iterative procedure of the equivalent-damping ratio of the damped braced frame. Results show that the trend tends to stabilize after some iterations giving the final value of the equivalent-damping ratio.

Finally, it should be underlined that the proposed design method allows accurately estimating the target displacement dp accounting for the real seismic demands of the damped braced frame. In fact, the pushover analysis of the dual RC-damped brace system explicitly considers the effects of frame-damped braces interactions and, particularly, the force demands applied to the frame due to the damper yielding and strain hardening. This allows evaluating the effects of frame-damped braces interactions on the deformation capacity of beams and columns in terms of chord rotation (Appendix of EN 1998-3 [28]) and, thus, on the target displacement dp to be considered in the analysis. This is a significant advantage over other procedures available in literature since they neglect the frame-damped braces interactions and their effects on the deformation capacity of the structure.

Tables 1 and 2 show the final parameters of the idealized bilinear SDOF systems for the bare frame (Table 1) and the damped braced frame (Table 2). Table 3 defines the parameters of MDOF system for the damped braced frame. Table 4 shows the design parameters of the damped braces at each storey.


0.142kNs2/mm
1.422-
86.0kNs2/mm
60.5mm
1.721kN/mm
1.806s


15.93%
62%-
2s
173mm
1.12s
4.48kN/mm
0.06-
2.76kN/mm


86.00mm
2.76kN/mm
238kN
201kN
8.53-
10.08mm


FloorMassDisplacementInclination
[kNs2/mm]brace [kN][kN][kN][kN/mm]

10.06460.073356.69201.3245.8407.418
20.06310.2403521.4194.6237.6172.549
30.06200.4673541.1173.2211.4112.424
40.06160.7723567.4132.1161.364.102
50.04571.0003564.864.7679.0542.004

The seismic response of the RC structure with damped braces is finally evaluated by means of nonlinear time history analysis. For this purpose, according to the Italian Code [54] a group of seven time histories is applied and the average of the response quantities from all the analyses is used as the design value of the seismic effect. The description of the seismic motion is made using recorded accelerograms and this allow accounting for characteristics like frequency, duration, and energy of real earthquake ground motions. The European Strong-motion Database (ESD), the SIMBAD database (Selected Input Motions for displacement-Based Assessment and Design), and the Italian Accelerometric archive (ITACA) [55, 56] are used for selecting the recorded accelerograms. The suite of accelerograms observed the rules recommended in the seismic standards. In fact, the mean of the zero period spectral response acceleration values (calculated from the single time histories) is not smaller than the value of agS for the site in question, where S is the soil factor and ag is the design ground acceleration on type A ground. Furthermore, in the range of periods of interest no value of the mean elastic spectrum, calculated from all time histories, is less than 90% of the corresponding value of the target elastic spectrum. The recorded accelerograms considered in the numerical analysis are summarized in Table 5. In Figure 8, the spectrum compatibility for the selected acceleration records is represented.


Waveform IDEarthquake NameDateDirectionPGAMagnitudo
[m/s2]

1243XIzmit13.09.99NS0.7145.8
6326YSouth Island21.06.00EW1.1426.4
4676XSouth Island17.06.00NS3.9206.5
292XCampano-Lucano23.11.80NS0.5886.9
368XLazio-Abruzzo07.05.84NS0.6285.9
7142YBingol01.05.03EW2.9186.3
6331XSouth Island21.06.00NS0.5136.4

Figure 9 shows height-wise distribution of axial displacement of damped braces obtained from the nonlinear time history analysis. The results for the selected records are plotted in Figure 9(a) while Figure 9(b) compares the mean value to the ultimate value (corresponding to failure). The results highlight that the proposed design procedure is effective in controlling deformations and avoiding undesired soft-storey mechanisms of failure. Figure 10 shows the hysteresis loops of dampers under the Izmit earthquake ground motion. It can be noted that the proposed design method is able to assure very large plastic strains in all the dampers thus increasing the energy dissipation capacity during the earthquake ground motion.

Figure 11 compares the height-wise distributions of storey drift ratio obtained in the following cases: (1) existing RC structure; (2) structure retrofitted with hysteretic dissipative braces sized by the preliminary design; (3) structure retrofitted with hysteretic dissipative braces sized by the final design. The average value from seven nonlinear time history analyses at the Life Safety Limit State is plotted. Results show that the existing structure has a soft-storey mechanism at the 4th floor level. The structure retrofitted with damped braces sized by the preliminary design maintains unchanged the fundamental mode shape even after retrofit thus evidencing a nonuniform distribution of storey drift along the height. On the contrary, the final design procedure uses the damped braces to give a more uniform height-wise distribution of storey drift thus improving the seismic behaviour of the structure.

