A Soil-Dependent Approach for the Design of Novel Negative Stiffness Seismic Protection Devices

Conventional base isolation (BI) techniques require a great reduction in the fundamental frequency of the system in order to mitigate the structural dynamic responses due to earthquake excitations. However, the resulting base displacements are large and can cause utility connection problems, rendering BI inadequate for retrofitting. This paper proposes a vibration control system (VCS) that can be used as a supplement to the conventional BI to increase the effective damping, and thus reduce the required base displacements. A novel passive negative stiffness (NS)-based vibration absorber, based on the KDamper, is implemented in parallel to a BI. The design of the VCS follows a constrained optimization approach that accounts for geometrical and manufacturing limitations. The NS is realized with a realistic displacement-dependent mechanism that generates controlled NS. The VCS is designed for various soil-types in order to determine its effectiveness and soil-structure-interaction (SSI) effects are accounted with respect to the soil-type. The earthquake excitation input is selected according to the EC8 by generating a database of artificial accelerograms for each ground type. Finally, the VCS is compared to a conventional BI, and based on the numerical results obtained, the VCS is an effective alternative to BI and a possible retrofitting option.


Introduction
The fundamental principle of BI is the decoupling of the superstructure from its base. This way the structure fundamental frequency is reduced, and as a result, the earthquake induced forces in the superstructure are significantly mitigated, as it essentially behaves as a rigid body. However, the earthquake induced displacements are decoupled from the structure, resulting in large base displacements. As a result, significant technological demands and implications are imposed on the BI devices to be used, a fact that affects the structural overall performance. In addition, proper utility connections (waterworks, gas fittings) are needed. An adequate separation distance between adjacent buildings is necessary in order to prevent structural pounding/collisions [1]. Furthermore, structural sensitivity in horizontal loads is increased. Perhaps the most important drawback is that BI is essentially prohibitive for retrofitting existing structures. An interesting approach that simultaneously controls base displacements and superstructure accelerations is presented in [2][3][4], where seismic isolation devices are implemented at various elevations.
These facts have motivated researchers to develop alternative control strategies that can be combined with BI, in order to seismically protect the structure, and at the same time provide possible retrofitting options. Perhaps the most popular and mature approach for vibration control is the Tuned Mass Damper (TMD) system. The TMD consist of an additional mass attached to the primary structure with a positive stiffness element and a linear damper, usually tuned with the fundamental frequency of the primary structure 1.
The VCS is implemented as a supplement to BI and provides a novel passive retrofitting option for building structures; 2.
The optimal VCS parameters are selected by a constrained optimization procedure that accounts for geometrical and manufacturing limitations; 3.
Different set of optimized VCS parameters are obtained with respect to the underlying soil, resulting in a soil-dependent design procedure; 4.
The effect of the soil-structure-interaction (SSI) is taken into consideration with the use of nonlinear spring stiffnesses with respect to the considered soil type; 5.
A realistic mechanism is proposed for the realization of two-dimensional NS.

Nonlinear Dynamic Modeling of Controlled Building Structure
Conventional base isolation (BI) is an effective way to mitigate the dynamic responses of the superstructure at the cost of large base displacements. This paper proposes a vibration control system (VCS) that combines the BI with an extended version of KDamper in order to retain the required base displacements in reasonable ranges with the extraordinary damping properties that the KDamper offers.
More specifically, the building structure-superstructure is mounted on a stiffness connection with low damping (BI). Additionally, the internal mechanism of the extended KDamper is implemented as a supplement, in parallel to the BI, in order to increase the apparent damping behavior of the BI. The proposed VCS implemented in the base of a multi-story structure is illustrated in Figure 1.

