1 Introduction

On 26th January and 3rd February, 2014, two strong earthquakes of magnitude Mw6.1 and Mw6.0, respectively, struck the island of Cephalonia in western Greece (Fig. 1, Table 1). The epicentre of the first earthquake was located approximately 9 km southwest of the town of Lixouri, followed by numerous strong aftershocks. Recorded peak ground accelerations from 8 strong motion instruments reached 0.57 g at Lixouri and 0.43 g at Argostoli. The second earthquake took place 8 days later very close to the town of Lixouri, on a different fault relative to the first event. The second event induced very high accelerations around Lixouri, leading to peak recorded motions of 0.76 g in the village of Havriata close to the fault. Both events are associated with strike-slip faulting in nearly vertical dextral faults, which are part of the so-called Cephalonia Transform Fault (CFT) zone that extends to the nearby island of Lefkada, following the underwater western valley of the island. It is the same fault system that generated the earthquake sequence of 1983 in Cephalonia and the 2003 earthquake in Lefkada (Giarlelis et al. 2011; Sextos et al. 2010; Vintzileou et al. 2007). Numerous studies  of the earthquake sequence have been published, among others, by Ganas et al. (2015), Papadopoulos et al. (2014) and Theodoulidis et al. (2015).

Fig. 1
figure 1

Map of the region showing approximate locations of the fault lines and the main events of the earthquake sequence at hand

Full size image
Table 1 Table 1
Full size table

Unlike other destructive earthquakes in Greece (Anagnostopoulos et al. 1987; Mylonakis et al. 2003; Margaris et al. 2010; Giarlelis et al. 2011), most structures in the area exhibited a satisfactory performance, despite the proximity to the fault and the severity of the recorded ground motions (GEER/EERI/ATC report 2014, Karantoni & Pantazopoulou 2016,). Despite the damage suffered by some buildings, especially in the Paliki peninsula in the western part of the island, in most cases the damage was treatable and only a few collapses or irreparable damage were observed. According to the data gathered through the EPANTYK research program on structural repairs (Nakos et al. 2006), the majority of buildings in Cephalonia were designed following older seismic regulations (e.g. 78% of the 27,400 buildings listed were built before 1985 when an update to the first Greek Seismic Code of 1959 went into effect). This means that most buildings were designed for lateral actions much lower than the requirements of the modern seismic design regulations i.e. the Greek Seismic Code 2000 (Earthquake Planning & Protection Organization 2003) and Eurocode 8 (CEN 2004), but also much lower than those induced by the two earthquakes at hand.

Unlike structural damage, geotechnical damage was more intense. In the western part of the island landslides were observed. Moreover, quay walls, especially in the port of Lixouri, suffered relatively large displacements. The damage to the road network was also severe and extensive (GEER/EERI/ATC report 2014).

In the capital city of Argostoli damage was limited to a few structures including a school complex located at the center of the city, about 10 km from fault. One of the buildings of this complex is the focus of the study presented herein. It is a two-storey reinforced concrete structure designed following an older generation of codes which suffered light damage. As the structure is located very close to the ARG 2 station that provided accelerograms for the event, it offers a suitable case study to investigate the causes of the limited damage that occurred in Argostoli. To this end, a series of analyses were performed. These include (a) structural assessment following the analysis procedures of the recently published Greek Code for Structural Interventions (Earthquake Planning & Protection Organization, 2022) and (b) non-linear static and time-history analyses with and without consideration of soil-structure interaction (SSI) effects. These analyses highlight the interplay of soil, foundation and superstructure in modifying seismic demand. It is shown that soil-structure interaction may strongly affect inelastic seismic response and alter the dynamic behaviour of structures even for moderately soft soil conditions. Furthermore, the numerical results demonstrate that the minor damage experienced by the building (despite the intensity of the ground motion), is justified.

2 The seismicity of Cephalonia

The seismicity in the area of Cephalonia is extremely high, as can be observed from the history of seismic events in the region during the last 6 centuries (during which the island was mostly under Venetian rule), a period for which there are reliable reports. The main earthquake events in the region are presented in Table 2. Of particular interest are the earthquakes of 1658, 1741, 1767 and 1972, since they present similarities to the 2014 event, regarding the region that was mainly affected, which is the south-western part of the island (Paliki peninsula).

