The loading protocol required by the cyclic pushover analysis has to be properly selected in order to simulate the load and deformation histories that the structure undergoes during an earthquake. Unfortunately, no loading protocol is able to represent exactly the deformation histories as experienced by the structural members in earthquakes. Indeed, the damage actually cumulated during an earthquake by the structure depends on different aspects, such as the intensity and frequency content (magnitude, distance, and soil type dependence) of the ground motion, the configuration, strength, stiffness, and modal properties of the structure or the deterioration characteristics of the structural systems and its components. Based on these considerations, to come up with a compromise loading history which is conservative but statistically representative of ground motions and structural configurations, several loading protocols have been proposed in the scientific literature (e.g. Clark et al. [26], Krawinkler et al. [27]), or included in codes (e.g. ATC-24 [24], AISC [25], ICBO-ES [28], ASTM [29], FEMA [30]).

In this paper, two loading protocols are considered: the ATC-24 loading protocol [24] and the SAC loading protocol [25], which are depicted in Figs. (1a and b), respectively. The ATC-24 loading protocol was specifically proposed for the evaluation of seismic performance of components of steel structures. It assumes the yielding displacement D_y as the reference parameter for increasing the amplitude of cycles. The loading history is composed of at least six elastic cycles (i.e. with amplitude lower than D_y), followed by three cycles each of amplitude D_y, 2D_y and 3D_y, followed by pairs of cycles whose amplitude increases in steps of D_y until severe cyclic deterioration occurs.

The SAC loading protocol was developed based on a statistical study [27] performed on the number and amplitudes of storey drift cycles of two steel moment frames. This protocol adopts the storey drift angle rather than yielding deformation as the amplitude control parameter. Compared to the ATC-24 protocol, the SAC protocol contains smaller (elastic) cycles, two cycles of an intermediate amplitude, but slightly fewer cycles of larger amplitude. In particular, it is composed of three initial groups of cycles, which push the structure up to a storey drift angle equal to 0.0025 rad, 0.0050 rad and 0.0075 rad, respectively. Six loading cycles are included in each of those groups. The following four loading cycles lead the structure to a storey drift angle θ equal to 0.01 rad. The protocol concludes with further loading cycles that increase the storey drift angle attained in the previous loading cycle of 0.005 rad until failure.

The base shear V_b versus top displacement D_t relationship obtained by the CPA is cyclic. To determine a single pushover curve that can be compared to that obtained by the monotonic pushover analysis, the cyclic response needs to be enveloped. To this end, the so-called backbone curve is determined according to the three following steps: (1) evaluation of the top displacement corresponding to the first yielding of the structure; (2) determination of the elastic branch; (3) determination of the plastic branch. In order to evaluate the yielding displacement D_y (step 1), the performance curve in terms of base shear and top displacement of the considered structure is required. To this end, a monotonic pushover analysis is run at first. In this paper, it was conducted by a distribution of lateral forces that is proportional to the first mode of vibration. Given the performance curve, the tangent stiffness is evaluated at each step. The top displacement corresponding to a tangent stiffness lower than 80% of the initial elastic stiffness is assumed as the yielding displacement D_y. As long as the structure is pushed towards a top displacement lower than D_y, the structure remains elastic. Thus, the elastic branch of the backbone curve (step 2) retraces the cyclic response. Since in a loading protocol several cycles are performed at a given amplitude of the displacement, it is necessary to identify for each considered amplitude of displacement D_i the first loading branch and the first unloading branch in order to evaluate the following points of the backbone curve (step 3).

Fig. (2). (a) SAC Loading protocol (b) Cyclic response and determination of the generic point of the backbone curve.

The points of the backbone curve following the yielding (step 3) are determined as the intersection between the first unloading branch of the cycles at amplitude D_i-1 and the first loading branch of the cycles that go to a level of displacement D_i larger than D_i-1.