Superstructure Modeling and SSI Effects
The modeling of the multi-story residential concrete building structure is based on the following assumptions:

•
The effect of soil-structure-interaction (SSI) is taken into consideration with the use of nonlinear elastic springs; • The slabs and grinders on the floors are rigid as compared to the columns; • The concrete columns of the superstructure are inextensible and provide the lateral stiffness of the structure; • The total superstructure mass is concentrated at the floor levels as the sum of the floor masses and half of the columns mass at either side of the slab; • The superstructure is considered to remain within the elastic limit during the dynamic analysis.
An N-story building structure is considered with N dynamic DoFs, that correspond to the relative to the ground floor displacements, and are collected in the array u str . The equations of motion of the controlled structure with the proposed VCS are expressed in a matrix form as follows: M ..
The matrices and vectors entering Equation (1) are formed as: where M N , K N , and C N are the superstructure mass, stiffness and damping matrices, assuming it is mounted on a fixed base, k 1 is the lateral stiffness of the first floor, and I is the (N + 3) × 1 influence vector associated with the ground motion Xg. The nonlinear NS spring k NS depends on the value of the NS stroke u NS (relative displacement between the terminals of the negative stiffness element). The stiffness of the foundation system (k SSI ) that is derived due to the SSI effects depends on the foundation relative displacement u F . The controlled structure is mounted on a mat foundation, as in the case of a conventional BI structure. The effect of the SSI is taken into consideration with the use of nonlinear spring stiffnesses (k SSI ). The k SSI is calculated assuming a rigid beam subjected to horizontal displacement at the top. The height of this element is measured from the bottom of the mat foundation to the bottom end of the implemented bearing ( Figure 1). As a consequence, the calculated stiffness (k SSI ), is a resultant of both the rotational and horizontal displacement of the soil footing and is expressed at the top of the rigid beam element.
The soil stratum is considered to be isotropic/homogeneous. The interface between foundation-soil is considered to be 'fully bonded' with the foundation elements considered attached to the ground. A Matlab-based code is incorporated to model the original superstructure and the proposed VCS as a series of masses, linear dampers, and linear and non-linear positive and negative stiffness elements.

VCS Dynamic Properties and Nonlinear Behavior
The fundamental advantage of the VCS is that in order to be effective, it does not require significant reduction in the fundamental frequency of the system, as in the case of BI systems, due to the increase of the effective damping that the KDamper configuration offers. Thus, the nominal system frequency f 0 is a freely selectable parameter, the value of which can be selected with respect to the structural system to be controlled in its initial undeformed (static) state.
Although theoretically the values of k NS , k PS , and k R are selected according to Equation (6a) to ensure the static stability, due to various reasons, such as temperature variations, manufacturing tolerances, or non-linear behavior, k NS , k PS , and k R may present variations in practice, since almost all NS designs result from unstable nonlinear systems. An increase of the absolute value of k NS and/or a decrease of the values of the positive stiffness elements k PS and k R by a factor ε NS , ε PS , and ε R , respectively, may result in the system being unstable. This happens when the determinant of the stiffness matrix (structure-VCS) equals zero: Contrary to the KDamper concept, which foresees variation only in the NS element k NS , in the proposed VCS, a variation in all stiffness elements is taken into account. Assuming that the nominal VCS frequency f 0 , and the value of the negative stiffness element k NS are known, using Equations (6a) and (7), the rest of the stiffness elements result as: where parameters a, b, and c are defined as: The NS element of the VCS configuration is realized with a displacement-dependent mechanism, proposed in [29] as illustrated in Figure 2. A vertical conventional positive stiffness element (coil spring) is attached to the additional mass m D , and is connected to the base floor with an articulated mechanism. The generated NS is calculated according to the following procedure. Initially, the potential energy due to the deformation of the vertical spring is given, and consequently, the value of the elastic nonlinear force and the equivalent generated NS are calculated: where k V is the value of the positive vertical stiffness element, l VI and l V is the length of the vertical spring in its initial and deformed state, and c I is a non-dimensional design variable that correlates the constant geometric properties of the configuration, i.e., the length a of the rod, the total distance b between the masses and the l VI . The value of c I takes values in the range [−0.1, −0.01] in order to achieve a near to linear behavior. Having selected a value for c I , and the variation of the NS element value ε NS , Equation (7), the parameters a and k V are obtained by solving the system of equations presented below:

Earthquake Ground Motion Representation
The structure's displacements and accelerations are within specified limits for the given fundamental structure period and damping ratio, according to the seismic design codes. The parameters that mainly control these limits are the ground conditions (soil type) and the expected seismic intensity. However, the selection of the VCS parameters is not possible by following such approaches due to complexity of the VCS and the geometrical and manufacturing limitation imposed for a realistic design. Therefore, analysis in the time-domain is necessary. Ground motions can be generated, among others, from: synthetic (seismological models), real earthquakes (not all soil types, not smoothed spectra), and artificial accelerograms, compatible with a specific design response spectrum [32,33]. In this paper, a database of design response spectrum (EC8) compatible ground acceleration excitations for various soil types will be generated using the SeismoArtif Software [34]. More specifically, 10 artificial accelerograms are generated for each of the following soil types: (1) firm (ground type A), (2) medium (ground type C), and soft (ground type D). For each soil type, the following seismic properties are assumed: spectral acceleration 0.36 g, spectrum type I and importance class II. Figure 3 presents the mean acceleration response spectrum of all the artificial accelerograms and soil types, as compared to the respective EC8 design response spectrum. An accurate match is observed in all cases.

Constrained Optimization for the Selection of VCS Parameters
The fixed parameters of the VCS are the additional mass m D and the stability factors ε NS , ε PS , and ε R . The independent design variables sought in the optimization problem are: (1) the nominal VCS frequency f 0 , (2) the value of the NS element k NS , and (3, 4) the value of the artificial dampers c NS and c PS . The optimization problem is formed for the mean of the maximum dynamic responses for the 10 artificial records of the database of the respective soil type. The objective function and the geometrical and manufacturing constraints are selected from the geometrically non-linear time domain responses, as described below:

1.
Assign values to the fixed parameters. According to previous work of KDamper [31], an additional mass of 5% is efficient. In this paper, in an effort to drastically reduce the additional mass, m D is selected as 0.1%. The stability factors of the positive stiffness elements k PS , k R and of the NS element k NS are selected as 5%; 2.
Selection of soil type and thus the respective artificial accelerogram database; 3.
Set the objective function (OF) as the minimization of the mean of the maximum relative to the ground base displacements of the structure; 4. Set an acceleration filter (AF) as a constraint for the structure absolute acceleration, expressed as a percentage of the mean PGA of the artificial accelerograms of the database of the selected soil type;

5.
The NS element value and stroke are set as constraints, indicated from the proposed design of the NS configuration presented in [29] (50% lower compared to [28,35]); 6.
Set an upper limit for c NS , c PS with respect to the superstructure mass. This constraint is based on previous works [29] as well as on manufacturing restrictions; 7.
The nominal SBA frequency f 0 varies in the range [0. 15 1.5

] (Hz) (Constraint 6);
The VCS design process is schematically represented with a flowchart in Figure 4. For the optimization process, the harmony search algorithm (HS) [36] is adopted. After the optimal system parameters are obtained, the proposed NS mechanism is designed, and consequently, nonlinear dynamic analysis is performed with nonlinear NS, in order to verify the effectiveness and realistic design of the proposed VCS. The conversion criteria mentioned in the outer loop of the flowchart presented in Figure 4  The first one ensures the effectiveness of the VCS in mitigating the seismic effects on the superstructure at the desired level (AF). The second one safeguards the realistic design of the proposed NS mechanism with conventional structural elements.