Table 2 Earthquakes that caused extensive damage in Cephalonia, 1469 A.D. – present (Papazachos and Papazachou. 2003; Spyropoulos 1997)
Full size table

The most severe seismic event in the recent history of the island is the seismic sequence of August 1953, which caused a great number of casualties and extensive structural damage, and led to the legislation of the first national seismic design code in Greece (Greek Seismic Code 1959).

3 Engineering characteristics of strong ground motion

The permanent strong motion network in Cephalonia and Ithaki Islands, monitored by ITSAK and the Geodynamic Institute of NOA, consists of 8 triaxial accelerometric instruments (Theodoulidis et al. 2015). In the town of Argostoli the mains shock was recorded by ARG 2 station, installed at the ground level of a 3-storey building housing the Prefecture of Ionian Islands. The station is founded on soil with shear wave velocity VS, 30 = 440 m/s. For this station, the significant duration of ground motion, i.e. the time span where 5% to 95% of the total energy is released, was 7 s for the first event and 5–7 s for the second one.

The horizontal acceleration records of the ARG Station at directions NS and EW are presented in Fig. 2. From these, the elastic horizontal acceleration spectra of Fig. 3 were computed. As can be concluded from this figure, for the town of Argostoli the second event had lower intensity in terms of accelerations than the first one. In the same figure, the response spectra for a typical structure in Argostoli following the EC8, the Greek seismic code 2000 (GSC-2000), still in force for new structures in parallel with EC8 and the 1959 Greek seismic code (GSC-1959) provisions, are plotted. For EC8 and GSC 2000, the reference peak ground acceleration is agR = 0.36 g, the importance factor is considered as γI = 1, the behaviour factor is taken q = 1 and the ground type is assumed to be B (corresponding to the shear wave velocity of the soil below the ARG 2 station). The latter means that for EC8 the soil factor is S = 1.2 while there is no such factor for GSC 2000.

Fig. 2
figure 2

Time histories of the horizontal accelerations recorded at the Argostoli Station (ARG 2) at directions NS and EW from the main events of the 26/1/2014 and 3/2/2014

Full size image
Fig. 3
figure 3

Elastic horizontal acceleration spectra from the Argostoli Station (ARG 2) records, at directions NS and EW from the main events of 26/1/2014 and 3/2/2014 compared to seismic codes' requirements

Full size image

The equivalent response spectrum of the 1959 Seismic Code is plotted as a horizontal (period-independent) line. Most of the structures in Cephalonia were designed following that code, i.e., based exclusively on pseudo-dynamic considerations. A constant distribution of the lateral load with a value of (ε W) was considered, where ε is the seismic coefficient of the 1959 code and W is the weight of the structure. The equivalent demand in the realm of the modern codes would be a constant response spectrum with Rd = 1.75 (q × ε), with 1.75 being an empirical conversion factor between modern and old codes, relating reinforced concrete provisions based on allowable stresses and ultimate strength concepts (Anagnostopoulos et al. 1987). Should a value of 1.50 be adopted for the behaviour factor (a reasonable approximation for older generation RC structures in Greece built without capacity design considerations) and factor ε taken at 0.08 (which is the lowest value specified by the 1959 code for the region), then Rd = 0.21.

From the response spectra of Fig. 3 it is evident that the spectral values in terms of acceleration were significantly higher than the requirements of the older codes, but also higher than those of the current codes in the range of periods between 0.05 s to 0.3 s, approximately, which are critical for the low rise buildings  on the island. Furthermore, for   Argostoli area the first event induced much higher accelerations than the second one.

4 Building description

The school complex is located in the center of Argostoli, approximately 50 m from the ARG 2 accelerometer (Fig. 4a). It also lies very close to the building housing the Archaeological Museum of Cephalonia which suffered heavy structural damage. The school complex comprises four structurally independent buildings separated by seismic joints. The construction of the structures was carried out gradually, based on studies by a distinguished structural engineer of that era, Dimitrios (Takis) Pavlatos. Pavlatos provided guidelines and ingenious structural systems for the reconstruction of numerous structures in Cephalonia after the disastrous earthquakes of 1953, and was also involved in the preparation of the guidelines of the first Greek seismic code of 1959. The complex initially housed the Korgialenio Boarding School for Girls and later was used as a public school.

