3.2.1. Regularity in plan and elevation

The criteria for regularity in plan were verified according to EN 1998–1:2004[2]. The slenderness of the building corresponds to λ = 1,0, since the building is square in the horizontal plane, and therefore λ is smaller than 4,0. The ratio between the torsional radius and the radius of gyration of the floor mass at each storey level in both horizontal directions (r_i/l_s) ranges between 1,18 and 1,25. Since this ratio surpasses 1,0, the corresponding criterium is satisfied. Additionally, the buildings possess double symmetry, ensuring that the centre of gravity aligns with the centre of rigidity. This results in an eccentricity (e₀) of zero in the orthogonal directions. Consequently, the specified criterion of e₀ ≤ 0,3∙r_i is also fulfilled. Since these three conditions are met, the buildings are categorised as regular in plan in both directions.

The buildings also meet the qualitative criteria to be considered regular in elevation according to EN 1998–1:2004[2]. The prEN 1998–1-1:2023[3] refers to a quantitative criterion preconizing that “both the lateral stiffness and the mass of the individual storeys remain constant or decrease gradually by no more than 20% relative to the storey below, without abrupt changes, from the base to at least one storey below the top storey”. Since interstorey stiffness and mass of the individual storeys decrease gradually by no more than 10%, the buildings are classified as regular in elevation.

To categorize a building as having "torsional flexibility", the verification involves the comparison of two uncoupled vibration frequencies: ω_θ/ω_x, and ω_θ/ω_y. Here, ω_x and ω_y denotes the uncoupled angular natural frequencies of the system in the x and y directions, respectively, while ω_θ represents the uncoupled circular natural frequency of torsional vibration. If the ratios ω_θ/ω_x, and ω_θ/ω_y exceed 1,0, the response primarily exhibits translational characteristics, classifying the structure as non-torsionally flexible. Conversely, if ω_θ/ω_x, and ω_θ/ω_y are below 1, the response is predominantly influenced by torsional behaviour. In the current study, the angular frequency ratios ω_θ/ω_x, and ω_θ/ω_y exceed 1,0, thus categorizing the structures under consideration as non-torsionally flexible.

3.2.2. Behaviour factor applied in the analysis

To build up the design spectra for elastic analysis, and regarding the structural systems in question, the behaviour factor q and its components are defined per EN 1998–1:2004[2] and prEN 1998–1-2:2023[3] for structures regular in plan and elevation.

The present work considers the analysis of the DCM ductility class, which is preconised in EN 1998–1:2004[2]. For the structures of interest for the current work, i.e., multistorey and multi-bay regular frame structures, the behaviour factor adopted is the basic value of the behaviour factor, defined by expression (1):(1)$q_0=3,0\bullet \frac{{\alpha }_u}{{\alpha }_1}$where α₁ is the value by which the horizontal seismic design action is multiplied in order to first reach the flexural resistance in any member in the structure, while all other design actions remain constant, and α_u is the value by which the horizontal seismic design action is multiplied, in order to form plastic hinges in a number of sections sufficient for the development of overall structural instability, while all other design actions remain constant. For multistorey, multi-bay frames the ratio∙α_u/α₁ takes the value of 1,3, therefore q₀= 3,9. The upper limit value of the behaviour factor q is expressed as depicted in expression (2):(2)$q=q_0\bullet k_w\ge 1,5$where k_w is the factor reflecting the prevailing failure mode in structural systems with walls. The parameter k_w takes the value of 1,0 for frame systems. Considering the aforementioned, the adopted behaviour factor (q) assumes a value of 3,9.

The new generation of Eurocode 8, prEN 1998–1-2:2023[3], introduces a behaviour factor composed by the product of the partial behaviour factor components q_S, q_R, and q_D. The parameter q_R is the behaviour factor component accounting for overstrength due to the redistribution of seismic action effects in redundant structures, q_S is the behaviour factor component accounting for overstrength due to all other sources, and q_D is the behaviour factor component accounting for the deformation and energy dissipation capacity.

3.2.3. Response spectra

Table 7 refers to the maximum response spectral acceleration at the constant acceleration range (S_α) and the different values for behaviour factor q. For reliability differentiation, EN 1990:2002[11] establishes consequences classes (CC), which define quantitatively the consequences of failure or malfunction. A Consequence Class CC2 is assumed, defined as having medium consequences for the loss of human life, with considerable impacts on economic, social, and environmental aspects. For the consequence class CC2 the design spectrum does not require amplification related to reliability differentiation.

3.3.1. Control of second-order effects

The second-order effects (P-Δ), which are associated with ULS effects, are specified through an interstorey drift sensitivity coefficient (θ) as shown in Eq. (3) based on EN 1998–1:2004[2]:(3)$\theta =\frac{P_{\mathit{tot}}}{V_{\mathit{tot}}}\bullet \frac{d_r}{h}$

According to prEN 1998–1-2:2023[3], second-order effects (P-Δ) are associated with SD limit states and are defined by Eq. (4):(4)$\theta =\frac{P_{\mathit{tot}}}{q_R\bullet q_S\bullet V_{\mathit{tot}}}\bullet \frac{d_{r,\mathit{SD}}}{h_s}$where P_tot and V_tot are the total cumulative gravity load and seismic shear, respectively, at the storey under consideration; h_s and h are the storey height.

EN 1998–1[2] and prEN 1998–1-2[3] both specify that when the interstorey drift sensitivity coefficient (θ) has a value between 0,1 and 0,2, an approximate adjustment for P-Δ effects can be made by increasing the seismic forces used in the analysis by a factor of 1/(1-θ). On the other hand, when θ is between 0,2 and 0,3, a second-order analysis is necessary.

3.3.2. Limitation of the interstorey drift ISD

In a current design with a force-based approach, the interstorey drifts are the product of elastic interstorey drift from analysis and the behaviour factor q. According to EN 1998–1:2004[2], the interstorey drift verification is included regarding the serviceability condition as damage-limitation (DL). The prEN 1998–1-2:2023[3] considers two limit states for the verification, the damage limitation (DL) and the significant damage limit state (SD).

Since the seismic analysis in a force-based approach is linear-elastic and based on the design response spectrum, the elastic spectrum with 5% damping divided by the behaviour factor q is used. The values of the displacements (d_s) to be used in the verification are calculated from the analysis (d_e) and multiplied by the behaviour factor q, as shown in Eq. (5):(5)$d_s=q\bullet d_e$

The design interstorey drift d_r, is calculated as the difference of the average lateral displacements’ d_s induced by the reduced seismic action at the top and bottom of the storey under consideration as depicted in Eq. (6):(6)$d_{r,i,}=d_{s,i+1}-d_{s,i}$

The prEN 1998–1-2:2023[3] presents a similar formulation, where λ_NS is the coefficient accounting for the sensitivity of ancillary elements to interstorey drift, the subscript LS refers to limit state and h_s has the same meaning as h.(8)$\frac{d_{r,\mathit{LS}}}{h_s}={\lambda }_{\mathit{NS}}$