A design method for seismic retrofit of reinforced concrete frame buildings using aluminum shear panels

Despite significant progress in research and development of aluminum shear panels in recent decades, their implementation for seismic retrofit of existing reinforced concrete (RC) buildings can still be significantly extended. Their application is limited by the general lack of relatively simple and effective design criteria and proper guidelines. This paper develops a design method for the seismic retrofit of reinforced concrete buildings using aluminum multi-stiffened shear panels as dampers. Both the nonlinearity in the structure and the dampers-structure interaction are considered to give an optimal distribution of the shear panels over the height of the building. The analytical laws refer to dissipative aluminum shear panels recently tested and analyzed by the authors. The proposed procedure has been described in detail. Its applicability has been demonstrated by analyzing two typical RC buildings having drift capacity-to-demand ratios ranging from 0.505 to 0.624. The design value of the panel-to-frame stiffness ratio has been found to range from 0.594 to 1.432 as a function of the lateral stiffness of the existing building. The verification of the proposed procedure has been carried out by checking the validity of the design assumptions. The first one (i.e., the mode shapes remain the same before and after retrofit) has been checked using the modal assurance criterion that gives values ranging from 0.992 to 0.998. The second one (i.e., uniform yield drift distribution over the building height) has been checked by comparing the yield drifts with their average value giving a standard deviation ranging from about 11 to 15%. The effectiveness of the design method has been finally validated through nonlinear time-history analysis for different seismic accelerograms and hysteresis models. The results show that the seismic retrofit design procedure is effective in significantly reducing inter-story drift (maximum inter-story drift ratio demands ranging from 1.04 to 2.07%) thus satisfying the acceptance criteria of the building, and avoiding drift concentration and consequential weak story collapse.


Introduction
Most of the reinforced concrete buildings designed according to old codes may have inadequate stiffness, strength, ductility, and energy dissipation capacity and, therefore, may be extremely vulnerable to seismic excitation. During earthquake strong ground motions, these buildings are subjected to large seismic energy inputs that can produce the loss of human life as well as structural and economic damage. Traditional seismic structures resist earthquakeinduced forces through the energy dissipation of the structural members that cause damage, loss of life, and property. This structural damage may be so great that the repair is too expensive and time-consuming, and the demolition and subsequent reconstruction are more economically advantageous. To avoid such damage, control systems (passive, active, and semi-active) have been introduced in structural engineering. Among them, passive control systems have been proven to be more practical, and passive isolation and energy dissipative devices have been developed and applied all over the world in building and bridge structures [1][2][3][4][5]. Energy dissipative devices (EDDs) are commonly used as sacrificial elements to concentrate the dissipation of seismic input energy and prevent damage to the bearing structure of the building. They mitigate the seismic risk and can be replaced if damaged by an earthquake, thus minimizing the costs related to the loss of functionality of the building. Current EDDs can be grouped into two categories: displacement-dependent devices and velocity-dependent devices. The first type comprises fluid viscous dampers [6][7][8]. The second type includes friction dampers [9,10] and hysteretic dampers that can be further divided depending on the material (i.e., steel [11], aluminum [12], shape memory alloys [3][4][5][6][7][8][9][10][11][12][13], the structure type (i.e., brace [14], shear link panel [15], stud panel, wall panel [16]), and the yielding mechanism (i.e., shear, yielding ring, flexural plate, axial, extrusion, torsional bar, X-shaped metallic plates [17], triangular plates [18], honeycomb [11] and cushions [19,20]). In the latest decades, metal shear panels (MSPs) have been widely documented as an efficient alternative for seismic retrofit of RC buildings, showing many advantages, such as high strength and stiffness, relatively low weight, suitability for prefabrication, ease of replacement, large ductility and excellent hysteretic behavior. The application of transversal stiffeners placed into the panel delays their inelastic buckling and increases their energy dissipation capacity without pinching even up to a shear strain of 20% [12,21]. However, still few works in literature have been dedicated to the effectiveness of aluminum shear panels for seismic retrofit of steel and reinforced concrete buildings. In general, despite significant progress in research and development of metalyielding dampers in recent decades, their implementation for mitigating seismic risk in existing reinforced concrete buildings can still be considerably extended. One of the reasons for this situation is the general lack of relatively simple and effective design criteria and proper guidelines. Among the existing design methods, some of them require iteration or nonlinear time-history analysis and, thus, are still very complex for practical use. Other ones neglect the inelastic behavior of the main structure and/or the mechanical interaction between the metal-yielding dampers and the main structure. Consequently, there is a general need for simpler, more practical, and static methods suitable for practical use. To bridge this gap, the current paper proposes a design method of aluminum multi-stiffened shear panels that accounts for both the nonlinearity in the structure and dampers-structure interaction and gives an optimal distribution of the shear panels over the height of the building. The analytical laws refer to dissipative aluminum shear panels recently tested and analyzed by the authors. The proposed procedure has been described in detail and then applied to two typical RC buildings. Its verification has been carried out by checking the design assumptions. Its effectiveness has been validated using nonlinear response-history analysis.