To clarify the procedure, Fig. (2) shows an example of the construction of the backbone curve. The SAC loading protocol applied in the CPA is shown in Fig. (2a) and the cyclic response is plotted in Fig. (2b) with the grey line. The first four groups of loading cycles of this loading protocol push the structure to top displacements lower than D_y, which is indicated by the dashed grey line in Fig. (2a). So far, the backbone curve (black line) follows the cyclic response (grey line), as shown in Fig. (2a). The following group of cycles pushes the structure to top displacements larger than D_y. For the sake of clarity, in this example, the top displacement equal to 0.594 m is considered. The red line in Fig. (2a) indicates the first unloading branch of this group of loading cycles. The cyclic response obtained during this unloading branch is plotted in red in Fig. (2b). The blue line in Fig. (2a), instead, identifies the first loading branch of the group of loading cycle that pushes the structure to a top displacement larger than that attained by the previous cycle (0.594 m), in this case, 0.693 m. The corresponding cyclic response is plotted in blue in Fig. (2b). The intersection between the unloading branch of the previous loading cycle (red line) and the loading branch of the following loading cycle (blue line) is the point of the backbone curve and is indicated by the grey dot. With this procedure, all the points of the backbone curve following the first yielding are obtained in Fig. (2b).

The considered frame is three span wide and three storey high and it is the outermost one in the transversal direction (Fig. 3a). Beams and columns of the case study frames are modelled as beam with hinges elements. The middle part of the element is elastic, while the inelasticity is concentrated in the plastic hinge segments located at the two ends. Within the plastic hinge length, the cross section is subdivided into fibres.

Because of the adopted design procedure, a significant inelastic behaviour is expected in beams while a moderate inelastic behaviour is expected in columns. For this reason, the degradation of stiffness and strength due to nonlinear cyclic behaviour is considered only for beams. Further, as reported in [35], the I-shaped cross section, commonly used for the MRF beams, starts deteriorating at the location of maximum bending moment due to the initiation of flange local buckling in the compression zone, which may also induce web deformation. Despite the interaction between flange buckling and web deformation, however the formation of local plastic mechanism at the level of flange in compression is a complex phenomenon and it leads to the general shape of the plastic mechanism. Instead, the local plastic mechanism for the web is sometimes partially formed. To simulate the degradation of stiffness and strength in beams due to local buckling of I-shape flanges, the numerical model proposed by Bosco and Tirca [23] is assigned to flange fibres of beam.

According to this model, each flange of the I-shape cross section is divided into n_f,s segments and n_f,l layers (30 x 4), while the web is discretized into n_w,l layers (30), as depicted in Fig. (4). The uniaxial model by Menegotto and Pinto (Steel02) [36] is assigned to fibres in order to simulate the response of steel. According to this model, the relation that provides the normalised stress σ^* as a function of normalised strain ε^* is:

(9)

Fig. (4). I-shape beam modelling.

where b is the strain hardening ratio assumed equal to 0.0030. The parameter R influences the shape of the transition curve and varies as a function of the plastic excursion ξ_pl of the previous loading path. It accounts for the Bauschinger effect and is updated in the analysis by the following equation:

(10)

Herein, R₀ is the value of parameter R during the first loading, while cR₁ and cR₂ are experimentally computed. In the proposed model, the coefficient R₀ is set equal to 20. Coefficients cR₁ and cR₂ are assumed equal to 0.925 and 0.15, respectively, as proposed in OpenSEES manual [37]. No isotropic hardening is considered. The yield strength F_y and the Young modulus E are equal to 235 MPa and 210000 MPa, respectively. This material model is able to take into account the accumulated plastic deformation at each point of load reversal. Accordingly, each hysteretic loop follows the previous loading path for a new reloading curve, while deformations accumulate. The low-cycle fatigue material is combined with Steel02 material assigned to fibres of the plastic hinge zones. The fatigue material (uniaxialMaterial Fatigue), which is already implemented in OpenSees, uses an accumulative strain model to predict damage in accordance with the Miner’s rule. The relationship between the plastic strain amplitude experienced at each cycle and the number of cycles to failure is that proposed by Coffin and Manson. The fatigue ductility exponent m is equal to -0.5. To simulate the gradual stiffness and strength degradation caused by local buckling, the fatigue coefficient ε₀ is assigned to flange fibres according to a linear distribution [23]. In particular, a minimum value ε_0,min is assigned to fibres located at the edges of both flanges and a maximum value (ε_0,min + Δε₀ ) is assigned to flanges fibres located at the intersection with the web (Fig. 4). The term ε_0,min is related to the initiation of local buckling, and it is here assumed equal to 0.029, according to the value suggested in [23]. The term Δε₀ controls the rate of degradation. Particularly, high values of Δε₀ imply smoother stiffness and strength degradation, while low values lead to a more abrupt reduction of stiffness and strength. The value of Δε₀ can be calculated according to the following equation given in [23].

