Seismic codes divide buildings into three categories: low (N < 5), medium (5 ≤ N ≤ 15) and high-rise (N > 15) depending on the number of storeys (N). Given the range of number of storeys within medium-rise category (5 ≤ N ≤ 15), the effect of N was investigated for N of 5, 7, 10, 12 and 15 for two column sizes: 0.5 X 0.5 m and 0.5 X 1m, as detailed in Table 2.

The FE results, plotted on Figs. (4-7), indicate that the increase in the number of storeys N causes an increase in lateral deflection and an increase in the difference in lateral displacement between flexible and fixed-based structures. For example, at abs max at the 5^th level, the lateral deflection of S5 to S15 flexible cases is amplified from 0.0257 m to 0.0679 m for C 0.5 X 0.5 m and from 0.0126 m to 0.0518 m for C 0.5 X 1m under El-Centro (1940). Note that even though a shear wall-based structure design should be used for structures having more than 10 storeys as well as for structures under Northridge (1994) earthquake, we modelled these cases on moment frame structures to compare with the behaviour of the frame structures under El-Centro (1940) excitation.

As shown in Figs. (4-11), an increase in N results in an increase in lateral deflection and levelling shear force results under both C sizes. For C 0.5 X 0.5 m, midrise structures having 5 ≤ N ≤ 15 are divided into two categories: (1) 5 ≤ N < 10 and (2) 10 ≤ N ≤ 15. This categorization is modified for C 0.5 X 1 m to become 5 ≤ N ≤ 10 and 10 < N ≤ 15 under El-Centro (1940) and 5 ≤ N < 7 and 7≤ N ≤ 15 under Northridge (1994). As detailed in Table 3, the ratio of flexible to fixed-based structures lateral deflection is lower than one for the first category and greater than one for the second category. Therefore, SSI is beneficial to the first category and detrimental to the second. In the second category, the seismic behaviour of frame structures founded on silty sandy soil is similar to the behaviour of structures rested on soft soils [3, 14, 35]. In the literature, Farghali et al. [14], Shehata et al. [3] and Nadar et al. [35] obtained an increase in storey displacement and drift with SSI cases. In addition, they showed that the storey number amplifies SSI effects. This amplification was described by higher storey displacement responses, mostly affected at the lower and the upper storeys. In this study, this behaviour can be clearly detected under Northridge (1994) excitation at abs max where the lateral displacement of the different structures increases with N until the 5^th and the 6^th level. The lateral displacement then changes direction and re-increases with the increase in N to hit the maximum displacement at the upper storey level (Figs. 4b-7b).

In this study, results show that within the second structures’ category, as N increases, the ratio of flexible to fixed based structures’ base shear decreases. In general, the structure's base shear tends to increase or decrease depending on the stiffness of the structure and the properties of the soil. Therefore, when the lateral deflection decreased, we observed an increase in base shear values for fixed based-structures and a decrease in base shear values for flexible structures when C increased from 0.5 X 0.5 m to 0.5 X 1 m (Table 3). In Figs. (8-11), the results show that for the same level, the increase in N amplifies the ratio of flexible to fixed cases shear forces. For example, for C 0.5 X 0.5 m, at the 5^th level, the level shear force ratio increases from 0.40 to 0.98 to 1.52 under El-Centro (1940) and from 0.51 to 1.01 to 1.14 under Northridge (1994) for S5 to S10 to S15 respectively. In Figs. (4c-7c), the results show the important effects of N and C on the performance of midrise structures. In fact, the increase in storey number and decrease in column size lead to shifts in inter-storey drift curves to life-threatening and hazardous categories. In addition, lateral deflections and therefore inter-storey drift results are more affected by the higher PGA Northridge (1994) earthquake than the lower PGA El-Centro (1940) earthquake even though both seismic loads have similar magnitudes. For example, based on the performance limit categories set by the Australian Earthquake code [54], S10, S12 and S15 structures shift from “life safe” category under El-Centro (1940) earthquake to “near collapse” and “collapse” categories under Northridge (1994) earthquake. Note that this code considers “life safe” as the acceptable limit category.

