A recently proposed direct Displacement-based procedure of nonlinear static (pushover) analysis on multi-storey reinforced concrete (r/c) buildings is verified here against the results of nonlinear response history analysis. An asymmetric, regular in elevation, torsionally flexible, multi-storey r/c building designed according to Eurocode EN 1998 is investigated. Taking fully into account the inelastic torsion and the higher mode effects, as well as the P-Delta effects, the proposed procedure applies a pattern of seismic floor enforced-displacements along the “Capable Near Collapse Principal Axes of the building”, aiming at the Near Collapse state. The envelope of the results of sixteen final non-linear static analyses on the investigated building shows that the main aspects of the spatial seismic action effects can be safely captured by the proposed procedure, especially regarding the inelastic interstorey drift ratios, as well as the plastic mechanism of the building.

1. Introduction

The main weakness of the nonlinear static (pushover) procedure proposed by the contemporary seismic codes, as is Eurocode EN 1998-1, is its inability to properly predict the seismic interstorey drift ratios over the height of multistorey r/c buildings. Indeed, the interstorey drift ratios of asymmetric multi-storey buildings are the most representative response measure that controls the distribution of the local inelastic deformations of the structural elements, in plan and in height of the multistorey building, and hence of the structural damage^[1,2]. Interstorey drift ratios are usually underestimated by the code-proposed pushover procedure due to the consideration that the building response is controlled only by the fundamental mode, ignoring the contribution of the higher and torsional modes that develop in the linear area and significantly affect the behavior in the non-linear area of response. Another weakness of the code-proposed pushover procedure is the use of the superposition technique on nonlinear analyses results, into the nonlinear area of response, to consider the spatial seismic action^[3]. Additionally, the lack of proposals by the seismic codes about the main loading directions of buildings as well as for the proper consideration of the torsional sensitivity of buildings in pushover analysis raises questions about the correct/rational application of the pushover procedure. The last two issues are directly affected by the lack of definition on the one hand of the real inelastic centre of stiffness and on the other hand of the real inelastic torsional radii of the multi-storey buildings, which are continuously alter in the non-linear response region. These issues have been investigated recently by Bakalis and Makarios^[4-8], Makarios and Bakalis^[9-11] and Bakalis et al.^[12]. Lastly, P-Delta effects should always be considered in the framework of non-linear analysis, an issue which is not emphasized enough in the seismic codes.

To deal with the inelastic torsion or/and the contribution of higher modes in the linear (and maybe into the nonlinear) response of buildings, a large number of improved pushover procedures have been developed the last two decades. These procedures use either an invariable loading vector or a variable one, usually consisting of monotonically increasing floor forces and torques (Forced-based procedures) but in some cases also of floor enforced displacements (Direct Displacement-based procedures). The first category includes the multi-mode procedures^[13-19], pushover procedures combined with some kind of linear dynamic analysis^[20-23] and pushover procedures that use dynamic or corrective eccentricities in order to apply the floor lateral loads^{[4-7,9,12,24,25]}. The adaptive pushover procedures^[26-31] belong to the second category. The effectiveness of the improved pushover procedures on the seismic assessment of buildings is discussed in several published research papers^[32-39]. The computational cost and the complexity of implementing some of the improved pushover procedures is another important issue to emphasize, as simplicity and clearness are key points in order to have a useful tool for the assessment of the seismic capacity of structures.

To provide a comprehensive treatment of all the previously mentioned weaknesses of the code-proposed pushover procedure on multi-storey reinforced concrete (r/c) buildings, a simple Direct Displacement-based pushover procedure has been recently proposed by Makarios and Bakalis^[10,11], Bakalis and Makarios^[8]. According to this procedure, three seismic enforced-displacements are applied at each floor-diaphragm with respect to an ideal inelastic principal reference system of the multi-storey r/c building: two floor enforced-translations and one floor-enforced rotation. These seismic, drift-based, floor enforced-displacements are the outcome of a large parametric analysis on multi-storey r/c buildings and aim directly at the seismic performance level of Near Collapse (NC). Using all possible sign combinations of the three floor enforced displacements, eight final pushover analyses are performed along each main loading direction and the envelope of a total of sixteen analyses provides an accurate prediction of the seismic demand, especially as regards the distribution in-plan and in-elevation of the seismic interstorey drift ratios. Key point of the process is the definition of an ideal inelastic principal reference system of the multi-storey r/c building ( $$ \mathrm{CR}_{\mathrm{sec}}\left(I_{\mathrm{sec}}, I I_{\mathrm{sec}}, I I I_{\mathrm{sec}}\right) $$ ) at the Near Collapse state, by considering that all the structural members have developed plastic hinges at their two ends. This reference system is called as “Capable Near Collapse Principal System of the multi-storey r/c building”. Also, the torsional sensitivity of the multi-storey r/c building is examined relative to this ideal principal system using an enchased formula. The proposed pushover procedure can be used for the seismic assessment of any asymmetric, torsionally flexible or not, ductile, multi-storey r/c building. Regularity in elevation of the building and the formation of whatever type of plastic mechanism (except the mechanism of soft-storey) are the mandatory conditions to apply the procedure. A possible practical implementation of the proposed pushover procedure into codification could concern the safe estimation of the seismic demands of asymmetric r/c buildings at the Near Collapse state, especially of the interstorey drift ratios at the stiff sides of torsionally flexible buildings.

It is noted that, the effectiveness of the proposed pushover procedure with enforced displacements has been fully verified in the first author’s doctoral dissertation ^[40]. Various regular in elevation, ductile, multistorey r/c buildings were examined there, with varying number of floors, structural type, static eccentricity, and torsional sensitivity. In this paper, the effectiveness of the proposed procedure is verified by the seismic assessment of an asymmetric, torsionally flexible, six-storey r/c building using the nonlinear response history analysis.

2. Application Steps of the Enforced-Displacement Pushover Procedure

The basic steps for the application of the enforced-displacement pushover procedure are summarized here. It is noted that the proposed procedure follows the rationale of Eurocode EN 1998-3 on the secant stiffness at yield of structural r/c elements ( $$ E I_{\text {sec }} $$ ), to properly simulate the extreme capable Near Collapse state of the building. Full details of the proposed methodology can be found in References^[8,10,11,40].