(11)

d being the height of the cross section and λ_f the slenderness of the flange evaluated as function of F_y and E:

(12)

In both equations (11) and (12), t_f and b_f are the thickness and the width of the flange, respectively. The shear length L_v and the wave length of beam flange during local buckling L_m are evaluated as follows [23]

(13)

(14)

t_w being the thickness of the web. Finally, the length of the plastic hinge is set equal to 0.22 L_v [14].

No degradation is considered for cross sections of columns. In fact, capacity design principles are applied in design and columns sustain a minor yielding during earthquake. Thus, it is possible to consider a significantly lower number of fibres for the cross sections of these members. In particular, five fibres are considered for each flange and five fibres for the web. Four additional fibres are considered to simulate the root fillets.

To consider a wide range of ground motion durations [33], three sets of ten artificial accelerograms generated by the computer program SIMQKE [34] were adopted as seismic inputs. Each accelerogram is defined by stationary random process modulated by means of a compound intensity function [38]. The earthquake rise time is 5.0 s, the parameter IPOW of the first branch and the parameter ALFA0 of the third one are assumed equal to 2.0 and 0.25, respectively. The stationary phase is followed by a latter 15 s phase with decreasing values of accelerations. Details regarding the envelope intensity function and the procedure for the determination of the length of the parts of the compound function may be found in [38]. The three sets are hereinafter named Long Set (LS), Medium Set (MS) and Short Set (SS). Each accelerogram of LS, MS and SS set is characterized by a total duration T_t equal to 90 s, 60 s and 27 s, and a duration of the stationary part T_s equal to 70 s, 40 s and 7 s, respectively. Figs. (5a, b and c) show an acceleration time history for each set. For each accelerogram, the value of the Arias Intensity AI has been calculated according to the following equation.

(15)

where a is the ground acceleration recorded at each time step t. The significant duration of each ground motion, denoted as 5-95% D_s, has been determined as the interval between the times at which 5% and 95% of the Arias Intensity of the ground motion have been recorded, i.e. the duration of time over which the 90% of the energy is accumulated [22]. Given the values set for the total duration and the stationary part duration, the mean value of 5-95% D_s over the ten accelerograms resulted to be equal to 63.5 s, 37 s and 7.5 s for LS, MS and SS sets, respectively. These values are selected so as to cover the range of effective durations as reported in [22].

The accelerograms of each set are compatible with the elastic response spectra reported in EC8 for an equivalent viscous damping ratio equal to 5%, PGA equal to 0.35 g and soil type C. The elastic spectra of the selected accelerograms (grey thin lines), the corresponding average response spectra (red line) and the elastic spectrum of EC8 (black thick line) are shown in Fig. (5).

The seismic response of the case study frame has been determined both in terms of global and local response parameters.

The assessment of the structural response at global level has been conducted considering the value of base shear V_b, top displacement D_t and peak ground acceleration (PGA). In the case of IDA, the value of every response parameter has been determined as follows. For every level of PGA and for each of the 10 ground motions, the maximum value of the relevant parameter during the time history is recorded. For each PGA, the mean value of the considered response parameter is calculated by averaging the values of the 10 ground motions. Consequently, every value of PGA corresponded to a value of D_t and V_b.

In the case of MPA, each point of the pushover curve (i.e. each couple of top displacement demand D_t and base shear V_b) is associated with the corresponding peak ground acceleration (PGA) according to the Capacity Spectrum Method [7], whereby the equivalent damping ratio ξ_eq is evaluated by the equation proposed for steel members in [39].

