Seismic retrofit strategies of frame structures are quickly evolving, and several significant contributions have been recently offered in terms of design approaches and modelling (Marano et al., 2007; Greco and Marano, 2016; Di Trapani et al., 2020; Aloisio et al., 2022), as well as of implementation of innovative protection techniques or upgrade of existing ones (Aloisio et al., 2022; Boggian et al., 2022; Lu and Phillips, 2022; Zhao et al., 2022).

Among the strategies based on the incorporation of supplementary damping devices, those making use of Added Damping and Stiffness (ADAS) steel dissipators have a well-established tradition (Bergman and Goel, 1987; Whittaker et al., 1991; Aiken et al., 1993; Martinez-Romero, 1993; Tsai et al., 1993; Dargush and Soong, 1995), updated to recent developments too (Mazza, 2014; Teruna, 2015; Moghaddasi and Namazi, 2016; Sorace et al., 2016; Mohammadi et al., 2017; Mazza and Imbrogno, 2019; Shojaeifar et al., 2020; Gandelli et al., 2021; Khoshkalam et al., 2022). This is a consequence of the plain working principle of these devices, based on the elastic-plastic behaviour of the constituting plates, as well as of their relatively easy installation. In spite of this, the design of ADAS dampers is not an easy task, as it requires a proper balance between the addition of energy dissipation and horizontal translational stiffness. The former allows remarkably reducing plastic demand on structural members. The latter is always beneficial in terms of lateral displacements but, as long as the dampers respond elastically, can cause a significant increase in base and storey shears, and thus in the stress states of the frame skeleton elements and foundations.

In view of this, an energy-based preliminary sizing criterion of ADAS dampers was proposed by the first two authors of this article (Sorace et al., 2016), aimed at achieving a joint control of stress states and displacements. According to this criterion, typically framed within a performance-based design approach (SEAOC, 1995; Priestley, 1997; Priestley and Calvi, 1997; Fardis. 2007; Priestley et al., 2007), the total number of plates is tentatively fixed by: 1) pre-estimating the energy dissipation capable of providing a target drop of base shear in retrofitted conditions; 2) computing the energy dissipation capacity by assuming a maximum displacement of the dissipators calibrated on a target reduction of storey drifts. By expanding and developing this method, a new sizing procedure is formulated in this article, where the energy dissipation capacity of the dampers is evaluated by expressly compensating the bracing-related increase of base shear, caused by the increase in lateral stiffness of the structure.

The distinguishing features of the procedure, as compared to other existing methods, can be synthesized in the two following points: 1) it provides a quick estimate of the energy demand on the protective system, based on simple analytical relations making use of spectral displacement, velocity, and pseudo-acceleration ordinates; 2) it is intrinsically not iterative, as these relations are linear.

Concerning the application of the procedure, the use of spectral quantities produces the best approximation for structures characterized by first translational modes along the two coordinate axes in plan with associated modal masses constituting a predominant portion of the total seismic masses. This condition is basically met by structures with a rather regular geometry in plan and elevation. However, in order to effectively apply the sizing procedure to irregular structures too, a modulating coefficient of the energy dissipation demand is included to compensate for higher translational modes and/or plan torsion response effects.

The formulation of the sizing criterion is presented in Section 2. A demonstrative application is offered in Section 3, Section 4, concerning the retrofit design of a representative case study, i.e., a 6-storey residential building with reinforced concrete (RC) structure situated in L’Aquila, Abruzzo, Italy. Based on the results of a seismic assessment analysis initially carried out on the building (Section 3), the application of the procedure is explicated and discussed in detail in Section 4.1, Section 4.2. The performance of the structure in retrofitted conditions is finally evaluated (Section 4.3), to compare the response of the designed ADAS-equipped dissipative bracing system with the hypotheses formulated at the sizing stage.

ADAS dissipators consist of an assembly of multiple X-shaped or isosceles triangle-shaped (in which case they are named T-ADAS) steel plates mounted in parallel, which act as short cantilever beams with cross section linearly varying along the span, similarly to the bending moment diagram.

