1.Introduction

This research aims to test the hypothesis that, in certain cases, the rotational components of seismic ground motion can significantly affect the response of a seismically isolated structure. The object of the research is buildings and structures with various seismic isolation systems; the subject is the influence of these rotational components on the torsional response of seismically isolated buildings.

Since the 1950s, observations of buildings and structures after strong earthquakes have revealed numerous structural failures caused by rotational motions. Such damages were documented after Agadir (1960), San Fernando (1971) [1], Armenian (1988), Loma Prieta (1989) [2], Northridge (1994) [3], Kobe earthquakes (1995) [4], etc. It is important to note that failures caused by rotational motions of earthquakes were observed in 42 % of all damaged buildings [5].

There are three primary factors which cause torsion of buildings and structures.

Structural Asymmetry. If the elements in the building are distributed in the plan such that the center of rigidity does not coincide with the center of mass (CM), then the translational seismic excitation causes torsional vibrations in the building. Buildings where the center of rigidity does not coincide with the CM are called asymmetric (or irregular) buildings, and the torsional motions induced in them are referred to as natural torsion [6]. Seismic Rotations. The second factor of torsion is seismic rotations. These result from the spatial nature of seismic excitation. This nature of the impact is determined by the wave nature of earthquakes. These rotations are defined by the parameters of the wave field corresponding to the seismic impact. This complex effect can be assessed by both indirect and instrumental methods. This factor induces torsion in symmetric buildings (buildings where the CM and center of rigidity coincide). This factor can be categorized as accidental torsion [6]. Nonlinear Inelastic Behavior. The third factor can be observed during inelastic analysis of structures. This effect is caused by the nonlinear stiffness of the building’s elements. The effect of plastic yielding cannot be perfectly identical for all elements under translational seismic excitation, especially for the peripheral elements of the building [7]. This paper focuses on building torsion caused specifically by seismic rotations (the second factor). As of 2024, there are very few recorded six-component earthquakes despite developing devices [8] registering rotational components of earthquakes.

There are two types of rotational seismic motion.

Coherent seismic motion (wave passage effect). This type of motion occurs when harmonic waves arrive at a given point on the surface with a frequency and without random amplitude fluctuations, but with a phase shift. Such wave motion can be studied analytically by solving the wave equations. It is possible to generate the rotational components of seismic excitation using translational components of seismic excitation [10]. This problem can also be approached with a different formulation: modeling the non-vertical propagation of seismic waves. By specifying a particular angle of incidence, one can simulate the rocking or torsional motion of the ground surface [15, 16]. Incoherent seismic motion. This is a type of motion where a seismic wave passing up to through inclusions in the soil (local inhomogeneities) undergoes random changes in amplitude and phase. The energy of a wave propagating through such soil is partially dissipated. This effect is called as the scattering effect. Incoherent ground motion is also termed the spatial variation of seismic motion. In general, this type of motion is defined resolving the task of soil-structure interaction (SSI) using special software, for example, ACS SASSI [11, 12]. These calculations are based on probabilistic methods using Abrahamson’s coherency function [13, 14].

In general, a ground motion is described by a six-component vector with three translational and three rotational components at each point on the ground surface. Assuming a rigid foundation it is possible to consider that seismic motions are averaged over ground volume and applied at one point as the time-dependent vector of seismic accelerations. The main assumption of this study is that a seismic excitation is six-component data which include three translational and three rotational components. This model of seismic ground motion is called integrated model. The integrated (generalized) model of seismic ground motion assumes: a linear-elastic layered half-space; decomposition of translational motion into -, -, and -wave contributions; plane wavefront propagation with depth-dependent attenuation functions. Applicability is limited to sites with approximately horizontal stratification and negligible nonlinear soil effects. For sites with significant lateral heterogeneity, site-specific wave propagation analyses are recommended [10].

The rigid foundation assumption implies that foundation flexibility, SSI effects (radiation damping) and spatial variability of ground motion across the foundation footprint are not explicitly modeled. For structures on soft soils (  < 600 m/s), foundation flexibility may alter the effective input motion and modify the rotational response. The averaging procedure is consistent with the kinematic interaction framework when foundation embedment is neglected and the mat is assumed rigid.