Fig. 4
figure 4

a Aerial view of the school complex and b Sketch of its configuration showing the individual buildings

Full size image

The building under investigation is Building 4 of the complex as shown schematically in Fig. 4b. It is a reinforced concrete building (Fig. 5) with a frame system of beams-walls- columns as shown in Fig. 6. It has a floor plan of an elongated rectangle with dimensions of approximately 10 × 32 m and consists of 2 levels, ground floor and first floor. Its total area is about 640 m2 while its current use includes a gym and office space on the ground floor and 4 classrooms on the first floor. It was built in two phases: in the late 70's, the ground floor was erected and in the mid 80’s a side extension as well as a second floor were added. The foundation is formed by spread footings (Fig. 7) connected with tie-beams.

Fig. 5
figure 5

Front and back view of the school building

Full size image
Fig. 6
figure 6

Ground and 1st storey layout along with Sect. 1–1 (the location of observed structural damage is denoted with red circles)

Full size image
Fig. 7
figure 7

a Exposed footing and b Sketch of dimensions of the footing of column C4

Full size image

The building can be classified as irregular: the addition of the 1st floor required strengthening of the columns, beams and slabs of the ground floor. However, beams and slabs were not strengthened; instead, new slabs were built on top of the  old ones using “inverted beams”, resulting in increased seismic mass at that level. Furthermore, the distance between the vertical elements is relatively large, of the order of 7.5 m, and the same applies to the floor heights which are about 4 m.

Observed damage was limited to small vertical cracks at the base of several vertical elements of the ground floor, i.e., columns C3, C4, C5 and structural walls W12 and W17 (Figs. 6, 8, 9). Careful examination of the cracks and removal of the concrete cover revealed that the corrosion of the reinforcement also contributed to the   damage, albeit to a small extent. Also, some infill walls on both floors especially those along the transversal direction of the building were detached from the surrounding frames (i.e., there was an out-of-plane action along the longitudinal direction). Overall, the observed damage was   minor compared to other neighbouring buildings, such as the Archaeological Museum at a distance of about 50 m, but larger than that of the majority of the structures in Argostoli that experienced only minor (non-structural) damage.

Fig. 8
figure 8

Minor cracks on structural wall W17

Full size image
Fig. 9
figure 9

Minor cracks a on structural wall W12 and b on column C4. Evidence of corroded, smooth longitudinal reinforcement along with apparent lack of transverse links

Full size image

Blueprints and original calculations are not available, therefore a systematic post-earthquake investigation conducted by the authors provided as-built structural information including dimensions, cross sections of structural members, steel reinforcement, material properties and dead loads. The reinforcement configuration and the rebar size were established through rebar scanning of a large number of structural elements, but also through investigative sections in order to verify the scans. Three rebar specimens were taken so as to evaluate steel quality. Regarding concrete quality, this was evaluated using destructive (testing of sample cores) and non-destructive (Schmidt hammer and ultrasonic tests) techniques.

The steel reinforcement used was of type St I which has a characteristic value fyk = 240 ΜPa. However, from the lab tests it was concluded that the reinforcement used in the building has a much higher strength. Characteristic yield strength values of fyk = 344 ΜPa and a mean value of fym = 377 ΜPa was found for small bar diameters (used in slabs and stirrups).

Larger diameter bars used in columns, walls and beams were not tested; their yield strength was evaluated from the correlation between the yield strength of smaller and larger diameter bars. fyk = 294 ΜPa and fym = 325 ΜPa were assumed.

Regarding concrete, destructive and non-destructive tests showed a characteristic strength fck = 12.9 ΜPa and a mean value fcm = 16 ΜPa. Also, concrete carbonation was checked by applying a phenolphthalein indicator solution. The carbonation depth was found to range between 25 to 50 mm, in accordance with the observed corrosion of certain steel rebars.

5 Geotechnical conditions

The building is located at a distance of 50 m from the 3-storey office building of the Prefecture of the Ionian islands, where the ARG 2 station is installed at   ground level. According to geological information, SPT testing and geophysical data using the Crosshole Seismic and Microtremor array methods, the station is founded on Pleistocene deposits (of clay type) with an average shear wave propagation velocity VS, 30 = 440 m/s. Below that depth, regional sandstone with VS = 1 km/s is encountered which is considered as seismic bedrock. However, the soil profile (Theodoulidis et al. 2015) indicates moderately soft soil conditions in the upper part, crucial for the estimation of the SSI effects, as shown in Fig. 10.

Fig. 10
figure 10

Soil profile at a nearby site to ARG 2

Full size image

Specifically, the shear wave velocity at the upper 10  m of the soil layer is approximately VS = 260 m/s. Accounting for the effect of cyclic degradation on the shear modulus by using standard G-γ curves (Ishihara 1996), the effective shear wave velocity is found to be   approximately 180 m/s.