2.1 End-section analysis of r/c elements

Given the geometry and reinforcement details of the end-sections of all structural elements, section analysis provides the Moment-Curvature ( M - φ ) diagrams which are then bilinearized to extract the yield moment $$ M_{\mathrm{y}} $$ of each end-section. These diagrams are converted to Moment-Chord Rotation ( M - θ ) diagrams by using the plastic hinge length $$ L_{\mathrm{pl}} $$ proposed in the Informational Annex A of EN 1998-3 (Eqs. (A.4) and (A.9)). In addition, Eqs. (A.10) or (A.11) in the same Annex of EN 1998-3 can be used for the calculation of the chord rotation at yield $$ \theta_{\mathrm{y}} $$ .

2.2 Secant stiffness at yield of r/c elements

The secant stiffness at yield of the end-sections of each structural element is calculated according to EN 1998-3 (Informational Annex A.3.2.4(5)):

(1) $$ E I_{\text {sec }}=M_{\mathrm{y}} \cdot L_{\mathrm{v}} / 3 \theta_{\mathrm{y}} $$

Where $$ M_{\mathrm{y}} $$ and $$ \theta_{\mathrm{y}} $$ are respectively the yield moment and the chord rotation at yield calculated in the previous step and $$ L_{\mathrm{v}} $$ is the shear span, usually taken equal to half the clear length of the structural elements. The secant stiffness at yield to be assigned to each structural member of the non-linear model of the building is the average of the corresponding stiffness of their two end-sections for negative and positive bending.

2.3 Non-linear model. Definition of an ideal inelastic principal reference system of the multi-storey r/c building

The secant stiffness at yield must be assigned to each structural member in the nonlinear model of the r/c building. This is mandatory in EN 1998-3^[41] for the verification in terms of deformations. The same rationale is adopted by the proposed procedure. This implies the formation of plastic hinges at both end-sections of all the structural members of the building model. But this is an ideal state, characterized by the authors as the extreme Capable Near Collapse state. By considering this state of full plastic mechanism at NC, the verification of the building is always on the safe side, because the building is more flexible, and the displacements and deformations are slightly enlarged at the NC state. To simulate the locations of all possible plastic deformations in the nonlinear model, concentrated plastic hinges should be inserted at the critical end-sections of each structural member.

In the previously formed non-linear model, the following must be determined: (a) the inelastic Centre of Stiffness ( $$ \mathrm{CR}_{\mathrm{sec}} $$ ) of the multistorey r/c building and, hence, the in-plan location of the inelastic vertical Principal Axis ( $$ I I I_{\mathrm{sec}} $$ ) passing through it, (b) the inelastic horizontal Principal Axes ( $$ I_{\mathrm{sec}}, I I_{\mathrm{sec}} $$ ) of the multistorey r/c building, (c) the inelastic Torsional radii ( $$ r_{\mathrm{I}, \mathrm{sec}}, r_{\mathrm{II}, \mathrm{sec}} $$ ) of the multistorey r/c building. All the previously mentioned can be determined by performing a set of linear analyses, using the well-known methodology for the identification of the vertical fictitious elastic axis (torsional optimum axis) of the multi-storey building^[42-52]. $$ \mathrm{CR}_{\mathrm{sec}} $$ is the pole of rotation into the floor-diaphragm closest to the height of $$ 0.8 H_{\text {tot }} $$ , where $$ H_{\text {tot }} $$ is the total building height. Thus, the above methodology approximately defines an inelastic , ideal, 3D Principal Reference System $$ \mathrm{CR}_{\mathrm{sec}}\left(I_{\mathrm{sec}}, I I_{\mathrm{sec}}, I I I_{\mathrm{sec}}\right) $$ for the application of non-linear methods of analysis aiming at the NC state, which is the “Capable Near Collapse Principal System of the multi-storey r/c building” so named by the authors. The term inelastic was selected because of the non-linear characteristics ( $$ M_{\mathrm{y}} $$ , $$ \theta_{\mathrm{y}} $$ ) used in the calculation of the secant stiffness at yield (Eq. 1). Therefore, all the above inelastic properties of the multi-storey r/c building are strength dependent. The inelastic static eccentricity of the multi-storey r/c building along the $$ I_{\mathrm{sec}} $$ or $$ II_{\mathrm{sec}} $$ axis is equal to the distance along the same direction between the location of the center of mass CM in each floor and that of the vertical Principal axis $$ I I I_{\mathrm{sec}} $$ . The two main loading directions coincide with the directions of the horizontal Principal Axes $$ I_{\mathrm{sec}} $$ and $$ II_{\mathrm{sec}} $$ .

The (mean) torsional sensitivity of the multi-storey r/c building is verified by the following equation:

(2) $$ r_{\mathrm{I}, \text { sec }} \text { or } r_{\mathrm{II}, \text { sec }} \leq 1.10 \cdot r_{\mathrm{m}} $$

where $$ r_{\mathrm{m}}=\sqrt{J_{\mathrm{m}} / m} $$ is the radius of gyration of the floor closest to the $$ 0.8 H_{\text {tot }} $$ level, $$ J_{\mathrm{m}} $$ is the mass moment of inertia of the same floor about a vertical axis passing through its geometric centre and $$ m $$ is its mass. If the equation is fulfilled, then the multi-storey building is characterized as torsionally flexible. In the above equation, a higher limit value of 1.10 was adopted^[5,8], compared to the value of 1 used in the linear area^[53], to take account of the increased torsional sensitivity observed in several cases of single-story buildings in the non-linear area, initially characterized as torsionally stiff (in the linear area).

