Introduction and motiovation of the study

In the new generation building codes, design and assessment procedures are strongly based on the evaluation of the seismic action, which currently represents the main hazardous event from which preserve new and existing reinforced concrete (RC) buildings. To optimally quantify the seismic performance of buildings, the general synopsis aims to compare the two main actors of the structural design or assessment, that is, capacity and demand. Such dualism characterizes the entire processes of structural design and assessment, in which the comparison between the two entities is the result of several steps of an articulated path consisting of the quantification of the loads, the definition of the structural geometries and materials, the choice of the modelling assumptions, and the method of analysis (all with the related uncertainties). Especially when talking about seismic actions, the method of analysis assumes a key role in the performance definition, considering the possibility to employ linear and nonlinear approaches. According to modern building code provisions (e.g., Italian Building Code, NTC18, 2018 [1]), linear methods consist of analysing an elastic numerical model simulating the real building through a specific load pattern, which depends on the acceleration (or displacement) derived by an elastic spectrum, usually considered as inelastic by employing a behaviour factor established on the basis of the type of investigated structure and an expected ductility performance. Linear methods can be differentiated in static and dynamic, where for the first one the only effect of the fundamental mode of vibration is considered, while for the second one the combined effect of all vibration modes presenting a participation mass greater than 5% is accounted for. Given some limitations to apply the static option, the dynamic approach results the more eligible, where accelerations (or displacements) of interest are derived from the well-known Response Spectrum Analysis (RSA). Regarding nonlinear methods, most robust tools can be exploited to explore the performance of structural systems in post-elastic field, accounting for both geometrical and mechanical nonlinearities. Also in this case, new generation codes endorse both static and dynamic methods, where to difference of linear methods, the seismic action is differently simulated. Nonlinear dynamic analysis (usually named nonlinear time history analysis, NTHA) consists of submitting to structure a seismic excitation represented by a time history of ground motions, in order to reflect the actual acceleration, velocity, and displacement experienced by the structure during an earthquake. This allows for a more realistic simulation of the dynamic behaviour of the structure under seismic forces. Although the effectiveness of the results provided via NTHAs, the current practice does not match with this method, considering the theoretical hurdles in several aspects characterizing both demand (i.e., selection of ground motion records, number of accelerograms, efficiency, and sufficiency) and capacity (i.e., definition of the cyclic behaviour of structural elements and set up of a correct numerical model). On this basis, international and national building codes (e.g., NTC18, 2018) are currently promoting the nonlinear static analysis as most credible candidate to emulate the effects given by a dynamic approach. In fact, a more extensive and user-friendly framework is offered to engineers and practitioners, which can develop a series of steps to assess the seismic performance of buildings (i.e., through the N2 method, [2]) and then, to compare capacity and demand by computing the so-called capacity/demand (C/D) ratio. The method consists of examining the seismic behaviour of buildings by applying a static load pattern distributed over the height and by monotonically increasing its magnitude until the overall collapse mechanism of the structure is achieved. The structural response is returned in the form of a capacity curve, which is a scalar force–displacement relationship typically presented in terms of base shear (Vb) and top displacement (δR), this latter recorded at a specific control node (CN).

As well-known in the scientific literature, nonlinear static analysis (or pushover) methods cannot properly replace dynamic approaches for manifold reasons, as explicated in Sect. "Conventional and non-convetional nonlinear static analysis methodologies: state-of-the-art". Nevertheless, the code framework aims to provide different rules for employing this analysis method, such as by considering different load profiles, different directions and verses of analysis, different eccentricities. In this way, the real structural behaviour of the investigated building can be identified within a range of scenarios (especially with regard to the post-elastic field), for which to the more disadvantageous combination is conferred the final assessment and it allows to take decisions (e.g., design of retrofit intervention for existing structures). According to this concept, the recent release of the Italian Building Code (NTC18, 2018 [1]) increased the number of possibilities to perform pushover analysis (and then, to enlarge the range of scenarios), especially for including more cases of new and existing structures (e.g., irregular buildings). Hence, the following new prescriptions were added:

  1. (1)

    To consider alternative CNs (usually the centre of the mass of last storey is accounted for), as for example by monitoring corner points when the considered structure exhibits significant coupling among translations and rotations.

  2. (2)

    To use a load profile proportional to the storey shear force obtained by an RSA. The code specifies that this load patter is universally applicable, and it can be employed even if the participating mass (M[%]) in a given direction is less than 75%. In addition, it is mandatory to use this load profile in the case in which the fundamental period of vibration (T1) of the structure exceeds 1.3 times the value of the corner period (Tc).

  3. (3)

    To account for the spatial variability of the seismic motion in all main directions, a combination of the effects of pushover analysis could be performed. In particular, the overall performance should be computed by combining the results in the considered main direction to a quote (i.e., 30%) of the results in the perpendicular main direction. Still, the analysis needs to be reiterated, alternating the reduction coefficients among the components.