4. Application to a RC School Building

4.1. Description, In Situ Measurements, and Laboratory Tests

The real case study is a three-storey school building in Vibo Valentia (Calabria-Italy). The building is composed of three reinforced concrete framed structures named “A”, “B”, and “C” in Figure 12. The building was designed in 1962 according to the provisions of an Italian Code dating back to 1937 [57]. The site belonged to the first seismic category zone, whose seismic intensity coefficient was C=0.07. The allowable stress design method was used in design. Structure A has been selected as case study (Figure 13). The building has an L-shaped floor plan with dimensions of 17.70 x 35.50 m (Figure 14). The following investigations were carried out: (1) soil investigations including sampling and testing, (2) geometrical measurements, and (3) determination of mechanical properties of materials by testing of samples taken from the structure. The soil deposits primarily comprise silt sandy loam, sand slightly silty clay, and micaceous sand. The mechanical properties of soil and the ground type according to soil classification of Eurocode 8 [28] give a soil classified as ground type B. Due to the extensive measuring and testing, the full knowledge level KL3 [28, 54] is attained, which allows a Confidence Factor CF=1. The geometry is known from original outline construction drawings integrated by direct visual survey. The structural details are obtained from original construction drawings together with in situ inspection. Information on the mechanical properties of the construction materials is taken from a comprehensive in situ testing. Collected results give the following mean values of strength: =35.2 MPa for concrete and =408 MPa for steel.

4.2. Seismic Assessment

The seismic performance evaluation is developed using the procedure reported in current Italian Code [54] and Annex B of EN 1998-3 [28]. The RC framed structure is modelled in SAP2000 finite element computer program [58]. Figure 15 shows the fundamental mode shapes of the existing structure and the corresponding dynamic properties. The parameters of the elastic design response spectra used for seismic assessment are plotted in Table 6. The Limit States of Immediate Occupancy (IO), Damage Limitation (DL), Life Safety (LS), and Collapse Prevention (CP) are defined according to Eurocode 8 [28]. The nonlinear analysis is carried out using a fibre hinge model implemented in the SAP2000 [58]. The concrete is modelled with the stress-strain relationship originally proposed by Mander et al. [53]. The steel is modelled with an elastic-plastic-hardening relationship. The rigid elements are placed at beam-column connections to prevent the development of plastic hinges inside the connections. The capacity of ductile and brittle members is estimated in terms of chord rotation and shear strength, respectively. The deformation capacity of beams and columns is defined in terms of the chord rotation according to Appendix of EN 1998-3 [28]. Figure 16 shows the pushover curves for two directions (X and Y), two lateral force distributions (first mode and uniform), and accidental eccentricity of 5%. The comparison between capacity and demand for the Life Safety Limit State is carried out with the procedure implemented in Annex B of Eurocode 8 [28] and Italian Code [54]. The synthesis of the seismic safety verification is shown in Table 7, in which are plotted both the capacity of the existing building in terms of peak ground acceleration (PGA) for different limit states and the corresponding safety index IR (ratio between capacity and demand in terms of PGA). The main deficiencies of the existing building may be summarized as follows: (1) insufficient stiffness for IO and DL Limit States, (2) poor shear capacity of brittle components, (3) torsional effects in X-direction that activate a partial failure mechanisms, (4) inadequate member chord rotation capacity for the Life Safety Limit State, and (5) inadequate seismic gap from adjacent building structures.


Limit StateIODLLSCP

Probability of exceedance 0.810.630.100.05
Return Period (years)12020118982475
Peak ground acceleration PGA/g0.0860.1120.3150.418
Amplification factor 2.2762.2762.4482.485
Transition Period (s)0.2930.3150.3800.412


IO X-Dir.DL X-Dir.LS X-Dir.IO Y-Dir.DL Y-Dir.LS Y-Dir.