2.4 Proposed vectors of floor enforced-displacements

Figure 1 and Figure 2 illustrate the proposed patterns in elevation of the seismic enforced interstorey drift ratios $$ \gamma_{\mathrm{I} ; \mathrm{II}} $$ and of the seismic floor enforced-rotations $$ \psi_{R, I I I} $$ used in the framework of the proposed procedure for the verification of the building at the performance level of NC. These enforced displacements are the outcome of a large parametric analysis in various regular in elevation, ductile, multi-storey r/c buildings designed according to EN 1998-1 for Ductility Class High, by performing nonlinear response history analysis (N-LRHA) and always considering the influence of P-Delta effects. The parameters examined were the structure type (as defined in EN 1998-1^[3]), the number of floors, the inelastic statistic eccentricity and the torsional sensitivity. The proposed values of Figure 1 and Figure 2 are mean values determined through a statistical processing.

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Figure 1. Proposed seismic enforced interstorey drift ratios $$ \gamma_{I, \mathrm{i}} $$ or $$ \gamma_{I I, \mathrm{i}} $$ (rad) in elevation for the NC performance level, at the location of the vertical axis $$ I I I_{\mathrm{sec}} $$ and along the horizontal directions $$ I_{\mathrm{sec}} $$ and $$ II_{\mathrm{sec}} $$ (all cases of inelastic static eccentricity and torsional sensitivity). NC: near collapse.

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Figure 2. Proposed seismic floor enforced-rotations $$ \psi_{R, I I I, \mathrm{i}} $$ (rad) about vertical axis in elevation for the NC performance level (all cases of inelastic static eccentricity). NC: near collapse.

In Figure 1 and Figure 2, $$ H_{\mathrm{i}} / H_{\mathrm{n}} $$ is the relative height of the examined level ( $$ H_{\mathrm{i}} $$ , measured from the ground level) to the total building height $$ H_{\mathrm{n}} $$ . Linear interpolation can be performed for intermediate number of stories. The parameters used in Figure 1 are the structural type and the number of stories. In Figure 2, the torsional sensitivity is an additional parameter (TF for torsionally flexible buildings and TS for torsionally stiff buildings). Figure 1 and Figure 2 provide proposed patterns of $$ \gamma_{\mathrm{I} ; \mathrm{II}} $$ and $$ \psi_{R, I I I} $$ along each horizontal direction $$ I_{\mathrm{sec}} $$ and $$ II_{\mathrm{sec}} $$ , separately. Then, the unique values of the i -floor enforced-rotations $$ \psi_{R, I I I, \mathrm{i}} $$ are obtained as the average of the corresponding values along each horizontal direction. The proposed values of seismic enforced interstorey drift ratios $$ \gamma_{\mathrm{I} ; \mathrm{II}} $$ (rad) in elevation are converted to floor enforced translational displacements $$ \psi_{I ; I I} $$ using the following equations:

(3a) $$ \psi_{I ; I I, 1}=\gamma_{\mathrm{I} ; I I, 1} \cdot H_1 \quad i=1 $$

(3b) $$ \psi_{I ; I I, i}=\gamma_{\mathrm{I} ; I I, \mathrm{i}} \cdot H_{\mathrm{i}}+\psi_{i-1} \quad i=2, \ldots, n $$

Where $$ H $$ is the floor height. It is noted that the value of $$ \gamma_{I, I I, \mathrm{i}} \cdot H_{\mathrm{i}} $$ gives directly the i -floor seismic drift along the horizontal direction $$ I_{\mathrm{sec}} $$ or $$ II_{\mathrm{sec}} $$ . Τhe proposed values of floor enforced-displacements given by Eq. (3) are further adapted to the building under seismic evaluation following the procedure of the next step.

2.5 Temporary nonlinear static analyses along the ideal principal axes and of the multi-storey building

The proposed patterns of floor enforced interstorey drift ratios $$ \gamma_{\mathrm{I} ; \mathrm{II}} $$ can be adapted more specifically to the examined r/c multi-storey building by the envelope of two temporary sets of pushover analyses (with lateral floor forces and considering the P-Delta effects) along the horizontal directions $$ I_{\mathrm{sec}} $$ and $$ II_{\mathrm{sec}} $$ . In the first set of pushover analyses (2 analyses along the $$ I_{\mathrm{sec}} $$ axis and more 2 analyses along the $$ II_{\mathrm{sec}} $$ axis, considering the two signs (±) of application of the floor lateral forces), the floor lateral forces are applied at the location of the vertical $$ I I I_{\mathrm{sec}} $$ axis having a pattern according to the first uncoupled fundamental translational mode along each horizontal ideal direction, $$ I_{\mathrm{sec}} $$ or $$ II_{\mathrm{sec}} $$ .

The second set of pushover analyses is similar to the first (2 analyses along the $$ I_{\mathrm{sec}} $$ axis and 2 analyses along the $$ II_{\mathrm{sec}} $$ axis, considering the two signs (±) of application of the floor lateral forces) with the difference that only the 80% of the base shear is used, while the remainder 20% of the base shear is applied as a concentrated lateral load at the top of the vertical $$ I I I_{\mathrm{sec}} $$ axis. In both sets, the value of the base shear that is distributed in elevation can be considered equal to one (unit base shear). The target displacement of the two temporary sets of pushovers can be obtained from Table 1. This Table contains the results of a large parametric analysis on multi-storey r/c buildings about the mean value of the seismic drift ratio $$ \gamma_{\mathrm{t}, \text { top }} $$ of the building top at the location of the vertical $$ I I I_{\mathrm{sec}} $$ axis, along the directions of the $$ I_{\mathrm{sec}} $$ and $$ II_{\mathrm{sec}} $$ axes. For intermediate structural types of dual systems (consisting of frames and walls) or coupled (via beams) wall systems, as well as for various number of storeys, linear interpolation is performed. These values are valid for all cases of inelastic static eccentricity and torsional sensitivity. The final values of the proposed interstorey drift ratios, which will be used to calculate the floor enforced translational displacements of the proposed procedure along the $$ I_{\mathrm{sec}} $$ and $$ II_{\mathrm{sec}} $$ axes (Eq.3), are calculated as the average of the proposed values of Figure 1 and of those computed from the envelope of the two temporary sets of pushover analyses^[40]. It should be noted that, in^[8], the envelope of the $$ \gamma_{\mathrm{I} ; \mathrm{II}} $$ values from the three previous sources was initially proposed for the safe calculation of the enforced translational displacements. However, the difference is small and in the first author’s doctoral dissertation the average values were preferred, mainly for the most efficient adjustment of the $$ \gamma_{\mathrm{I} ; \mathrm{II}} $$ values at the lower floors of multi-storey r/c buildings.