The foundation of the structure consists of spread footings connected with tie beams, as mentioned previously, and lies at a depth of 1.1 m. Regarding their size, they can be divided into two groups: "large" ones, with dimensions of approximately 1.4 × 2.8 m supporting structural walls and "small" ones, with dimensions of approximately 0.9 × 0.9 m supporting columns. Following the practice of the time, the footings were basically designed for gravity loads and this reflects on their small size compared to the current state of practice.

Closed-form solutions of dynamic stiffness of spread footings are available by regression analysis based on finite element and boundary element method analysis data (Mylonakis et al 2006). These solutions come as the product of static stiffnesses multiplied by dynamic stiffness modifiers. The latter depend on the frequency corresponding to the period associated with the dominant response of the structure according to NIST GCR 12–917-21. When using the period of the fixed based structure (0.23 s) the dynamic stiffness modifiers are found to practically equal to unity for the specific structure and soil conditions. Therefore using the above mentioned solutions, the soil data (Vs = 180 m/s, ρs = 1.8 Mg/m3, vs = 0.33), and the foundation properties, the stiffness of each footing can be approximated by Κz: 170 ~ 365 ΜN/m for the vertical mode, Κx and Κy: 139 ~ 302 ΜN/m for the horizontal modes and, Κrx and Κry: 27 ~ 379 kN m/rad for the rocking modes depending on the size of the footings. Regarding damping, due to limitations of the software an additional viscous damping of 3% was added to the whole system.

6 Structural assessment

A variety of analyses were performed for the structural assessment of the building following the provisions of the Greek Code for Structural Interventions (GCSI-2022)—a state of the art code that introduces performance-based design that is complementary to the Greek Annex of EC8-3. These include: (a) modal response spectrum analyses to assess the design according to modern codes. (b) pushover analyses indented to investigate probable failure mechanisms and provide information as to strength and deformation capacities. (c) Non-linear time-history analyses for a more accurate evaluation of seismic demand. Regarding the supports, two alternatives are considered: (a) fixed-base conditions and (b) flexible-base conditions.

7 Linear elastic analysis

7.1 Numerical modelling

The structure is treated as a space frame subjected to combined gravitational and earthquake loading. The beam-column-structural walls frame system is modelled with linear elements located along the centroidal axes of the members. The slabs are considered rigid in their own plane (diaphragms). Regarding the gravitational loads of the structure, these were taken according to the observations of the post-earthquake investigation conducted by the authors. Apart from dead loads, a live load of 0.6 kN/m2 for the classrooms were considered in all floors.

The analysis considers cracked properties for the members. The effective stiffness of each element was established from the secant stiffness at yielding, according to GCSI-2022 and EC8-3. It is calculated from the moment—curvature diagram of the section as Keff = My L/ 6 θy where My and θy are the values of moment and chord rotation, respectively, of the end section of the element at yielding. Resulting stiffness values are at 20% and 30% of the uncracked for beams and columns, respectively. Infill walls are taken into account as diagonal struts connecting the joints of the frames only where there are no openings. The thickness of the diagonal strut is the same as the thickness of the wall, while its effective width, b, is taken as b = 0.15 L, where L is the length of the diagonal strut following GCSI-2022. Τhe uncracked secant modulus of elasticity Ew of the infill wall is taken as Ew = 750 fm for clay bricks, as proposed by Paulay and Priestley (1992), where fm is the compressive strength of masonry.

7.2 Modal analysis

The dynamic analysis considers the natural modes of the structure encompassing almost 100% of the total effective mass in each direction. For fixed-base case, fundamental periods were estimated at 0.28 s along the transverse “weak” axis (Υ-Υ) with a participation factor of 82%, and 0.23 s along the longitudinal “strong” axis of the building (Χ-Χ) with a participation factor of 84%. For the flexible-base case, the periods were determined at 0.35 s with a participation factor of 86%, and 0.29 s with a participation factor of 89% for the transverse and the longitudinal directions, respectively. So, taking into account SSI results to approximately a 24–27% increase in the fundamental periods, which is anticipated given the medium soft soil conditions.