In this paper, the amount of distortion and rocking components were calculated based on Trifunace et al. [58, 59] relationships. These components are a function of the maximum foundation rocking angle and lateral deflections at the top of flexible and fixed-based structures. The results indicate that as N and C increase, foundation rocking angle increases (Fig. 12). This increase is more ostensible under Northridge (1994) than under El-Centro (1940) and more under C 0.5 X 1 m than under C 0.5 X 0.5 m. In addition, results show that these effects are more pronounced for second structures’ category. This is in line with Torabi and Rayhani [37] who found that rigid slender structures are highly affected by SSI effects displayed in their foundation rocking angle. For illustration, foundation rocking angle increases from 0.11 ° to 0.44 ° for C 0.5 X 0.5 m and from 0.10 ° to 0.78 ° for C 0.5 X 1 m for S5 to S15 flexible structures under the influence of Northridge (1994) earthquake. The increase in foundation rocking angle is reflected by an increase in the amount of rocking component. While the amount of distortion component depends on lateral deflection values at top of flexible and fixed-based structures. For example, under Northridge (1994) earthquake for C 0.5 X 1 m., foundation rocking angle increases from 0.10 ° to 0.78 ° for S5 to S15. Therefore, the lateral deflection at the top of S5 structure for C 0.5 X 1 m equal to 0.0699 m is divided into 0.029 m rocking component and 0.045 m distortion component, while S5 fixed-base structure lateral deflection is equal to 0.167 m and is due entirely to distortion component.

Response spectrum curves, plotted on Figs. (13 & 14), are usually used in seismic codes designs to calculate base shear values as a function of the structure-foundation-soil frequency. In this study, the results show that response spectra amplitude S_a is higher for waves beneath structures compared to FF waves even though these waves are not affected by the variation of N by more than 10% under C. Note that when C increases, the acceleration at the base of the structure-foundation slightly increases. This is directly related to the natural frequencies of the structures and soil. Note that the obtained natural frequencies/ fundamental periods from ABAQUS differ from the periods calculated using different codes (Table 4). Seismic codes underestimate the building fundamental period that is a function of the height of the building and not the characteristics or geometries of the beams, columns, etc. that form the structure. For example, EC-8 [56] and ASCE-7 [55] estimate the fundamental period of S10 and S10-e-1m structures as 0.961 s and 0.995 s respectively, while the fundamental period obtained from ABAQUS for S10 case is equal to 1.47 s for C 0.5 X 0.5 m and 1.282 s for C 0.5 X 1 m and for S10-e-1m case, it is equal to 2.041 s for C 0.5 X 0.5 m and 2.023 s for C 0.5 X 1 m (Table 4). This is similar to the results obtained by Shehata et al. [3] who proved that codes are conservative and underestimate the structural period that is a function of SSI. In addition, as shown in Table 4, the results are in accordance with Farghali et al. [14], Shehata et al. [3] and Nadar et al. [35] who obtained an increase in the structural time period with SSI cases.

SSI effects are divided into inertial and kinematic interactions. Inertial interactions are related to the structure and foundation parameters; i.e. linked to the increase in the mass of the structure-foundation system. On the other hand, kinematic effects are related to the structure’s base motions that depend on the soil properties and the earthquake characteristics (magnitude and PGA). The significant contribution of the number of storeys may be due to the extra mass produced by an increase of N that causes an increase in inertial effects. This mass alters the dynamic characteristics of the structure-foundation-soil system and affects the energy absorbed by the structure [36]. The increase in C is expected to decrease the structure’s lateral deflection. Nevertheless, even though fixed-based structures’ lateral deflection decreases with the increase in C, flexible structures’ lateral deflection only decreases in S5, S7 and S10 under El-Centro (1940) while only decreasing in S5 under Northridge (1994). In addition, a decrease in the ratio of flexible to fixed-based structures base shear is obtained in all simulated cases when C increases.

In this study, we observed that as N and C increase, inertial effect in the form of absorbed energy by the structure increases and causes excessive lateral deflection. This absorbed energy depends not only on N and C but also on the earthquake’s characteristics. The lateral defection results are in accordance with wave acceleration results summarized in Table 5 . As N and C increase, the mass of the structure increases. Therefore, bigger column sizes C 0.5 X 1 m cases exhibit higher accelerations at top of the fixed structures than C 0.5 X 0.5 m, reflecting the energy absorbed by the structure. For example, for the S15-C0.5X0.5m case under Northridge (1994), the maximum acceleration increases from 5.31 m/s ² at the base to 6.92 m/s ² at the top in the flexible case while increasing to 8.69 m/s ² at the top of the fixed case. In addition, results indicate that the maximum wave acceleration value decreases from fixed to flexible cases under all scenarios, reflecting SSI effects.