2.6 Enforced-displacement combinations to consider the spatial action of the earthquake

To consider the spatial character of the seismic action, the three enforced-displacements $$ \psi_{I, \mathrm{i}} $$ , $$ \psi_{I I, \mathrm{i}} $$ and $$ \psi_{R, I I I, \mathrm{i}} $$ should act simultaneously on the location of the vertical $$ I I I_{\mathrm{sec}} $$ axis inside each i -floor but with an appropriate way along each main loading direction $$ I_{\mathrm{sec}} $$ and $$ II_{\mathrm{sec}} $$ . Therefore, the floor enforced translational displacement along the examined (main) loading direction keep its full value inside the combinations while the second floor enforced translational displacement, along the other (normal) direction, is inserted in the combinations with 30% of its full value. Considering the two signs (±) of action of the floor enforced-displacements, eight possible combinations of them are obtained for each one of the main loading directions $$ I_{\mathrm{sec}} $$ and $$ II_{\mathrm{sec}} $$ . These combinations are presented in Table 2 and Table 3, in which the "+" or "-" symbol indicates with emphasis the combined values of seismic enforced-displacements.

2.7 Estimation of the seismic demand

The sixteen final pushover analyses of Table 2 and Table 3 are performed using as target displacement the value of the enforced translational displacement at the top of the vertical axis $$ I I I_{\mathrm{sec}} $$ , along each main loading direction $$ I_{\mathrm{sec}} $$ and $$ II_{\mathrm{sec}} $$ . Therefore, the target displacement is equal to the value $$ \psi_{I ; I I, \mathrm{n}} $$ calculated by Eq. (3b), where n is the top floor. Τhe envelope of the response of the sixteen pushover analyses of the proposed procedure provides a safe prediction of the seismic demand at the NC state, especially as regards the interstorey drift ratios. The floor displacements are computed slightly conservatively in multi-storey buildings with more than three floors and are corrected according to the next step.

2.8 Correction of the floor displacement profiles

Since the goal of the proposed pushover procedure with floor enforced-displacements is to capture the maximum interstorey drift ratios of each floor-diaphragm, the estimates for the floor seismic displacement profiles will be slightly conservative, especially from the middle height of a multi-storey building to its top^[1]. The abovementioned issue can be observed in multi-storey buildings with more than three floors. This is inevitable, since the floor enforced-displacements $$ \psi_{I ; I I} $$ are calculated sequentially with Eq. (3), from the base of the building to its top, through the proposed interstorey drift ratios $$ \gamma_{\mathrm{I} ; \mathrm{II}} $$ which estimated conservatively in the parametric analysis with N-LRHA and express the maximum seismic interstorey drift ratios observed in the different floors of the building. However, these maximum values of interstorey drift ratios do not occur simultaneously.

To correct this conservatism, the values of $$ \gamma_{\mathrm{t}, \text { top }} $$ in Table 1 can be used as a weighting factor on the value of the enforced translational displacement $$ \psi_{I ; I I, \mathrm{n}} $$ at the top of the building calculated by Eq. (3). Therefore, the following reduction factor $$ S_u $$ is proposed to apply totally on the floor displacement profiles along the $$ I_{\mathrm{sec}} $$ and $$ II_{\mathrm{sec}} $$ axes^[40]:

(4) $$ S_u=\frac{\gamma_{\mathrm{t}, \mathrm{top}} \cdot H_{\mathrm{n}}}{\psi_{\mathrm{n}}} $$

Where $$ H_{\mathrm{n}} $$ is the total height of the building, $$ \gamma_{\mathrm{t}, \text { top }} $$ is taken from Table 1 and $$ \psi_{\mathrm{n}} $$ is calculated by Eq. (3). The use of the reduction factor $$ S_u $$ is proposed for multi-storey buildings with four or more floors. The decrease shown on the floor displacement profiles by this factor is about 7-15%, where the largest values apply to high-rise buildings.

3. Numerical Example of a 6-storey Dual Building

3.1 Building characteristics

The six-storey r/c building shown in Figure 3 is examined to verify the proposed pushover procedure on torsionally flexible multi-storey buildings. The construction materials are concrete C25/30 and steel B500c with mean strengths f_cm = 33 MPa and f_ym = 550 MPa, respectively. The building has a polygonal plan shape consisting of perimetric and interior frames and of perimetric walls. A 15 cm thick rigid floor-diaphragm is extended outside the perimeter of the floor plan forming a continuous cantilever 1 m wide. The cross-sections of all the structural elements do not change in elevation. Εach floor has 3.5 m height, giving a total building height of 21 m. The six-storey building is regular in elevation but asymmetric in-plan, mainly due to the placement of walls at the building perimeter (two walls along x and y directions and a skew one). The x and y walls have rectangular cross-sections of dimensions 1.5/0.3 m and 0.3/1.8 m while the skew one has a rectangular section 1.5/0.3 m having a barbell at one end of dimension 0.5/0.5 m. All the columns are square of dimensions 0.5/0.5 m. The beams have a T-section of dimensions 30/60/150/15, where the effective flange width is considered different in linear and nonlinear analysis (1.5 m in nonlinear analysis and according to EN 1998-1 and EN 1992-1^[54] in linear analysis). The mean gravitational distributed load on all floors -including self-weights and masonry- (dead G + live Q) is about 12 kN/m², giving a translational mass of 450 tn and a mass moment of inertia about vertical axis of 34,000 tn·m² for each floor. The latter is artificially increased to 60,000 tn·m² in each floor, in order to have a torsionally flexible building (on limit).

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Figure 3. Nonlinear model of the six-storey r/c building.

3.2 Building design

The building is designed according to the provisions of EN 1992-1 and EN 1998-1 for Ductility Class High (DCH). Modal response spectrum analysis was performed in the linear building model with PGA=0.24 g (where g is the gravity acceleration), soil category D and total behavior factor q =4.