It is well-known that rotational components of ground motion can be generated using theoretical methods. The first integrated model was introduced by Newmark [9]. He defined rotational seismic motion around vertical axis as a function of translational components of ground motion scaled by S-wave phase velocity. Rotational component of an earthquake can be expressed as follows:

                                                                    (1)

where  is translational components of ground motion;  is S-wave phase velocity.

Newmark’s method was applied and developed by many researchers: Luco [17], Shibata et al. [18], Lee and Trifunac [19, 20], Todorovska et al. [21], Basu et al. [22]; in USSR and Russia by Rasskazovsky [23], Hachiyan [24], Nikolaenko and Nazarov [25], and Nazarov et al. [10].

Nazarov did a comparative analysis using works of researchers mentioned above [26]:

Table 1. Comparison of results on the rotational characteristics of seismic ground motion calculated by mentioned researchers [26].

Researcher

Newmark

Trifunac

Hachiyan

Rasskazovsky

vs, m/s

3000

1000

300

250

500

 rad/min

–

2E – 4/0.7’

(2 – 4) E – 4/0.7’ – 1.5’

4E – 3/3.43’

5E – 4/1.72

 rad/s*s

1.05E-2

(3-6)E-2

(1.5-2.0)E-1

–

–

where  is the S-wave velocity,  is angular velocity of ground motion,  is angular acceleration of ground motion. The table shows that authors have obtained similar scientific results. The data in table shows that authors’ results differ proportionally to the shear wave velocity in the soil

In 2000–2020, new parameters characterizing earthquake intensity have emerged. During the period when only translational ground motions could be measured by accelerometers and seismometers, parameters such as Peak Ground Acceleration (PGA), Peak Ground Velocity (PGV), and Peak Ground Displacement (PGD) were used. The PGA parameter characterizes earthquake intensity and remains in use as of 2024 due to its simplicity. However, it is important to mention this parameter provides limited information on the actual earthquake intensity.

Sbaa et al. [27] measured the ground rotational velocity along the direction of translational acceleration on the island of Kefalonia (Greece) and compared the peak characteristics of rotational velocity (Peak Ground Rotational Velocity (PGRV)) and translational acceleration (Peak Ground Translational Acceleration (PGTA)). Here, PGTA is the maximum value among all translational components of ground acceleration, and PGRV is the maximum angular velocity among all rotational components. Their comparison revealed a linear relationship between PGTA and PGRV on a logarithmic scale. A similar trend was obtained by other authors such as Liu et al. [28], Takeo [29], and Yin et al. [30].

A comparable result was obtained by Smerzini et al. [31] based on data from the Parkway Valley (New Zealand) and UPSAR (California) arrays. A linear relationship was also found between the PGTA for horizontal components and the PGRV about the vertical axis (torsion). Furthermore, an analogous relationship was discovered between the PGRV of rotational components about the horizontal axes (rocking) and the PGTA of the vertical component.

Subsequently, Ringler et al. [32] recorded three components of ground rotational velocities using proto-seismic magnetohydrodynamic (SMHD) sensors and attempted to use the PGTA/PGRV ratio to estimate the seismic wave propagation velocity and compare it with the velocity obtained from the PGTV/PGRD ratio (PGTV – Peak Ground Translational Velocity, PGRD – Peak Ground Rotational Displacement). In summary, the established linear relationship can be expressed as follows:

                                                                 (2)

                                                                  (3)

where  is -wave velocity.

This study employs the generalized model of seismic wave propagation proposed by Nazarov [10]. Nazarov decomposed the translational motion along each axis into three wave components (Fig. 1):

                                                                   (4)

                                                                  (5)

                                                                   (6)

Figure 1. Decomposition of wave motion [10].

Based on this decomposition, Nazarov derived formulas [10] that describe seismic rotations about three perpendicular axes. These formulas represent the most general case under the assumption of    and  waves propagating along each of the three orthogonal axes:

                                       (7)

                                       (8)

                                     (9)

where        and  are the amplitude functions of the monochromatic waves;  is the shear wave velocity beneath the foundation base, a key parameter governing the building’s rotation.