Numerical Results and Discussion
The examined original structure (OS) is a three-story residential RC building (Figure 5a The foundation system is designed based on the properties and approximate loading of a typical residential building. On the basis of empirical correlations, the initial, small strain, elastic modulus (E 0 ) of the soil was considered equal to 1800 S u . Based on the provisions of the EC8, the undrained shear strength S u varies with respect to the ground type. For ground type A (firm soil), the underlying soil is a rock-like formation and can be thus considered as a fixed base. For ground type C (medium soil), S u varies in the range [70 250] kPa, and for the analysis, is selected as the average, S u = 160 kPa. For ground type D (soft soil), S u is equal or less than 70 kPa, and this value is adopted for this paper. The soil stratum is considered to be isotropic/homogeneous and the same is applied for the mat foundation which is, however, considered to be elastic (Reinforced Concrete). The boundaries of the model have been strategically selected sufficiently far to avoid spurious boundary-effects on the system response. Static pushover analyses are subsequently undertaken in order to derive the non-linear soil-foundation stiffness relationships (k SSI ).
The mean PGAs of the artificial accelerograms of the database are 0.421, 0.545, and 0.614 (g) for the firm, medium, and soft soil, respectively. The acceleration filter (AF) is set to 80% of the respective mean PGA of the database, and the optimal VCS parameters are obtained following the procedure described in Section 3.2 and are presented in Table 1.  In order to confirm the effectiveness of the proposed VCS, a comparison is made with a conventionally base isolated structure (BI) with a base nominal frequency of f B = 0.4 Hz (T B = 2.5 s) and a BI damping ratio of ζ B = 5%. The system main dynamic responses, considering the max values of the dynamic responses for all the Artificial Accelerograms in the database, mean of 10 maxes for each soil category, of the original structure (OS), the conventionally base isolated one (BI), and the controlled structure with the proposed vibration control system (VCS) are collected and presented in the bar plots of Figures 6 and 7.    Figure 8 presents the base relative displacements expressed as a percentage of the BI values. The VCS system manages to retain the u B in the order of a few centimetres, 70-90% lower as compared to the BI. In Tables 2-4 are the main dynamic responses of the OS, BI and VCS for all the artificial accelerograms of the database for all the soil types examined. The reductions that refer to the superstructure dynamic performance (floor accelerations and interstory drifts) are with respect to the OR, and the reductions of the base displacements are with respect to the BI system.  The effectiveness of the VCS is examined also with real earthquakes. Two real ground motions are selected for each soil type. Chi-Chi and Niigata fall into the soft soil category, JMA and Northridge fall into the medium soil category, and Kocaeli and Tabas fall into the firm soil category. The categorization is based on the average shear-wave velocity for the upper 30-m depth of the station. The acceleration time histories are presented in Figure 9, and the details of the ground motions are presented in Table 5.   Table 6 present the controlled system main dynamic responses of the OS, BI, and VCS for the real earthquake records. The reductions presented in Table 6 that refer to the superstructure dynamic performance (floor accelerations and interstory drifts) are with respect to the OR, and the reductions of the base displacements are with respect to the BI system. It is observed that the reductions in the superstructure dynamic responses regarding both seismic protection approaches, BI and VCS, are in the same order of magnitude as in the case of the artificial accelerograms. For all the considered ground motions, artificial and real, the VCS manages to seismically protect the superstructure (more than 50% improvement) and at the same time retain the required base displacements in reasonable ranges, in the order of a few centimeters.

Conclusions
In this paper, a negative stiffness-based VCS is employed, as a supplement to a conventional BI, in the base of a multi-story residential RC building structure for seismic protection. The methodology for the implementation of the VCS is presented and the nonlinear dynamic model of the controlled structure is formed. A realistic displacementdependent mechanism that generates controlled nonlinear negative stiffness is proposed, and the SSI effects are accounted with the use of nonlinear springs, with respect to the considered soil category, and are coupled in series with the stiffness of the conventional BI (k BI ) providing the total stiffness of the base level (k R ). The fixed VCS parameters are selected and the independent design variables are presented. The optimal design of the VCS follows a constrained optimization procedure to be as realistic as possible. Various VCS set of optimized parameters are obtained for different soil types in order to determine the effectiveness of the proposed vibration control strategy. Based on a comparison with a BI and the numerical results obtained, the following conclusive remarks can be made:

1.
The proposed dynamic model of the controlled structure accounts for the nonlinear dynamic behavior of the NS element and the effect of SSI; 2.
The VCS design is realistic, employing small added masses and realistic constraints; 3.
The VCS is effective in seismically protecting the superstructure for all the considered soil types (firm, medium, soft), and retaining at the same time the base displacements significantly lower as compared to a BI; 4.
The superstructure dynamic behavior is greatly improved with the proposed VCS. More specifically, the floor accelerations and interstory drift are reduced 60-80% as compared to the original multistory structure; 5.
The required base displacements are retained in reasonable ranges, in the order of a few centimeters (2-8 cm), and render the VCS a possible retrofiting option. As compared to a conventionally BI structure, reductions of 70-90% are observed.