All the structural members of the linear model have been provided with their effective flexural and shear stiffness, equal to one-half of their respective geometric stiffness. The building is classified into the structural type of dual buildings, equivalent to frame buildings according to EN1998-1, along both the horizontal ideal principal axes $$ I_{\mathrm{des}} $$ and $$ I I_{\mathrm{des}} $$ . The translational uncoupled periods of the linear model are 1.15 sec along the $$ I I_{\mathrm{des}} $$ axis and 1.05 sec along the $$ I_{\mathrm{des}} $$ axis. The horizontal ideal principal axes $$ I_{\mathrm{des}} $$ and $$ I I_{\mathrm{des}} $$ of the linear model are rotated relative to the x,y-axes by 14.06^o counterclockwise. The two static eccentricities $$ e_{\mathrm{I}, \mathrm{des}} $$ and $$ e_{\mathrm{II}, \mathrm{des}} $$ along the $$ I_{\mathrm{des}} $$ and $$ I I_{\mathrm{des}} $$ axes (distance between $$ \mathrm{CR}_{\text {des }} $$ and CM in the floor-diaphragm closest to the $$ 0.8 H_{\text {tot }} $$ level (fifth floor) from the base) are about equal to $$ 0.06 \cdot L_{I, \text { des }} $$ and $$ -0.13 \cdot L_{I I, \text { des }} $$ respectively, where $$ L_{\mathrm{I}, \mathrm{des}} $$ and $$ L_{\mathrm{II}, \text { des }} $$ are the maximum plan dimensions along the two horizontal principal axes. Additionally, the building is characterized as torsionally flexible since the (mean) normalized torsional radii $$ r_{\mathrm{II}, \mathrm{des}} / r_{\mathrm{m}}=0.96 $$ is less than 1 (the other ratio $$ r_{\mathrm{I}, \mathrm{des}} / r_{\mathrm{m}} $$ is equal to 1.06), where $$ r_{\mathrm{m}}=11.55 \mathrm{~m} $$ is the radius of gyration of the floor-mass at the fifth floor. The as designed building has appropriate longitudinal/confinement steel reinforcement details that provide an overall high ductile behavior, spreading the ductility demands to the end-sections of all beams and to the base end-sections of all columns/walls (beam-sway mechanism). The numbering of the structural members in the mathematical model of the building, the dimensions of the cross-sections and the reinforcement details are presented in Figure S1 and in Table S1, Table S2, and Table S3.

3.3 Non-linear model of the building

Section analysis is performed by the module Section Designer of the FEM program SAP2000^[55] which is used as the analysis tool. Having the Μ-φ curves along the local axes of the end-sections of all the structural members, as well as their shear span $$ L_{\mathrm{v}} $$ and the plastic hinge length $$ L_{\mathrm{pl}} $$ by the Eq. (A9) of EN 1998-3 (with γ_el coefficient equal to 1.7), we can obtain the respective Μ-θ curves and subsequently calculate the secant stiffness at yield $$ E I_{\text {sec }} $$ of all the structural members of the nonlinear building model (Eq. (1)). The mean values in elevation of the ratio $$ E I_{\mathrm{sec}} / E I_{\mathrm{g}} $$ for each type of structural member of the nonlinear model are shown in Table 4, where $$ E I_{\mathrm{g}} $$ is the geometric stiffness. These values vary along the two local axes of columns and walls (2 and 3 axes) due to the different shear span $$ L_{\mathrm{v}} $$ in four columns with cantilever bending mode and due to the different strength $$ M_{\mathrm{y}} $$ and shear span $$ L_{\mathrm{v}} $$ along the strong (axis 3) and weak (axis 2) directions of walls. We can observe the low values of the members’ bending stiffness, especially those of beams. These $$ E I_{\text {sec }} $$ values are assigned to all the structural members of the nonlinear building model.

The “Capable Near Collapse Principal System, $$ I I I_{\mathrm{sec}}\left(\mathrm{CR}_{\mathrm{sec}}\right), I_{\mathrm{sec}}, I I_{\mathrm{sec}} $$ ” of the six-story r/c building is next determined according to Section 2, at the floor-diaphragm closest to the level $$ 0.8 H_{\text {tot }}=16.8 \mathrm{~m} $$ from the building base, i.e., at the fifth floor with height equal to 17.5 m from the ground. Figure 3 shows the in-plan location of the inelastic center of stiffness $$ \mathrm{CR}_{\mathrm{sec}} $$ and the orientation of the horizontal ideal inelastic principal axes $$ I_{\mathrm{sec}}, I I_{\mathrm{sec}} $$ in the nonlinear building model.

The latter are turned relative to the x, y-axes by 11.5^o counterclockwise. The inelastic static eccentricities are equal to $$ e_{\mathrm{R}, \mathrm{Isec}}=2.3 \mathrm{~m} $$ and $$ e_{\mathrm{R}, \text { IIsec }}=2.63 \mathrm{~m} $$ and their normalized values are equal to $$ e_{\mathrm{R}, \mathrm{Isec}} / L_{\mathrm{Isec}}=0.09 $$ and $$ e_{\mathrm{R}, \mathrm{IIsec}} / L_{\mathrm{IIsec}}=-0.13 $$ , where $$ L_{\mathrm{I}, \mathrm{des}} $$ and $$ L_{\mathrm{II}, \text { des }} $$ are the maximum plan dimensions along the axes $$ I_{\mathrm{sec}} $$ and $$ II_{\mathrm{sec}} $$ , respectively. The (mean) normalized inelastic torsional radii are equal to $$ r_{\mathrm{II}, \mathrm{sec}} / r_{\mathrm{m}}=0.95 $$ and $$ r_{\mathrm{I}, \mathrm{sec}} / r_{\mathrm{m}}=1.06 $$ , where $$ r_{\mathrm{m}}=11.55 \mathrm{~m} $$ is the radius of gyration of the floor-mass at the fifth floor. Therefore, the nonlinear building model is characterized as torsionally flexible according to Eq. (2), because the smaller of the two torsional radii ratios (0.95) is lower than 1.10. We come to the same conclusion by examining the uncoupled fundamental modes of the non-linear model, with periods of $$ T_{\text {IIsec }}=2.23 \mathrm{~sec} $$ , $$ T_{\text {Isec }}=2.00 \mathrm{~sec} $$ for the translational ones and $$ T_{\mathrm{R}, \mathrm{IIIsec}}=2.11 \mathrm{~sec} $$ for the torsional one. In Table 5, the periods and the modal participating mass ratios of the first nine mode-shapes of the nonlinear model of the building are presented. Additionally, the building is classified according to EN1998-1 into the structural type of dual buildings, equivalent to wall buildings along the horizontal axis $$ I_{\mathrm{sec}} $$ and to frame buildings along the other horizontal axis $$ II_{\mathrm{sec}} $$ . This difference in structural classification along the $$ I_{\mathrm{sec}} $$ between the nonlinear (dual equivalent to wall buildings) and linear (dual equivalent to frame buildings) building models is mainly due to the greater reduction of the lateral stiffness of the frames relative to that of the walls because of the consideration of the secant stiffness at yield ( Table 4).