To evaluate the influence of kinematic effects on the foundation input/wave acceleration, the accelerations beneath the structure-foundation system are compared to the FF response. In Figs. (15 & 16), we notice that accelerations beneath structure-foundation systems are always higher than accelerations of FF under both seismic events, showing the effect of SSI. In general, the motion at the base of the structure-foundation system is divided between translation and rotation. The translational component is related to the base slab averaging while the rotational component is related to the rocking of the foundation. Kramer [1] argued that the variation between the base motion and FF is linked to the inability of the foundation to match FF deformation, i.e. kinematic interaction. Then, depending on SSI, the motion at the base of the structure-foundation system can be greater or weaker than FF motion. Similar to the results of Rayhani and El Naggar [45], the effect of kinematic interaction was manifested in this study by the amplification of the wave below the structure compared to FF motion (Table 5). On the other hand, the effect of soil type used (silty sand) was detected by the attenuation of the wave when it reached FF under both earthquakes. Therefore, the results found in this paper are not aligned with Farghaly and Ahmed [14], Hokmabadi et al. [28], Tabatabaiefar et al. [41] and Fatahi and Tabatabaiefar [46] who used clayey soil and obtained an amplification of the wave at FF under different seismic loads.

Building codes consider, through simplified methods, that the base excitation is the same as FF motion. Kim and Stewart [60] provided an analytical solution that calculates the ratio of the response spectral ordinate imposed on the foundation to the free-field (FF) spectral ordinate “RRS” for surface shallow foundation. This relationship was later reflected in FEMA-440 [61] and ASCE 41-06 [62] as follows:

(3)

where: b_e refers to the effective foundation size (ft) and T_eq refers to the effective period of the foundation-structure system considering any lengthening due to foundation flexibility or structural yielding.

To evaluate the effectiveness of this relationship on FE results, the RRS of the different simulated models were calculated. As illustrated in Table 6 , the use of the relationship in Eq. (3) on surface shallow foundation structures underestimates the RRS ratio, especially for low frequency structures (S5 and S7). In fact, RRS varies between 1.4 and 1.72 for El-Centro (1940) simulations and between 1.6 and 2.05 for Northridge (1994) simulations while RRS calculated using Eq. (3) is around 1. Therefore, the high RRS value indicates the important contribution of the kinematic effect. Results show that even though the peak frequencies of the simulated models are almost the same for each earthquake, the first and second mode natural frequencies of flexible S10 and S12 cases are very close to the peak response frequencies obtained from Fourier Analysis. This causes kinematic effects to be the least significant in S10 and S12 models having the lowest RRS values. For illustration, for C 0.5 X 0.5 m under Northridge (1994), RRS is equal to 1.91, 1.81, 1.66, 2.03 and 2.01 for S15-e-1.5m, S12-e-1.2m, S10-e-1m, S7-e-0.7m and S5-e-0.5m respectively. As N increases and C decreases, the structure’s fundamental period increases [63]. This is in line with Luco and Wong [64] and Velestos et al. [65] who found that SSI effects are more significant for high frequencies (short period) than low frequencies (long period) of excitation structures. In addition, results are in line with Aviles and Perez-Rocha [66] who proved that SSI effects are larger for tall and slender structures than for short and squat structures of the same period. As C increases to 0.5 X 1m, RRS decreases to 1.83, 1.6, 1.94, 2.05 and 2.0 for S15-e-1.5m, S12-e-1.2m, S10-e-1m, S7-e-0.7m and S5-e-0.5m respectively. Therefore, the deviation in the natural frequencies of the structure and soil leads to the deviation in SSI effects between the different models. As a result, as N and C increase, inertial effect dominates the kinematic effect and causes the categorization within medium rise building category.

Fig. (15). Variation of frequency content with the amplitude for C 0.5 X 0.5 m under a) El-Centro (1940) and b) Northridge (1994) at FF and at below the centre of the structure-foundation system-effect of structure’s number of storeys.

Fig. (16). Variation of frequency content with the amplitude for C 0.5 X 1 m under a) El-Centro (1940) and b) Northridge (1994) at FF and at below the centre of the structure-foundation system-effect of structure’s number of storeys.

Fig. (17). Variation of lateral deflection a) at max top and b) at abs max and c) Variation of inter-storey drift at max top with storey number for C 0.5 X 0.5 m-effect of raft size.

Fig. (18). Variation of lateral deflection a) at max top and b) at abs max and c) Variation of inter-storey drift at max top with storey number for C 0.5 X 0.5 m-effect of raft thickness.

Fig. (19). Variation of maximum foundation rocking angle and rocking and distortion components -effect of raft size.

