Ordinary (customary) light gauge steel framing is a convenient construction technology; however, given the studs low axial capacity, it has been considered mainly for low-rise buildings in low seismicity regions. This paper investigates the boundaries of application of this type of steel framing (in moderate and high seismicity areas) by designing the structures of three representative 5, 7 and 10-storey buildings in order to examine their feasibility. The necessity of using built-up columns consisting of two or more sections is investigated. The axial capacity of studs is estimated by code-type analyses. The critical buckling stress is determined with well-known closed-form expressions; as they do not contemplate the flexibility of the screwed connections (in built-up sections), the obtained results are complemented with those of the finite strip method and generalized beam theory formulations.

　　According to the United Nations (2022), the population of cities is growing faster than that of the whole world, as towns and rural areas are experiencing a slowdown; future projections show that the world urban population will increase from almost 6 billion in 2020 to nearly 7.5 billion by 2070. Therefore, cities will tend to grow more vertically than horizontally; this trend highlights the need to develop construction technologies that are suitable for multi-storey buildings. In this context, light-gauge steel framing is a convenient construction solution; it consists basically of vertical wall-like panels with supporting cold-formed studs, and composite slabs (concrete-steel or timber-steel). The main advantages of steel framing compared to traditional steel and concrete-based solutions are its fast erection, high construction quality, rather moderate cost, and lightness; the latter provides relevant benefits in poor foundations soils and in seismic areas, as the seismic forces are proportional to mass. However, given the rather low axial capacity of cold-formed steel studs, steel framing has been considered mainly for low-rise buildings only; even code limitations on the number of storeys apply (ASCE/SEI 7-22 2022). On the other hand, horizontal actions generate additional axial forces on the extreme studs (chord studs) of each wall panel; therefore, wind and earthquakes also prevent increasing the building height. To overcome this limitation, built-up or tube sections have been proposed for such end studs.

　　In the context described in the previous paragraph, the objective of this paper is to investigate the boundaries of application of light-gauge steel framing construction for low to mid-rise buildings located in seismic areas. In order to maintain the main aforementioned advantages of this construction technology, customary (habitual) and inexpensive features are not modified; this refers basically to ordinary steel, thin plates, and also thin wall panels. To perform this study, steel framing structures of three representative five, seven and ten-storey housing buildings located in moderate and high seismicity areas (PGA?=?0.2?g, 0.3?g, 0.4?g peak ground acceleration) are preliminary designed to analyse their feasibility. The demanding internal axial forces on the supporting members (steel studs) of these prototype buildings are approximately obtained by simplified (although reliable and reasonably accurate) hand calculations. Several options are considered for the abovementioned extreme (chord) studs by combining Cee profiles: single, back-to-back, toe-to-toe (face-to-face), nested, and a stud pack (4C component section Phan et al. 2021, 2022). The axial capacity of such members is estimated by code-type analyses following the forthcoming European regulations; all the feasible buckling modes are considered: member (flexural and flexural–torsional), local and distortional. The slenderness is obtained as the square root of the ratio between the yield and buckling stresses or loads. The critical stress is determined with well-known closed-form expressions; as these formulations do not contemplate the flexibility of the screwed connections (in built-up sections), the obtained results are complemented with those of FSM (finite strip method) (Li et al. 2014) and GBT (generalized beam theory) (Schardt 1994; Silvestre and Camotim 2002).

　　Cold-formed steel (CFS) framing is commonly used as load-bearing structures for 1 to 2-storey buildings, particularly in North America, where this technology has benefited from the timber framing tradition; in Europe, the development of steel framing apparently stems from traditional and composite constructions. Report LGSF (2022) discusses the worldwide prospects of this technology.