Finally, point hinges of type M₃ (pure bending) and P - M₂ - M₃ (biaxial bending with axial force) are inserted at the end-sections of each beam and column/wall of the nonlinear model, respectively, with constitutive laws according to Mander et al.^[56] for the unconfined/confined concrete and according to Park and Paulay^[57] for the steel reinforcement. The backbone curves of the response of the fiber hinges at the end-sections of the structural elements (determined from the abovementioned constitutive laws) and the Takeda and the Kinematic hysteresis model for the concrete and the steel rebars respectively are embedded in the SAP2000 FEM program used for the nonlinear analysis. Additional information can be found in SAP2000 analysis manual.

It is noted that accidental eccentricity was not considered in nonlinear analysis. As was observed in the extended parametric analysis of regular in elevation, ductile, asymmetric, r/c multi-storey buildings implemented in the doctoral dissertation of the first author^[40], the value of the accidental eccentricity proposed by the seismic codes has only a very minor impact on the nonlinear response in the deep nonlinear area (Near Collapse).

3.4 Seismic demand

Τhe inertial characteristics of the nonlinear model of the six-storey building are concentrated at the geometric centre of each i -floor $$ \left(m_i=450 \mathrm{tn}, J_{m, i}=60000 \mathrm{tn} \cdot \mathrm{~m}^2\right) $$ . Figure 4 shows the 14 pairs of uncorrelated artificial accelerograms -formed by 11 unit-normalized accelerograms of 25 sec duration- which are used for the nonlinear response history analysis (N-LRHA) of the six-storey building, to provide the seismic demand. The artificial accelerograms were constructed by SeismoArtif^[58] aiming to fit as close as possible (±10%) the unit PGA, 5% damped, design elastic acceleration spectrum of EN 1998-1 for soil category D ( Figure 5). The significant (strong motion) duration of the accelerograms is about 14-18 sec and the Arias Intensity is 28 m/s on average, characteristics like those of the Hellenic tectonic faults^[59]. The characteristics of the 11 unit-normalized artificial accelerograms are shown in Table 6, as reported by SeismoArtif. Next, the two seismic components of each pair are scaled to PGA = 0.4 g during analysis in order the building to attain the Near Collapse state. Additionally, each pair is rotated successively per 15° in order to find the worst seismic load scenario^[47] by the 24 analyses obtained for each pair. Also, the four sign (±) combinations of the two seismic components of each pair were examined. The envelope of the response of 14 x 24 x 4 = 1344 N-LRHA is considered as the seismic demand, at every control location, to compare with the response effects of the proposed pushover procedure at the Near Collapse (NC) state. The average response from the N-LRHA is also recorded.

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Figure 4. 14 pairs of unit-normalized artificial accelerograms.

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Figure 5. Mean spectrum of artificial accelerograms vs EN 1998-1 design elastic spectrum ( ξ =0.05, PGA=0.4 g, soil D ).

3.5 Proposed interstorey drift ratios and floor enforced-rotations

For the implementation of the proposed procedure of pushover analysis, the initially proposed interstorey drift ratios $$ \gamma_{I, \mathrm{i}} $$ , $$ \gamma_{I I, \mathrm{i}} $$ at the in-plan location of the vertical axis $$ I I I_{\mathrm{sec}} $$ are obtained from Figure 1 for dual buildings along both the axes $$ I_{\mathrm{sec}} $$ and $$ II_{\mathrm{sec}} $$ . Also, the proposed floor enforced-rotations $$ \psi_{R, I I I, \mathrm{i}} $$ are obtained from Figure 2 for dual torsional flexible buildings along both the axes $$ I_{\mathrm{sec}} $$ and $$ II_{\mathrm{sec}} $$ . These proposed values of enforced-displacements are presented in Figure 6. The initial values of the proposed interstorey drift ratios will be reviewed through the methodology described in the next step.

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Figure 6. Proposed values of (a) interstorey drift ratios $$ \gamma_{\mathrm{I} ; \mathrm{II}} $$ (rad) and (b) floor enforced-rotations $$ \psi_{R, I I I} $$ (rad).

3.6 Two temporary sets of pushover analysis to calculate the final proposed floor enforced translations

Two temporary sets of pushover analysis (with forces) are performed to find revised values of interstorey drift ratios $$ \gamma_{I, \mathrm{i}} $$ , $$ \gamma_{I I, \mathrm{i}} $$ . The pattern of floor lateral forces in the two sets, which are applied at the in-plan location of the vertical axis $$ I I I_{\mathrm{sec}} $$ along the (±) directions of the horizontal $$ I_{\mathrm{sec}} $$ and $$ II_{\mathrm{sec}} $$ axes, is depicted in Figure 7.

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Figure 7. Pattern of floor lateral static loads on the in-plan location of the vertical $$ I I I_{\mathrm{sec}} $$ axis and along the horizontal $$ I_{\mathrm{sec}} $$ and $$ II_{\mathrm{sec}} $$ axes: (a) first temporary set of (4) pushover analyses; (b) second temporary set of (4) pushover analyses.