Figure 1 shows the drawing of a T-ADAS damper, the geometry of a constituting plate and the P (t)–d (t) force–displacement hysteretic response cycle of the latter, idealized with a bilinear hardening shape. The mechanical parameters governing the cycle are: P _y , P _u = yielding and ultimate force, d _y , d _u = yielding and ultimate displacement, all intended with positive and negative sign, k _e , k _p = stiffness of the elastic and post-elastic response branch, and γ = k _p /k _e = strain hardening ratio of the post-elastic branch, normally assumed as equal to 0.03 (Dargush and Soong, 1995).

Based on the geometric symbols in Figure 1, and named f _y the yielding stress of steel, the expressions of the response cycle parameters are as follows: P y = f y B T 2 6 H (1) d y = P y k e (2) k e = E s B T 3 6 H 3 (3) k p = γ k e (4) P u = f y B T 2 4 H (5) d u = d y + P u − P y k p (6)

The preliminary sizing procedure of the dampers proposed in this study is aimed at evaluating the number of plates to be installed in the bracing system, n _pl , starting from a quick estimation of the energy dissipation capacity by which a pre-established reduction of lateral displacements and base shear of the structure is attained. For frame buildings with a substantially regular geometry in plan and elevation, this estimation is carried out in terms of spectral quantities, by referring to the periods of the first translational modes along the two coordinate axes in plan (which a predominant portion of modal masses is associated with, in the above-mentioned hypothesis of approximate regularity). Both in current and retrofitted conditions, these periods are assumed to be included in the wide period interval of the normative response spectra that corresponds to the constant pseudo-velocity branch, identified by the periods limits T _C and T _D , where the fundamental vibration periods of low rise through mid-to-high rise RC frame structures are situated.

For both axes in plan, X and Y, of the reference Cartesian coordinate system assumed in the analysis, named T _IN the fundamental period of the structure in current state, the procedure starts by fixing a tentative reduction, ΔS _D- , in the spectral displacement computed for T _IN . Consistently with the first mode-based quick estimation of seismic response adopted here, the difference between S _D (T _IN ) and ΔS _D- is put as equal to an assumed target value of the top lateral displacement of the structure, D _top,targ : S D T I N − Δ S D − = D t o p , t a r g (7) from which ΔS _D- derives: Δ S D − = S D T I N − D t o p , t a r g (8)

Then, named S _V,cost the value of the horizontal branch of the pseudo-velocity spectrum, the period in retrofitted conditions, T _FIN , given by the difference between T _IN and the period reduction ΔT corresponding to ΔS _D- , can be evaluated as follows: T F I N = T I N − Δ T = T I N − 2 π Δ S D − S V , c o s t (9)

As illustrated in Figure 2, a ΔS _A+ rise in pseudo-acceleration is generated when passing from T _IN to T _FIN , as a consequence of the increase in lateral stiffness induced by the incorporation of the dissipative bracing system. Therefore, its supplementary damping contribution must contextually generate a drop in the spectral ordinate, ΔS _A-,diss , so as to decrease base shear—or, when deemed sufficient, to limit its growth—as compared to current conditions. Consistently, ΔS _A-,diss is tentatively fixed in proportion to ΔS _A+ , by means of an energy dissipation demand coefficient, m _ed , to be calibrated on the design objective(s) fixed in terms of base shear. This condition is expressed as: Δ S A − , d i s s = m e d Δ S A + (10) where m _ed > 1 targets to reduce both lateral displacements and base shear, whereas m _ed ≤ 1 only the former. It can be noted that a moderate increase in base shear can be often accepted, since a significant portion of storey shears is absorbed, as axial forces, by the diagonal trusses of the bracing system. This allows remarkably reducing bending moments and shears in beams and columns, as targeted in the design, only causing an increase of axial forces in the columns belonging to the vertical alignments where the system is placed, as well as of the stress states in the underlying portions of the foundations. Therefore, in order to prevent an excessive addition of supplementary damping, m _ed values equal to or slightly greater than 1 (i.e., equal to 1.1–1.2), and in some cases even slightly smaller than 1, can be adopted, especially for regular structures. At the same time, greater m _ed values are normally needed for irregular structures, to compensate for possible significant contributions of the higher translational modes, and/or plan torsion response effects related to the rotational modes around the vertical axis of the reference coordinate system. Furthermore, m _ed values greater than 1 are normally needed for frame structures, with RC or steel skeleton, characterized by rather high fundamental vibration periods (e.g., greater than 1.5 s), since in these cases a greater increase in lateral stiffness—and thus in base shear—is induced by the stiffening action of the bracing system.