　　As horizontal actions are a major limitation for tall buildings, the first CFS framing load-bearing structures in mid to high-rise buildings are gravity load-carrying systems supplemented with additional elements to resist horizontal actions (wind and seismic); usually, these elements are RC stair (and elevator) cases (shafts). An example is the SFIA Matsen Tower (Ford 2016); it is a hypothetical 40-storey residence building designed for a non-seismic area. The next steps have focused on structures made entirely of CFS framing; however, recent examples are also located in non-seismic regions: two 7-storey Hotels in Chicago, and an 8-storey (seven storeys of cold-formed steel framing over a pre-cast concrete podium) apartment building in Green Bay (WI).

　　Horizontal actions on steel framing require the wall panels are strengthened to resist the ensuing shear forces. An overview of the proposed strategies can be found in Sharafi et al. (2018), Taranu et al. (2023) and Yilmaz et al. (2023); those that do not alter the CFS nature of the steel frames are: timber sheathing boards, X-bracing steel straps, additional bolting, and concrete or mortar filling, among others. Among these options, strap bracing (Fig. 4) is the most convenient and versatile, given its superior strength and stiffness; however, the subsequent tensile forces in the bracing straps generate over-compression in the chord (end) studs. This effect is highly limiting, and may require using built-up sections, made by joining two or more profiles. The influence of the joining technique and design in their performance can be analysed using (AISI S100-16 2016) prescriptions (reduced overall slenderness) or, more deeply, using the aforementioned FSM or GBT approaches.

　　An early attempt to explore the boundaries of application of CFS constructions under earthquake excitation can be found in Madsen et al. (2011); Nakata et al. (2012), where a two-storey building is preliminarily analysed. In Torabian et al. (2016), archetype mid-rise pure CFS buildings (4–20 storeys) located in a high seismic area are used to assess the height limits of steel frames, and to aid in the development of new CFS technologies; the main conclusion is that hold-downs and high-capacity studs are needed. A more recent study is the seismic analysis of a 10-storey building (Zhang et al. 2022) to be built and tested (full scale); this research is aimed to investigate the seismic performance of buildings exceeding the height limitation of 19.8?m (65 ft) set by ASCE/SEI 7-22 (2022).

　　The above studies belong to the US context; this paper presents a rather similar study for the European situation.

　　As discussed in Sects.?1 and 2, wind and earthquakes are major limitations for the use of steel framing for mid and high-rise buildings.

　　Earthquakes are one of the most damaging environmental effects on civil construction. Unless other actions (gravity, wind, thermal, etc.), strength is neither the only, nor the main requirement; in brief, an adequate seismic design of buildings pursues, among others, the following major qualities: (i) lightweight to mitigate the mass inertial effects, (ii) bilateral stiffness to prevent excessive inter-storey drift and collisions with adjoining buildings, (iii) bilateral strength to resist the equivalent seismic forces, (iv) plan symmetry (this is a mechanical issue, not a geometrical one), and torsional resistance and stiffness (to avoid highly damaging twisting motion), (v) uniformity along height to put off damage concentration in soft storeys, (vi) rigid diaphragms to guarantee solidarity between the lateral resisting systems, and providing a regular, simple and predictable seismic behaviour, (vii) adequate foundation to foil uplift, (viii) structural simplicity mainly to facilitate detecting hidden failure modes, (ix) structural redundancy to maintain a reserve of strength, and (x) ductility to allow for rather light design (as long as some damage is accepted). Steel framing structures are particularly well suited for issues i, v, vi (if a topping RC layer is provided), viii and ix; the remaining issues are discussed next.

　　Lateral stiffness and strength The rigidity and resistance in both horizontal directions can be reached with the measures described in Sect.?2, particularly cross-bracing with steel straps. As discussed in that section and in Sect.?6.6, this requires the over-strengthening of the end (chord) studs; if only CFS profiles are accepted, built-up sections need to be used. Obviously, this poses relevant limitations in the use of this construction solution in high seismicity areas.

　　Mechanical plan symmetry This type of symmetry is not related to any geometrical regularity, but to small eccentricity between the centres of gravity (G) and rotation (R). As an important number of braced wall panels can be placed in any horizontal direction, symmetric or near-symmetric arrangements can be achieved in all situations.