For the structural assessment, the response spectrum of EC8-3 (identical to GCSI-2022 requirements) is used. The spectrum is similar to the one described earlier (i.e., peak ground acceleration of agR = 0.36 g and soil factor, S = 1.2, for ground type B). However, an importance factor γI = 1.2 is considered, since the importance class is III (school building). In Fig. 11, the elastic response spectrum employed in the analyses is presented along with that of the Greek seismic code 2000 (according to which γI = 1.15) while the horizontal line corresponds to the equivalent response spectrum of the 1959 Seismic Code. In the same graph, the elastic horizontal acceleration spectra from the Argostoli Station (ARG 2) records of the main event of 26/1/2014, oriented in parallel to the L and T axes (N155 and N65 respectively) of the building, (i.e. rotated counter-clockwise by 25 degrees), are also plotted for comparison purposes. As already mentioned, the first event induced much higher accelerations than the second one for the majority of structures in Argostoli, therefore the behaviour of the structure at hand during the second event is not examined.

Fig. 11
figure 11

Elastic horizontal acceleration spectra from the Argostoli Station (ARG 2) records, from the main event of 26/1/2014, oriented parallel to the L and T axes of the building, compared to seismic codes' requirements

Full size image

An elastic analysis of the building was performed using a behaviour factor q = 1, mean values for the material strengths and safety factors equal to 1, in order to determine the extent and distribution of inelastic demand. The analysis results in the failure indices, λ, of each member:

λ = SΕ / Rm(1) where SΕ is the action effect (bending moment) due to the action of the seismic combination and Rm is the corresponding available resistance of the structural member. The analysis results, presented in Table 3, indicate that for most columns the failure index in bending and in shear lies in the range 3 to 5 with more than half having λ > 4. This means that the structural system has very inadequate strength against the design lateral loads.

Table 3 Failure indices, λ, of ground floor vertical elements due to bending and shear from the preliminary analysis
Full size table

8 Nonlinear analyses

8.1 Inelastic modelling

In order to investigate the seismic behaviour of the structure during the first earthquake (which induced much higher accelerations than the second one in Argostoli), non-linear static analyses were first performed. The possible collapse mechanism was investigated and the capacity curve of the structure was calculated. Non-linear dynamic time history analyses were also performed to more accurately estimate the seismic demands induced by the particular earthquake.

The building is modelled as a space frame under gravity and seismic loads using Seismostruct (2022). Seismostruct is a finite element software for structural assessment and retrofitting, capable of predicting the large displacement behaviour of space frames under static or dynamic loading, taking into account both geometric and material nonlinearities.

The beams, columns and shear walls of the building were modelled with inelastic plastic-hinge frame elements with concentrated inelasticity within a fixed length at each member’s end. Rigid diaphragmatic action is considered at all floors (Fig. 12). The nonlinear concrete model of Mander et al. (1988), as modified by Martinez-Rueda & Elnashai (1997), was adopted for the confined and unconfined concrete, while the Menegotto-Pinto model (1973) was adopted for the reinforcement bars. The mass was calculated using the dead and the live loads, assumed to act on the structure during the design earthquake. The thickness of the infill walls is 0.20 m and their compressive strength was estimated at around 1.5 MPa. Infill walls are modelled using the inelastic infill panel element, which is an equivalent strut nonlinear cyclic model proposed by Crisafulli (1997) for modelling the nonlinear response of infill panels in framed structures. Each panel is represented by six strut members, with each diagonal direction featuring two parallel struts to carry axial loads across two opposite diagonal corners, and a third one to carry the shear from the top to the bottom of the panel.

Fig. 12
figure 12

Model of the building in Seismostruct

Full size image

The struts act only when the diagonal is in compression, hence, their activation depends on the deformation of the panel. The axial load struts use the masonry strut hysteresis model, while the shear strut uses a dedicated bilinear hysteresis rule, as described by Crisafulli (1997). The thickness of the infill panel was taken equal to the thickness of the infill walls, while its equivalent width was calculated in accordance with the relevant provisions of GCSI (2017).

Regarding the supports, two alternatives are considered: (a) fixed-base conditions and (b) flexible-base conditions.

8.2 Non-linear static analyses (pushover)

Pushover analysis is a first step for the assessment of the structure. It is performed in both directions, in the positive and negative sign for each one, using two force distributions with height : rectangular and triangular. The adopted plastic hinge frame elements are force-based.