Metal shear panels as dissipative devices
Firstly applied in the late 1920s as cladding panels, the metal shear panels (MSPs) have then been widely used in the USA, Japan, and Canada to enhance the seismic behavior of new and existing building structures. Several different configurations have been developed and applied ( Fig. 1) using the MSPs both as shear walls and as damper devices. At first, the MSPs were designed and stiffened to be "compact", thus avoiding premature buckling and providing stable and large hysteretic cycles. Subsequent research [22][23][24] has highlighted the possibility of using also slender shear panels since the post-buckling behavior due to the diagonal tension field mechanism can provide considerable stiffness, strength, and ductility. However, it also increases the interaction with the structural members and stress concentration at the corners and may provide a poor hysteretic behavior with a pronounced pinching effect due to out-of-plane displacements produced by the shear buckling. To mitigate these effects, the shear strength of the MSPs should be weakened to allow the development of inelastic deformations at a relatively low level of lateral loads. Therefore, optimized shapes [25,26], perforated shear panels [27][28][29], and low-yield stress materials have been proposed in the literature. At first, mild steel Typical layouts for metal shear panels: a full bay type; b partial bay type; c bracing type; d pillar type and low-yield strength (LYS) steel shear panels have been developed [30][31][32][33][34]. Then, the low availability on the market suggested the use of pure aluminum. The 3003-O alloy was proposed as the base material by Rai et al. [35] while the heat-threaded EN-AW 1050A and AW1054 alloys were proposed and developed in Europe [36,37]. The pure aluminum shear panels (ASPs) should be designed to have a pure shear dissipative mechanism without buckling up to the required plastic deformation level. To this aim, the core plates should be sized to satisfy the slenderness requirement [29,[38][39][40] involving the ratio of elastic buckling limit to yielding limit. Nevertheless, undesirable out-of-plane strains of shear plates are often inevitable in the large strain field. To restrain buckling, different design solutions have been proposed in the literature, such as external restraining plates [41], concrete layers [42], and buckling-inhibited boundaries [43,44]. Nevertheless, traditionally transversal and longitudinal welded stiffeners remain the most popular solution to delay the negative effects of buckling phenomena in the plastic stages [44]. For this purpose, the metal multi-stiffened shear panels should be designed to ensure a pure shear dissipative mechanism, i.e. to avoid both the local shear buckling of the plates confined within longitudinal and transversal stiffeners and global shear buckling where, due to their inadequate inertia, the stiffeners are involved in the buckling mode.

Background on retrofit design procedures of hysteretic dampers
The conventional approach to seismic design is traditionally based on force-based methods and response reduction factors. The Japan Society of Seismic Isolation [46] developed a design method based on the so-called elastic response reduction curve. This approach suffers from several deficiencies since it does not address the inelastic response of the main structure under earthquake ground motions and corresponding damage potential. For this reason, simple design methods based on the elastic-plastic response reduction curve (EPRRC) have been proposed in the literature [47], highlighting their tendency to overestimate acceleration reduction. This situation has stimulated the development of displacement-based seismic design methods [e.g., coefficient method of FEMA-273 [48], capacity-spectrum method of ATC-40 [49], direct displacement-based design method (Lin et al. [50])]. In general, current guidelines and building codes [51][52][53] do not provide appropriate rules to estimate the reduction factors for the linear analysis and require nonlinear time-history analyses for equivalent damping ratios higher than 15%. Therefore, general practice to size and locate the energy dissipative devices (EDDs) is to select initial trial values based on the engineer's expertise and then run static or dynamic analysis to check if the acceptance criteria for structural members and dissipation devices are satisfied. Thus, an iterative trial-and-error process is implemented until the optimum number of EDDs to meet the given performance acceptance criteria is finally achieved. This approach is very complex when sophisticated current nonlinear static procedures are applied, and is timeconsuming and computation-demanding when the nonlinear time-history analysis is used that requires sophisticated hysteresis models and spectrum-compatible earthquakes. Moreover, the trial-and-error process is quite a complex task since multiple performance objectives should be met simultaneously. Finally, the results depend on the skill of the designer. All of this has discouraged the wide-scale application of supplemental energy dissipation devices in current practice and stimulated the development of effective design procedures [54][55][56][57][58][59][60][61][62][63][64][65][66]. To avoid the trial and error analysis, Guo and Christopoulos [67] presented a performance spectra-based method for structures equipped with passive supplemental damping devices (including both hysteretic and viscous dampers). Zhang et al. [68] developed a simplified design method based on the concept of uniform damping ratio (UDR) to maximize the use of each damper. Pan et al. [69] proposed an alternative strength-based design approach for dual BRB-reinforced concrete frames. Finally, many design methods were proposed based on the Direct Displacement-Based Design (DDBD) method originally proposed by Priestley et al. [70] and then applied to many types of structures including buildings equipped with energy dissipation devices [71][72][73][74]. Each of the design procedures available in the literature suffers from some limitations. Some of them [54,65,66,73] are valid for low-rise buildings and/or neglect both torsional effects and higher modes contribution and are based on the proportional stiffness criterion (i.e., lateral story-stiffness due to the hysteretic dampers proportional to that of the main structure). Other ones rely on the hypothesis that the main structure remains elastic. Often, they neglect the interaction of the EDDs with the main structure, do not take into account the key parameters that relate the main structure to the dampers, and fail to provide the best combination of the corresponding stiffness and strength ratios. To overcome these limitations, Di Cesare and Ponzo [57] developed an iterative procedure where hysteretic dampers are used to regularize the stiffness and strength over the height of the buildings, thus controlling the maximum inter-story drifts. Barbagallo et al. [59] proposed a multiperformance design method for BRBs based on the control of the story drift demand. All the aforementioned design procedures focus essentially on maximum response, while the structural damage due to seismic loading is also caused by the accumulated energy dissipation. This stimulated the development of energy-based methods for the design of energy dissipation devices as an alternative to conventional force or displacement-based design procedures. Choi and 106 Page 4 of 29 Kim [75] developed a procedure for the seismic design of structures with buckling-restrained braces based on hysteretic energy spectra and accumulated ductility spectra. Aliakbari et al. [76] developed an energy-based design method for metallic dampers aiming to obtain uniform distributions of drift and ductility. Habibi et al. [77] proposed a multi-mode design method based on modal pushover analysis and energy spectra. Rabi et al. [78] developed an energy-based method to design dissipative bracing systems using the principle of optimum strength distribution. Hareen and Mohan [79] presented a retrofit design method for passive dampers that uses target energy as design criteria. Despite their efficiency, the application of energy-based methods in current engineering practice is discouraged since some important issues are not sufficiently addressed, such as hysteretic energy spectra to define the seismic action, and additional parameters to account for cyclic seismic behavior.