　　Uplift prevention In rather slender buildings located in high seismicity areas, uplift is a serious risk for any building. Noticeably, lightweight is not a relevant advantage, as both gravity and seismic forces are proportional to the building weight; however, the uplift forces can be rather easily compensated with heavy foundations. Hold-down ties need only a moderate section (area), and can be easily embedded in the steel framing structure.

　　Ductility Steel framing buildings should be rather ductile, as the only employed material is steel, which is a highly ductile material. Although x-braced wall panels are not as ductile as moment-resisting frames, a minimum ductility is largely ensured. This is particularly relevant, given that several buckling instabilities (flexural, torsional-flexural, local and distortional) are possible, mainly in the chord studs. The dissipative structural elements are the brace straps; as the chord studs are over-compressed due to the axial tension in such bracing members, the design of these studs shall be based on conservative capacity equilibrium conditions. This strategy protects against the abovementioned fragile buckling instabilities.

　　Other particular seismic design recommendations, such as “no long cantilevers”, “no interrupted columns”, “no short columns”, “tying of footings and pile caps”, “strong column-weak beam”, “separation to adjoining buildings”, “compact plan layout” and “protection of non-structural components” (among others) are not related to steel framing technology, and do not require particular discussion.

　　The paper Torabian et al. (2016) contains a rather similar discussion.

　　The main remark of these considerations is that pure steel framing buildings are rather well suited for seismic resistance; the most critical issue is the high axial compressive demands on the chord studs.

　　Three 5, 7 and 10-storey residential buildings are used as case studies representative of common low to mid-height buildings. They are regular, the storey height is 3?m, and the plan layout is square 18?m?×?18?m. These buildings have 3 apartments (flats) per floor and a central hall, containing stairs and elevators; the wall panel distribution is the same on each floor. The building structure consists of prefabricated wall panels and floor and roof slab panels. In the walls, only CFS profiles (studs, tracks and diagonal straps) contribute to the load-bearing capacity (“all steel” approach AISI S240-15 2015). The wall panels are completed with insulation, EPS (Expanded polystyrene) lightweight mortar, mortar rendering and gypsum lining. Furthermore, a concrete layer 60?mm thick is poured on site above the metal structure of floor and roof panels, where the sliding between the metal frame and the concrete layer is prevented by shear connectors; this layer is reinforced (with a welded wire mesh) to provide some in-plane strength and prevent shrinkage cracks. Figure?1 displays an elevation of the 10-storey building, and a plan view of a generic storey. The global reference axes (x, y, z) used in Fig.?1 are maintained along this paper.

　　Fig. 1

　　Prototype buildings

　　Figure?1b describes the aforementioned three apartments; the internal partitions (not represented) are made with dry walls. Figure?1a depicts the openings of a fa?ade of the building with 10 levels; Fig.?1b shows the correspondence with the architectural distribution. In fact, the architectural layouts of these buildings and their impact on the wall panels distribution have been carefully considered in order to have realistic and representative examples.

　　As described in Sect.?2 and Sect.?4.1, the wall panels are composed of vertical supporting studs framed by top and bottom tracking profiles; the slab panels have a similar composition, although they are topped with a 60?mm RC layer. Figure?2 displays views of wall (Fig.?2a) and slab (Fig.?2b) panels.

　　Fig. 2

　　Steel framing structure

　　The structural capacity of the wall panels represented in Fig.?2a but without the braces is sufficient for gravity loads, but their lateral strength and stiffness are not enough to resist relevant wind and seismic effects. For that reason, some wall panels (all of them apart from those with openings) are braced with steel straps located at both wall sides (front and rear, Fig.?2a). Figure?3 presents several bracing arrangements.

　　Fig. 3

　　Strap-braced wall panels

　　Figure?3a shows a basic X (diagonal) bracing, and Fig.?3b, c describe similar arrangements with one and two horizontal bridging members, respectively. The purpose of these members is to provide buckling restraint in the wall plane. It is assumed that this bridging system is capable of sufficiently blocking the intermediate in-plane displacements of the studs, or that it is adequately reinforced with complementary elements (additional straps or blocking elements) to fulfil this function.

