Introduction

In recent decades, many researchers have conducted studies on structural vibration control to improve the safety and serviceability of high-rise buildings or towers against earthquakes and strong winds. The architectural requirements necessitate sometimes special horizontal or/and vertical irregularities in buildings to meet their functional requirements; these irregularities require a sensitive deal with these structures, especially in seismic design. Steel buildings, compared to reinforced concrete (RC) buildings, are characterized by lightweight and speed in implementation; however, the disadvantage of the steel structures is the low value of damping. The 3D modeling with soil–structure interaction (SSI) effect will give a realistic representation of the seismic behavior of steel structures, and the tuned mass dampers (TMDs) are considered easy and effective mitigation techniques for earthquake resistance especially when implemented in proper locations in the structure.

Kim and Adeli (2005) used a hybrid control system consisting of a passive supplementary damping system and a semi-active tuned liquid column damper (TLCD) system to control the seismic behavior of the irregular steel high-rise buildings and concluded that the hybrid damper-TLCD control system significantly reduces the responses of irregular buildings subjected to various earthquake ground motions.

Abd-el-rahim and Farghaly (2010) studied the seismic response of different irregular in plan building models and concluded that T-shape plan models have the worst behavior when subjected to an earthquake in increasing the base shear and top displacements, and they also suggested that structural separation of the T, L, or U shape in plan is a good solution to decrease the straining actions resulting from an earthquake.

Farghaly and Ahmed (2012) studied the effect of different arrangements of TMDs on the seismic response of a 3D symmetrical tall building model and concluded that the response of the structure can be dramatically reduced using groups of TMDs with a specific arrangement in the model.

Pnevmatikos and Hatzigeorgiou (2014) studied the seismic response of active or semi-active control for irregular buildings based on eigenvalues modification, by proposing a reduction of the seismic response of irregular structures by control devices equipped with the “pole placement” control algorithm and the calculation of suitable poles, and concluded that this control strategy can achieve a sufficient reduction of the response of one story or to multi-story irregular buildings.

Bigdeli et al. (2014) studied the vibration control of 3D irregular buildings using a developed neuro-controller strategy and their results showed that the proposed control algorithm is effective in structural control.

Georgoussis et al. (2015) proposed an approximate method for the seismic analysis of multi-story asymmetric setback buildings with mass and stiffness irregularities.

Nezhad and Poursha (2015) studied the seismic response of vertically irregular building frames with stiffness, strength, combined-stiffness-and-strength and mass irregularities by either non-linear response history analysis (NL-RHA), or consecutive modal pushover (CMP) or modal pushover analysis (MPA) method, and their results showed that the CMP and MPA methods can accurately compute the seismic demands of vertically irregular buildings, however, less accurate especially in estimating plastic hinge rotations for weak or weak-and-soft top and middle stories.

Stathi et al. (2015) proposed a reliable index (ratio of torsion, ROT) capable of assessing the torsional effect in plan irregular buildings and found that the proposed index provides a reliable prediction of the magnitude of torsional effect for all test examples considered.

Kaveh et al. (2015) determined the optimum parameters of tuned mass dampers to minimize the dynamic response of multi-story building systems under seismic excitations using the Charged System Search (CSS) optimization algorithm.

Hirde and Aher (2016) studied the performances of T-shape and L-shape irregular buildings located in a severe earthquake zone and identified the most vulnerable building among them to be the T-shape and retrofitted it with X steel bracings.

Ercolino et al. (2016) performed non-linear static analyses to investigate the influence of the infill panels on the seismic response of an existing irregular RC building and found that the infill panels change the behavior of the building in terms of strength and stiffness at different seismic intensity levels and brittle failures in structural elements could be caused by either the local interaction with infills or the failure of the strength.

Farghaly (2017) studied the TMD control of an L-shape in plan RC high-rise building considering soil–structure interaction and concluded that the length-to-width ratio affects the response and that when the soil gets stronger the best TMDs distribution is on the top at the model corners, while when the soil gets softer, the best TMDs’ distribution is through the elevation of the model.

Vijayanarayanan et al. (2017) presented a simple procedure to estimate story stiffness using only the properties of the fundamental lateral translational mode of oscillation (i.e., the natural period and associated mode shape) and provided simplified analytical expressions to identify the stiffness irregularity in both low- and mid-rise buildings.