In order to address and quantify the effects of varying the geometry of the raft foundation, a baseline case of 15 storey midrise concrete frame structure was considered. For a 1.5 m raft thickness, the raft size was varied between 1.3, 1.5 and 2B (S15-xB) with xB referring to the width of the structure corresponding to 20 m (1.3B), 22.5 m (1.5B) and 30 m (2B) (B=15 m). Afterwards, for a 20 m width raft size, raft thicknesses were varied between 1, 1.5 and 2 m (S15-e-y) with e-y referring to the thickness. Note that the effects of raft size and thickness were tested for 0.5 X 0.5 m and 0.5 X 1 m column sizes (Table 2).

Increasing the dimension of the raft foundation alters the dynamic characteristics of the structure-raft-soil system. In fact, the structure-raft-soil system absorbs more energy from the seismic load when the contact between the raft foundation and surrounding soil increases. This energy is then transferred to the structure and affects the seismic structural response. The FE results show that for the effect of raft dimension, column size strongly affects the behaviour of S15 flexible structures. This trend is more ostensive under raft size than under raft thickness.

The results, in Figs. (17 & 18) and in Tables 7 and 8, show that increasing the size of the raft amplifies the ratio of lateral deflection of flexible to fixed base δ_flexible /δ_fixed S15 structures for C 0.5 X 0.5 m and attenuates this ratio for C 0.5 X 1 m. Furthermore, the results indicate that for the same 1.3B raft size (including different raft thicknesses), δ_flexible (C 0.5 X 0.5 m) is lower than δ_flexible (C 0.5 X 1 m) while for 1.5B and 2B raft sizes, δ_flexible (C 0.5 X 0.5 m) is greater than δ_flexible (C 0.5 X 1 m). The decrease in lateral deflections is reflected in inter-storey drifts results where structures move to lower and safer categories (Fig. 17c). The results show that increasing the raft thickness from 1 to 2 m only increases δ_flexible /δ_fixed from 1.9 to 1.98 and from 1.59 to 1.64 for C 0.5 X 0.5 m under El-Centro (1940) and Northridge (1994) respectively. Nevertheless, δ_flexible /δ_fixed is slightly affected by the change in raft thickness for C 0.5 X 1 m under both earthquakes. As a result, raft thicknesses within the same column size slightly affects lateral deflection and inter-storey drift results (Fig. 18).

Results indicate that accelerations of the seismic waves at the base and top of the flexible structures decrease with the increase in column size under both seismic loads (Tables 9 and 10). In addition, for fixed structures, as the column size increases, the accelerations of the seismic waves increase while lateral deflections at top of these structures decrease. Therefore, we note that increasing the column size in the direction of the earthquake load is beneficial to structures. The contribution associated with column size is more apparent under El-Centro (1940) than under Northridge (1994). In addition, it is more ostensive for 1.5B and 2B raft sizes cases than for 1.3B raft size case. The extra mass coming from both columns and raft sizes as well as the extra contact coming from the size of the raft foundation leads to an important inertial effect rise ending in a reduction in the structures’ lateral deflections. This is in line with Nguyen et al. [33] who obtained a reduction in lateral deflection curves for flexible structures rested on clayey soil when they increased the raft sizes from 1.1B to 2B for 4 different earthquake loads.

The numerical results show that as the column size increases and the raft size and thickness decrease, foundation rocking angle increases (Figs. 19-20). In addition, under El-Centro (1940) for both column sizes, the increase in raft dimension leads to a decrease in the amount of distortion component and an increase in the amount of rocking component. However, the opposite behaviour is obtained under Northridge (1994) earthquake. It is worth noting that minimal changes in the amount of distortions and rocking components are obtained for the effect of raft thickness caused by the small changes obtained in foundation rocking angles. For illustration, under Northridge (1994) earthquake for 1.3B case, foundation rocking angle increases from 0.44 ° to 0.78 ° when increasing column size from C 0.5 X 0.5 m to C 0.5 X 1 m. Therefore, the lateral deflection at the top of the structure for C 0.5 X 0.5 m equal to 1.034 m is divided into 0.346 m rocking component and 0.688 m distortion component. On the other hand, the lateral deflection at the top of the structure for C 0.5 X 1 m increases to 1.055 m and is divided into 0.338 m rocking component and 0.718 m distortion component. In addition, for 1.3B cases, increasing the raft thickness from 1 m to 2 m for C 0.5 X 1 m only attenuates the rocking angle from 0.24 ° to 0.23 ° under El-Centro (1940) earthquake while the rocking angle is attenuated from 0.8 ° to 0.6 ° under Northridge (1994) earthquake, thus reflecting the impact of the seismic excitation. Since foundation rocking angle is not related to the structure’s failure, an increase in its value can have a beneficial effect on flexible structures. For the effect of raft size for C 0.5 X 0.5 m and the effect of raft thickness, the increase in foundation rocking angle is accompanied by a reduction in levelling shear force and lateral deflection.