Excluding point (1), the above prescriptions extend the concepts of linear analyses to nonlinear ones, with the main aim to enlarge the set of available choices in performing pushover analysis and then, to obtain more and more conservative results especially for the more complex cases (e.g., irregular buildings). If from one hand, this principle follows a practical need (i.e., design and assessment should be always on the safe side), on the other hand there are no demonstrations that the new prescriptions are reliable, especially when irregular structures are investigated. Hence, with the aim to assess the new code provisions, the authors of this paper worked within the framework of the RELUIS (Consorzio della Rete dei Laboratori Universitari di Ingegneria Sismica e Strutturale) project, Work Package n. 11, funded by the Italian Department of Civil Protection, for deeply investigating the effectiveness of each point in the cases in which irregularities arises. Concerning point (1), Uva et al. [3] investigated the influence of varying the position of CN in pushover analysis results for a consistent sample of RC archetype buildings presenting different plan configuration and different number of storeys. The results demonstrated that the variation of CN led to a different representation of the capacity curves, resulting in a shift of δR (and then, a shift of capacity curves). Obviously, this can bring different safety levels, with different outcomes in terms of C/D ratio and a certain effect on the overall results (especially when buildings are irregular). On the other hand, it is worth reminding that different CNs do not imply different structural behaviours. Still, the CN should represent the point in which condensing the entire mass of the storey (schematization of the building like a multi-degree-of-freedom, MDOF, lollipop) and then, assuming for example a corner CN does not represent a physically based mass distribution in the storey. Regarding point (3), Ruggieri and Uva [4] conducted a study to investigate global and local seismic performances of some archetype low-rise RC buildings presenting increasing plan-irregularity, by using bi-directional pushover analyses, which consisted in a combination of load profiles in both main horizontal directions, using the coefficients used in RSA (i.e., unitary in the considered main direction and 0.3 in the perpendicular one). Comparing the results with the ones obtained by using conventional load profiles (i.e., inverse triangular- and uniform-like), the new suggested approach did not assume a real conservative stance, as the response often falls within the range of scenarios identified by the conventional load profiles. Similar outcomes were derived by Cantagallo et al. [5], which compared the effects of bi-directional pushover analysis with NRHAs on two RC existing buildings, one regular and one irregular. Results in terms of interstorey drift ratios (IDRs) showed that, although some differences were observed among bi-directional and conventional load profiles (especially for the irregular case), results of NRHAs were more conservative.

The above experiences revealed that the new code prescriptions did not provide a valuable improvement to the current practice for pushover analysis, especially for points (1) and (3). The aim of this paper is concluding the evaluation of the new code prescriptions by investigating the effectiveness of point (2). It is clear that, the application of a load profile proportional to the storey shear force obtained by an RSA on regular structures provides comparable output (e.g., capacity curve) to the one obtained by an inverse triangular load pattern or an unimodal load profile, because first mode of vibration drives the dynamic behaviour of the structure. Instead, some chances to observe differences among the RSA-based load profile and the conventional ones might exist for irregular structures, in which first mode of vibration loses importance and higher modes assume significance in the dynamic response. With this goal in mind, it is worth mentioning that the point (2) was also investigated in Ruggieri and Uva [4], and the results obtained on plan-irregular archetype buildings suggested that no improvements were achieved with respect to conventional load profiles. Still, the paper concluded that probably, a possible application of the load profile proportional to in-height irregular buildings could lead more significant results. This is the starting point of the present paper, in which the effectiveness of the load profile derived by an RSA was assessed. To this end, a sample of archetype buildings characterized by increasing in-height irregularity was firstly designed and after investigated. The sample was characterized by two subset of buildings presenting same geometry but different design levels to simulate new and existing buildings, i.e., one was design according to the new Italian building code (NTC, 2018 [1]) and one was designed according to the older Italian building code (Regio Decreto n. 2229, RD2229, 1939 [6]). Hence, pushover analyses were carried out, by using conventional (i.e., inverse triangular- and uniform-like) and the focused (i.e., derived from RSA) load profiles. The obtained results were after compared with NRHAs, looking at global (capacity curves) and local (IDRs) performances of buildings and showing if the proposed option by code can really improve or not the output obtained by the traditional load profiles.

Conventional and non-convetional nonlinear static analysis methodologies: state-of-the-art

The nonlinear static analysis yet represents the most practical viable option to the nonlinear dynamic one (i.e., NTHA), even if several concerns characterized the methodology itself and the reliability of the results. The application of the methodology is nowadays well-framed in several international codes (e.g., NTC18, 2018), which provide rules for defining the shape of some load patterns to apply in the analysis. Generally, the shape of load profiles can be summarized according to:

$$\overline{F }=\overline{\Psi }\cdot \lambda \left(t\right)$$
(1)

where F represents the vector of the forces constituting the load profile, Ψ is a parameter representing the shape of the load profile and it identifies the distribution of the forces along the height, λ is a time-dependent parameter defining the amplitude of the horizontal forces and it assumes increasing monotonic values for each analysis step (identified at the time t). According to NTC18, the application of pushover analysis must be carried out by using two different load profiles (extracted by two different lists), which allow to identify a range of structural behaviours (especially for the post-elastic branches) in which the ground truth shall be defined. The first option is represented by a load profile proportional to the first mode of vibration, named unimodal, and its shape depends on the eigenvector of that mode (Φ1) in the considered direction applied to the mass matrix, [M]:

$$\overline{F }=\left[M\right]\cdot \overline{{\Phi }_{1}}\cdot \lambda \left(t\right)$$
(2)