He et al. (2017) proposed a new type of tuned mass damper with tuned mass blocks, orthogonal poles, and torsional pendulums (TMDPP) to simultaneously control the translational responses and the torsional angle of asymmetry structures. The damping capacity of the TMDPP was verified by the time history analysis of an eccentric structure, multidimensional earthquake excitations were considered, and the results showed that the performance of TMDPP is superior to the traditional TMD.

Bekdaş and Nigdeli (2017) proposed a metaheuristic-based optimization approach for the optimum design of tuned mass dampers (TMDs) implemented to seismic structures considering soil–structure interaction (SSI), where two metaheuristic algorithms (namely the harmony search algorithm and the bat algorithm) were employed, and the bat algorithm was found to be advantageous in minimizing the optimization objective and finding a precise optimum value.

Mirzai et al. (2017) proposed optimum parameters of tuned mass dampers (TMDs) using the gravity search algorithm (GSA) and particle swarm optimization (PSO) to reduce the responses of tall and typical buildings.

Rahman et al. (2017) proposed a new type of passive mass damper system for controlling wideband earthquake vibrations, called Multiple Wall Dampers (MWD) which does not require additional mass for the damping system, because the boundary wall mass of the building was used as a damper mass, and the obtained structural responses under different earthquake forces demonstrated that the MWD is one of the most capable tools for reducing the responses of new and existing multi-storied buildings.

Nazarimofrad and Zahrai (2018) developed a mathematical model for the seismic control of an irregular plan multi-story building using two active tuned mass dampers (ATMDs) at the center of mass on the top floor with fuzzy logic control algorithm considering SSI effect.

Amini et al. (2018) showed that the simple adaptive control method (SACM) is successful in reducing the response of asymmetric buildings with rotationally non-linear behavior under seismic loads, and furthermore, the results of the SACM were very close to those of the LQR algorithm.

Gupta and Dhawale (2019) performed a comparative study of RC buildings with and without vertical irregularity subjected to earthquake and wind loading.

Kamgar et al. (2019) compared different optimization criteria for the optimal design of the TMD system considering the effects of SSI in a high-rise building, employed the whale optimization algorithm (WOA) to optimize the parameters of the TMD system, and concluded that the soil type and selected objective function efficiently affect the optimal design of the TMD system.

Naderpour et al. (2019) investigated the effectiveness of a hybrid control strategy, combining base isolation and non-traditional tuned mass dampers (TMDs) (i.e., TMDs with dashpots directly connected to the ground) in suppressing the seismic vibrations of high-rise buildings.

Kontoni and Farghaly (2019a) investigated the mitigation of the seismic response of a cable-stayed bridge with soil–structure interaction (SSI) effect subjected to four different earthquakes by using tuned mass dampers and spring dampers with different placements in four different mitigation schemes.

Kontoni and Farghaly (2019b) studied the effect of base isolation and tuned mass dampers (TMDs) on the seismic response of reinforced concrete (RC) high-rise buildings considering soil–structure interaction; a comprehensive study of the combination of TMDs with three different base-isolator types for three different soil types and under five different earthquakes was conducted to show the most suitable hybrid passive vibration control system.

Kontoni and Farghaly (2020) investigated the TMD effectiveness for a steel high-rise building of 15 stories under the dynamic load of wind or four different earthquakes taking into consideration the effect of soil–structure interaction (SSI) and the TMDs were found to be a successful passive resistance method.

Kaveh et al. (2020a) investigated the optimum design of a tuned mass damper inerter (TMDI) to control a 10-story base-excited shear building by selecting the H2 and H∞ norms of three different objective functions to tune two different single and double inerter TMDI configurations using a metaheuristic technique, and concluded that the TMDI is most effective when the optimization goal is to reduce the floor accelerations.

Kaveh et al. (2020b) studied the optimum design of a tuned mass damper inerter (TMDI) to control a 10-story base shear building under seismic loading by minimizing the H∞ norm of the roof displacement transfer function as the objective function for robust building control, and they recommended to optimize and design the TMDI using the MDOF model and not the SDOF model.

Kaveh et al. (2020c) compared the H2 and H∞ algorithms for the optimum design of tuned mass dampers (TMDs) under near-fault and far-fault earthquake motions and concluded that the H∞ objective function is superior to the H2 objective function under NF and FF earthquake excitations.

Fahimi Farzam et al. (2021) presented a comprehensive literature review on the optimal design of various well-known control devices for optimal structural control.