Proposed design method of aluminum shear panels
As aforementioned, the use of low-yield stress material allows the panels to yield even for low deformation levels. Therefore, pure aluminum shear panels (ASPs) may be used to increase the stiffness, damping, and energy dissipation capacity of building framed structures. To this aim, different layouts may be applied to both steel and RC structures, namely full bay type, partial bay type, bracing type, or pillar type (Fig. 1). The proposed design method is presented regarding the partial bay type (Fig. 1b) applied to RC building framed structures. However, as will be evident in the following, the method is generalizable to other arrangements. The single steps of the proposed design method are described in the following paragraphs.

Dual system (RC structure-metal shear panels)
Once the aluminum shear panels are added, the building becomes a dual system where both the main structure and the shear panels contribute to the global behavior. Practically, the total structural system may be decomposed into two subsystems, namely the frame system and the shear panel system (Fig. 2). The gravity loads act on the main structure, while the seismic actions are resisted also by the aluminum shear panels. As aforementioned, many design methods available in the literature require the main structure to work elastically. However, many existing RC buildings crack and then yield exhibiting inelastic deformations even for low lateral displacements. Moreover, usually, the singledegree of freedom (SDOF) system equivalent to the existing building is defined based on the pushover analysis of the bare frame system, thus neglecting the mechanical interaction between the metal shear panels and the main structure. On the contrary, the proposed design method accounts for both the inelastic behavior of the main structure and its interaction with the aluminum shear panels. As far as the main structure is concerned, a fiber plastic hinge model is considered thus automatically accounting for the P-M interaction and capturing the nonlinear hysteretic effects. The cross-section is discretized into a series of axial fibers which expand longitudinally along the hinge length. A specific stress-strain relationship is considered for each fiber depending on the material: confined concrete for the core, Fig. 2 Decomposition of the total structural system into two subsystems: frame system and shear panel system unconfined concrete for the cover, and steel for the longitudinal bars. As far as the aluminum shear panels are concerned, the partial bay-type configuration (Fig. 1b) is considered. The effectiveness of this solution has been extensively proved by full-scale experimental tests and local and global numeric analysis developed in previous studies [12,16,80,81]. The aluminum plate shear element is comprised of aluminum in-fill plates bounded by a column-beam system. A surrounding hinged steel frame system (one for each floor) is considered in this paper to install the aluminum shear panels into the RC structure. All the members of the surrounding steel frame should be designed such that they remain fully elastic for the forces transferred by the internal panels. Specifically, both columns and beams should have adequate stiffness and strength to sustain the boundary stresses associated with the tension field in the shear panel surface. For example, UPN180 coupled profiles made of Fe430 steel have been used for the constituent members in previous studies [12,80,82]. As far as the mechanical interaction between the aluminum shear panels and the main structure, the shear panels may be considered continuously and rigidly connected to the RC beams by adequate connection systems so that no slip among the connected parts occurs. For example, coupled UPN200 profiles connected by M16 threaded bars have been used to transfer the forces absorbed by the panels to the RC beams in previous studies [12,80,82]. As far as the nonlinear modeling of the aluminum shear panel and surrounding steel frame, refined FEM nonlinear models have been developed and extensive numerical analyses have been carried out [12,16,80], also including geometric imperfections. As an alternative, the well-known strip model [23] represents the behavior of the shear panel as a series of inclined strips having the same panel thickness and oriented in the same direction as the principal tensile stresses. Its effectiveness, when applied to slender full bay-type shear panels, has been demonstrated in several studies [16,83,84]. Its validity has been extended to compact shear panels, considering two series of strips, one for tension and the other for compression diagonal [83] that may be completely neglected if the early shear buckling phenomenon occurs. However, the implementation of these models in current practice may be very complex and time-consuming. Therefore, simplified numerical models have been proposed that account for the plate-frame interaction and consider the behavior of the steel plate and frame separately. In particular, the model developed by Sabouri-Ghomi [29] is based on some simplifying hypotheses that allow it to be used easily in generalpurpose computer programs. The shear plate is considered as simply supported along its boundaries and the effects of global bending on shear buckling stress are neglected. The columns are considered rigid enough to neglect their deformation and assume that a uniform tension field is developed. The bending on the floor beam due to the tension field is neglected. Both the aluminum shear plate and the surrounding steel frame are considered to exhibit linearly elastic and perfectly plastic behavior. Finally, it should be observed that in the case studies herein examined, a hinged steel frame system is used to install the shear panel. Therefore, the stiffness and energy dissipation capacity of the surrounding steel frame can be neglected when compared to those of the aluminum shear panel.