PGA/gPGA/gPGA/gPGA/gPGA/gPGA/g

+Modal + Ecc0.1431.4020.1971.4700.3511.0230.0940.9220.1280.9550.4321.259
- Modal + Ecc0.1511.4800.1991.4850.3320.9680.0900.8820.1260.9400.4341.265
+Modal - Ecc0.1481.4510.1971.4700.3210.9360.0900.8820.1260.9400.4351.268
- Modal - Ecc0.1431.4020.1991.4850.3250.9480.0900.882
0.1280.9550.4311.257
+Uniform + Ecc0.1551.5200.2041.5220.3701.0790.0940.9220.1280.9550.4431.292
-Uniform + Ecc0.1521.4900.1971.4700.2370.6910.0900.8820.1260.9400.4451.297
+Uniform - Ecc0.1491.4610.1971.4700.3400.9910.0900.8820.1260.9400.4461.300
-Uniform - Ecc0.1511.4800.2021.5070.3420.9970.0900.8820.1260.9400.4421.289

4.3. Retrofit Design and Seismic Assessment

The seismic retrofit of the RC building is carried out using hysteretic dissipative braces that are allocated so to limit the underpinning area of the existing foundation, minimize the torsional effects, and keep most areas of the building operational during the retrofit construction (Figures 17 and 18). Some external and internal views of the building after retrofit are shown in Figures 19 and 20. The seismic retrofit has required some local interventions including strengthening of columns next to the steel braces by steel angles and strips (Figure 18), shear strengthening of the unconfined joints, fibre reinforced polymer (FRP) shear, and bending reinforcement of some beams at the first floor. The size of the seismic gap between structures A and B has been increased up to 3 cm to avoid structural pounding.

At first, the preliminary design of the damped braces is carried out using the procedure described in Section 2.1. The lowest capacity curve of the RC bare structure is selected among those plotted in Figure 16. A design displacement of 35 mm is chosen in X-direction to avoid seismic pounding with the adjacent structure B (Figure 12). A design displacement of 62 mm is selected in Y-direction. Both values are checked and, if necessary, modified as the design progresses considering the response of the braced RC building. The seismic retrofit design is carried out using two different design solutions, one based on buckling-restrained braces (BRB) and the other based on steel hysteretic dampers (HBF). The steel hysteretic dampers were finally used during the construction phase of the retrofit project (Figures 19 and 20). Tables 812 show all the results of the preliminary design procedure described in Section 2.1.

(a) SDOF system of bare frame

Directionx-xy-y

1.1381.476kNs2/mm
1.4171.183-
50.0073.85kNs2/mm
39592535kN
38802454kN
92158mm
3162mm
35.2962.43mm
38862454kN

(b) SDOF system of damped braces

Directionx-xy-y

2020mm
5%5%-
dy22mm
1010-
0.20.2-
8.5758.575-
0.060.06-
(%)16%16%-

(a) SDOF system. Bare frame + damped braces

Directionx-xy-y

20.8624.73kN/mm
10431826kN
7191260kN
9.63%11.14%-
0.780.73-
2.862.86sec
303303mm
0.610.95sec

(b) MDOF system

Directionx-xy-y

50.0073.85mm
20.8524.73kN/mm
x 10431826kN
7191260kN
8.588.58-
5.838.61mm


Distribution of base shear along the height

FloorMass
kNs2/mm
Direction x-xDirection y-y
Eigen vectorBrace inclinationEigen vectorBrace inclination
kNkNkNkN/mmkNkNkNkN/mm

10.6220.3315513671912531132.800.3314723912601847950.54
20.5820.66549256583888694.000.6654644910211457721.31
30.4291.00049284326497388.681.00046498572816403.98

(a) Distribution of BRB along the height, direction x-x

FloorBRB typeSteel Brace
kN/mmkNmmmmmmmm2kN/mmkN/mm(kN/mm

148/40210417164034001760193.7/105770688.47160.921287.33
234/40153301162536602035193.7/84670481.92116.13929.04
327/40123239158537002115168.3/63060303.8387.55700.44

(b) Distribution of BRB along the height, direction y-y

FloorBRB typeSteel Brace
kN/mmkNmmmmmmmmmmmm2kN/mmkN/mmkN/mmkN/mmkN/mm

148/4021041716404220391125802271193.7/105770469.65533.55145.11150.691172.06
234/4015330116254310400026852375193.7/84670365.25412.93107.83111.64870.26
327/4012323915854310400027252415168.3/63060235.82266.0980.8484.12653.25

(a) Distribution of HBF along the height, direction x-x

FloorDissipative BracesSteel BraceHBF type
mm2mmkN/mmkNmmkN/mmmmkN/mm

1(1-2-3-4)Φ193.7/20109082.0351126420/404180.5822.352.6221.915
2(1-2-3-4)Φ193.7/14.280042.29734300/403000.57515.902.8715.563
3(1-2-3-4)Φ168.3/1049812.44428240/402380.57512.453.0212.098