Farghaly and Kontoni (2022) studied the mitigation of seismic pounding (due to five different earthquakes) between two adjacent RC twin high-rise buildings, founded on raft foundation supported on piles inside a liquefaction-prone soil, considering SSI. Contact pounding elements between the two buildings (distributed at all floor levels and at the raft foundation level) were used. The three mitigation methods investigated were the base isolation, the tuned mass damper (TMD) method, and the pounding tuned mass damper (PTMD) method (using PTMDs connected between the two buildings), and it was found that the PTMD method was more efficient than the other two methods in mitigating the earthquake-induced pounding risk.

Fahimi Farzam and Hojat Jalali (2022) proposed a roof-top tandem tuned mass damper inerter (TTMDI) device for the structural control of passive control of buildings under seismic loads.

Hojat Jalali and Fahimi Farzam (2022) proposed an inerter-connected double tuned mass damper for the passive control of buildings under seismic excitation.

Farghaly and Kontoni (2023) investigated the mitigation of the seismic pounding between two L-shape in plan high-rise buildings (HRBs) subjected to earthquake in three different direction cases, including the SSI effect, and found that the most effective technique to mitigate the seismic pounding and help in seismically protecting these adjacent HRBs is the use of a combination of pounding tuned mass dampers (PTMDs) all over the height (at the connection points) together with tuned mass dampers (TMDs) on the top of both buildings.

Kontoni and Farghaly (2023a) investigated the enhancement of the earthquake resistance of RC and steel high-rise buildings by bracings, shear walls and TMDs considering SSI.

Kontoni and Farghaly (2023b) investigated the seismic control of T-shape in plan steel high-rise buildings (HRBs) with SSI effect using six different plan arrangements of (2, 4, 8, 8, 12, 18) tuned mass dampers (TMDs) applied on the top plan of the fixed HRB model, while for the SSI HRB model, these TMDs distributions were applied on the top plan and additionally on two intermediate plans along the HRB height and this three-level TMDs’ distribution was required for the plan arrangement of 2 or 4 TMDs.

Kontoni and Farghaly (2023c) investigated different seismic mitigation schemes of passive tuned mass dampers (TMDs) to mitigate the service and seismic loads affecting an offshore wind turbine (OWT) including the pile–soil–structure interaction (PSSI) effect.

Hojat Jalali et al. (2023) presented a comprehensive study on control algorithms for a building under earthquake ground motions including far-fault and near-fault records, with the presence of soil–structure interaction (SSI) effects.

Zakian and Kaveh (2023) presented an overview of seismic design optimization of structures, focusing on common solution methods, types of optimization problems and goals of optimization, and concluded that, in contrast to regular buildings, there is still a research gap needing to be bridged regarding the optimal design of irregular buildings under earthquake excitations.

The architectural requirements may impose irregularities in the vertical and horizontal directions in the structural design of a high-rise building (HRB). Seismic control of vertically and horizontally irregular steel high-rise buildings subjected to earthquakes needs special devices implemented in proper locations of these structures, to improve the irregular building’s performance under seismic loads. In this paper, two irregular steel HRBs modeled in 3D and equipped by bidirectional TMDs were studied, without and with the soil–structure interaction (SSI) effect, and the seismic response control of each irregular steel HRB was investigated under three different earthquakes, for two different mitigation schemes of TMDs.

Description of numerical models

HRB models

Two 3D steel high-rise building (HRB) models were analyzed by the SAP2000 version 17 (2015) software to show the effect of the vertical irregularity and the horizontal (plan) irregularity with the SSI effect on the seismic response of such structures and investigate the use of TMDs as a control system to mitigate the harmful effects of different frequency earthquakes.

Figure 1a shows the vertically irregular steel HRB model of height 72 m from the ground surface with a step-pyramid shape (each of the 6 square plan-groups had 4 floors with a group height of 12 m) and with steel members designed according to the Egyptian code of steel structures (ECP-205, 2008). Figure 1b shows the vertically and horizontally irregular (L-shaped in-plan) HRB model of height 72 m from the ground surface with a stadium shape (each of the 6 L-shaped plan-groups had 4 floors with a group height of 12 m) and with steel members designed according to the Egyptian code of steel structures (ECP-205, 2008). Figure 1c and d shows the structural plan of each steel HRB model.