Fig. (5). Examples of acceleration time history: (a) Long Set (LS), (b) Medium Set (MS), (c) Short Set (SS) and elastic spectra of the selected accelerograms: (d) Long Set (LS), (e)) Medium Set (MS), (f) Short Set (SS).

(16)

where ξ₀ is the inherent viscous damping, here assumed equal to 5%, and μ is the ductility demand evaluated at each step as the ratio of the displacement demand at the considered step over the yielding displacement D_t,y.

When the CPA is considered, instead, the Capacity Spectrum Method [7] is adopted with reference to the backbone curve.

The seismic response at local level was evaluated based on the distribution of storey drift along the height of the frame. In order to investigate a structural behaviour that ranges from the almost elastic to the strongly inelastic, two limit states were considered in the IDA, i.e. the attainment of a maximum storey drift equal to 2% and 6%. Each limit state corresponds to a value of peak ground acceleration (PGA_2% and PGA_6%) and top displacement (). Fixing the top displacement at the value corresponding to the attainment of the considered limit state in the IDA, the corresponding distribution of storey drift obtained by the IDA has been compared with that evaluated by the CPA and the MPA when either or . In particular, in the case of the CPA, the top displacement corresponding to the drift limit state in the IDA was picked from the cyclic analysis, not the backbone curve.

Further, the seismic response has been investigated at cross section level. To this end, the values of bending moment M and curvature χ allowed the evaluation of the degradation of stiffness and strength in beams. In particular, the damage of the beams is measured by means of a damage index DI computed for each cross section within the plastic hinge zone of each I-shape beam. The DI is computed as the ratio of the number of flanges fibres in which the Miner’s damage index is equal to one (i.e. the number of fibres reaching fatigue) to the entire number of flange fibres within a given cross section [23]. When DI = 1.0, flanges of I-shape beam’s cross section are not able to contribute to the plastic resistance of the cross-section in the plastic hinge zone. A value of DI = 0.375, that corresponds to 80% of remaining bending moment capacity of the beam, was assumed to indicate the beam’s failure due to low-cycle fatigue [23]. In the case of the CPA, the DI is evaluated at the end of each loading cycle corresponding to the attainment of the top displacement imposed by the adopted loading protocol. For the sake of clarity, suppose that every top displacement D_t prescribed by the loading protocol is attained by a loading cycle composed of 5 loading branches. The number of damaged fibres is counted at the end of each loading branch. Then, once the fifth loading branch is completed, all the fibres that damaged after each loading branch are summed and the DI is evaluated. Instead, in the IDA the DI is evaluated at the end of the each time-history (i.e. of each accelerograms scaled at a given PGA). The obtained values of the DI of each cross-section are averaged over the number of accelerograms and are related to the maximum top displacement for the selected PGA. In order to have a comprehensive view of the damage in the frame, the following parameters were evaluated: (1) the average DI_n evaluated at each n-th storey as the average value over the ending cross sections of the beams of the n-th storey, (2) the DI_av evaluated as the average of the DI_n over the n storeys; (3) the DI_max evaluated as the maximum DI over the entire frame. The three response parameters were evaluated by the IDA and compared to those obtained by the CPA. No damage index can be evaluated when the MPA is carried out because there is no cyclic loading.

Fig. (6) compares the seismic response predicted by the CPA considering two different loading protocols, namely the SAC loading protocol (red line) and the ATC-24 loading protocol (black continuous line). The cyclic responses thus obtained are plotted in terms of base shear and top displacement in Fig. (6a). In Fig. (6b) the backbone curves corresponding to the cyclic responses obtained by the two loading protocols are plotted alongside with the performance curve obtained by the MPA (black dashed line). For each value of top displacement demand, the corresponding peak ground acceleration is determined by means of the Capacity Spectrum Method in Fig. (6c). Despite the different loading sequences and the different control parameter, the two loading protocols (SAC and ATC-24) lead to almost the same backbone curve (Fig. 6b). Since the backbone curves provided by the application of the two loading protocols are extremely close, the differences in the values of the peak ground acceleration corresponding to a fixed top displacement demand are negligible as well (Fig. 6c). The comparison between the results obtained by the MPA and those by the two cyclic pushover analyses show that, given a value of top displacement, the MPA leads to larger values of both base shear and peak ground acceleration compared to the CPA. This is due to the fact that the MPA is not able to take into account the effect of the cumulated damage, that is considered by the CPA, and neglects the corresponding reduction of stiffness and strength caused by the damage.