Evaluation of seismic capacity
To account for the interaction between the main structure and the metal shear panels, the nonlinear static (pushover) analysis is carried out on the dual system. Only in the first step of the design procedure, the pushover analysis is carried out using the bare frame model since the properties of the metal shear panels are unknown. The pushover analysis under the first-mode distribution of the lateral forces gives the story pushover curves (i.e., story shear force vs story drift). By decomposing the dual systems into two subsystems (Fig. 2), the i-th story shear force ( V i ) is written as the sum of the shear force in the frame system ( V f i ) and the shear force in the panel system ( V p i ). This allows plotting the story pushover curve (i.e., story shear force V f i vs story drift f i ) of the main structure for each story (Fig. 3a). These curves are then idealized using the well-known Takeda model [85] (Fig. 3a, b). To this aim, the initial stiffness K f 0,i is assumed to be equal to the tangent stiffness (Fig. 3b) and the following hypotheses are made: parameter is the yield story drift ( f y,i ) which is calculated using the energy equivalence method. By way of example, Fig. 4a, b show the story pushover curves of a frame system and the corresponding trilinear idealizations, respectively. Finally, a simplified trilinear model has been identified assuming the same value of the yield drifts for all stories, given by: where f y,i is the i-th story yield drift and N is the number of the stories of the building. Moreover, the same target drift capacity ( f t ) and ductility ratio ( f = f t ∕ f y ) are considered for all stories giving the simplified curves of Fig. 4c.

Equivalent SDOF system of the main structure
After the simplified trilinear idealization, the frame system is represented as a simplified multi-degree-of-freedom system (MDOF) (Fig. 5) with lumped masses at different heights ( h i ). Each story is characterized by its simplified trilinear pushover curve, but all the stories have the same yield drift ( f y ), target drift capacity ( f t ), and ductility ratio ( f ). The analogy between the fundamental vibration mode of the MDOF system and an equivalent single-degree-of-freedom (SDOF) system allows calculating the equivalent height ( h eq ) and mass ( m eq ), given by: where m i , u i and h i are, respectively, the mass, modal displacement, and height of the i-th story. The secant period ( T f ) is defined based on the first mode period ( T f ) of the bare frame, and secant ( K f ) and initial ( K f 0 ) stiffness, as follows: The drift angle demand ( f d ) can be estimated as follows where T f and f 0 are, respectively, the secant period and the viscous damping ratio of the equivalent SDOF system, S D and S A are the spectral displacement and acceleration. The drift angle capacity f t is given by: where the target drift capacity f t can be selected based on the seismic capacity of the existing building calculated from the pushover analysis. The drift capacity-to-demand ratio (i.e., f t f d ) is defined as the drift reduction factor (namely R p ).