Fig. 1.
figure 1

3D models and structural plans of the studied two irregular steel HRBs (dimensions in mm)

Full size image

Two cases were tested for each model, the first was for the fixed base model (without SSI) and the second for raft foundation with 3D soil elements, as shown in Fig. 2 (with SSI). The effect of SSI was taken into consideration in the raft foundation with a 2 m projection at each side of the models and a thickness of 2 m (top and bottom reinforcement mesh of 7#22/m). The soil was a medium type soil (E = 30 MPa, ν = 0.40, ρ = 1.95 Mg/m3). The equations used to find the parameters (stiffness Kx, Ky, Kz and damping Cx, Cy, Cz) of the 3D soil elements (Fig. 2) can be found e.g., in Newmark and Rosenblueth (1971), Dowrick (2009), etc.

Fig. 2.
figure 2

3D soil element representing the soil under the raft foundation

Full size image

Just as a case, in Fig. 1a the raft foundation with 3D soil elements (with SSI effect) is shown, while in Fig. 1b, the fixed base model (without SSI effect) is shown. However, herein the two different irregular steel HRBs were analyzed both without and with the SSI effect.

Mitigation schemes using tuned mass dampers (TMDs)

The tuned mass damper (TMD) device is a control system to mitigate the response of buildings subjected to earthquakes. The parameters of TMDs as per Zahrai and Ghannadi-Asl (2008), are shown below

$${\alpha }_{\mathrm{opt}.}=\frac{1}{1+\mu }\sqrt{\frac{2-\mu }{2}}$$
(1)
$${\upzeta }_{\mathrm{opt}.}=\sqrt{\frac{3\mu }{8(1+\mu )}}\sqrt{\frac{2}{2-\mu }}$$
(2)
$${k}_{d}=4{\pi }^{2}\mu {\alpha }^{2}\frac{{m}_{s}}{{T}_{s}^{2}}$$
(3)
$${c}_{d}=4\pi \mu \alpha \zeta \frac{{m}_{s}}{{T}_{s}},$$
(4)

where αopt. is the optimum frequency ratio, ζ is the damping ratio, ζopt. is the optimum damping, kd is the spring stiffness, cd is the damping, md is the damper mass, and μ is the mass ratio μ = md /ms (ratio of the damper mass md to the building mass ms). Using the values of αopt. and ζopt., optimum values of damping cd and stiffness kd of the TMD can be calculated.

Two different mitigation schemes using TMDs were investigated in this research and Fig. 3 shows the distribution of TMDs in each irregular HRB model. The first mitigation scheme, as shown in Fig. 3a, used 4 TMDs at the top of the vertically irregular (step-pyramid-shaped) HRB model (placed at the corners of the HRB plan), and used 8 TMDs at the top of the vertically and horizontally (L-shaped in-plan) irregular (stadium-shaped) HRB model. The second mitigation scheme, as shown in Fig. 3b, used four groups of 4 TMDs at each change of the plan area of the vertically irregular HRB model (4 TMDs: at the top, at 12 m, at 24 m, and at 36 m from the top of the model, see also Fig. 1a), while used four groups of 8 TMDs (8 TMDs: at the top, at 12 m, at 24 m, and at 36 m from the top of the model) for the vertically and horizontally (L-shaped in-plan) irregular (stadium-shaped) HRB model. The type of TMD used herein is a bidirectional TMD, as shown in Fig. 3c. The relative mass of the designed TMDs was 10% in total for each TMDs floor group.

Fig. 3
figure 3

The two mitigation schemes using bidirectional TMDs: a first mitigation scheme (4 TMDs for the vertically irregular HRB, or 8 TMDs for the vertically and horizontally irregular HRB: at the top of the model); b second mitigation scheme (4 TMDs for the vertically irregular HRB, or 8 TMDs for the vertically and horizontally irregular HRB: at the top, at 12 m, at 24 m, and at 36 m from the top) (dimensions shown in mm); c bidirectional TMD

Full size image

Loads and seismic loads

The loads affecting the models are the own weight of steel members with welding connections and the live load on each floor of the models, and partitions of gypsum board built over the beams of the models. The two models were subjected to three different earthquakes which have different accelerograms, as shown in Fig. 4. These earthquakes were the 1940 El Centro earthquake that occurred in California with a magnitude of 6.9, the 1961 Hollister earthquake that occurred near central California with a magnitude of 5.6, and the 1999 Chi–Chi earthquake that occurred in Taiwan with a magnitude of 7.6. Herein, in one case, the earthquake affects the steel HRB models both in the x- and y-directions, while in the other case, the earthquake affects the steel HRB models only in the x-direction of the models (Fig. 1).