To examine the results more accurately, the seismic response is investigated at cross section level. Fig. (7) shows the bending moment M – bending curvature χ cyclic response of the left end cross section of the first span beam of the three storeys. Each M-χ loop is determined by the CPA applying both the considered loading protocols until the final top displacement. The stiffness and strength degradation of the cross sections is larger at lower storeys. Indeed, the cyclic response of the first storey is characterised by a larger loop than the third storey. The same tendency is found also for the ending cross sections of the other beams. In particular, at the third storey all the cross sections, with the exception of the outermost cross sections of the left and right beams, remain elastic and the cyclic response is linear. However, regardless of the cross section, the response is almost independent from the applied loading protocol. To delve deeper the analysis on stiffness and strength degradation, Fig. (8) plots the value of the average damage index D_n at each storey (Fig. 8a), the average damage index D_av of the frame (Fig. 8b) and the maximum damage index D_max of the frame (Fig. 8b). Each of the damage indexes is evaluated at the attainment of every peak top displacement required by both the loading protocols. The dashed line is the threshold corresponding to the damage of 80% of fibres of the cross section, i.e. the failure of the considered cross section. At the end of the two loading protocols, i.e. for a top displacement equal to 693 mm and 700 mm for SAC and ATC-24 protocols respectively, the cross section that reached the largest damage index was the left end cross section of the first span beam of the first storey, where the DI was equal to 0.508 and 0.525, respectively. The largest mean D_n was recorded at the first storey and, at the end of the loading protocol, was equal to 0.496 for SAC protocol and 0.515 for ATC-24 protocol. The threshold value of DI was overcome at the second storey, as well. Instead, at the third storey the DI was on the average significantly lower than 0.375. At the end of the loading protocols, the average and the maximum indexes of the frame reached values equal to 0.319 and 0.508 for the SAC protocol, and 0.332 and 0.525 for ATC-24 protocol. According to the results presented so far, the differences in the seismic response due to the loading protocol are basically negligible. This is due to the fact that the two protocols differ mainly for the number of loading cycles in the elastic range, while the differences in the loading sequences in the inelastic range are negligible. Thus, the rate of degradation due to the cumulated damage is almost the same for both the loading protocols. Hence, the results by the CPA that will be discussed in the following paragraphs are only those obtained by the SAC loading protocol.

Fig. (6). Seismic assessment by CPA, applying SAC loading protocol and ATC-24 loading protocol, and by MPA.

Fig. (7). Bending moment vs Curvature of the left end of the first span beam evaluated by CPA with SAC and ATC-25 loading protocol at: (a) storey 1, (b) storey 2, (c) storey 3.

Fig. (8). Damage Index evaluated by CPA with SAC and ATC-25 loading protocol: (a) average DIn at each storey, (b) average DIav, (c) maximum DImax..