According to Nazarov’s theory, two very important special cases follow:

If the considered    and  waves have a plane wavefront and attenuate with depth according to  then Equations (7)–(9) are simplified to [10]:

                          (10)

                           (11)

                                                        (12)

If the monochromatic    and  waves have a plane wavefront, do not change with depth, and their stationary amplitude functions within the considered local model can be considered constants, i.e.,  then Equations (7)–(9) take the following form [10]:

                                                                (13)

                                                                (14)

                                                                 (15)

Note that Equation (15) is equivalent the Newmark’s (Equation (1)) model, which was proposed as early as 1969 in his work [9]. The formulas of Rasskazovsky, Khachiyan, Luco, and Trifunac are also special cases of Nazarov’s generalized model. Thus, the model developed by Nazarov provides a more general description of seismic rotations according to the integrated model of seismic ground motion and explains the variability of the results presented above in Table 1.

In summary, seismic rotations can be obtained by direct and indirect methods. The direct method is an instrumental one. This method does not provide exhaustive or precise data, the required equipment is quite large and not always available to seismologists, who provide initial data for structural. Therefore, indirect methods, which derive rotational motions from translational ones, are of significant interest for dynamic analysis.

Indirect methods are divided into two types: obtaining rotational accelerograms from data from special arrays (analyzing fields of strong seismic motions considering their spatial micro-characteristics, known as the Multiple Station Procedure (MSP)), and obtaining rotational motions from a single ground motion recording point (known as the Single Station Procedure (SSP)) [33].

The MSP method requires data from instrument arrays (recordings at multiple points). This method is accurate for generating rotational fields; however, few such arrays exist worldwide. Consequently, it is primarily used for research rather than practical engineering applications [34–38].

The SSP method requires data from only a single station (three translational accelerograms). It is more commonly used because array records are limited in most regions. Rotational ground motion is a spatial derivative of translational motion. In the SSP method, the rotational motion caused by a specific plane wave is given by the time derivative of the corresponding translational motion, scaled by the wave velocity. The main principle is the decomposition of complex seismic motion into    and  waves, as developed by several authors [19, 39–42].

Thus, a key conclusion is that rotational motions (angular accelerations) can be obtained from three translational motions recorded at a single point:

                                                            (16)

                                                           (17)

                                                           (18)

where      are the translational accelerograms recorded at a single point;  is the shear wave velocity beneath the foundation base.

Subsequently, the rotational motion will be used about the vertical axis, obtained using Equation (18), generated from the two translational accelerograms of the seismic impact recorded at a single point. In this study, the rotational motion about the vertical axis was generated using Equation (18) from two translational accelerograms, implemented in the “Odyssey” software v.2. The “Odyssey” software was developed by Y.P. Nazarov, E.V. Poznyak and A.V. Filimonov; its theoretical basis is documented in the peer-reviewed source Nazarov et al. [10].

This paper focuses on base-isolated structures, as they are particularly susceptible to torsion due to their flexibility. Few researchers have studied the torsion of symmetric base-isolated structures subjected to rotational earthquakes. Almazan and de la Llera [43] investigated a symmetrical building on pendulum bearings, applying accidental eccentricity to simulate torsion. Ryan and Chopra [44] studied a base-isolated building subjected to rocking motions, which did not cause significant amplification of accelerations and displacements. Other researchers, such as Shimazaki [45], have also investigated base-isolated structures. A recent observational review of the torsional response of civil engineering structures during earthquakes is provided by Guéguen and Astorga [46].

In light of the foregoing, research specifically dedicated to the influence of the rotational components of seismic ground motion on the torsional response of seismically isolated buildings has not yet been conducted to a sufficient extent. Meanwhile, an increasing number of seismic-resistant building projects incorporate various seismic isolation systems in their design. Therefore, the authors consider this research topic to be relevant and important for the future analytical verification of the isolated buildings’ seismic resistance.

The main goal of this research was to determine the seismic isolation systems most affected by rotational ground motions and to perform both qualitative and quantitative evaluations of this effect.