Fig. 4
figure 4

Three earthquake accelerograms

Full size image

Results and discussion

Two different 3D irregular steel HRB models were analyzed. The first steel HRB model of a step-pyramid shape is a vertically irregular HRB (with six different plan areas along its height) without and with the SSI effect subjected to three different earthquakes. The second steel HRB model of stadium shape is a vertically and horizontally (L-shaped in plan) irregular HRB without and with the SSI effect subjected to three different earthquakes. Herein, in one case, the earthquake affects the steel HRB models both in the x- and y-directions, while in the other case, the earthquake affects the steel HRB models only in the x-direction of the models (Fig. 1). The responses under seismic loads of the two 3D irregular steel HRB models were recorded to study the behavior of each model and investigate the most effective TMDs mitigation scheme used in such structures.

Vertically irregular (step-pyramid-shaped) steel HRB

Figures 5, 6, 7, 8 represent the displacements and top accelerations of the vertically irregular (step-pyramid-shaped) steel HRB model subjected to three earthquakes using two different TMDs mitigation schemes, without considering the SSI effect (Figs. 5 and 6) or with the SSI effect (Figs. 7 and 8). Figures 9 and 10 represent the base shear forces and base moments, and the base axial forces, respectively, of the vertically irregular (step-pyramid-shaped) steel HRB model under three earthquakes, without and with the SSI effect, and for two different mitigation schemes. The abbreviations used in Figs. 5, 6, 7, 8, 9, 10 are: “No” which means “No TMDs”, “one dir” which means one-direction earthquake (in the x-direction of the HRB plan), “two dir” which means two directions earthquake (both in the x- and y-directions of the HRB plan), “4TMD” which means “4 TMDs” at the top of the vertically irregular HRB, and “4 × 4TMD” which means 4 groups of “4 TMDs” for the vertically irregular HRB (at the top, at 12 m, at 24 m, and at 36 m from the top).

Fig. 5
figure 5

Displacements in x- and y-directions of the vertically irregular (step-pyramid-shaped) HRB model subjected to three earthquakes, with TMDs mitigation schemes, and with fixed base (without SSI)

Full size image
Fig. 6
figure 6

Top accelerations in x- and y-directions of the vertically irregular (step-pyramid-shaped) HRB model subjected to three earthquakes, with TMDs mitigation schemes, and with fixed base (without SSI)

Full size image
Fig. 7
figure 7

Displacements in x- and y-directions of the vertically irregular (step-pyramid-shaped) HRB subjected to three earthquakes, with TMDs mitigation schemes, and with SSI effect

Full size image
Fig. 8
figure 8

Top accelerations in x- and y-directions of the vertically irregular (step-pyramid-shaped) HRB model subjected to three earthquakes, with TMDs mitigation schemes, and with SSI effect

Full size image
Fig. 9
figure 9

Base shear forces and base moments of the vertically irregular (step-pyramid-shaped) HRB model with TMDs mitigation schemes, without and with SSI effect

Full size image
Fig. 10
figure 10

Base axial forces of the vertically irregular (step-pyramid-shaped) HRB model with TMDs mitigation schemes, without and with SSI effect

Full size image

Figure 5 represents the displacements of the vertically irregular (step-pyramid-shaped) steel HRB model subjected to three earthquakes using two different TMDs mitigation schemes, without considering the SSI effect (i.e., with fixed base). Figure 5ia–iiia shows the displacements in the x-direction (without SSI), where the effects of the one or two directions (in x- and y-directions) earthquakes are mostly nearly similar, so that for the two different TMDs mitigation schemes (for earthquakes in one or two directions), the lowest x displacements occur in the 4 × 4TMDs mitigation scheme which are less than the “no TMDs” cases by nearly 1.75 times for the El Centro earthquake, by nearly 5 times for the Hollister and by nearly 2.5 times for the Chi–Chi, respectively. Figure 5ib–iiib shows the displacements in the y-direction (without SSI), where in the “no TMDs” case, the effect of one-direction earthquake is less than the two directions earthquake case. The lowest y displacements occur at the 4 × 4TMDs mitigation scheme, and the y displacements at the top of both mitigation schemes are similar but are different along the height of the HRB model for the top only 4TMDs and for the 4 × 4TMDs cases. For one-direction earthquake, there is no remarkable reduction in the y displacements, but in two directions earthquakes, the 4 × 4TMDs or the top only 4TMDs cases reduced the y displacements by nearly 1.75 times than the “no TMDs” in the two directions El Centro earthquake case, and by nearly 3 and 4 times less than in the two directions Hollister and Chi–Chi earthquakes, respectively.