The fibre-based damage accumulation model introduces the parameter Δε to control the rate of degradation. This type of modelling allowed the investigation of the effect of the rate of degradation on the seismic response. To this end, the seismic response of the case study steel frame has been determined by the CPA considering three numerical models with values of Δε equal to 0.08353, 0.04176 and 0.0. The first value is provided in [23] based on the analysis of experimental results, and it is here assumed as the reference value. The second and the third values are selected to show clearly the changes in the structural response due to a reduction of Δε to 50%, or even to zero. The cyclic responses in terms of base shear and top displacement of the considered numerical models are shown in Fig. (9), whereby the corresponding backbone curves are plotted as well. Until the attainment of a top displacement approximately equal to 300 mm (i.e. 3% of interstorey height), the damage due to the cumulated fatigue has not occurred yet and the differences among the three backbone curves are almost negligible. After this value of top displacement, stiffness and strength tend to decrease almost simultaneously in each curve. Indeed, the degradation has occurred and some fibres, or even some cross sections, have collapsed. However, the reduction of stiffness and strength, denoted by the softening branch in the backbone curve, is related to the value of Δε. The value of Δε=0.0 represents the lower limit case. In fact, it makes the distribution of the fatigue coefficient ε₀ constant along the flanges of the cross sections (Fig. 4). Hence, the damage occurs in all the fibres of the flanges simultaneously and the web of the considered cross section is the only resisting part. In this case every cross section can experience only two opposite scenarios: the entire cross section is effective (before the damage) or only the web is effective (after the damage the flanges suddenly collapse). Because of this, the moment resistance drops from that of the entire cross section to that provided by the web; for instance, Fig. (10a) shows the cyclic response of the left end cross section of the first span beam. This behaviour of the cross sections is reflected on the global response, which is denoted by the abrupt reduction of stiffness and strength shown in Fig. (9a). On the contrary, when Δε=0.08353 the fibres of the flanges collapse progressively. Thus, in addition to the two possible scenarios listed above, a third intermediate scenario could develop in the cross sections, i.e. the flanges are partially effective, because after the damage occurs only some fibres have collapsed, and the web is still effective. For this reason, the reduction of the moment resistance is very soft (see the example of the M- χ loop in Fig. (10c) and the slope of the backbone curve decreases smoothly, as shown in Fig. (9c). When Δε=0.0416, the behaviour is intermediate between those two limit cases.

Fig. (9). Seismic assessment by CPA applying SAC loading protocol considering a rate of degradation Δε equal to (a) 0.00, (b) 0.0417176, (c) 0.08353.

Fig. (10). Bending moment vs Curvature of the left end of the first span beam evaluated by CPA with SAC protocol considering a rate of degradation Δε equal to (a) 0.00, (b) 0.0417176, (c) 0.08353.

At cross section level, the rate of degradation affects the evaluation of the damage index as well. Indeed, when Δε is null, all the fibres of the flanges collapse simultaneously and the DI of a generic cross section can assume two alternative values: zero (before the damage) or 1 (after the damage), and the increase is abrupt. If Δε is larger than zero, the DI gradually increases from zero to 1, and the larger Δε the smoother the increase. Figs. (11a, b and c) show the average DI_n at each storey, the average DI_av and the maximum DI_max, respectively. In these plots, the red, grey and black lines refer to the results obtained by assuming Δε equal to 0.08353, 0.0416 and 0.00, respectively. Fig. (11a) shows that the largest values of DI_n are recorded at first and second storey, pinpointed by diamonds and dots, respectively. Regardless of Δε, the damage occurs at a top displacement equal to 300 mm. However, when Δε is null (black line) the average DI_n index rises from zero up to (almost) 1 abruptly, while the increase of DI_n is softer when Δε is larger than zero. Note that, given a value of the top displacement, the larger Δε the smaller the number of the collapsed fibres, thus the smaller DI_n. Only at the third storey (pinpointed by triangles) the cross sections do not fail and DI_n is below the limit value of 0.375. Nevertheless, also in this case the most sudden increase of DI_n is attained when Δε is null. The same tendency is confirmed by the average DI_av (Fig. 11b), and the maximum DI_max (Fig. 11c) over the entire frame. It is worth noting that the value assumed for the rate of degradation influences the evaluation of the structural response. Indeed, a rough fibre discretization of the cross sections and a null value of Δε leads to an overestimation of the cumulated damage and an underestimated stiffness and strength. Instead, a fibre modelling able to consider the damage accumulation properly, i.e. with Δε larger than zero, allows a more accurate prediction of the effect of the cumulated damage on the structural performance. Based on these results, the following results are presented assuming Δε equal to 0.08353.

Fig. (11). Damage index evaluated by CPA applying SAC loading protocol considering a rate of degradation Δε equal to 0.00, 0.0417176, 0.08353: (a) average DIn at each storey, (b) average DIav, (c) maximum DImax..

In this section the effectiveness of the CPA is evaluated by comparing the seismic response of the case study frame determined by the CPA with those obtained by the MPA and IDA, the last being assumed as the target response.