To achieve this goal, the following tasks were completed:

a review of the literature on the research topic and the identification of the primary methods for accounting for the rotational component of seismic excitation in seismic analysis; the generation of accelerograms incorporating the rotational component of seismic ground motion; the creation of two mathematical models of seismically isolated buildings with different seismic isolation systems (elastomeric and pendulum bearings); the implementation of calculations, followed by a comparative analysis of the structural responses; the formulation of conclusions regarding the influence of the rotational component of seismic excitation on seismically isolated buildings.

2.Methods

2.1.The Motion Equation of a Base-Isolated Building by Lead Rubber Bearings

In order to assess the contribution of the rotational component of seismic impact, it is necessary to formulate a system of differential equations for base-isolated building by lead rubber bearings. It is assumed that a building oscillates as rigid body during an earthquake. Bearings have a bilinear diagram “force – displacement.”

Model of base-isolated building located on lead rubber bearings is shown in Fig. 2:

Figure 2. 3DOF model of base-isolated building located on lead rubber bearings subjected to torsional earthquake.

The model shown in Fig. 2 has three degrees of freedom: two translational displacements and a torsional angle around the vertical axis. The system of differential equations of motion for a base-isolated building on lead rubber bearings is as follows:

                                                 (19)

where   are the accelerograms of the translational components of the seismic impact recorded at a single point;  is the generated rotational component of the seismic motion;    are the total forces in the bearings;  is a number of the bearing;    are the forces in -th bearing along  and  axis (Fig. 3);  is the total moment from the bearing forces;    are the total forces and moment from viscous damping;  is the sum of torsional moments from each bearing;  is a mass of the building;  is the moment of inertia around the vertical axis;  is a building length;  is a building width;  is a center of rigidity of the base isolation system;    are the eccentricities of the base isolation system. Point  is the initial location of -th isolator at  point  is initial location of -th isolator at    is the building’s rotation angle.

Figure 3. Idealized “force-displacement” relationship for the i-th bearing along the x and y axis.

The initial conditions for the system are:

                        (20)

Coordinates in the model:

At  point  point

At  point  point

Let  be the displacement vector of a bearing,  be the radius vector of a bearing relative to the CM.

                    (21)

                               (22)

The moment of force for each bearing about the CM is given by:

                                                                     (23)

                         (24)

The total moment is:

                                                                     (25)

The total stiffness of the bearings is defined as:

                                                       (26)

                                                    (27)

The natural frequencies of the system are:

                                                 (28)

Sensitivity of natural frequencies to bearing layout and stiffness distribution: the translational frequency  is primarily governed by the total lateral stiffness; the torsional frequency,  depends on individual bearing stiffnesses and their radial distances from the center of rigidity. Possible resonance was checked against the predominant earthquake periods that were adopted as input for the analysis (see Figs. 6–8).

The eccentricities of the base isolation system are:

                                                   (29)

Model assumptions:

The vertical component of seismic impact is neglected due to the high vertical stiffness of the bearings. Rotation angles about the horizontal axes are neglected due to the high corresponding torsional stiffness. The torsional stiffness of the bearings themselves can be neglected. Coriolis accelerations can be neglected due to the small angular velocities of the ground motion during an earthquake. The effect of a biaxial yield surface (where the force point on the hysteresis curve traces an ellipse under simultaneous two-directional loading) is neglected [55].

The system of differential equations can be rewritten as:

                                                    (30)

The initial conditions are:

                                  (31)

Thus, a model has been developed for the dynamic analysis of a building supported on an arbitrary number of elastomeric bearings with arbitrary plan coordinates. This model is capable of calculating the eccentricity for any given arrangement of bearings or can accept eccentricity as a predefined input parameter for subsequent analysis. The system of differential equations for mass oscillations was implemented in the Mathcad software (v. 15.0) package and solved using the 4th-order Runge–Kutta numerical method. For the purposes of this study, eccentricities were not considered.