Figure 6 represents the top accelerations of the vertically irregular (step-pyramid-shaped) steel HRB model subjected to three earthquakes using two different TMDs mitigation schemes, without considering the SSI effect (i.e., with fixed base). Figure 6ia–iiia represents the top accelerations of the model in the x-direction (without SSI) for the El Centro, Hollister, and Chi–Chi earthquakes. The trend of the top x accelerations in the “no TMDs” case for the three different earthquakes is similar in the one-direction earthquake cases which are increased compared to the top x accelerations in the two directions earthquake cases by nearly 1.6 times for all earthquakes. The top x acceleration for the model with 4TMDs is less than the “no TMDs” case by nearly 4, 7.5 and 2.7 times for the El Centro, Hollister, and Chi–Chi, respectively, where no significant difference is observed between the use of the 4 × 4TMDs through the model or the only top 4TMDs. Figure 6ib–iiib represents the top accelerations of the model in the y-direction (without SSI) for the El Centro, Hollister, and Chi–Chi earthquakes. The trend of the top y accelerations for the different earthquakes is similar in the two directions earthquake cases, where the y accelerations in the “no TMDs” case are increased than the one-direction earthquake cases by nearly 2.5 times in all earthquakes. The top y acceleration in the one-direction earthquake cases for the model with TMDs is less than the “no TMDs” case by nearly 3 times, where for all earthquakes nearly the same trend occurs. The TMDs mitigation schemes were more effective in the y-direction accelerations than in the x-direction accelerations.

Figure 7 represents the displacements of the vertically irregular (step-pyramid-shaped) steel HRB model subjected to three earthquakes using two different TMDs mitigation schemes, considering the SSI effect. Figure 7ia–iiia shows the displacements in the x-direction (with SSI), where the effects of the one or two directions earthquakes are nearly similar, the x displacements for the two different TMDs mitigation schemes (for earthquakes in one or two directions) are less than the “no TMDs” cases, and the lowest x displacements occur at the 4 × 4TMDs case. Figure 7ib–iiib shows the displacements in the y-direction (with SSI), where the effect of one-direction earthquake in the “no TMDs” case is less than the two directions earthquake case, and the lowest y displacements occur at the 4 × 4TMDs case. The y displacements at the top of both models are similar but are different along the height of the model for the top only 4TMDs and for the 4 × 4TMDs cases. For one-direction earthquake, there was a reduction in the y displacements, but in the two directions earthquakes, the 4 × 4TMDs and the top only 4TMDs significantly reduced the y displacements compared to the “no TMDs” case in the two directions El Centro, Hollister, and Chi–Chi earthquakes.

Figure 8 represents the top accelerations of the vertically irregular (step-pyramid-shaped) steel HRB model subjected to three earthquakes using two different TMDs mitigation schemes, considering the SSI effect. Figure 8ia–iiia represent the top accelerations of the model in the x-direction (with SSI) for the Chi–Chi, Hollister and El Centro earthquakes. The trend of the top x accelerations for the different earthquakes in the “no TMDs” case is similar in the one-direction earthquake cases which are increased compared to the top x accelerations in the two directions earthquake cases for all earthquakes. The top x acceleration for the model with 4TMDs is less than the “no TMDs” case for the El Centro, Hollister, and Chi–Chi respectively, where no significant difference is observed between the use of the 4 × 4TMDs through the model or the only top 4TMDs. Figure 8ib–iiib represents the top accelerations of the model in the y-direction (with SSI) for the El Centro, Hollister, and Chi–Chi earthquakes. The trend of the top y accelerations for the different earthquakes is similar in the two directions earthquake cases, where in the “no TMDs” case, the y accelerations are increased than the one-direction earthquake cases in all earthquakes. The top y acceleration in the one-direction earthquake cases for the model with TMDs is less than the “no TMDs”, where for all earthquakes, nearly the same trend occurs. The TMDs mitigation schemes were more effective in the y-direction accelerations than in the x-direction accelerations.

The top accelerations for the vertically irregular steel HRB model with the SSI effect are higher than the corresponding values of the fixed base case, and thus, the TMDs mitigation schemes are more effective in reducing the top accelerations in the SSI than in the fixed base case.