　　As the horizontal structural elements (floor and roof slabs, Fig.?2b) do not significantly contribute to the building lateral seismic capacity, only the vertical members (wall panels, Fig.?2a) are described herein. All of them are either 100 or 150?mm thick, and the tracks are made of channel 105?×?40?×?1.5 or 155?×?40?×?1.5 profiles, respectively. The internal (middle) studs are Cee single or built-up sections (pairs) spaced between 400 and 600?mm, the external (chord) studs are Cee built-up sections (pairs or foursomes) (Fig. 6), and the diagonal braces are straps (Figs.?2a, 4).

　　Fig. 4

　　Braced (red) and unbraced (white) wall panels in both directions

　　Figure?4 displays, similarly to Fig.?1b, a building plan view detailing the location and length of the braced wall panels. Figure?4 is aimed to deliver deeper information about the vertical elements that provide lateral stiffness and strength.

　　Figure?4 shows that there are several adjacent wall panels; however, their chord studs are considered to work independently (that is, they are not connected along their length). In other words, the building is designed as a series of uncoupled shear panels in each direction; such panels are connected to floor slabs that behave as rigid diaphragms. Thus, the horizontal shear forces on each floor are distributed between the panels proportionally to their stiffness. Figure?4 also shows that the braced wall panels in x and y directions are similar (in number and length). The panels in y direction are symmetrically distributed; conversely, the distribution of panels in x direction is rather asymmetric. This choice is intentional, and aims to reveal circumstances that can occur in real life (because of architectural requirements), even in externally symmetrical buildings. As mentioned in Sect.?3, this might impair their seismic performance; this effect is expected to be moderate.

　　Abstract

　　1 Introduction

　　2 State-of-the-art of steel framing for mid to high-rise buildings

　　3 Seismic performance of steel framing buildings

　　4 5, 7 and 10-storey prototype buildings

　　5 Simplified formulation for the global structural analysis of the prototype buildings

　　6 Seismic analysis of the prototype buildings

　　7 Member buckling analysis

　　8 Main results of the global structural analyses of the prototype buildings

　　9 Conclusions

　　Abbreviations

　　References

　　Acknowledgements

　　Author information

　　Ethics declarations

　　#####

　　This section presents a simplified formulation for the global structural analysis of the prototype buildings described in Sect.?4. This analysis consists in obtaining approximately the internal forces in the building structural elements; Sect.?7 contains a local structural analysis providing the strength of the most critical structural elements, thus allowing to verify whether they exceed or not the demanding forces determined in this section. In fact, this process is only performed for the most critical members, mostly the chord studs of the first-storey fa?ade braced wall panels.

　　Given the particular characteristics of the seismic action, its effects are discussed in an individual section (Sect.?6).

　　The prototype buildings are designed for the major actions, namely: gravity, wind and seismic. The actions and their combinations are determined according to European regulations (EN 1990 2005). Gravity loads are divided in permanent (dead) and variable (live), being commonly represented as G and Q, respectively; wind forces are variable, named as WX and WY (W stands for wind). Finally, seismic actions are accidental, being ordinarily denoted as EX and EY (E stands for earthquake). The combinations involve safety and combination factors; for this simplified analysis, the following six major combinations are considered:

　　(1)

　　The vertical actions are determined according to the current European regulations (EN 1991-1-1 2002). The permanent loads are distributed between the levels (slabs); the estimated values are 2430?kN for the storeys and 2592?kN for the roof. The variable loads arise from an average distributed force of 2?kN/m2; the total force for each level is 648?kN. The internal forces in the studs are determined by assuming that the load on each slab is distributed isostatically (i.e. as if all the joists supports were hinged) between all the supporting studs; this criterion is common, and leads to small inaccuracies only.