2.2.The Motion Equation of a Base-Isolated Building by Pendulum Bearings Equipped with Plastic Dampers

Rutman developed [47] a pendulum bearing consisting of the following components (Fig. 4): a support frame connected to the substructure (1), a frame connected to the superstructure (2), a tie rod (3), a plastic damping rod (4), a spherical plain bearing (5):

Figure 4. Pendulum bearing [48].

This type of bearings tunes the building to a different fundamental frequency, thereby filter out the energy-intensive frequencies of the seismic impact. However, to eliminate resonant phenomena due to the multi-frequency nature of the seismic impact, damping is necessary. Plastic dampers (PDs) are implemented to these bearings which exhibit a bilinear force-displacement relationship. The total internal force in this type of bearings is composed of the restoring force of the pendulum tie rod  and the resisting force of the PDs

                                                                       (32)

The resulting combined force-displacement diagram is shown in Fig. 5:

Figure 5. Combined force-displacement diagram for pendulum bearings.

Technical characteristics of the pendulum bearing: load capacity: 100–1000 tons. Effective natural frequency: 0.2–0.5 Hz. Maximum allowable horizontal displacement: 0.5 m. Rod-shaped damping elements made of mild steel, when subjected to plastic deformation within 2 %, remain operational for approximately 500 loading cycles. This directly indicates the limiting deformation (2 %) and cyclic durability (i.e., degradation).

As previously mentioned, damping is achieved through the deformation of curved rods connected to the support structure via a separator and a holder (Fig. 4). The required activation forces of the PDs are determined by varying key parameters of the rods, such as their length, diameter, and number.

Rutman et al. [49] derived a system of differential equations describing the motion of a base-isolated structure with six degrees of freedom. The mathematical model of the pendulum seismic isolation system is based on exact kinematic relations that account for geometric nonlinearity under large horizontal displacements and a shifted center of gravity. The deformation of each rod was defined as the difference between its initial length (Z0) and its current length  This expression is nonlinear with respect to the horizontal displacements and the rotation angle. The force in the rod is calculated as  and then projected onto the coordinate axes scaled by  Thus, the model is capable of describing the system’s behavior under significant horizontal displacements without linearization (See [49]).

Taking into account assumptions mentioned for lead rubber bearings model (LRBs-model) above (with the exception of the first item as the vertical seismic component is accounted for here), the system of differential equations can be written as follows:

                                        (33)

The initial conditions are:

            (34)

Thus, a model has been developed for the dynamic analysis of a building supported on an arbitrary number of pendulum bearings with arbitrary plan coordinates. This mathematical model was coded in the MATLAB software package (v. R2021a). The program is designed for the analysis of a building supported on  bearings with specific  and  coordinates and a pendulum tie rod length  The length of the pendulum tie rod, as well as the PDs with a specific bilinear force characteristic, will be the defining initial data for specifying the bearing properties.

2.3.Seismic Excitations Used in Dynamic Analysis

The rotational component of seismic motion is a result of the elastic deformation of the soil mass; consequently, the softer the soil, the greater the rotations will be. This component depends on the shear wave velocity beneath the foundation base. For this reason, a series of rotational accelerograms were generated for a range of shear wave velocities. Given that the input data consisted of records from the strongest and most destructive earthquakes, it is challenging to assign specific soil conditions to each original translational accelerogram. Therefore, for each pair of translational accelerograms (along the  and  axes), a corresponding rotational accelerogram was generated. The investigated range of shear wave velocities spanned from  = 10000 m/s (representing a hypothetically very stiff/rock-like soil) down to  = 250 m/s (representing a soft soil, such as loam or clay).

According to EN 1998-1 (Eurocode 8) ground type classification, the investigated Vs range corresponds to the following categories: Ground type A (Rock or other rock-like geological formation, with at most 5 m of weaker material at the surface) for Vs > 800 m/s; Ground type B (Deposits of very dense sand, gravel or very stiff clay, 360–800 m/s) for range 360 m/s < Vs < 600 m/s; Ground type C (Deep deposits of dense or medium-dense sand, gravel or stiff clay, 180–360 m/s) for Vs < 360 m/s. The selected range thus covers the full spectrum from hard rock (Type A) to stiff clay and dense sand/gravel deposits (Type C). Lower Vs values (