The supplementary damping-related reduction in base shear, ΔV, is obtained by multiplying ΔS _A-,diss by the seismic mass M of the building. Hence, the tentative energy dissipation capacity to be assigned to the protective system is computed as: E d = 4 ∆ V ∆ S D − = 4 M Δ S A − , d i s s ∆ S D − = 4 M m e d Δ S A + ∆ S D − (11)

By separately applying relations (Eq. 7) through (Eq. 11) for the X and Y axes, the corresponding energy dissipation capacities, E _d,X , E _d,Y , are obtained. Then, the total numbers of plates of the sets of ADAS dampers to be installed in X and Y, n _p,X , n _p,Y , are evaluated by dividing E _d,X , E _d,Y by the estimated total energy dissipated by each plate, E _d,p,t , under the input ground motions assumed in the design analyses. E _d,p,t is calculated by computing, at a first step, the E _d,p,c energy dissipated by a plate in a cycle bounded by the maximum estimated (positive or negative) displacement, d _max,e (symbolically indicated in the graph of Figure 1 too), given by: E d , p , c = 2 P y ∙ 2 d max ⁡ , e − d y (12)

Then, E _d,p,t is obtained by summing to E _d,p,c the energy dissipated in all remaining cycles with smaller amplitudes, E _d,p,r . E _d,p,r was evaluated to be about 10 times greater than E _d,p,c in a recent extensive numerical research (Gandelli et al., 2021). At the same time, a similar value was preliminarily estimated in this study for simpler frame structures than the case study one, not reported here for brevity’s sake, by extending the findings of a previous study of the authors carried out for a special infrastructure (Sorace et al., 2016). Based on this approximate E _d,p,r value, the tentative evaluation of E _d,p,t results as follows: E d , p , t = E d , p , c + E d , p , r = 1 + 10 E d , p , c = 11 E d , p , c (13)

Thus, n _p,X , n _p,Y are finally evaluated as follows: n p , X = E d , X E d , p , t (14a) n p , Y = E d , Y E d , p , t (14b)

For the practical application of Eqs 12, 13, d _max,e must be preliminarily quantified. This can be done by relating it to the maximum inter-storey drift (ID _m ) of the structure, and fixing a target design value for the latter, ID _m,t . By considering that the supporting diagonal braces of the ADAS devices are remarkably rigid at the horizontal translation, and thus lateral displacements are mainly determined by the plastic deformation of the dissipators (Sorace et al., 2016), at the sizing stage d _max,e can be approximately assumed to coincide with ID _m,t .

The latter is calibrated on the specific characteristics of each case study and the severity of the basic earthquake level considered in the performance assessment analysis (formally defined in the next section). By way of example, in order to limit damage to a fully repairable level in masonry infill and partition panels of RC frame buildings, when they are built in contact with the structural members, named IH the inter-storey height, ID _m,t could be tentatively put as equal to 0.5% of IH, i.e., the limit fixed by the Italian Technical Standards (Italian Ministry of Infrastructure and Transport, 2018) as basic requirement for the attainment of the Immediate Occupancy (IO) performance level. The same drift limitation is assumed by internationally prominent performance-based seismic Standards and Guidelines, among which SEAOC Vision 2000 (SEAOC, 1995), FEMA 356 (ASCE, 2000), ATC 58-2 (ATC, 2003), ASCE/SEI 41 (ASCE, 2017), as repairability threshold for unreinforced masonry infill and partition walls.

The examined case study is a residential building, located in L’Aquila, Abruzzo, Italy, designed in compliance with the 1986 edition of the Italian Seismic Standards, where a maximum pseudo-acceleration spectral ordinate of 0.07 g (g = acceleration of gravity) was prescribed for the corresponding seismic zone. As illustrated in Figure 3, referred to the ground storey, the building has a 21.05 × 11.45 m² sized rectangular plan, with the first side parallel to the X axis and the second one to the Y axis of the assumed Cartesian reference system. The building has six above-ground storeys in elevation, with inter-storey heights equal to 3.4 m for the ground storey and 3.04 m for the remaining ones.