Especially when the first vibration mode is not completely decoupled from other vibration modes, this load profile can be also derived from the formulation of static load profile to employ in linear static analysis, where the load pattern can be approximated to an inverse triangular-like distribution, and the magnitude of the normalized forces (αi) at the ith floor can be computed as:

$${{\alpha }}_{i}={\text{z}}_{i}\cdot \frac{{\text{W}}_{i}}{\sum_{j}{\text{z}}_{j} \cdot {\text{W}}_{i}}$$
(3)

where zi and zj are the progressive height of the ith and jth masses (i.e., floors) and Wi and Wj are the corresponding weight of the ith and jth masses (i.e., considering the gravity acceleration). Then, the load profile can be computed as:

$$\overline{F }=\left[M\right]\cdot \overline{{{\alpha } }_{1}}\cdot \lambda \left(t\right)$$
(4)

where α1 indicates the shape of the inverse triangular load profile. The second option regards the mass proportional load profile, where the forces are directly proportional to the building masses, according to the following expression:

$$\overline{F }=\left[M\right]\cdot \overline{I }\cdot \lambda \left(t\right)$$
(5)

where I is the identity vector. In the case in which the masses associated to all floors present the same value, the shape of the load pattern is uniform. The load profiles in Eqs. (2), (4), (5) represent the conventional cases for pushover analysis, in which unimodal- or inverse triangular-like load profiles provide lower values of Vb, higher values of δR, and higher demands to upper storeys (due to a higher application of the resultant force), while the mass proportional- or uniform-like load profiles provide higher values of Vb, lower values of δR, and higher demand to lower/ground storeys (due to a lower application of the resultant force). Using both options allows to define different safety levels (i.e., C/D ratios), and the worst result can drive further actions (see recent works by [7] and [8]). Nevertheless, the effectiveness of the results obtained by the above conventional load profiles is limited to regular cases, which present a certain behaviour in post-elastic field (e.g., buildings conceived according to the principles of capacity design and hierarchy strength). The above features are not usually proper of existing structures, which can present several sources of irregularity and a structural capacity ruled by brittle failures. Hence, conventional nonlinear static analysis methods require updates, in order to better capture the seismic performance. With this scope, over the last 30 years the scientific community provided alternative options to run pushover analysis, considering two main issues: (a) load profiles that account for higher and coupled modes of vibration, (b) load profiles that account for unexpected and unpredictable progressive stiffness variations in the post-elastic field (combined with the variations of dynamic properties). The answers to the above problems are the multimodal and adaptive nonlinear static analysis, respectively.

Concerning multimodal pushover methods, the main aim of the tool is prevalently to predict the dynamic behaviour of high-rise (or characterized by a long first period of vibration) and irregular buildings (e.g., in terms of mass or stiffness), which are characterized by a first mode of vibration not able to fully describe the dynamic behaviour of the investigated building. This aspect sparked significant interest among researchers, leading to the development of various proposals, especially for capturing the torsional behaviour of RC buildings [9]. On the other hand, given the nonlinear nature of the problem, the scientific literature proposed two primary approaches: (i) a combination of significant modes into a singular load profile; (ii) a combination of effects after running a series of unimodal-like pushover analyses, assumed as proportional to the most influencing modes characterizing the dynamic behaviour of buildings (e.g., M[%] higher than 5%). Regarding the use of a multimodal profile, one of the first approach was proposed by Kunnath [10], which developed the modal combination method consisting in summing and subtracting load profiles proportional to the first two vibration modes. Hence, author defined new participation factors dependent on seismic demand and on the elastic period of the considered modes. A similar approach was provided by Barros and Almeida [11], which defined a participation factor to employ in a modal combination, for purpose of defining a new load profile. The proposed method was assessed on a three-dimensional (3D) RC building, on which authors varied some features to introduce irregularity. Kaatsız and Sucuoğlu [12] proposed the generalized pushover analysis, elaborated for identifying a load profile to apply on a 3D torsionally coupled system. The method was developed for simulating maximum IDRs under seismic action. Porco et al. [13] proposed a unique load profile from the modal decomposition of the maximum displacements vector obtained by the complete quadratic combination (CQC) applied to the RSA results. The contributions of all considered modes were weighted using new participation factors. The concepts at the base of this latter approach, developed by the authors of this paper and tested on a real-life irregular structure, were very similar to the ones proposed by NTC18 (2018) and object of this study, even though some substantial differences can be denoted (i.e., weighting of the modes, load profile derived from displacements and not from storey shear force). Regarding the combination of the effects given by more unimodal load profiles, the pioneering work of Chopra and Goel [14] opened the perspectives of the modal pushover analysis, which combined the results by more pushover analyses through two combination rules, that is, CQC and square root of sum of square (SRSS). For this analysis typology, some refinement by the same authors were provided (e.g., [15,16,17]), accounting for the different combination rules and for the influence of higher modes in tall buildings. In the end, some alternative multimodal load profiles were provided by several authors, which are herein only mentioned for the sake of brevity [18,19,20,21,22].

Regarding to adaptive approaches, Bracci et al. [23], Gupta and Kunnath [24], and Elnashai [25] provided the first versions of this approach, which were definitely developed by Antoniou and Pinho [26, 27]. In their works, authors proposed the force and the displacement adaptive pushover in which the analysis was run for steps, by identifying some specific limits of IDR to update the load profile according to the observed stiffness modification. The method was tested on stick-models designed to represent a various range of buildings. Several versions of the adaptive pushover were lately provided, as reported in Sürmeli and Yüksel [28], which combined the principles of the multimodal pushover with the ones of the adaptive pushover, in order to produce a stepwise load pattern able to capture the variation of dynamic characteristics and the structural damages. Amini and Poursha [29] proposed an adaptive force-based multimode pushover to predict seismic behaviour of mid-rise buildings, constituted by several consecutive steps. Jalilkhani et al. [30] proposed a multimode adaptive displacement-based pushover, where the main novelty of the proposed approach was constituted by the continuous upgrade of the load pattern distribution at the formation of a new plastic hinge. Barbagallo et al. [31] proposed an over-damped multimodal adaptive nonlinear static analysis, which consisted in modifying the value of the equivalent viscous damping ratio according to the observed increment of energy dissipation.