(b) Distribution of HBF along the height, direction y-y

FloorDissipative BracesSteel BraceHBF type
mm2mmkN/mmkNmmkN/mmmmkN/mm

1(5-6-8)Φ193.7/1689282.71692420/404180.5822.353.2921.651
2(5-6-8)Φ193.7/12.571122.96505300/403000.57515.903.5415.414
3(5-6-8)Φ168.3/8.844073.045304240/402380.57512.453.6211.960
1(7)Φ193.7/17.596822.42840420/404180.5822.353.0021.771
2(7)Φ193.7/12.571122.655562300/403000.57515.903.2315.463
3(7)Φ168.3/8.844072.73339240/402380.57512.453.3112.009

The fundamental modal shapes and corresponding dynamic properties of the damped braced structure are plotted in Figure 21. It can be noticed that the first two modes are two flexural mode shapes with more than 85% modal mass participation. Practically, the seismic design retrofit with damped braces reduces the lateral-torsional coupling effects and, thus, the mode shapes vary considerably if compared with the existing RC building. That is one of the reasons why the design procedures based on the pushover analysis of the RC bare frame [2934] may fail in this case study. On the other side, the optimal design procedures based on an equivalent shear-bar model [31, 4247] neglect the torsional effects and, thus, their application to asymmetric in plan buildings deserves further study.

Figure 22 shows the pushover curves of the damped braced structure and the performance points for the different limit states for two directions (X and Y) and two lateral force distributions (first mode and uniform). When comparing Figures 16 and 22, it becomes clear the increase of stiffness, strength, and ductility due to the hysteretic dissipative braces. However, the structure is not ductile enough to sustain large plastic deformation under the first mode distribution in Y-direction. In fact, the premature failure occurs since the chord rotation capacity at Life Safety Limit State is reached in the first storey columns. The damped braces increase the axial forces in the columns and this effect reduces their deformation capacity and prevents the collapse mechanism to fully develop. This result may be evidenced only if the effects of frame-dampers interactions are included in the analysis.

However, it should be highlighted that the pushover analysis is not the appropriate method of analysis in this case. In fact, the code procedure [28, 54] based on pushover analysis is equivalent to the capacity spectrum method [50]. The Elastic Demand Response Spectrum (EDRS) is generally represented by the 5%-damped response spectrum. This approach is too much conservative since the hysteretic steel dampers have low yield displacement (2 mm) and high energy dissipation capacity. On the other side, the use of highly damped spectrum with equivalent viscous damping in the capacity spectrum method may not give accurate results. Thus, the seismic assessment requires the nonlinear response history analysis (NRHA). For this purpose, the well-known Bouc-Wen model [59] is used for the hysteretic dampers. The seismic motion consisted of two simultaneously acting accelerograms along both horizontal directions. A group of three pairs of time histories is considered for the Life Safety Limit State and another group of three pairs of time histories is applied for the Collapse Prevention Limit State. The envelope of the response quantities from all the analyses is used as the design value of the seismic effect. The artificial accelerograms are used to define the input ground motion [60]. The choice of accelerograms follows the rules recommended in the seismic standards [28, 54]. The Life Safety verification compares the member chord rotation demand in beams and columns to the corresponding chord rotation capacity. The Collapse Prevention verification compares the displacement ductility demand of the hysteretic dampers to the corresponding displacement ductility capacity.

Figure 23 shows the hysteresis loops of some steel dampers at the Collapse Prevention Limit State. The damper in the hysteretic dissipative brace 7 (see Figure 17) at the third storey does not reach the inelastic region. This result is common to all the dampers of the third storey thus implying that these dampers are useless since their energy dissipation capacity is not activated.

Table 13 shows the results obtained with the procedure described in Section 2.2 (Final Design). In Figure 24 the hysteresis loops of the dampers in the dissipative brace 7 at the Collapse Prevention Limit State are plotted. Similar results are obtained for all the dissipative braces. It can be noticed that the proposed design method is able to give very large plastic strains in all the dampers during the earthquake ground motion thus increasing the energy dissipation capacity of the structure.