Equivalent SDOF system of the aluminum shear panels
The parameters of the equivalent SDOF system of the main structure and the corresponding drift angle capacity ( f t ) and demand ( f d ) are the starting point to define the equivalent SDOF system of the aluminum shear panels. To this aim, the stiffness ratio r p = K p K f (i.e., panels to bare frame ratio to meet the target drift reduction factor R p = f t f d ) is defined based on the following formula [46,86]: where p = K f K f , c is the cracking ductility ratio of the main structure, s is the stiffness ratio between the surrounding steel frame and the shear panel ( s = 0 if hinged steel frames are used to connect the shear panels to the structure), a = 25, λ = 0.5, R = 0.6. In Eq. (6), the ductility demand p of the aluminum shear panels is unknown. Therefore, an iterative procedure is developed assuming, in the first step, that the ductility demand p is equal to the ductility capacity of the shear panels. This procedure finally gives the ductility ratio p of the equivalent SDOF system (Fig. 6a) and the corresponding design stiffness and shear force of the aluminum shear panels, given by:

Optimal distribution of the aluminum shear panels along with the height
The design stiffness and shear force of the shear panels are distributed along with the height of the building to convert the SDOF to the MDOF system (Fig. 6b). The following simplifying hypotheses are made for the conversion to the (6) 6 Sheal panels modeling. a Equivalent SDOF system; b simplified MDOF model; c shear panel system MDOF system: (1) the equivalent viscous damping of the MDOF system is the same of the SDOF system; (2) yield drift angle, drift angle demand, and ductility demand of the MDOF system are uniformly distributed over the height. After some steps, these hypotheses provide the optimal lateral stiffness of the shear panels for each i-th story of the building [46]: where V i is the design shear force and K f ,i is the secant stiffness of the main structure. The corresponding design shear force of the aluminum shear panels is given by:

Design of the aluminum shear panels
As highlighted in Sect. 2, if the aluminum shear panels are properly designed and stiffened, the buckling phenomena in the plastic field are delayed up to the target displacement. Therefore, the yielding spreads all over the panel thus increasing the energy dissipation capacity, and a pure shear dissipative mechanism is activated. For that to happen, the stiffeners should be designed to avoid both local and global shear buckling. The local shear buckling concerns the part of the plate enclosed within longitudinal and transversal stiffeners. The global shear buckling can include the stiffeners themselves if their inertia is inadequate. Equations (8) and (9) give the distribution in elevation of the story stiffness and strength of the aluminum shear panels, respectively. Their layout in-plan may be selected by accounting for different issues, such as the capacity to prevent torsional effects in the seismic behavior, impact on architectural functionality, and full operation of the building during the retrofit implementation. If shear panels of the same type are used, the stiffness k p i and strength v p i of the single panel can be immediately derived as follows: where n i is the number of the shear panels to arrange on the i-th story. Substituting Eqs. (8)- (9) in Eq. (10) gives: Equation (11) shows that the stiffness k The multi-stiffened pure aluminum shear panels are designed using the partial bay type configuration (Fig. 1b). Their geometry is evaluated as follows. At first, the design stiffness k p i of the shear panel is used to evaluate the width (b w,i ) and thickness (t w,i ) of the panel at the i-th story. To this aim, the stiffness of the shear panel is expressed using the following formulation [23,29] that neglects the pre-buckling shear resistance of the plates when the slenderness ratio of the applied shear plates is quite large: In Eq. (12), E is the Young modulus of the aluminum, t w,i , b w,i , and d i are, respectively, the thickness, width, and depth of the panel (Fig. 7). The shear strength v p i of the panel is used to design its intermediate stiffeners. To this aim, the factor ρ v for shear buckling is used that is defined as the normalized design shear resistance [44], as follows: The value of ρ v,i is then used for designing the spacing of longitudinal and transversal web stiffeners in such a way as to ensure that the yielding strength is lower than the elastic buckling strength. For this purpose, the factor ρ v for shear buckling is expressed as a function of the slenderness parameter w , as follows [45]: where η depends on the hardening ratio ( f aw f ow ), as follows: In Eq. (15), w is a slenderness parameter given by: where f v is the yielding shear strength of the aluminum, τ cr is the Eulerian shear buckling stress, k τ is the shear buckling coefficient, a is the spacing of the intermediate stiffeners (Fig. 7). The shear buckling coefficient k τ for squared subpanels and rigid stiffeners is given by: where the sum of the first two terms is the shear buckling coefficient based on the Timoshenko theory, while k st accounts for the longitudinal stiffeners as follows: 3 4 . In Eq. (19), I st is defined as the sum of the second moment of area of all the longitudinal stiffeners. For each of them, a collaborating area 15t w wide is considered. The only unknown parameter in Eqs. (17)- (19) is the spacing (a) of the intermediate stiffeners. For each panel, its value (a i ) is evaluated so that the value of ρ v,i calculated using Eq. (15) is equal to the design value given by Eq. (14). To this aim, the design method should include a trial and error process, which determines the required spacing of the stiffeners. Finally, the second moment of inertia of the ribs is checked to be high enough to avoid the development of global buckling phenomena of the whole shear panel due to its small out-of-plane flexural stiffness. For this purpose, the minimum second moment of inertia of the ribs to avoid global buckling is defined using the following formulation proposed in the literature [45]: where: The flowchart in Fig. 8 clarifies the design process. It should be underlined that at the beginning of the design procedure both the geometry and layout of the aluminum shear panels are unknown. In the first step, the pushover analysis is carried out on the RC frame system (the first step in Fig. 8) thus neglecting both the contribution of the aluminum shear panels and their interaction with the main structure. Then, the design procedure is developed and the layout and geometry of the metal shear panels are determined. Subsequently, the pushover analysis of the dual system is carried out and its nonlinear response is decomposed as the sum of two subsystems (namely RC frame system and shear panel system) (Fig. 8), thus giving the story pushover curves of the RC frame that account for the interaction with the shear panels. Finally, the parameters of the simplified trilinear idealization of the story pushover curves are calculated and their values are compared with those obtained in the previous step. The procedure is iterated until their difference is smaller than a preset threshold of 10%.
5 Implementation of the proposed design procedure