The structure is constituted by a RC frame skeleton. Columns are of five types, named CT1 through CT5, having the following cross sections and steel reinforcing bars: CT1—300 × 600 (side parallel to X×side parallel to Y) mm², 8ø14; CT2—300 × 550 mm², 8ø14; CT3—500 × 300 mm², 6ø14; CT4—300 × 400 mm², 4ø14; and CT5—300 × 300 mm², 4ø12. The transversal reinforcement is constituted by 200 mm-spaced ø6 stirrups for all columns. By way of example, the column types–vertical fixed lines association on the ground storey is the following (Figure 3): CT1—2/B, 5/B, 2/C, 3/C, 4/C, 5/C (columns with acronym printed in red); CT2—3/B, 4/B, 3/D, 4/D (grey); CT3—2/A, 3/A, 4/A, 5/A, 2/D, 5/D (green); CT4—1/A, 6/A, 1/B, 6/B, 1/C, 6/C, 1/D, 6/D (black). The cross sections of columns CT1 through CT4 are also drawn in Figure 3. The association for the entire structure is recapitulated in Table 1. On all storeys, perimeter beams (named BT1 in Figure 3) have out-of-depth section sized 300 × 500 (base×height) mm², with constant reinforcement given by 3ø16 top bars and 2ø16 bottom bars, and 200 mm-spaced ø6 stirrups; internal beams parallel to X (BT2) have in-depth sections sized 600 × 250 mm², with a maximum of 8ø16 top bars and 6ø16 bottom bars, and ø6 stirrups with spacing varying from 70 mm to 200 mm; internal beams parallel to Y (BT3) have in-depth sections sized 800 × 250 mm², with a maximum of 8ø20 top bars and 5ø20 bottom bars, and ø6 stirrups with spacing varying from 150 mm to 200 mm. The flights of stairs, embedded in the 3/C-3/D and 4/C-4/D alignments in plan, are borne by inclined beams (BT4) having 300 × 400 mm² section, with constant 4ø14 top and 4ø14 bottom bars, and 200 mm-spaced ø6 stirrups. The foundations are constituted by a mesh of inverse T-shaped RC beams. Their base sections are 2,600 mm-wide in Y direction and 1,600 mm-wide in X, with 1,200 mm (Y) and 800 (X) mm-high webs, and 600 mm (Y) and 400 mm (X) mm-high and 700 (X and Y) mm-wide webs. Floors are made of 200 mm-high RC joists, parallel to X axis, interposed clay lug bricks, and a 50 mm thick upper RC slab. Both for the elevation and foundation structures, reinforcing bars are made of FeB44k steel, with yield stress and tensile strength values greater than 430 MPa and 540 MPa, respectively; and concrete is 25/30 class, with cylindrical and cubic compressive strength of 25 MPa and 30 MPa, respectively. The dead and live loads assumed in the analyses are as follows: 5 kN/m² (dead) and 2 kN/m² (live) on first through fifth floor; 3.8 kN/m² and 1.2 kN/m² on sixth (roof) floor. The live loads on the stairwell are equal to 4 kN/m².

A view of the finite element model of the structure, generated by SAP2000NL software (CSI, 2022), is displayed in Figure 4, where the Cartesian reference system is traced out too. Elastic frame elements were used to model columns and beams, consistently with the choice of evaluating the effectiveness of the sizing procedure within a conventional elastic analysis approach, where seismic performance in retrofitted conditions is assessed in terms of reductions of stress states in structural members and inter-storey drifts, as compared to current state. A Rayleigh-type inherent viscous damping was assigned to the finite element structural model.

A modal analysis carried out by this model highlighted two main translational modes in current state, whose shapes are plotted in Figure 4 too, with vibration period, T _IN , equal to 1.21 s along X and 1.18 s along Y, and effective modal masses of 86.4% and 85.7%, respectively, of the total seismic mass M of the building, equal to 1,675 kN/g. Both values are greater than the 85% threshold representing the minimum percentile fraction required by the Italian Technical Standards to develop a modal superposition analysis. Period and effective mass of the first rotational mode around the vertical axis Z are equal to 1.05 s and 81.1%, respectively. As a consequence of the substantial structural regularity in plan and elevation, a total of nine modes is needed to activate summed modal masses greater than 90% of M along X, Y, and around Z, and a total of eighteen modes to attain over than 99% of M.