　　The static wind forces (pressure and suction) in each direction are determined according to the current European regulations (EN 1991-1-4 2005) by assuming a roughness length for terrain category IV. Then, the storey shear forces are obtained by equilibrium conditions; inside each storey, such forces are distributed between all the wall panels in the corresponding direction (Fig.?4) according to their lateral stiffness. The axial compressive force in the corresponding chord stud of each panel is determined as the vertical projection of the tensile force in the steel strap brace. In turn, this force is obtained through a simplified structural analysis of the panel, neglecting the contribution of the internal studs to its lateral stiffness (Sect.?6.1). Noticeably, both gravity and wind axial compressive forces on the studs are assumed to be applied on the chord stud central axis (passing through the section centroids); in other words, these elements undergo a centred compression only.

　　The seismic design of the prototype buildings is performed, in a simplified way, according to the upcoming version of the European seismic design code (EN 1998-1 2023); whenever required, that regulation is supplemented with the American document (AISI S400-20 2020). The vertical seismic action is not considered.

　　EN 1998-1 (2023) proposes two 2-D static design strategies: Lateral Force and Response Spectrum Methods; they consist basically in considering the contribution of one and several modes, respectively. Several conditions must be met (in each direction) to allow the use of the most simplified approach (Lateral Force Method): (i) buildings not taller than 30?m, and (ii) fundamental period in the considered direction (T1) not exceeding neither 1.5?s nor four times the corner period TC (Sect.?6.4, Table 2). Obviously, the first condition is fulfilled (Fig. 1); regarding the limitations for the building fundamental period, the obtained results (Table 11) also fulfil the required conditions. Hence, the Lateral Force Method is adopted in this research; in this formulation, the base shear force in the considered direction is given by:

　　(2)

　　In Eq.?(2), m is the mass corresponding to the building seismic weight (Sect.?6.5), and λ is a dimensionless coefficient being equal to 1 if the building has more than two storeys and T1 is greater than two times the corner period TC, and 0.85 otherwise. Sr(T1) is the design spectral ordinate (Sect.?6.4) corresponding to period T1 (Sect.?6.5). Table 3 displays the adopted values of λ in each direction.

　　In Eq.?(2), Fb represents the equivalent static effect of the design accelerograms in every horizontal direction; it shall be distributed between all the storeys proportionally to their masses and height (with respect to the building base). Then, the demanding seismic internal forces in the chord studs are determined similarly than in the wind analysis (Sect.?5.4); the main difference is that the seismic shear storey forces are assumed to be applied in the centre of gravity of each storey (G), and this point exhibits actual and accidental eccentricity with respect to the corresponding centre of rotation (or rigidity, R). This issue is discussed more deeply next.

　　Table 1 displays the actual eccentricity in the y direction (Fig.?4) between the centres of mass (G) and rigidity (R) of the first storey of the prototype buildings.

　　Table 1 Eccentricity ey (m) between the centres of mass (G) and rigidity (R)

　　In Table 1, Sα,475 refers to the site seismicity (Sect.?6.2); the indicated values correspond to the final situation, once the structure has been finally designed (Sect.?8).

　　The real eccentricity in Table 1 shall be complemented with the accidental eccentricity; EN 1998-1 (2023) states that?±?5% must be taken in any direction. By adding the actual and accidental eccentricities, the forces at each braced wall panel are determined proportionally to their stiffnesses; noticeably, both translational and rotational geometric compatibility conditions need to be imposed (rigid diaphragm effect).

　　On the other hand, although regularity in the plan is not specifically required, this is equally considered, as it prevents twisting (torsional) behaviour, having proven to be extremely damaging. Three types of conditions are basically entailed for each level and direction: geometrical symmetry (including compact configuration), rigid diaphragm, and torsional stiffness. The first two conditions are largely fulfilled, while the third one involves the contentment of two inequalities: ey?≤?0.3 ry and ry?≤?ls; the subindexes correspond to verification for the x direction. In these expressions, ry is the square root of the ratio between torsional and lateral stiffness (“torsional radius”), and ls is the radius of gyration of the floor mass in plan (square root of the ratio between the polar mass moment of inertia -with respect to point G- and the floor mass). Noticeably, condition ey?≤?0.3 ry refers to a small eccentricity between points G and R, and ry?≤?ls is equivalent to the first 3-D mode of the building being translational. These inequalities are satisfactorily fulfilled in all the analysed situations.