(a) Distribution of BRB along the height, direction x-x

FloorBRB typeSteel Brace
[kN/mm][kN]mmmmmm[mm2][kN/mm][kN/mm](kN/mm]

148/40210417164034001760193,7/105770688.47160.921287.33
227/40123239158536602075193,7/63540358.2791.56732.51
314/4060119156037002140168,3/42060202.1546.27370.14

(b) Distribution of BRB along the height, direction y-y

FloorBRB typeSteel Brace
[kN/mm][kN]mmmmmmmmmm[mm2][kN/mm][kN/mm][kN/mm][kN/mm](kN/mm]

148/4021041716404220391125802271193,7/105770469.65533.55145.11150.691172.06
227/4012323915854310400027252415193,7/63540272.81307.8384.7887.88684.43
314/406011915604310400027502440168,3/42060157.31177.3043.4344.83350.26

Figures 25 and 26 show the height-wise distributions of storey drift ratio at the Life Safety Limit State along X and Y-direction, respectively. The envelope value from three nonlinear time history analyses under bidirectional ground motions is plotted. The following cases are compared: (1) existing RC structure; (2) structure retrofitted with hysteretic dissipative braces sized by the preliminary design; (3) structure retrofitted with hysteretic dissipative braces sized by the final design. The existing structure shows higher values of first storey drift due to the soft-storey mechanism failure. The structure retrofitted by damped braces sized by the preliminary design shows lower values of storey drift but maintains unchanged their nonuniform distribution along the height. The proposed design method (Final design) is successful in giving a more uniform height-wise distribution of storey drift thus improving the seismic behaviour of the structure.

5. Conclusions

The paper proposes a design method for seismic retrofit of RC buildings with hysteretic dissipative braces. This method is based on the displacement-based design and on the adaptive version of the capacity spectrum method and explicitly accounts for the effects of frame-damped brace interactions. The lateral forces for seismic design are distributed along the height of the building according to the inelastic state of the structure. The optimal distribution of dampers is selected through an iterative procedure. The accuracy of the proposed method is validated by comparison with nonlinear time history analysis. This study has led to the following conclusions:(1)The proposed method has been applied to two comprehensive case studies (a plane RC frame and a real school building with RC framed structure) showing its effectiveness to address the main issues of seismic design of damped braces: effects of frame-damper interactions, higher modes contribution, effect of soft-storey irregularities, and torsion effect in asymmetric building.(2)The effects of frame-damped braces interactions have been addressed through the adaptive pushover analysis of the dual RC-damped brace system. This allowed accurately estimating the target displacement accounting for the real seismic demands of the damped braced frame. These effects have shown to be particularly important since the introduction of damped braces leads to an increase of the axial forces in the columns that reduces their deformation capacity. In one of the cases here examined this has led to the premature failure of the structure since the chord rotation capacity at the Life Safety Limit State is exceeded. Clearly, these results cannot be highlighted by design procedures neglecting the frame-damped braces interactions.(3)The issues of local weaknesses of certain stories (weak storey) have been discussed in detail. It has been demonstrated that the proposed design method is successful to arrange the damper stiffness and strength along the height of building so as to give a more uniform height-wise distributions of storey drift, reduce the storey drift demand below the drift capacity and assure very large plastic strains in all the dampers. The design methods based on optimal dampers distributions can produce the uniform distribution of peak storey drift. However, they suggest that no damper should be inserted in case of so called “upper-deformed type” frames. This gives a concentration of shear forces in the first storey columns below the damped braces that might prematurely produce the collapse of the structure. This effect cannot be evidenced by these methods since they are based on an ideal shear beam model neglecting the frame-damped braces interactions. On the other side, the conventional design procedures based on the proportional stiffness criterion used for the preliminary design has evidenced a non-uniform distribution of storey drifts and local ductility demands along the height.(4)The location of the damped braces in the building plan is of key importance in current practice. Many existing buildings are designed without consideration on seismic loading and, thus, they are often irregular in plan and/or elevation with the consequent lateral-torsional coupling. The proposed procedure has been applied also to buildings with plan irregularities (represented by a L-shaped in plan RC school building). Many design procedures in the literature are based on lumped-mass models, and, thus, they may be rigorously applied only to symmetric-in-plan buildings. On the other side, the conventional design procedures based on the proportional stiffness criterion uses the hypothesis that the mode shapes of the structure may be considered unchanged even after inserting the damped braces. This hypothesis is not well verified in case of irregular buildings since the damped braces are positioned in plan to improve the seismic behaviour of the building and significantly reduce the lateral-torsional coupling effect.(5)Results of the nonlinear time history analyses have shown that the proposed method can be used in current practice as a substitute to the traditional force-based design method for seismic retrofit of RC buildings with damped braces.

Finally, it is important to emphasize that the equivalent viscous damping of the damped braced structure is a very important design parameter. Thus, future developments are required to calibrate this parameter on the basis of hysteretic responses obtained from nonlinear cyclic analyses of damped braced frames.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors wish to express their gratitude to Eng. Gennaro Di Lauro and all the professional team belonging to Aires Ingegneria for their important support during the on-site activities.

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Copyright © 2018 Massimiliano Ferraioli and Angelo Lavino. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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