Seismic assessment of the benchmark structures
Two 6-story reinforced concrete framed buildings are considered in this section. The plan view of the benchmark structures, namely Building A and Building B, are plotted in Fig. 9. The buildings are regular in both plan and elevation, and have a beam span of 5.0 m and an interstory height of 3.0 m. The first structure (i.e., Building A) is designed for a medium-risk seismic zone in compliance with an old Italian building code [87]. The second structure (i.e., Building B) is designed for gravity loads with no seismic provisions. As far as materials are concerned, steel FeB38k and concrete C20/25 are considered for design. The cross-sections and the corresponding reinforcement in the main structural elements are shown in Figs. 10, 11 and Table 1. The computer program SAP2000 [88] is used for the analysis. Table 2 shows the main modal properties   of the benchmark structures. The seismic performance assessment is carried out according to the procedure of Eurocode 8 [89] also implemented in the Italian Building Code [53]. The demand response spectrum of the Italian Code [53] is used based on the following parameters: soil type C, topographic factor T 1 , and reference life V R = 75 years. The capacity response spectrum is calculated from the nonlinear static (pushover) analysis. For this purpose, a fiber plastic hinge model is developed using SAP2000 computer program [88]. For concrete fibers, the well-known Mander's stress-strain model [90] is used. For steel fibers, a simple elastic-plastic-hardening relationship is considered. The biaxial moment-rotation relationships are obtained by integrating stresses over the cross-section and multiplying for the hinge length defined based on the Italian Code provisions [53]. End length offsets are defined from the connectivity of columns and beams. A rigid zone factor of 0.5 is used at the beam ends to model the panel zone elastic stiffness. As far as the mechanical interaction between the aluminum shear panels and the main structure, the shear panels are considered continuously and rigidly connected to the RC beams by adequate connection systems. Therefore, the panel-to-frame connections are modeled by considering that no slip among the connected parts occurs. A conventional pushover with two distributions (uniform and modal) is carried out also considering an accidental eccentricity of 5%. The seismic assessment is performed for the limit states of Immediate Occupancy (IO), Damage Limitation (DL), Life Safety (LS), and Collapse Prevention (CP). The drift ratio is checked for IO and DL limit states, while the chord rotation is checked for DL, LS, and CP limit states. Figures 12, 13 show the pushover curves and the performance points corresponding to the different limit states. For the sake of brevity, only the pushover curves are plotted that are obtained using the modal distribution of the lateral forces. Table 3 shows the results of the seismic assessment for the different limit states, namely the performance point (target displacement d and corresponding base shear T normalized by the seismic weight W), capacity peak ground acceleration (PGA c ), demand peak ground acceleration (PGA d ), and corresponding safety index ζ PGA = PGA c /PGA d . The outcomes show that the IO and DL limit states are satisfied, while the LS and CP limit states are not satisfied. This highlights that both the stiffness and strength of the structure are adequate, but the energy dissipation capacity should be enhanced through appropriate seismic retrofit measures.

Design of the aluminum shear panels
The features of the multi-stiffened pure aluminum shear panels tested in [45] are considered in the analysis. The panels are made of aluminum alloy EN-AW 1050A. The following mechanical features are used: conventional yield strength f 0.2 = 21 N/mm 2 , ultimate strength f u = 97.44 N/mm 2 , ultimate deformation ε u = 13%, elastic modulus E = 70,000 N/mm 2 . The aluminum shear panels are arranged in the partial bay-type configuration (Fig. 1b) and sized using the design procedure described above. Figures 14, 15, 16, 17 shows the pushover curves (a), their trilinear idealization (b), and the simplified trilinear curves (c) obtained for buildings A and B at the first step of the design procedure. In Figs. 18, 19, 20, 21 are plotted the corresponding curves obtained at the last step of the design procedure. The parameters of the equivalent SDOF systems as defined by Eqs.
(2)-(6) are summarized in Table 4. A drift angle capacity f t = 0.02 is selected based on the seismic capacity of the existing building calculated from the pushover analysis. This gives drift capacity-to-demand ratios (R p ) ranging from 0.505 to 0.624 in Table 4. The results show very  Fig. 15 Pushover of the RC frame system (Building A, Y-direction, first step of the design procedure). a Story pushover curves; b trilinear idealization; c simplified trilinear model  Fig. 19 Pushover of the RC frame-shear panels systems (Building A, Y-direction, last step of the design procedure). a Story pushover curves; b trilinear idealization; c simplified trilinear model similar values for the equivalent mass and height obtained for the two buildings. Building B is designed for gravity loads only and this leads to a lower lateral stiffness performance if compared to Building A. Therefore, both the fundamental vibration period ( T f ) and secant period ( T f ) of the equivalent SDOF systems are significantly higher. The optimal stiffness ratio to meet the target drift reduction factor R p is calculated from Eq. (6). The values obtained, ranging from 0.594 for Building A to 1.432 for Building B in the X-direction (Table 4) ). Finally, it should be underlined that for Building B in Y-direction, the lateral stiffness is very low at the 5th story and even negative at the 6th story. This is the reason why no panel is placed in these stories (Fig. 23) (Table 8).