These floor forces correspond to a unit base shear that is distributed in elevation according to the translational building modes along the $$ I_{\mathrm{sec}} $$ and $$ II_{\mathrm{sec}} $$ axes in both sets but with an additional top force in the second set equal to 20% of the base shear. The target displacement at the top of the vertical axis $$ I I I_{\mathrm{sec}} $$ , along both the $$ I_{\mathrm{sec}} $$ and $$ II_{\mathrm{sec}} $$ axes, is calculated directly from Table 1 by double linear interpolation for multi-storey buildings with six floors and with a dual structural system: $$ u_{\mathrm{t}}=\gamma_{\mathrm{t} \text { top }} \cdot H_{\text {tot }}=\frac{\left(0.0235-\frac{0.0235-0.0205}{5}\right)+\left(0.026-\frac{0.026-0.024}{5}\right)}{2} \cdot 21.00=0.0245 \cdot 21.00=0.515 \mathrm{~m} $$

It is noted that the envelope of the displacement results from N-LRHA provides a displacement at the top of the vertical axis $$ I I I_{\mathrm{sec}} $$ equal to 0.574 m along both the $$ I_{\mathrm{sec}} $$ and $$ II_{\mathrm{sec}} $ axes, which is greater from the previous one (the mean displacements from N-LRHA at the same point are about 0.40 m along both the horizontal principal directions). Additionally, by performing the methodology of EN 1998-1 Informational Annex B with the NC seismic action of PGA = 0.4 g we get 0.54 m and 0.60 m for the corresponding displacements along the $$ I_{\mathrm{sec}} $$ and $$ II_{\mathrm{sec}} $$ axes.

The envelope of the interstorey drift ratios $$ \gamma_{I, \mathrm{i}} $$ , $$ \gamma_{I I, \mathrm{i}} $$ resulted from the eight total pushovers of the two temporary sets is shown in the 2^nd and 3^rd column of Table 7. Τhe average of these values with the corresponding ones initially proposed by Figure 1 (depicted in Figure 6 and shown in the 4^th column of Table 7 ) are considered as the final interstorey drift ratios (5^th and 6^th column of Table 7 ) for the calculation of the floor enforced translational displacements $$ \psi_{I, \mathrm{i}} $$ , $$ \psi_{I I, \mathrm{i}} $$ using Eq. 3. The three enforced displacements of the proposed pushover procedure, which act at the location of the vertical $$ I I I_{\mathrm{sec}} $$ axis inside each floor-diaphragm, are shown at the last three columns of Table 7.

3.7 Proposed pushover procedure for the estimation of the seismic demand. Verification

The sixteen pushovers of Table 2 and Table 3 are performed with target displacement along the main loading directions $$ I_{\mathrm{sec}} $$ and $$ II_{\mathrm{sec}} $$ equal to $$ \psi_{l, 6}=0.556 \mathrm{~m} $$ and $$ \psi_{I I, 6}=0.553 \mathrm{~m} $$ ( Table 7). The envelope of the response from the 16 pushover analyses of the proposed pushover procedure can be considered as an estimation of the seismic demand.

The effectiveness of the proposed pushover procedure on the prediction of seismic demand will be verified against the results of N-LRHA, which is the benchmark method. Additionally, the response effects by the proposed pushover procedure compare with those resulted by the N2^[3,60] and Extended N2 pushover procedures^[21]. In the latter procedure, the response effects resulted by the conventional N2 pushover adopted by EN 1998-1 are corrected, in-plan and in elevation, using corrections factors determined from the results of a 3D modal response spectrum analysis. It is noted that, in the framework of the N2 pushover the lateral forces are applied on the location of CM inside each floor (without accidental eccentricity) along the horizontal ideal principal axes $$ I_{\mathrm{sec}} $$ and $$ II_{\mathrm{sec}} $$ of the six-storey building. The target displacement (at CM of the top floor) was considered equal to that resulted from Ν-LRHA.

The patterns of interstorey drift ratios in elevation, at the flexible and stiff sides of the building as well as at the location of CM and that of the vertical $$ I I I_{\mathrm{sec}} $$ axis ( $$ \mathrm{CR}_{\mathrm{sec}} $$ ) inside each floor, are presented in Figure 8. The (%) errors committed on the estimation of the seismic demand (N-LRHA) by the proposed and the Extended N2 pushovers are recorded in Table 8, where the (–) sign indicates unconservative estimates. It is obvious from Figure 8 that the N2 pushover procedure seriously underestimates the interstorey drift ratios at the stiff sides of the building, all over its height. It also underestimates the interstorey drift ratios at any location in the upper half of the building.

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Figure 8. Interstorey drift ratios: Proposed pushover procedure vs Ν-LRHA and Extended N2 pushover.

On the contrary, N2 pushover overestimates in general the interstorey drift ratios at any location (except the stiff sides) in the lower half of the building (2^nd and 3^rd floor). We can also see from Figure 8 and Table 8 that the seismic interstorey drift ratios at the flexible and stiff sides of the six-storey building are safely or marginally estimated by the proposed pushover procedure. These estimates are more balanced than those of the Extended N2 pushover, which provides in general more conservative estimates but also some unconservative ones along the $$ II_{\mathrm{sec}} $$ axis at the first floor. At the in-plan location of $$ \mathrm{CR}_{\mathrm{sec}} $$ , the seismic interstorey drift ratios are in general slightly underestimated by the proposed procedure due to the smaller value of the target (enforced) displacements used in pushover analysis compared to the corresponding Ν-LRHA displacements. But at the in-plan location of CM the interstorey drift ratios are in general marginally estimated.

According to section 2, the floor displacements computed by the proposed pushover procedure must be totally reduced by the factor $$ S_u $$ of Eq. (4): $$ S_u=\frac{\gamma_{\mathrm{t}, \mathrm{top}} \cdot H_{\mathrm{n}}}{\psi_6}=\frac{0.0245 \cdot 21.00}{0.553}=\frac{0.515}{0.553}=0.931 $$

Where $$ \gamma_{\mathrm{t}, \mathrm{top}} \cdot H_{\mathrm{n}}=0.515 \mathrm{~m} $$ is the proposed top target displacement of the temporary pushovers from Table 1 previously calculated and $$ \psi_{\mathrm{I} ; \mathrm{II}, 6}=0.556 \text { or } 0.553 \mathrm{~m} $$ is the enforced translational displacement of the building top ( Table 7). In Figure 9, the floor displacements profiles resulted by the proposed pushover procedure compare with the corresponding ones by N-LRHA and Extended N2 pushover.