Both multimodal and adaptive pushover methodologies provided by the scientific literature usually neglected one main aspect when simulating the seismic action, that is, the spatial variability of the motion. With this regard, it is worth mentioning some studies that incorporated spatial effects of the motion in bi-directional pushover analysis. Cimellaro et al. [32] proposed a new version of this approach, consisting in applying two simultaneous unimodal load profiles in both main directions on 3D structures. The capacity curve, recorded in one direction, intrinsically embedded the effect of the action in the perpendicular direction. Other works about the same topic deserves attention but they are herein only mentioned for the sake of synthesis [33] [34, 35]).

Definition of the sample of archetype buildings and set up of nonlinear analysis

Design and numerical modelling of the sample of archetype RC buildings

The sample of archetype RC buildings is constituted by eighteen buildings, presenting the same in-plan configuration, but characterized by different number of storeys, different design levels, and different degrees of in-height irregularity (Fig. 2). The base configuration presents a rectangular shape with sides of 14 m in X direction (four bays of 3.5 m) and 9 m in Y direction (two bays of 4.5 m). The sample of buildings was selected to be representative of low- (3 storeys), medium- (5 storeys), and high-rise (7 storeys) buildings, where each storey presented height of 2.70 m (Z direction). To introduce in-height irregularity, the prescription provided by NTC18 (2018) was taken as reference, which reported the following statement: “mass and stiffness remain constant or vary gradually, without sudden changes, from the base to the top of the construction (the variations in mass from one horizontal to another do not exceed 25%, the stiffness does not reduce from one horizontal to the one above it by more than 30% and does not increase more than 10%)”. According to the provided limits, the three above identified configurations were tripled by removing from some of the storeys the 25% (i.e., one bay in X direction) and the 50% (i.e., two bays in X direction) of the storey mass. As stated, the remotion of building portions was established according to the number of storeys (Ns), using the following formulation:

$${N}_{s,d}=\frac{\left({N}_{s} - 1\right)}{2}$$
(6)

where Ns,d indicate the number of storeys from which the structural parts were deleted, starting from the top of the building. Thus, by employing Eq. (6), irregular cases were characterized by removing parts from the last storey for the 3-storeys buildings, parts from the last two storeys for the 5-storeys buildings, and parts from the last three storeys for the 7-storeys building (see Fig. 2). Following the proposed approach, two main aspects were highlighted: (a) removing the 25% of the mass allowed to identify the in-height irregular building on the threshold provided by NTC18 (2018), and its structural performance can be compared with the extreme cases (i.e., regular case and strongly in-height irregular case); (b) using Eq. (6), the removed parts of building amounted to less than a half of the storeys, identifying practical cases (e.g., building with superelevation) and, at the same time, providing a main source of irregularity that can rule the seismic performance of the structure. The nine structural configurations were after designed exploiting two different building codes for simulating new and existing structures, that is, NTC18 (2018) and RD2229 (1939), respectively. The total number of buildings amounted to eighteen (constituting the entire sample), where each subset was constituted by buildings designed with the same technical standard. A common aspect of both design approaches was the not consideration of staircases or shear walls, in order to simplify the archetype buildings for further investigations (obviously, new or existing buildings having 7 storeys cannot ignore the presence of RC walls). Analogously, effect of masonry infills was neglected.

Concerning to the subset of the nine new buildings, a “Low” ductility class was adopted. Structural materials were set by using a concrete class C28/35 (cylindrical compressive strength equal to 28 N/mm2) and a steel reinforcement class B450C (tensile yielding stress of 450 N/mm2). Floor slabs were disposed as a checkboard and they were characterized by a classical RC ribbed typology, which consisted in joists of constant dimensions (height 20 cm, width 10 cm, and spaced 50 cm), hollow clay masonry blocks, and a concrete layer of 4 cm thick. Live service loads were considered according to a residential usage. Regarding the seismic action (useful for design, RSA, and ground motion records selection), the considered response spectrum was the one used in Ruggieri and Uva [4], for the municipality of Bisceglie, Puglia Region, Southern Italy (low‐medium seismicity site). Finally, according to the principle of capacity design, a behaviour factor of 3.9 was accounted for. Regarding to the subset of the nine existing buildings, a simulated design was performed according to admissible stress method. In detail, structural materials were assumed by setting the compressive admissible stress of concrete equal to 5 N/mm2 and tensile admissible stress of steel reinforcement equal to 140 N/mm2, typical of smooth bars. Floor slabs were disposed in the weak direction, while slab typology was the same for new buildings. Overall, dead loads were assumed equal to 5 kN/m2, while live loads were fixed to 2 kN/m2. To design columns, the simple axial stress was considered, by defining sections of columns through the ratio between the vertical load and the admissible stress of concrete. Instead, to design beams, section sides and simple steel reinforcement were derived by evaluating the maximum bending moment stress in simple schemes of fixed ends beams. In both design approaches, shear reinforcement of beams and columns was defined according to the minimum requirements of codes. A summary of the typical sections of beams and columns and the related steel reinforcement (at the end sections) is reported in Tables 1 and 2 for new and existing buildings, respectively, where the differences among the two design procedures can be appreciated. Obviously, new buildings were designed accounting for the seismic actions (i.e., after running RSA), while existing buildings were designed only accounting for gravity loads (without applying partial safety coefficients in load combinations) and they present a more discretized size of structural elements (e.g., section reduction of columns along the height).