Figure 9 represents the base shear forces and base moments of the vertically irregular (step-pyramid-shaped) steel HRB model under three earthquakes, without and with the SSI effect, and for two different mitigation schemes. Figure 9ia–iiia shows the base shear forces in the x- and y-directions for three different earthquakes. The maximum base shear forces appear in the fixed base cases, while the SSI effect reduces the base shear values for all three earthquakes. Figure 9ib–iiib shows the base moments in the x- and y-directions for three different earthquakes. The highest values of the base moment appear in both the x- and y-directions for the fixed cases. The mitigation schemes using top TMDs or TMDs through the model (i.e., along the upper half-height of the HRB model) give a significant reduction of the base moments for one- or two-direction earthquake than in the no TMDs case, and this trend is constant for each earthquake used.

Figure 10 represents the base axial forces under three different earthquakes with different mitigation schemes. In the fixed and SSI cases and for the mitigation schemes, the change of the axial force values is not significant, considering the increase of the total axial load as a result of the added loads of the TMDs control systems.

Vertically and horizontally irregular (stadium-shaped) HRB

Figures 11, 12, 13 and 14 show the displacements, the top accelerations, the base shear forces and base moments, and the base axial forces, respectively, for the vertically and horizontally (L-shaped in-plan) irregular stadium-shaped HRB model with and without TMDs control system, and also without and with the SSI effect. The abbreviations used in Figs. 11, 12, 13, 14 are: “No” which means “No TMDs”, “one dir” which means one-direction earthquake (in the x-direction of the HRB plan), “two dir” which means two directions earthquake (both in the x- and y-directions of the HRB plan), “8TMD” which means “8 TMDs” at the top of the vertically and horizontally irregular HRB, “4 × 8TMD” which means 4 groups of “8 TMDs” for the vertically and horizontally irregular HRB (at the top, at 12 m, at 24 m, and at 36 m from the top), and “ssi” means “SSI effect”, while when “ssi” is not mentioned means “fixed model”; for example: “No one dir” means “No TMDs, one-direction earthquake, fixed model”, while “No one dir ssi” means “No TMDs, one-direction earthquake, SSI effect”.

Fig. 11
figure 11

Displacements in x- and y-directions of the vertically and horizontally irregular stadium-shaped HRB model with TMDs mitigation schemes, without and with SSI effect

Full size image
Fig. 12
figure 12

Τop accelerations in x- and y-directions of the vertically and horizontally irregular stadium-shaped HRB model HRB model with TMDs mitigation schemes, without and with SSI effect

Full size image
Fig. 13
figure 13

Base shear forces and base moments of the vertically and horizontally irregular stadium-shaped HRB model with TMDs mitigation schemes, without and with SSI effect

Full size image
Fig. 14
figure 14

Base axial forces of the vertically and horizontally irregular stadium-shaped HRB model (under Chi–Chi, Hollister, or El Centro) with TMDs mitigation schemes, without and with SSI effect

Full size image

Figure 11 shows the displacements in the x- and y-directions for the vertically and horizontally (L-shaped in-plan) irregular stadium-shaped HRB model with and without TMDs control system, and also without and with the SSI effect. Figure 11ia–iiia shows the displacements in the x-direction, for different model cases under three different earthquakes, where the two directions earthquake affects the x displacements more than the one-direction earthquake. The x displacements in the fixed base case are less than the x displacements with SSI effect, the TMDs mitigation schemes generally affect the fixed base model by nearly 6 times than the SSI model, and the 4 × 8TMDs and 8TMDs reduce the x displacements at SSI model both with the same efficiency, where it has been reduced by about 3 times than the no TMDs case. Figure 11ib–iiib shows the displacements in the y-direction, for different model cases under three different earthquakes, where the two directions earthquake affects the y displacements more than the one-direction earthquake for both fixed and SSI cases. The TMDs mitigation schemes generally affect both fixed base and SSI cases, the 4 × 8TMDs and 8TMDs are effective in reducing the y displacements at the SSI model, but in the 4 × 8TMDs case reduced by twice than the 8TMDs case, this is for the El Centro earthquake and a similar trend holds for the Hollister and Chi–Chi earthquakes.

Figure 12 shows the top accelerations in the x- and y-directions for the vertically and horizontally (L-shaped in-plan) irregular stadium-shaped HRB model with and without TMDs control, and without and with SSI effect. Figure 12ia–iiia shows the accelerations in the x-direction, for different model cases under three different earthquakes, where the two directions earthquake with SSI effect gives the greater values than the rest cases. The TMDs whatever their configurations reduce the top accelerations nearly by the same values (nearly 16 times than the no mitigation scheme); this trend is repeated for each earthquake used in this study. Figure 12ib–iiib represents the top accelerations in the y-direction, where the highest top acceleration is recorded in the two directions earthquake without a mitigation scheme, for all earthquakes. The highest values of accelerations occurred for the El Centro earthquake, the mitigation scheme reduced the top y acceleration by nearly 12, 4 and 7 times in the two directions earthquakes of El Centro, Hollister, and Chi–Chi, respectively.