In this paper, the results show that, in line with Nguyen et al. [33], S_a is slightly affected by the increase in raft size (Figs. 21 & 22). In addition, S_a is slightly affected by the increase in column size. This is related to the natural frequencies of the tested models that did not quite differ between the different tested models. In fact, 2B to 1.3B C 0.5 X 0.5 m and C 0.5 X 1 m, S_a rises only 1.57% and 2.17% under El-Centro (1940) and 2.84% and 2.79% under Northridge (1994) respectively. Therefore, the slight increase in S_a leads to a small increase in the base shear results. The obtained results indicate that base shear ratio of flexible to fixed-based structures increases with the increase in raft dimensions under both column sizes (Figs. 23 & 24) and (Tables 7 and 8). It is worth noting that this increase is more apparent between 1.3B and 1.5B cases than between 1.5B and 2B cases. Therefore, bigger raft sizes result in safer design when SSI is considered. The bigger the contact between the raft and the soil and the larger the column size, the more SSI effect divided between inertial and kinematic is pronounced. For example, under the influence of El-Centro (1940), base shear ratio of 1.3B and 2B increases from 1.0 to 1.2 for C 0.5 X 0.5 m and from 0.95 to 1.01 for C 0.5 X 1 m (Table 7). While base shear ratio of 1 m and 2 m raft sizes also under El-Centro (1940) increases from 0.98 to 1.02 for C 0.5 X 0.5 m and from 0.93 to 0.96 for C 0.5 X 1 m (Table 8).

The accelerations response spectrum curves show that similar to the effect of structure’s number of storeys, the accelerations beneath structure-foundation systems are always higher than acceleration at FF under both seismic events, therefore showing the effect of SSI (Figs. 25 & 26). In addition, the fundamental periods obtained from ABAQUS are larger than the fundamental periods obtained from the different seismic codes (Table 4). Finally, the RRS obtained from FE is greater than the calculated RRS based on Eq. (3) (Table 6).

To assess which frequencies have detrimental effects on the tested S15 structure, the ratio of spectral acceleration at the base of the structure to the spectral acceleration at FF condition S _a,FE / S _a,FF was calculated. Results indicate that even though the peak frequencies of the different simulated models are very close to FF condition for both seismic loads, the response spectra ratio (S_a,FE/S_a,FF) is in the order of 2.29 over the frequency range 0.44-2.59 Hz under El-Centro (1940) and it is in the order of 3.5 over the frequency range 0.1-1.41 Hz under Northridge (1994). As shown in Fig. (27), SSI has detrimental effects on S15 flexible structures cases since the natural frequencies of the simulated models are within these frequencies’ ranges. Similar results were obtained by Rayhani and El-Naggar [45] who obtained that seismic SSI has unfavourable effects on horizontal ground motions for frequencies between 3 and 6 Hz.

As a result, based on overall structural stability and failure, engineers should optimize their design between column size and raft dimension. An increase in raft dimension for the adopted soil properties attracts more shear force and lateral deformation for C 0.5 X 0.5 m. For C 0.5 X 1 m, the raft dimension attracts more shear force and less lateral deformation. In addition, under both column sizes, foundation rocking angle decreases when the raft dimension increases.

Fig. (20). Variation of maximum foundation rocking angle and rocking and distortion components -effect of raft thickness.

Fig. (21). Acceleration response spectrum with 5% damping ratio under a) El-Centro (1940) and b) Northridge (1994) - effect of raft size.

Fig. (22). Acceleration response spectrum with 5% damping ratio under a) El-Centro (1940) and b) Northridge (1994) - effect of raft thickness.

Fig. (23). Variation of level shear force with storey number under a) El-Centro (1940) and b) Northridge (1994)-effect of raft size.

Fig. (24). Variation of level shear force with storey number under a) El-Centro (1940) and b) Northridge (1994)-effect of raft thickness.

Fig. (25). Variation of frequency content with the amplitude under a) El-Centro (1940) and b) Northridge (1994) at FF and at below the centre of the structure-foundation system- effect of raft size.

Fig. (26). Variation of frequency content with the amplitude under a) El-Centro (1940) and b) Northridge (1994) at FF and at below the centre of the structure-foundation system- effect of raft thickness.

Fig. (27). Spectral ratio ( S a,FE / S a,FF ) for 1.3 B cases under a) El-Centro (1940) and b) Northridge (1994).