Fig. (12) shows the seismic response of the designed frame in terms of PGA and top displacement. The red continuous line and the red dashed line plot the results provided by the CPA (SAC loading protocol and Δε=0.08353) and MPA, respectively. A preliminary comparison between the two nonlinear static analysis shows that, for fixed values of PGA, the MPA leads to lower top displacements than the CPA. Indeed, the MPA neglects the effect of the cumulated damage, thus it is not able to account for the corresponding reduction of stiffness and strength. In the same plot, the three dotted lines show the maximum top displacement evaluated by the IDAs. The black, grey and white dots show the results obtained by the Long Set (LS), Medium Set (MS) and Short Set (SS), respectively. Particularly, for each of the three input sets, given the PGA, the corresponding top displacement is the average of the 10 maximum top displacements of the 10 accelerograms. The comparison of these three plots shows that the duration of the input accelerograms affects the seismic response. In fact, given a value of PGA, the SS leads to top displacements significantly lower than those attained under the MS or the LS. Thus, as long as the effective duration of the input accelerogram is short (7.5 s), the stiffness and strength degradation due to the cumulated damage is not significant. On the contrary, when the effective duration of the seismic input becomes significant (37 s and 63.5 s), the degradation of the cross sections turns into a critical issue. Based on these results, the CPA becomes extremely effective when accelerograms of medium or long duration are considered. In such cases, instead, the MPA would underestimate the top displacement demand. Instead, if short inputs are considered, the MPA is able to provide a reasonable prediction of the seismic response.

Figs. (13a, b and c) compares the heightwise distributions of storey drifts evaluated by the CPA and the MPA with those provided by the IDA with the long, medium and short input sets, respectively. For each set of accelerograms, the storey drift related to the IDA is the mean of the maximum values obtained by the 10 ground motions. Two limit states are considered in every IDA: the attainment of a maximum storey drift angles equal to 2% and 6%. The corresponding distributions of storey drifts of the CPA and the MPA are those corresponding to a top displacement equal to that leading the IDA to the considered limit state. At 2% limit state, regardless of the input set, the CPA overestimates the drifts of the first two storeys and underestimates that of the third storey, while the MPA provides a better estimation of the drifts at lower storeys, but underestimates, even more, the drift at the top storey. However, both the analyses indicate correctly the storey where the drift concentration is larger (second storey). When 6% limit state is achieved and the input set has long or medium duration, the CPA still provides a conservative estimate of the drifts, while the MPA significantly underestimates the storey drifts at every storey. Only in the case of the short set, CPA and MPA lead to almost equal drift estimates. However, also in this case the CPA is slightly more accurate than the MPA.

Fig. (12). Peak Ground Acceleration vs Top displacement obtained by CPA, MPA and IDA with Long, Medium and Short sets.

To validate the CPA at local level, Fig. (14) compares the average DI_av and the maximum DI_max of the entire frame evaluated by the CPA to the values provided by the IDAs with LS, MS and SS. The damage indexes DI_av and DI_max are calculated as described in Section 4.3. In both cases, the SS (white dots) has led to almost no damage in any cross-section and the average and the maximum DI are null. The largest damage is attained when the duration of the input is 90 s (black dots), in accordance with-the results shown in the previous figures. In this case, the damage index keeps a value on average lower than 0.375, owing to the beams of the third floor that basically remain elastic and decrease the average damage index. Instead, the maximum value of the damage index overcomes the limit value, due to the concentration of damage that cause failure of the beams of the first and second floors. The average and the maximum damage indexes estimated by the CPA (red line) are plotted along with those of the IDAs. This shows that the CPA is able to estimate accurately both the average and the maximum damage index caused by long duration accelerograms, in agreement with the results obtained in terms of global response parameters.

Fig. (13). Storey drift distribution obtained by CPA, MPA corresponding to the top displacement leading to a maximum drift in the IDA equal to 2% and 6%.: (a) Long set, (b) Medium set (c) Short set.

Fig. (14). Damage index evaluated by CPA (applying SAC loading protocol and Δε=0.08353) and IDA with Long, Medium and Short sets. (a) average DIav, (b) maximum DImax.