Figure 13 represents the base shear forces and base moments for the vertically and horizontally (L-shaped in-plan) irregular stadium-shaped HRB model with and without TMDs mitigation schemes, and also without and with the SSI effect. The mitigation schemes with TMDs affect the HRB model with and without the SSI effect. Figure 13ia–iiia shows the base shears in x- and y-directions in the fixed base and the SSI cases with different TMDs mitigation cases, where all TMDs mitigation schemes (whatever 8TMDs or 4 × 8TMDs) significantly reduce the base shears in the SSI case (compared to the no TMDs case) for the El Centro, Hollister, and Chi–Chi earthquakes, and also for the fixed base, the TMDs mitigation schemes significantly reduce the base shears in both directions x and y (compared to the no TMDs case) for the El Centro, Hollister, and Chi–Chi earthquakes. Figure 13ib–iiib shows the base moments in the x- and y-directions in the fixed base and SSI cases with different TMDs mitigation schemes, where all TMDs mitigation schemes (whatever 8TMDs or 4 × 8TMDs) significantly reduce the base moments in the SSI case (compared to the no TMDs case) for the El Centro, Hollister, and Chi–Chi earthquakes, respectively, and also for the fixed base case, the TMDs mitigation schemes significantly reduce the base moments in both x- and y-directions (compared to the no mitigation scheme) for the El Centro, Hollister, and Chi–Chi earthquakes.

Figure 14 represents the base axial forces under three different earthquakes (El Centro, Hollister, or Chi–Chi) with different TMDs mitigation schemes, where in the fixed and SSI cases and for the TMDs mitigation schemes, the change of the axial force values is not significant, considering the increase of the total axial load as a result of the added loads of the TMDs control systems.

Conclusions

Two 3D vertically irregular or both vertically and horizontally irregular steel HRB models were studied without and with the effect of SSI, subjected to three different earthquakes (El Centro, Hollister, and Chi–Chi) to show the seismic response of such structures and suggest the most effective mitigation scheme using TMDs. Two different mitigation schemes using bidirectional TMDs were investigated in this research. The first mitigation scheme used 4 TMDs at the top of the vertically irregular (step-pyramid-shaped) HRB model, while used 8 TMDs at the top of the vertically and horizontally (L-shaped in-plan) irregular (stadium-shaped) HRB model. The second mitigation scheme used four groups of 4 TMDs at the top and through the upper half-height of the vertically irregular HRB model (4 TMDs: at the top, at 12 m, at 24 m, and at 36 m from the top of the model), while used four groups of 8 TMDs at the top and through the upper half-height (8 TMDs: at the top, at 12 m, at 24 m, and at 36 m from the top of the model) for the vertically and horizontally (L-shaped in-plan) irregular (stadium-shaped) HRB model. The seismic response of the 3D irregular steel HRB models, applying time history analysis, was judged by the lateral displacements, the top accelerations, the base shears, the base moments, and the base axial forces of the HRB models, and the following conclusions can be drawn:

  • Without any TMDs mitigation scheme, the lateral displacements and the top accelerations for the steel HRB models with the SSI effect are higher than the corresponding values of the fixed base case, while the higher values of the base moments appear in the fixed base case.

  • The proposed two TMDs mitigation schemes are effective in controlling the seismic response of the presented irregular steel HRBs in the fixed and SSI cases.

  • The two TMDs mitigation schemes in the vertically irregular steel HRB and in the both vertically and horizontally irregular steel HRB significantly reduce the lateral displacements and the top accelerations.

  • The two TMDs mitigation schemes are more effective in reducing the top accelerations in the SSI than in the fixed base case, on account of the higher top accelerations in the SSI case.

  • The distribution of the TMDs at the top and at different height levels along the upper half-height of the irregular steel HRBs significantly reduces the base shear forces and base bending moments.

  • The implementation of TMDs at the HRB plan corners on the top of the HRB and also at different floor levels along the upper half-height of the irregular steel HRB, significantly reduces the seismic response of the vertically and horizontally irregular steel high-rise buildings.