Table 1 Dimensions and steel reinforcements (end sections) of structural elements typologies for new archetype RC buildings. For beams, steel reinforcements are reported for top and bottom, while for columns steel reinforcements are reported for X and Y directions, respectively. For transversal reinforcements stirrup spacing is indicated. Still, deep (external frames) and wide shallow (internal frames) present two and four stirrups arms, respectively
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Table 2 Dimensions and steel reinforcements (end sections) of structural elements typologies for existing archetype RC buildings. For beams, steel reinforcements are reported for top and bottom, while for columns steel reinforcements are reported for X and Y directions, respectively. For transversal reinforcements stirrup spacing is indicated. Still, deep (external frames) and wide shallow (internal frames) present two and four stirrups arms, respectively
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After, the archetype RC buildings were simulated in the structural software SAP2000 [36] as 3D models constituted by beams and columns represented by frame elements fixed at the base with external restraints. Gravity and live loads were applied as distributed loads on shells with null area, mass, and stiffness. A rigid floor constraint was applied to all storeys. To capture the nonlinear behaviour, a lumped plasticity approach was employed, where specific plastic hinges were placed at the end sections of all the frames. It is worth specifying that only ductile mechanisms were accounted for (shear failures were neglected for avoiding convergence issues and to fully focus on the problem under investigation), by simulating the behaviour of beams through simple bending stress and the behaviour of columns through a combination of axial and bending stresses in both main directions. Constitutive law of plastic hinges was set as quadrilinear, characterized by four distinct branches. The first branch simulated the cracking of concrete, followed by a subsequent hardening branch representing the yielding of longitudinal steel reinforcement. The third branch implied a softening phase, indicating the strength degradation of the section. Finally, the fourth branch corresponded to the residual moment, set at 20% of the yielding moment. The yielding (θy) and ultimate (θu) rotations were estimated following the formulations outlined in the Annex of NTC18 (2018), by differentiating the expressions for some coefficients that assume different values for new and existing structures. The limit values of chord rotations were defined in accordance with different limit-states, i.e., immediate occupancy with a deformation equal to θy, life safety with a deformation equal to ¾θu, and near collapse with a deformation equal to θu, as reported in Fig. 1. Although the above criteria were useful for assessing the structural performance of structures in compliance with the current code (i.e., NTC18, 2018), for the case at hand they were used only for identifying if different failure behaviours occurred by using different load profiles in nonlinear static analysis. Still, according to the libraries of the software used, Taked rule was used for simulating the hysteretic behaviour of structural elements. Figure 2 shows the numerical models of a subset of the sample of RC archetype buildings, also considering the same geometry between the two subsets.

Fig. 1
figure 1

Moment-Rotation constitutive law employed to simulate the nonlinear behaviour of structural elements. My and Mu indicate yielding and ultimate moments, respectively. Green, yellow, and red dots indicate the achievement of the immediate occupancy, life safety and near collapse limit-states, respectively, in terms of rotation

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Fig. 2
figure 2

Geometrical configuration of a subset of RC archetype buildings (in the plane X–Z), accounting for increments in terms of number of storeys (from top to bottom) and in-height irregularities (from left to right)

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Computation of conventional and RSA-based load profiles

Once numerical models were available, load profiles for running pushover analysis were defined. First, the two conventional load profiles, i.e., uniform- and inverse triangular-like (indicated as U and T, respectively), were derived for each geometry and, to this scope, Eqs. (4) and (5) were involved. It is worth noting that by applying such formulations to in-heigh irregular archetypes presenting a consistent mass reduction at some storeys, some variations on the shape were obtained, especially when normalizing forces to the unit (see Fig. 3). For the case at hand, the load pattern proportional to the first vibration mode (i.e., unimodal) was not considered for different reasons: (a) for regular structures, the T load profile provides the same output of the unimodal load profile; (b) the strong in-height irregularity could sensibly reduce the participating mass in some main directions, until presenting values lower than the threshold to use the unimodal load profile (i.e., 75%); (c) according to NTC18 (2018), the T and the unimodal load profiles are alternative (i.e., practitioners can use one rather than the other). Still, some differences could occur among new and existing buildings, considering a certain difference in terms of mass (although in each building, the mass is constant for all storeys).

Fig. 3
figure 3

Load Profiles (normalized to the unit) for new archetype buildings in X direction

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After, the load profiles proportional to the storey shears of an RSA (indicated also as RSA-based) were computed. According to NTC18 (2018) recommendations, the first step consisted of running RSAs on the buildings by employing the design response spectrum at ultimate limit-state (e.g., life safety). This latter can vary from new to existing buildings, considering different values of the behaviour factor. For existing buildings (for new, the behaviour factor is defined in Sect. "Design and numerical modelling of the sample of archetype RC buildings") a behaviour factor of 1.5 was assumed. Moving to RSA, the shear response in each direction reflected the maximum likely seismic response derived from the contribution of all vibration modes having M[%] exceeding 5%. As well-known, the total number of modes to consider must be related to a total M[%] equal to or higher than 85%. Still, the shear forces derived from each considered mode at the end of the analysis shall be combined by employing a CQC approach. Following the above rules, RSA-based load profiles were computed by using the reference response spectrum mentioned in Sect. "Design and numerical modelling of the sample of archetype RC buildings" and after they were compared with other profiles. The results for new buildings are reported in Figs. 3 and 4 considering X and Y direction, respectively, where load profiles are indicated as LP, RSA-based load profile is indicated as D, number of storeys and direction are indicated with the corresponding number and letter in the title of graphs, while regular buildings and increasing in-height irregular ones are indicated as Reg, Irr25, and Irr50, respectively. Analogous results for existing buildings are reported in Appendix A, in Figs. 12 and 13.