Checking the design assumptions
Some simplifying assumptions are made in the design procedure. First, the mode shapes are assumed to remain  unchanged despite the insertion of the panels. Then, the yield drift ratios are considered uniformly distributed over the building height. To check the first design assumption (i.e., mode shapes unchanged), the comparison between the mode shapes before and after retrofit is carried out using the modal assurance criterion (MAC) [91]. The following values of the MAC index are found. For Building A, MAC = 0.992 for the flexural X mode shape and MAC = 0.998 for the flexural Y mode shape. For Building B, MAC = 0.998 for the flexural X mode shape and MAC = 0.970 for the flexural Y mode shape. Therefore, the level of correlation is high and the mode shapes remain practically unchanged after inserting the shear panels. To check the second design assumption (i.e., uniform yield drift distribution over the building height), Fig. 24 shows the story shear forces in the aluminum shear panels (ASPs) obtained from the pushover analysis of the retrofitted building. The results show that the aluminum panels tend to yield simultaneously and, thus, respond according to the hypotheses and predictions of the design method. Specifically, the standard deviation of the story yield drift values         is about 13% and 12% for Building A, and 11% and 15% for Building B, respectively in the X-and Y-directions.

Nonlinear dynamic performance verification
The effectiveness of the retrofit design procedure is finally verified through the nonlinear dynamic analysis. Two accelerograms are simultaneously applied along with the horizontal directions. A group of seven pairs of time histories is considered in the analyses. The design value is defined as the mean of the response quantities. The earthquake ground motions have been selected from accelerometric databases [92] and scaled corresponding to the target elastic spectra (Table 9). Figure 25 shows the scaled acceleration spectra and their spectrum compatibility. The dynamic analysis should account for the complex hysteretic behavior of the aluminum shear panels. Under earthquake loadings, stiffness and strength degradation, and pinching effects may occur. When the plastic deformation increases, the local buckling of aluminum web plates due to the residual strain may cause slip-type hysteresis loops. Therefore, the aluminum shear panels are idealized by nonlinear shear link elements with pivot hysteresis type. Four hysteresis models have been considered in the analysis to cover the range of possible behaviors. Figure 26 shows the different models herein considered: (a) semi-slip model (pinching of 50%); (b) semi-slip model (pinching of 90%); (c) slip model; (d) semi-slip model (pinching of 50%) with stiffness and strength degradation. It should be highlighted that the stiffeners have been designed to avoid both global and local buckling and, therefore, pinching in hysteresis loops should be limited. Therefore, an excellent hysteretic behavior of the ASPs can be considered in the analyses, and the load-displacement behavior should be well described by the semi-slip models shown in Fig. 26a, d. The effectiveness of the seismic retrofit strategy based on aluminum shear panels is proved by comparing the results of the nonlinear dynamic analysis for the existing and retrofitted buildings. The inter-story drift ratio (IDR) is used as an indicator of the distribution of seismic demands and damage mechanisms under earthquake ground motions. The results refer to the semi-slip model with stiffness and strength degradation (Fig. 26d). Figures 27,28,29,30 show the variation in the peak inter-story drift ratio (IDR) response over the height of the building. The results are plotted in both the X-and Y-directions and refer to existing Building A (Fig. 27), retrofitted Building A (Fig. 28), existing Building B (Fig. 29), and retrofitted Building B (Fig. 30). Both peak IDR value for each earthquake record and mean (μ) and mean (μ) + standard deviation (σ) values are plotted. For existing Building A, Fig. 27 shows that the profile over the height of the building strongly depends on the earthquake record, especially in the Y-direction. The IDR demand is very high, especially for story 1 in the X-direction (mean value equal to 7.54%), and for stories 1 and 5 in the Y-direction (mean value equal to 7.49% and 5.10%, respectively). The peak IDR response largely exceeds the maximum allowable transient drift that many seismic codes define as around 4% for the performance objective of collapse prevention. Practically, drift concentration and consequential weak story collapse occur in both directions. On the contrary, a more uniform distribution of the IDR response is observed over the height of the retrofitted Building A (Fig. 28) when compared to the existing Building A (Fig. 27)  the Y-direction. The above comparison of the results of the nonlinear dynamic analysis can be summarized as follows: the seismic retrofit strategy based on aluminum shear panels is effective in significantly reducing inter-story drift and avoiding drift concentration and consequential weak story collapse. According to the current generation of seismic codes [53,89] the effectiveness of the design method should be evaluated by checking the chord rotation in all the structural elements. The chord rotation capacity for the CP limit state is calculated based on formula A.1 of EN 1998-3 [89]. For columns, the chord rotation may be reasonably defined as the deflection at the end of the shear span divided by the shear span length, that is the interstory drift divided by the inter-story height. Therefore, the inter-story drift capacity in uniaxial bending (along X and Y directions) is evaluated from the chord rotation capacity. The inter-story drift capacity in bi-axial bending is defined based on the hypothesis that the interaction diagram is circular if the components of chord rotation along the sides of the section in bi-axial bending are normalized to the corresponding chord rotations in uniaxial loading. This gives the CP limit domain. Figure 31 (for Building A) and Fig. 32 (for Building B) show the limit domain obtained for a corner column at different stories and the target limit domain corresponding to the target drift of 2%. For the sake of verification, these capacity limit domains are compared to the corresponding time-history demand.
The results show the effect of the hysteresis model used for the ASPs. For Building A, the demand exceeds the capacity only in the columns of the first story for the hysteresis models (b) and (c). For Building B, the seismic capacity is exceeded in the column of the 5th story for models (b), (c), and (d), and in the column of the 6th story for model (c). However, it should be observed that this situation occurs only for a single earthquake ground motion, while the safety verification is satisfied since the mean of the inter-story drift demands is used as the reference value. In general, the chord rotation verification is satisfied for all the structural members.