In Fig. 13 the capacity curve is presented for both directions and triangular distribution of lateral forces, with and without considering soil-structure interaction (SSI). The bilinear approximation of the capacity curve is also shown, as well as the target displacement according to N2 method of EC8 for limit state of "Significant Damage" (SD). Evidently, the capacity of the structure in the transverse direction is lower than in the longitudinal direction, while the opposite happens with the target displacement, which is higher in the transverse direction.

Fig. 13
figure 13

Capacity curve and bilinear approximation for triangular lateral distribution of forces with and without accounting for soil-structure interaction (SSI) in both principal directions of the building (the dots indicate points of first yielding according to EC8)

Full size image

In Fig. 14 the ADRS diagram for triangular lateral distribution of forces is presented together with the EC8 design spectrum and the spectra of the record of the ARG Station in both directions for multiple viscous damping values.

Fig. 14
figure 14

ADRS diagram for triangular lateral distribution of forces, EC8 design spectrum and the spectra of the record of the ARG Station in both directions for multiple viscous damping values

Full size image

The results show the development of plastic hinges in a number of elements starting at a roof displacement of about 22 mm including columns C1–C5 where light cracks were observed.

The target displacement estimated according to EC8 for limit state of "Significant Damage" (SD) without taking into account soil-structure interaction (SSI), ranges from 35 to 75 mm for a triangular distribution of lateral forces in both directions of the building. When soil-structure interaction (SSI) is taken into account, the target displacement is naturally higher and ranges from about 75 to 110 mm. For these displacements, some structural elements have yielded, including those that had been actually damaged by the earthquake. However, the analyses also show the development of plastic hinges to some members that showed no signs of damage in the post-earthquake investigation. Given the uncertainties associated with the numerical prediction of the observed damage (steel and concrete material properties, reinforcement ratio, epistemic uncertainty, SSI and record-to-record variability), it is anticipated that the matching is mostly qualitative.

The target displacement, if calculated using the spectrum of the actual earthquake in both directions, is smaller than that calculated using the design spectrum of EC8, ranging from 15 to 22 mm if SSI is not considered and from 21 to 28 mm when SSI is accounted for.

8.3 Non-linear time history analyses

To explore the validity of the results of the pushover analyses and better understand the seismic behaviour of the structure, non-linear dynamic time-history analyses were performed. As already mentioned, the horizontal components of the record in Argostoli (rotated counter-clockwise by 25 degrees to align with the principal axes of the building), were imposed on the numerical model.

Figure 15 presents the roof displacement time history at the node of column C3 from the non-linear time-history analysis with simultaneous excitation in both horizontal directions with and without taking into account soil-structure interaction (SSI). . The maximum displacement is 27 mm in the longitudinal direction and 25 mm in the transverse when soil-structure interaction is not taken into account. If SSI is considered, maximum displacements are naturally higher, equal to 32 mm and 43 mm in the longitudinal and transverse direction respectively. The maximum displacements from the dynamic analyses are in good agreement with the displacement demands resulting from the pushover analyses, calculated using the spectrum of the actual earthquake.

Fig. 15
figure 15

Roof displacement (node at top of column C3) time history from non-linear dynamic analysis with and without taking into account soil-structure interaction (SSI)

Full size image

Figure 16 presents the inter-storey drift ratio, γ = dr /h (h being the storey height) of the first floor, in which damage to the infill walls was observed. According to EC8-1, the inter-storey drift ratio for buildings having non-structural elements of brittle materials attached to the structure, should be limited to γ = 10 ‰. However, for the Greek Seismic Code 2000 this limit is set lower, at 5 ‰. The results show that for the top storey, if SSI is not taken into account, the maximum inter-storey drift ratio is equal to 3.4 ‰ in the longitudinal direction, while in the transverse direction the maximum value is 3.0 ‰. Taking into account SSI, the inter-storey drift ratio decreases to 3.1 ‰ in the longitudinal direction and increases to 4.5‰ in the transverse. For the ground floor the maximum inter-storey drift ratio values are 3.0 ‰ and 2.9 ‰ for the longitudinal and transverse direction, respectively without SSI and increases to 4.5 ‰ and 5.8 ‰ respectively with SSI. This may explain better the observed detachment of infill walls with out of plane action in both floors along the longitudinal direction.