Fig. 4
figure 4

Load Profiles (normalized to the unit) for new archetype buildings in Y direction

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Looking the obtained load profiles, some aspects should be highlighted. First, as expected, comparing results obtained from new and existing buildings no substantial differences were observed, also considering that the main difference regarded the storey mass, which is a detail lost during the normalization. The other expected outcome regarded the shape of U load profile when increasing the in-height irregularity. In fact, this load profile presented a sharp reduction of the normalized force (of 25% and 50% of the unitary force), when irregularity arises. The most interesting results can be denoted for T and D load profiles. Concerning T, for all cases, the high irregularity (i.e., 50%) brought to increase the forces at the middle storeys and decrease the forces at the top storeys, especially for the X direction. This outcome is mainly due to a reduction of the M[%] of the first vibration mode in the considered direction, and then, the related increment of M[%] in higher modes (especially for the 7 storeys cases). Nevertheless, it is worth noting that this trend is more accentuated in Y direction, considering that the reduction of M[%] in this direction is faster when the irregularity increases (see results of eigenvalue analysis in Sect. "Eigenvalue and pushover analysis"). Different shapes were observed for D load profile, where the maximum force was always observed at the last storey while, by increasing the irregularity, medium–high storeys presented the lower forces. This shows the occurrence of a sort of inversion, proper of a load profile that accounts for higher modes (i.e., RSA-based). In the end, the differences among load profiles can imply different results in terms of capacity (global and local), which is assessed in the next Sections.

Ground motion records selection and application

For purpose of comparison among nonlinear static and dynamic analyses, an input of 7 ground motion records, with two horizontal components, was selected from the European-Strong-Motion-Database [37]. The number of records was selected according to the provisions by NTC18 (2018), which received the indications by Eurocode 8 [38] about NTHAs. According to this latter, ground motion records were selected considering that the mean spectrum should not present values lower than the 10% and higher than the 30% of the target spectrum.

Figure 5 shows elastic ground motion spectra for all individual records, their mean spectrum and the target spectrum, all for 5% damping (T denotes the period, g is the acceleration of gravity). The selected ground motion records were employed for running NTHAs, by subdividing the 14 components in the two groups, and by using each group for performing analysis on the numerical models in a specific main direction (i.e., X and Y). Obviously, it is well-known that NTHAs should be run by contemporary using two components in the two main directions but, for the case at hand, it is convenient comparing results of mono-directional pushover analysis with the ones by NTHAs carried out in the same direction. The NTHAs outcomes were compared in both global and local terms with the ones obtained by pushover analyses, also exploiting the mean of results when local analysis is focused (i.e., evaluation of IDRs).

Fig. 5
figure 5

Selected ground motion records, mean and target spectra

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Results

Eigenvalue and pushover analysis

For the entire sample of archetype buildings, eigenvalue analysis was performed. The obtained results in terms of fundamental period of vibration in the two main directions (X and Y, indicated as T1,X and T1,Y, respectively) and the related participating masses (M[%]X and M[%]Y, respectively) were reported in Table 3, for both new and existing buildings.

Table 3 Results of eigenvalue analysis on the sample of archetype buildings
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As expected, existing archetype buildings presented higher values of T1,X and T1,Y than new ones, due to an overall higher flexibility of structural components. Instead, more significant was an aspect already mentioned in Sect. "Computation of conventional and RSA-based load profiles" where, by increasing the in-height irregularity, M[%]Y decreased with a higher velocity than M[%]X, up to obtain values lower than 60%. This observation brought to highlight two main outcomes: (a) the presence of in-height irregularity empowered the contribute of higher modes to the dynamic behaviour; (b) the reduction of M[%] in the main modes can exclude the use of some traditional load profiles (i.e., unimodal/inverse triangular-like).

Although the response of eigenvalue analysis showed some values of M[%] lower than 75% (i.e., lower limit provided by NTC18, 2018 for running nonlinear static analysis with some load profiles), pushover analyses were performed on all archetype buildings, by using all load profiles (i.e., T, U, and D), in both main direction (X and Y). The results for new buildings are reported in Figs. 6 and 7 (for X and Y direction, respectively), while analogous results for existing buildings are reported in Appendix A, in Figs. 14 and 15. As expected, the shape of the capacity curves followed the imposed moment-rotation constitutive laws, with some differences in global behaviour due to a different design level. Concerning new archetype buildings, the results show that U load profiles provided the higher values of Vb, while no substantial differences can be appreciated in terms of displacement among load profiles, due to buildings designed by using capacity design rules (i.e., very high displacement capacity). Of interest was the comparison between D and T load profiles where, although in some cases a consistent difference among load profile shapes was observed (e.g., cases with 50% of in-height irregularity), capacity curves were very closed. When slight differences can be observed, the T load profiles provided the lower values of Vb, which implied that the traditional load profiles identified the larger range to identify the real structural behaviour. In any cases, for new buildings, it seems that the RSA-based load profile was not necessary, because it reproduced the same structural behaviour given by the triangular-like load profiles. The abovementioned results confirmed the results obtained in Ruggieri and Uva [4], where the pushover analysis results obtained by employing D load profiles provided similar results to the ones obtained by T load profiles on in-plan irregular archetype RC buildings, designed according to NTC18 (2018).

Fig. 6
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Capacity curves for new archetype buildings in X direction

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Fig. 7
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Capacity curves for new archetype buildings in Y direction

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Different outcomes can be observed for existing structures. The influence of the low design level was glaring, with a very short post-elastic branch before collapse and this behaviour can be appreciated for all archetype buildings and by using all load profiles. Also in this case, of great interest is the comparison among the traditional load profiles and the RSA-based one. First, as expected, the U load profile provided higher values of Vb and lower values of δR than other cases. Significant is also observing the trend of softening branches, which in some cases is different among U load profiles and D and T ones, and this suggests a different global failure mechanism. Nevertheless, D and T load profiles continued to be closed, even if some differences occurred in cases of high in-height irregularities. In fact, for 5- and 7-storeys buildings with 50% of irregularity, in both main directions, D load profiles provided curves between and well-separated from curves derived by T and U load profiles, suggesting possible cases in which RSA-based load profile can be useful.