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Figure 9. Floor displacement profiles: Proposed pushover procedure vs Ν-LRHA and Extended N2 pushover.

The (%) errors committed on the estimation of the seismic demand are recorded in Table 9. We observe that, at the in-plan location of the vertical $$ I I I_{\mathrm{sec}} $$ axis ( $$ \mathrm{CR}_{\mathrm{sec}} $$ ) the floor displacements are slightly underestimated by the proposed procedure. This is due to the smaller value of the target (enforced) displacements used in pushover analysis compared to the corresponding Ν-LRHA displacements and due to the above reduction by the factor $$ S_u $$ . But at the in-plan location of CM, the floor displacements are in general marginally estimated. Slightly conservative or marginal estimates of the floor displacements from the proposed pushover procedure are observed on the stiff and flexible sides of the six-storey building. The Extended N2 pushover provides in general more conservative estimates of the floor displacements on the flexible sides of the building but also some unconservative ones on the stiff side of the first floor of the building along the $$ II_{\mathrm{sec}} $$ axis.

The capacity curves of the sixteen pushovers with enforced-displacements are shown in Figure 10. The slope of the initial (elastic) branch and the ultimate and yield values of the base shear and top displacement of the building are different in each curve, depending mainly on the (±) action of the floor enforced-rotations. These curves are first bi-linearized and then can be used to connect the seismic capacity (pushover results) with the seismic demand (earthquake action) by using the informational Annex B of EN 1998-1 or any other acceptable method. By knowing the pattern of floor enforced translational displacements $$ \psi_{I ; I I} $$ in elevation and the corresponding target displacement on the building top, the backwards application of EN1998-1 Annex B gives the maximum (capable) earthquake in PGA terms that the building can attain (NC state). In our case, this PGA is equal to about 0.40 g, which is the same value used in Ν-LRHA.

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Figure 10. The 16 capacity curves of the proposed pushover procedure.

Finally, the plastic mechanism of the six-storey building (which is of beam-type) is safely predicted by the envelope of the sixteen pushovers of the proposed procedure. As regards the plastic chord rotations developed at the end-sections of the structural elements, they are fully compatible with the computed interstorey drift ratios which have been in general safely estimated.

4. Conclusions

An asymmetric, torsional flexible, regular in elevation, ductile, six-storey r/c building was seismically accessed here by a recently proposed pushover procedure with floor enforced-displacements to verify its effectiveness.

According to the proposed pushover procedure, a pattern of seismic floor-enforced displacements (two translational displacements and a rotational one) properly combined with each other to consider the spatial seismic action is applied and a set of sixteen pushovers is performed. The envelope of the sixteen pushovers provides a safe estimate of the seismic demand at the Near Collapse (NC) state of the building. Appropriate seismic, drift-based, floor enforced-displacements are proposed that lead the building at the attainment of the NC state. The proposed values of the floor translational displacements are further revised through two temporary sets of pushovers. All analyses in the framework of the proposed pushover procedure are implemented with respect to an ideal, inelastic, 3D principal reference system $$ \mathrm{CR}_{\mathrm{sec}}\left(I_{\mathrm{sec}}, I I_{\mathrm{sec}}, I I I_{\mathrm{sec}}\right) $$ , which is called “Capable Near Collapse Principal System of the multi-storey r/c building”. This system is determined by applying the torsional optimum axis methodology to the nonlinear model of the building in which all structural members have been supplied with the secant stiffness at yield $$ E I_{\text {sec }} $$ . Additionally, a new criterion is used for the verification of the torsional sensitivity of the building in the non-linear area.

To verify the effectiveness of the proposed procedure on the prediction of the seismic demand, nonlinear response history analysis (N-LRHA) with 14 pairs of artificial accelerograms has been performed to provide the seismic demand at the attainment of the NC state of the six-storey r/c building. From the comparison of the response computed by the proposed pushover procedure and by the (envelope of) N-LRHA, the main conclusions are:

The seismic interstorey drift-ratios on the stiff and flexible sides of the building are safely estimated by the proposed procedure. In other locations, such as those of CM and $$ \mathrm{CR}_{\mathrm{sec}} $$ , the interstorey drift ratios are marginally estimated or slightly underestimated. Hence, the main objective of the proposed procedure, which is the complete control of the distribution -in plan and in elevation- of the seismic structural damage at the NC state, is fully achieved.

The seismic floor displacement profiles on the stiff and flexible sides of the building are safely estimated by the proposed procedure. In other locations, such as those of CM and $$ \mathrm{CR}_{\mathrm{sec}} $$ , the floor displacements are marginally estimated or slightly underestimated.

The plastic chord rotations at the end-sections of the structural members are also safely estimated in general by the proposed procedure, following the safe estimation of the seismic interstorey drift ratios.

The plastic mechanism of the building at the NC state is fully captured by the envelope of the sixteen pushover analyses in the framework of the proposed procedure.

Therefore, the proposed enforced-displacement pushover procedure is a simple and effective tool for the seismic assessment of asymmetric, torsional flexible, regular in elevation, ductile multistorey r/c buildings. The effectiveness of the proposed pushover procedure with enforced displacements has been fully evaluated in the first author’s doctoral dissertation^[40] by examining various regular in elevation, ductile, multistorey r/c buildings with varying number of floors, structural type, static eccentricity and torsional sensitivity. In other words, the proposed procedure can be successfully used to seismically assess any type of asymmetric multistory r/c building, torsional flexible or not, provided that are regular in elevation and develop a beam-type plastic mechanism (and not a floor-type one).

Conflicts of Interest

Triantafyllos K. Makarios is an Editorial Board member of JBDE, and other authors declare no conflict of interest.

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© The Author(s) 2023.