　　As discussed in Sect.?1, the building is located in a seismic area; three levels of PGA (peak ground acceleration) are considered: 0.2?g (moderate seismicity), 0.3?g and 0.4?g (high seismicity). This parameter broadly represents the maximum expected acceleration at the engineering bedrock (soil type A) for 475?years return period; conversely, in EN 1998-1 (2023), the site seismicity is mainly characterized by the maximum response spectral acceleration (5% damping) Sα,475, corresponding to the constant acceleration range (plateau) of the horizontal elastic response spectrum. Assuming that, approximately, the ratio between PGA and Sα,475 is equal to 2.5, the considered site seismicity levels in terms of Sα,475 are 0.5?g, 0.75?g and 1?g.

　　The soil type (site category in EN 1998-1 2023) is an inherent component of the site seismicity; in this study it is soil B. In EN 1998-1 (2023) this site category refers to stiff soil, and corresponds to vs,H ranging between 400 and 800?m/s and H800 deeper than 5?m. Parameters H800 and vs,H are the engineering bedrock depth (characterized by a shear wave velocity of 800?m/s), and the weighted harmonic average of the shear wave velocity in soil depth H; H is equal to the lowest of H800 and 30?m. Noticeably, this soil would be rated as C in the American documents.

　　EN 1998-1 (2023) classifies the seismicity in four levels in terms of Sα,475: very low (Sα,475?nfirm that Sα,475?=?0.5?g corresponds to moderate seismicity, while Sα,475?=?0.75?g and Sα,475?=?1?g are referred to as high seismicity.

　　The behaviour factor (q) in the European regulations corresponds to the response modification factor (R) in the American documents. With regard to European regulations, in recent years there has been an urgent need to address the dissipative design of steel framing structures, since it was not properly covered in previous versions of Eurocode 8 (Landolfo et al. 2022; Fiorino et al. 2017; Hatami et al. 2010). Estimating q from cyclic and incremental tests on individual panels (Kasaeian et al. 2020; Hatami et al. 2010) results in considerably high values (up to 5). Conversely, proper numerical pushover and IDA analyses for an illustrative set of prototype buildings undergoing a sufficiently representative collection of ground motions provide smaller values (Fiorino et al. 2017); this work results in a moderate value of q?=?2.5, which is taken by the new version of the Eurocode (“Structural type g” in EN 1998-1 2023) for ductility class DC3. As a matter of fact, for Sα,475?=?0.75?g and Sα,475?=?1?g, such ductility class is strictly required (Table 11.3 of EN 1998-1 2023). The conditions for obtaining such a level of ductility are fulfilled, and therefore, q?=?2.5 is assumed in this study for all the prototype buildings. Noticeably, in the American documents this ductility class (DC3) corresponds broadly to “special buildings”.

　　The design spectra are based on the forthcoming version of the European seismic design code (EN 1998-1 2023).

　　The design spectrum according to EN 1998-1 (2023) considers an elastic response spectrum referred to as Se(T); it has five branches:

　　1.

　　Short horizontal branch (extremely short periods, T?≤?TA) Se(T)?=?Sα/FA.

　　2.

　　Linear growing branch (very short periods, TA?≤?T?≤?TB) Se(T)?=?[Sα/(TB?TA)] [η (T?TA)?+?(TB?T)/FA].

　　3.

　　Constant branch (plateau, short periods, TB?≤?T?≤?TC) Se(T)?=?η Sα.

　　4.

　　Hyperbolically decreasing branch (mid periods, TC?≤?T?≤?TD) Se(T)?=?η Sβ Tβ/T.