The differences and the similarities observed among the load profiles can be also assessed by looking at the different evolution of the plasticization in the structures when subjected to increasing actions. As for example, taking into consideration the most irregular existing archetype structures, i.e., 5- and 7- storeys, with 50% of in-height irregularity, and fixing the observation at a common value of δR (i.e., 10 cm), Fig. 8 shows the failure mechanisms occurred with different load profiles, mainly driven by the strong in height irregularity. In detail, for both models, using the T (Fig. 8, left) and the RSA-based (Fig. 8, right) load patterns, the obtained plasticization was comparable, even though some differences could be appreciated in terms of deformed shape. On the other hand, a different outcome was observed by using the U (Fig. 8, centre) load profile, which increased the plasticization at the base elements (due to a lower position of the resultant of the forces).

Fig. 8
figure 8

Identification of the plasticization in a main frame for a fixed value of δR (i.e., 10 cm) for 5 (top) and 7 (bottom) storeys existing archetype buildings, X direction, by using T (left), U (centre), and RSA-based (right) load profile. Green dots individuate the exceeding of the IO (i.e., θ > θy), blue dots individuate the exceeding of the LS (i.e., θ > ¾ θu)

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Hence, although the options T and U always traced the higher range of seismic behaviour scenarios, the response obtained by D load profiles shall be assessed against NTHAs results. In fact, in order to observe if the given prediction by D load profiles can better match with the nonlinear dynamic analysis results, especially in the most irregular cases, NTHAs were performed on all models and outcomes are reported in the next Section.

Nonlinear time history analysis and comparison with nonlinear static analysis

NTHAs were performed on all archetype RC buildings, both new and existing, according to the criteria defined in Sect. "Ground motion records selection and application". The obtained results were compared with nonlinear static outcomes from the global and local point of views. From the global point of view, NTHAs outputs were overlapped on the capacity curves and reported in terms of Vb—δR, to assess which capacity curve better matched the dynamic responses. Some of the results are reported in Fig. 9, for the cases showing higher differences among T and D load profiles, that is, existing archetype buildings, 5-and 7-storeys, 50% of in-height irregularity, X and Y direction. The overall results are reported in Appendix A, in Figs. 16 and 17 for new buildings in X and Y directions, respectively, and in Figs. 18 and 19 for existing buildings in X and Y directions, respectively. Ground motion records results are indicated in the legend with the letter R and are displayed as black dots, produced by recording the maximum δR and the corresponding Vb, as proposed by several studies (e.g., [13] and references therein).

Fig. 9
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Comparison among capacity curves and NTHAs for 5 and 7 storeys existing archetype buildings, X and Y directions

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In general, the NTHAs points were displayed in the elastic/near post-elastic part of the capacity curves, both for new and existing buildings, and this is due to a considered seismic input of low-medium intensity. Nevertheless, the obtained results allowed to understand that, in most of the cases, the load profiles better approximating the dynamic response were the U ones. In fact, black dots tended to follow the trend of the red lines and, in some cases, to exceed the maximum Vb values. This achievement confirmed that, from the global behaviour point of view, the D load profiles did not sensibly improve the estimates given by nonlinear static procedures. Still, facing the safety problem in a code-oriented view, T load profiles almost always provided the lower Vb values, which could result the most conservative option among the available ones. Of interest were the results obtained for the most irregular existing archetype buildings, where the abovementioned trend is confirmed, and the expected improvements from D load profiles were not observed.

As last chance to evaluate the effectiveness of the RSA-based load profiles, the comparison among static and dynamic results were performed from the local point of view. To this scope, the results in terms of IDR were evaluated to estimate the demand values on each storey given by static and dynamic analysis approaches. In order to provide a comprehensive overview about the possible outcomes given by IDR examination, results were computed for two specific values of δR, one assumed in the elastic field (i.e., before the yielding of all capacity curves) and one assumed in the post-elastic field (i.e., after the yielding of all capacity curves and before the softening branch for existing structures). All results were provided for both new and existing archetype buildings and considering both main directions (i.e., X and Y). Graphs report the IDRs from T, U, and D load profiles, the values for each record (indicated in grey with R), and the mean of the records (indicated in black with mean R). In the titles of graphs, the indication about the reference δR was added, by specifying elastic value with El and post-elastic value through P-El. Figures 10 and 11 show the results about new and existing archetype buildings, respectively, X direction, P-El value of δR, while other results are reported in the Appendix A, Figs. 20, 21, 22, 23, 24, and 25.

Fig. 10
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Comparison in terms of IDRs among pushover and NTHAs analyses, for new archetype buildings in X direction, assuming a post-elastic value of δR

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Fig. 11
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Comparison in terms of IDRs among pushover and NTHAs analyses, for existing archetype buildings in X direction, assuming a post-elastic value of δR

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Looking the obtained results, some observations can be derived. It is worth noting that the scale of the abscissas for all graphs is different, to considering the variations of IDR profiles obtained by the different ground motion records. First, as expected, the IDR values derived by new buildings are higher than the ones obtained by existing buildings. For this reason, the examination of results can be performed by differentiating the two design levels. In the complex, U load profiles led to higher IDR demands at the ground level, while T and D load profiles led to higher demands in the upper storeys [5].