Conclusions
The general lack of relatively simple and effective design procedures suitable for practical use prevents the application of aluminum shear panels for seismic retrofit of existing reinforced concrete buildings. To overcome this limitation, this paper has developed a design method for aluminum shear panels that accounts for both the nonlinearity in the structure and the dampers-structure interaction and provides an optimal distribution of dampers over the height of the building. The analytical laws are obtained from recent analyses on multi-stiffened pure aluminum shear panels. The proposed procedure has been first described in detail and then employed in two benchmark buildings having drift capacity-to-demand ratios ranging from 0.505 to 0.624. The corresponding design values of the panel-to-frame stiffness ratio have been found to range from 0.594 to 1.432 as a function of the lateral stiffness of the existing building. The verification of the proposed design method has been carried out first by checking the validity of its design assumptions, and then by evaluating its effectiveness through nonlinear response-history analyses. The following conclusions may be drawn from the results of this study.
• The proposed design procedure is shown to be feasible and effective even in the presence of complex hysteretic behavior such as that of aluminum shear panels. The retrofitted building is treated as a dual system, thus accounting for the interaction between the two subsystems. • Based on the results from the nonlinear static analysis, the RC buildings retrofitted with aluminum shear panels are shown to respond according to the hypotheses of the design method. The mode shapes of the The yield drift ratios are practically uniformly distributed over the height of the building since the aluminum panels tend to yield simultaneously. Specifically, the standard deviation of the story yield drift values of the ASPs is about 13% and 12% for Building A, and 11% and 15% for Building B, respectively in the X-and Y-directions. • The results from the nonlinear dynamic analysis for different seismic accelerograms and hysteresis models have shown that the seismic retrofit strategy based on aluminum shear panels is effective in significantly reducing inter-story drift and avoiding drift concentration and consequential weak-story collapse. The maximum inter-story drift ratio demands of the retrofitted building (1.04% for Building A and 2.07% for Building B) are significantly reduced compared to the existing building, for which code-based drift limits (around 4% for the collapse prevention limit state) are greatly exceeded. • The retrofitted buildings respond according to the requirements of the Italian Code, thus confirming the effectiveness of the design method in mitigating the inter-story drifts and minimizing structural damage. The effect of the hysteretic behavior of the aluminum shear panels is highlighted, thus underlining the importance of the local buckling stiffeners that should be properly designed to avoid both global and local buckling and, thus, pinching in hysteresis loops.
Finally, it should be underlined that the proposed procedure is based on some simplifying hypotheses. First, the SDOF assumption neglects both higher-mode and torsional effects. Thus, it is valid for normal low-rise regular buildings but may fail for high-rise or irregular buildings. Then, the equal yield displacement hypothesis may fail for buildings exhibiting torsional or soft-story failure mechanisms. Moreover, comprehensive consideration of hysteresis of aluminum shear panels is lacking, since the design method neglects their cyclic behavior and tries to size the stiffeners to delay the inelastic buckling and increase the energy dissipation capacity without pinching. Finally, the connection of the ASPs with the existing frame and the need for new foundations would require further exploration.
Funding Open access funding provided by Università degli Studi della Campania Luigi Vanvitelli within the CRUI-CARE Agreement. No funding was received for conducting this study.

Availability of data and materials
The paper contains all the data used in the study.

Declarations
Conflict of interest The authors declare that they have no conflict of interest.
Ethics approval This article does not contain any studies with human participants or animals performed by any of the authors.
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