　　5.

　　Faster decreasing branch (long periods, TD?≤?T) Se(T)?=?η TD Sβ Tβ/T2.

　　In these expressions, FA?=?2.5, Tβ?=?1?s, Sα?=?FT Fα Sα,475 and Sβ?=?FT Fβ Sβ,475; FT is the topography amplification factor, and Fα?=?1.3 (1?0.1 Sα,475) and Fβ?=?1.6 (1?0.2 Sβ,475) are the short and intermediate period site amplification factors, respectively. It is assumed that FT?=?1 (there is no topography amplification). Sα,475 can be taken as ag 2.5 in EN 1998-1 (2023), and Sβ,475?=?fh Sα,475, where fh is equal to 0.3 and 0.4 for moderate and high seismicity levels, respectively. The corner periods TA, TB, TC and TD are given by TA?=?0.02?s, TC?=?Sβ Tβ/Sα, TB?=?TC/4, and TD?=?1?+?Sβ,475. Finally, the damping correction factor η is taken as 1, as the building damping ratio is assumed to be 5%.

　　Then, the reduced spectrum Sr includes the influence of the behaviour factor q, through Rq(T): Sr(T)?=?Se(T)/Rq(T). In the first branch Rq(T)?=?1.5, in the second branch Rq(T)?=?1.5?+?(q?1.5) (T?TA)/(TB?TA), and in the other branches Rq(T)?=?q. Finally, the minimum value of Sr(T) is β Sα,475, where β?=?0.08.

　　Table 2 and Fig.?5 display the parameters and plots of the three design spectra, respectively.

　　Table 2 Main parameters of the reduced spectraFig. 5

　　Reduced spectra (Sr) according to EN 1998-1 (2023). Behaviour factor q?=?2.5 and soil type B

　　Prior to any structural calculation, the building fundamental periods in the direction under consideration (T1,est) can be estimated according to the simplified empirical expression for framed wall buildings in EN 1998-1 (2023): T1,est?=?0.05 (s), where Hb is the building height (m). The obtained results for the 5, 7 and 10-storey prototype buildings are:

　　(3)

　　There is a big uncertainty involved in the determination of these values; therefore, it would not be surprising that further calculations (once the structural parameters are known) provide significantly different results. In this context, the building modal parameters (natural periods and mode shapes) are determined by solving the classical eigenvalue problem (K ? ω2 M) =?0, where K is the stiffness matrix (in each horizontal direction), M is the mass matrix, and ω and ? represent the eigenvalues (natural frequencies) and eigenvectors (modal shapes), respectively. Matrix M is diagonal, containing the masses of each storey; matrix K is generated as tri-diagonal, by assuming that the steel framing structure behaves as a shear building. Storey masses (Sect.?5.3) are determined according to the permanent load combined with 15% of the variable one: 2.576?×?105?kg; for the roof, the combination involves 30% of the variable load: 2.840?×?105?kg. These values correspond to the seismic weight. Storey stiffnesses in x and y directions are determined according to AISI S400-20 (2020); the contribution of the non-structural components to the building stiffness is neglected. This last consideration might introduce important inaccuracies, as the non-structural elements contribution can be significant; it has been adopted as, for moderate and strong seismic inputs, usually such elements are damaged, thus leaving the structure alone. On the other hand, the influence of the non-structural components in the building fundamental period would be highly difficult to assess, given the large number of cladding and partitioning elements. The periods determined from the modal analyses are discussed in Sect.?6.7.