Fig. 16
figure 16

Inter-storey drift ratio of top storey time history, in both directions of the building from non-linear dynamic analysis with and without taking into account soil-structure interaction (SSI)

Full size image

Figure 17 presents the curvature ductility demand at the base of column C3. Its maximum value is 1.3 without taking into account SSI and increases to 2.0 if SSI is taken into account. Overall, the results of the non-linear dynamic analyses justify the damage observed in the structure. The role of soil-structure interaction is clearly unfavourable for this building. However, the analyses also indicates damage to members that didn’t suffer any during the earthquake, as was the case with the pushover analyses too. This could be attributed to the actual strength of materials (mainly of steel bars) which may be higher than the one considered.

Fig. 17
figure 17

Time history of curvature ductility demand in base of column C3 from non-linear dynamic analysis with and without taking into account soil-structure interaction (SSI)

Full size image

Dividing the maximum curvature ductility demand in columns (maximum in both directions) from non-linear dynamic analysis taking into account soil-structure interaction (SSI) with the curvature ductility demand from non-linear dynamic analysis without taking into account SSI the ratio λμ = μSSI/μFIXED is obtained. The maximum value of this ratio is depicted for all columns on the longitudinal axes of the building in Fig. 18. Evidently, for most columns the demand is higher when SSI is taken into account.

Fig. 18
figure 18

Ratio of maximum curvature ductility demand in columns (in both directions) from non-linear dynamic analysis taking into account soil-structure interaction (SSI) to corresponding curvature ductility demand from non-linear dynamic analysis without SSI

Full size image

9 Conclusions

On 26th January and 3rd February 2014, two strong earthquakes of magnitude Mw6.1 and Mw6.0, respectively, shook Cephalonia island in western Greece and were followed by several aftershocks. High accelerations were recorded by 8 strong motion instruments on the island.

While the majority of the buildings in Cephalonia have been designed according to past generations of codes based on allowable stress design procedures and relatively low base shear coefficients, most of them demonstrated a satisfactory behaviour considering the severity of the earthquakes.

In the capital city of Argostoli, damage was limited to a small number of structures including a school complex which is the focus of the study presented herein. It is a two-storey reinforced concrete structure designed in the late 70's. A thorough post-earthquake investigation was carried out which revealed damage in the structure. Although the damage was relatively moderate, the structural assessment according to EC8-3 showed that seismic strengthening is required. Results from linear modal response spectrum analyses, non-linear static analyses and non-linear time history analyses are in meaningful agreement with the observed damage.

Based on the analyses at hand the following can be concluded:

  1. (1)

    Elastic analysis with the modal response spectrum method following EC8-3 (GCSI-2022) provisions showed that the building was rather under-designed relative to current standards. There is an indication of failure in most vertical elements of the ground floor.

  2. (2)

    Displacement demands resulting from static pushover analyses are in good agreement with results from more rigorous non-linear time history analyses. The rather small damage observed in some of the vertical elements on the ground floor is reproduced by both approaches. However, both methods predict failures in members where no damage was observed. Given the uncertainties associated with the numerical prediction of the observed damage (steel and concrete material properties, reinforcement ratio, epistemic uncertainty, SSI and record-to-record variability) it is anticipated that the matching is mostly qualitative.

  3. (3)

    Accounting for SSI shows that the inter-storey drifts are much higher in the longitudinal direction than the transversal one, therefore it may explain the infill wall detachment with out of plane action observed in both floors in the longitudinal direction.

  4. (4)

    Overall, despite the moderately soft soil conditions (VS = 180 m/s), the role of soil-structure interaction was found to be important due to the rather small size of the footings and the significant stiffness of the building. This is confirmed by the increase in the building periods due to SSI, by approximately 24% and 27% in the longitudinal and the transverse direction, respectively. Taking into account SSI results in significantly higher (and probably more accurate) predictions of displacements, inter-storey drift ratios and ductility demands that better reproduce the observed damage.

  5. (5)

    Furthermore, the curvature ductility demand in vertical elements computed from non-linear dynamic analyses taking into account SSI, is higher than the one obtained when SSI is ignored, in nearly all elements. This is additional evidence that SSI effects can lead to higher seismic demand.

  6. (6)

    Finally, although the building has been designed according to outdated seismic provisions and despite the severity of the event, the observed damage can be deemed low to moderate. This is aligned with the performance of other buildings in the town of Argostoli during the specific seismic sequence. However, the problem remains that substandard buildings, which form a large part of the building stock in seismic regions in Greece are prone to significant damage if excited by the demands prescribed in modern codes. The pre-earthquake strengthening of such buildings at a large scale is not a trivial task from a financial and social viewpoint, but is of importance particularly for structures such as the school building studied herein.