Concerning new buildings, the mean IDRs from the employed ground motion records were better approximated in elastic field by T and D load profiles, while a greater accurateness was provided by U load profile in the post-elastic field. This observation also confirmed the results obtained by comparing static and dynamic outcomes from the global point of view. Still, for models presenting high levels of in-height irregularity, a clear trend was not observed, with some cases in which U load profiles provided closer IDRs to the dynamic ones (e.g., middle storeys) and some cases in which T and D load profiles better approximated the NTHAs responses (e.g., first and top storeys). As main outcome, T and D load profiles provided better results for new buildings, while U load profiles provided better results when strong in-height irregularities were considered.

When coming to existing archetype buildings, different observations can be derived. First, as expected, very different IDR results were obtained varying the considered ground motion record, evidence which confirmed the necessity to increase the number of inputs for reducing the overall uncertainty (i.e., record-by-record variability). Nevertheless, by evaluating NTHAs results in mean, it seemed that U load profiles better approximated the IDRs from dynamic analyses at the medium-upper storeys, while T and D load profiles better predicted the dynamic responses at the lower storeys. In fact, taking into account the results for post-elastic values of δR, the first plasticization provided a reduction of the IDRs at the lower storeys, aspect well-predicted by employing T and D load profiles. In some cases, as in Y direction for post-elastic values of δR, although the mean IDR from ground motion records was closed to the IDRs derived by all load profiles (thanks to a ground motion record presenting very high values of IDRs), it seemed that pushover outcomes could be conservative, as typical of static approaches. Finally, seismic behaviour of models with high levels of irregularities were better predicted by using the U load profiles.

In the end, both for new and existing buildings, one aspect is extremely relevant among the results: there were not consistent differences among T and D load profiles outcomes. Hence, the outcomes derived from T and D load profiles can be almost always related, confirming the inability of the RSA-based load profile of improving the prediction of the seismic response of archetype buildings.

Conclusions

The paper presents a study to assess the effectiveness of extending the concepts at the base of linear analyses to the current approaches of nonlinear static analysis. The need for this study arises from the one of the main trends toward which new generation building codes are orienting in the last years. In fact, to improve the current practices in the field of nonlinear seismic analysis of new and existing structures, some national standards as the Italian building code (NTC18, 2018 [1]) provided new rules to employ in nonlinear static analysis, such as to use a load profile proportional to the storey forces derived by a response spectrum analysis (RSA). In addition, this new prescription was declared as appliable for all cases, as well as for irregular buildings (e.g., in-plan and in-height).

In order to demonstrate the reliability of the new code proposal, this study aims to complete the work started in Ruggieri and Uva [4], where the RSA-based load profile was tested on in-plan irregular archetype structures, showing negligible improvements with respect to traditional load profiles (e.g., inverse triangular-like one). For the case at hand, the new load profile was assessed on a sample of increasing in-height irregular archetype buildings, characterized by different design levels (i.e., to simulate new and existing buildings), different number of storeys (i.e., 3-, 5-, and 7-storeys), and increasing irregularity level (i.e., introducing area/mass reductions in the upper storeys of 25% and 50%). The results obtained by RSA-based load profile were compared with the ones derived by using uniform- and inverse triangular-like load profiles (i.e., conventional load profiles) and the ones obtained by nonlinear time history analyses (NTHAs). The following outcomes were obtained:

  • Introducing in-height irregularities led to obtained different load profiles, both for conventional and RSA-based load profile. In particular, the uniform-like (U) load profiles presented a sharp reduction of the storey forces, according to the imposed mass reduction. The inverse triangular-like (T) load profiles presented higher forces in the middle storeys, with a consistent variation from the usual shape given by the reduction of the participating mass of the first vibration mode in the considered direction. The RSA-based (D) load profiles presented a sort of inversion in the middle storeys, given by the consideration of higher modes in the load profile computation. Overall, consistent differences were obtained among the shapes of load profiles, without sensible variations among new and existing structures.

  • Results in terms of pushover analyses showed that for new buildings, U load profiles provided the higher values of base shear (Vb), while no substantial differences can be appreciated in terms of displacement among load profiles, due to the use of the capacity design rules. Still, no tangible differences were observed among capacity curves computed through T and D load profiles, which made this latter useless for purpose of global capacity assessment. When moving to existing structures, although different behaviours from the new buildings were observed (e.g., short post-elastic branch, different softening slopes), by increasing the in-height irregularity some differences were obtained among the load profiles. Nevertheless, comparing the pushover results with NTHAs, U load profile better predicted the dynamic response, while D did not show any improvement.

  • Results in terms of interstorey drift ratios (IDRs) showed that for new buildings, T and D load profiles better approximated the dynamic response in the elastic field, while U load profiles better approximated the dynamic response in the post-elastic field, confirming the observed results in global terms (i.e., capacity curves). Instead, in existing structures, U load profiles better approximated the IDRs from dynamic analyses at the medium-upper storeys, while T and D load profiles better predicted the dynamic responses at the lower storeys. For the purposes of this study, the main observation derived by the local capacity of buildings, both for new and existing, was an inconsistent difference among T and D load profile outcomes.

Based on the performed extensive numerical campaign, the obtained results, and the comparison among different analysis methodologies, the answer to the question proposed in the title can be provided: does it make sense to extend the concepts of response spectrum analysis to nonlinear static analysis? The obvious answer is NO. As demonstrated in this and previous studies, the RSA-based load profile does not provide a better estimate of the dynamic behaviour of structures and cannot be used for any case. Although nonlinear static analysis presents several advantages (e.g., easy to employ, low computational effort), further investigations should not be oriented to provide other load profiles for nonlinear static analysis, but to improve the current practice about NTHAs, making this kind of analysis more accessible to engineers and practitioners, especially when assessing the structural performance of existing structures.