　　The chord studs are non-dissipative elements that support the strap braces, being dissipative elements as the only source of ductility; therefore, they should be designed with protective capacity conditions intended to prevent their brittle (buckling) failure. These conditions consist in replacing the actual value of the axial force in the chord studs due to seismic effects by an amplified value of the maximum force that can be transmitted by the strap braces; in other words, designing the chord studs to withstand this axial force (adding the contribution of gravity forces) guarantees that the plastic capacity of the strap braces can be fully developed. The demanding internal axial force in the chord studs is equal to EN 1998-1-1 (2023) and Landolfo et al. (2022):

　　(4)

　　In Eq.?(4), EEd is the demand internal force due to combinations V or VI (Eq.?(1)), EEd,G is the contribution of non-seismic actions, and ENfy is the internal force corresponding to the axial yielding capacity of the brace straps. Again in this equation, ωrm is the material overstrength factor, ωsh is the factor accounting for hardening of the dissipative zone, and Ωd is the minimum design overstrength. For the steels considered in this study ωrm?=?1.45 can be assumed; ωsh is taken equal to 1.1, and Ωd is taken equal to 1 (EN 1998-1-1 2023).

　　The global seismic analysis must be performed iteratively, as prior to the first design of the structural members, it is not possible to calculate the building fundamental period. Given this circumstance, the seismic design codes (in this case, EN 1998-1 2023) provide empirical expressions for such a period that depend only on the structural configuration and the building height; these expressions have been utilized to obtain the first estimates (Eq.?(3)). The values of the fundamental period provide the spectral ordinates (Sect.?6.4); in turn, such ordinates allow obtaining the base shear forces and the demanding axial forces in the chord studs (Sect.?6.1) and all the structural members. These forces are to be combined with those from other actions (Eq.?(1)), thus allowing a proper structural dimensioning (Sect.?6.6). Then, a new value of the fundamental period can be determined (Sect.?6.5), and the iterative process continues. Iterations are stopped when the periods do not change significantly, when their change has no effect on the spectral ordinates (as they lie in the plateau range), or when the demanding axial forces in the chord studs exceed their capacities.

　　The most important results of this iterative process (for the 5, 7 and 10-storey buildings) are described in Table 3. The convergence criterion is based on the base shear (Fbx and Fby) and the demanding axial forces in the chord studs ( and , Table 11): when the differences between two consecutive iterations are less than 5%, the iterations are stopped. It is shown in Table 3 that iterations do not start from the initially estimated periods (Eq.?(3)), but from longer ones (typically, 1?s); this is because the periods in Eq.?(3) are too short, thus providing excessively high spectral ordinates that generate axial forces in the chord studs that cannot be withstood by them.

　　Table 3 Main results of the global seismic analyses for the prototype buildings

　　In Table 3, represents the 1st mode mass participation factor; obviously, subindexes x and y denote the horizontal directions (Fig.?4).

　　The drift displacements of the building storeys are obtained from the lateral stiffness of the wall panels; such stiffness is taken from simplified expressions in the American document (AISI S400-20 2020). EN 1998-1 (2023) prescribes that drift should be limited for two reasons: to prevent damage to non-structural components (mainly brittle cladding and partitioning elements connected to two consecutive floors), and to investigate the need for performing second-order analyses. Noticeably, recent studies indicate that geometric non-linear effects can be significant in cold-formed steel braced panels (Papargyriou et al. 2021). The first limitation is established as dr,SD?≤?0.01 hs, where hs is the storey height, and dr,SD is the drift for the significant damage (SD) limit state; this drift is calculated from the average lateral displacements ds obtained by multiplying the reduced displacements dr by q (for the range of periods of interest in this study). The second limitation is set as θ?≤?0.1, where θ (inter-storey drift sensitivity coefficient) is basically the ratio between the first and second-order moments, and is precisely defined as ; in this expression, Ptot and Vtot are the vertical gravity and horizontal seismic forces on the storey under consideration, respectively. Table 4 displays the obtained values of the first storey drift (dr,SD, mm) and the corresponding inter-storey drift sensitivity coefficient (θ).

　　Table 4 First storey reduced drifts dr,SD (mm) and inter-storey drift sensitivity coefficients (θ) in the prototype buildings

　　Table 4 shows that dr,SD does not exceed the bound (30?mm) in any case. Regarding θ, all its values are clearly below the bound (0.1). These trends show that drift is not excessive.

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