The seismic assessment study of the building was developed via time-history analysis by using as input a set of seven groups of three accelerograms. For each group, one accelerogram was applied in X direction, one in Y, and one in Z. The artificial ground motions were generated from the elastic pseudo-acceleration response spectra at linear viscous damping ratio of 0.05 assumed by the current Italian Technical Standards for L’Aquila, drawn in Figure 5. The three spectral graphs for the horizontal components and the three graphs for the vertical one are referred to the Serviceability Design Earthquake (SDE, characterized by 63% probability of being exceeded over the reference structural lifespan, V _R ), Basic Design Earthquake (BDE, with 10%/V _R probability) and Maximum Considered Earthquake (MCE, with 5%/V _R probability) hazard levels, for C-type (i.e., medium-dense sand) soil and T1-type (i.e., flat surface) topographic category. The corresponding peak ground acceleration values are: 0.156 g (SDE), 0.347 g (BDE), and 0.407 g (MCE), for the horizontal components; 0.045 g (SDE), 0.180 g (BDE), and 0.261 g (MCE), for the vertical one.

The results of the analysis in current conditions are synthesized in the top halves of Tables 2, 3. Table 2 reports the maximum storey shears along the two axes in plan, V _m,X and V _m,Y , and their ratios, ρ _V,X and ρ _V,Y , to relevant strength values, V _R,X and V _R,Y (where V _R,X , V _R,Y are given by the sums of the shear strength values of the ground storey columns). Table 3 lists the maximum inter-storey drifts, ID _m,X , ID _m,Y , and their ratios, ρ _ID,X and ρ _ID,Y , to IH.

As shown in Table 2, ρ _s,X , ρ _s,Y ratios greater than 1 are found for the first two, four and five storeys at the SDE, BDE, and MCE, respectively, reaching values of 1.19, 2.99, 3.66 (ρ _s,X ), and 1.16, 2.65, 3.38 (ρ _s,Y ) on the ground storey. Values notably below 1 come out for the top storey up to the MCE. The corresponding results in terms of stress states in the structural members highlight 22%, 72%, and 81% of columns, and 18%, 68%, and 76% of beams in unsafe conditions at the SDE, BDE, and MCE, respectively. By way of example of these results, the M ₁ –M ₂ biaxial moment interaction curves obtained at the BDE for the columns belonging to the 2/B vertical fixed line—where M ₁ , M ₂ are the bending moments around the two local reference axes of the column cross section—are plotted in Figure 6. Except for the top storey, these curves remarkably overcome the boundaries of the axial force-biaxial moment safe domains, traced out for comparison in the same graphs, reaching maximum values of the demand/capacity ratio equal to 2.23 (ground storey), 1.76 (first), 1.98 (second), 1.84 (third), and 1.65 (fourth).

Table 3 highlights that the ρ _ID,X ratios at the SDE are slightly greater than the above-mentioned IO performance level-related 0.5% IH limit (ρ _ID,X = 0.53%, 0.51% on the first and second storey, respectively), assessing a satisfactory performance in terms of drifts for this earthquake level. At the same time, ρ _ID,X and ρ _ID,Y ratios slightly smaller than 1 are observed at the BDE on the ground and third storey, and greater than 1 on the first and second storey, reaching ρ _ID,X peak values of 1.24% (first) and 1.17% (second), and ρ _ID,Y peak values of 1.15% (first) and 1.11% (second), implicitly assessing irreparable damage conditions of infills and partitions at these storeys. At the MCE, ρ _ID,X and ρ _ID,Y values greater than 1 are observed on the four lower storeys, with maximum values of 1.51 (ρ _ID,X ) and 1.42 (ρ _ID,Y ) on the first storey.

The use of supplementary damping devices with high elastic stiffness requires, unlike pure dampers or low-stiffness dampers, proper control of the stiffening effects produced by their installation in dissipative bracing systems adopted as retrofit measure for existing frame structures. This is the case of ADAS dampers, the elastic stiffness of which is normally so high that the system behaves like a corresponding traditional bracing system up to the first plasticization of the devices.

A sizing procedure for ADAS dampers aimed at expressly controlling the stiffening effects involved in their application is offered in this study, whose conceptual and practical implications are summarized below.

- The procedure allows estimating the energy dissipation capacity to be assigned to the protective system by means of simple analytical relations making use of spectral displacement, velocity and pseudo-acceleration ordinates.