IT201800007222A1 - Negative stiffness modular device for vertical seismic isolation - Google Patents

Description

MODULAR DEVICE WITH NEGATIVE RIGIDITY

FOR VERTICAL SEISMIC ISOLATION

Field of the invention

The present invention relates to a modular device that uses negative stiffness for vertical seismic isolation. The invention can be applied to obtain an insulation at the three-dimensional base of the structures using such vertical devices with negative stiffness.

Technical background

During seismic events, typically earthquakes, the earth vibrates and vibrations are transmitted to structures such as buildings, plants, monuments and the like. Numerous studies and solutions have been proposed to analyze seismic vibrations and protect structures from them, by means of insulation systems.

For decades, seismic isolation at the base has focused the attention of the structural engineering scientific community. Indeed, it plays a special role in mitigating the seismic effect on structures. The insulators at the base are in fact able to:

(i) shift the main structural natural frequencies away from the more dangerous seismic frequencies;

(ii) act as an internal fuse for the interaction forces;

(iii) dissipate part of the energy introduced;

(iv) provide additional damping.

Therefore, it is widely accepted (and standardized) that base insulation gives the best architectural impact, being nearly invisible, and has better seismic performance than other structural protection techniques.

The current standard EN 1998-1 (2003) concerns the design of seismically isolated structures in which the isolation system, below the main mass of the structure, aims to reduce the seismic response of the lateral force resistant system. Again, EN 15129 (2009) specifies that the insulators provide insulation against only horizontal seismic actions.

US patent 8857110 recognized to Constantinou et al. describes a negative stiffness device for the seismic protection of a structure. The device has an anchoring frame and a movement frame that can translate laterally with respect to the anchoring frame. The anchor frame and the movement frame have respective extendable components. A connection is pivoted to the extendable portion of the anchor frame. A compressed spring has a first end connected to the extension portion of the motion frame and a second end connected to the linkage. In a rest state, the compressed spring does not apply lateral force to the motion frame. In an active state, the compressed spring is configured to apply a lateral force to move the motion frame in the lateral direction of the seismic load. The spring force is amplified by the connection when the motion frame is moved sideways.

Although it has long been assumed that the consideration of the effect of the horizontal component of earthquakes is sufficient for the seismic reliability of structures, it has been shown in many cases (e.g. for near-field earthquakes) that the vertical component should be considered together. to the horizontal one. US 8857110 does not deal with the vertical component and its influence on the structures protected by the device described therein. This aspect is of paramount importance for conventional seismic isolation systems which are able to reduce the structural effects of the horizontal component of an earthquake, while the vertical component is transmitted directly into the structure. As a result, there is considerable and growing worldwide research interest in vertical and three-dimensional (3D) insulation systems to protect a wide range of structures and valuables such as assets and specialty equipment. In fact, they are not only unique and expensive, but also critical for safe operation (Basu et al., 2014 Perotti et al., 2014; Abdollahiparsa et al., 2016; Zhou et al., 2016). In the literature one can also find some application examples of 3-D base insulation systems which consist in modifying the design parameters for laminated rubber insulators (ID Aiken, 1989; Kelly, 1988; Okamura et al., 2011).

The GERB system (JM Kelly, 1990) was introduced in 1990 and consists of horizontally and vertically flexible steel coil springs. The spring is essentially undamped and therefore this system must be used in conjunction with additional shock absorbers. The system was applied in two buildings in California. The limits of this system can be identified in the coupling between the horizontal and vertical movement due to geometric non-linearities.

Suhara (Suhara, 2003) has developed a three-dimensional seismic isolation device that uses a laminated rubber isolator for the horizontal direction and a rolling pneumatic spring in the vertical direction. The problem with this device are the effects of pitch and rotation, so it must be used in combination with a special system for suppressing these oscillations.

3D seismic isolation systems for the protection of nuclear power plants are also available in the literature. Different solutions are proposed, such as (i) the 3-D base insulation of the entire building or (ii) the vertical insulation of selected critical structural (or plant) elements in parallel to the horizontal insulation at the base of the building ( Inoue, 2004; Morishita, 2004).

The protection of structures through constant load spring devices against near-field earthquakes, characterized by a vertical component of the same intensity as the horizontal ones, is investigated in (Asai, 2008).

Hybrid control solutions were also investigated for the development of a 3-D vibration isolation system that assembles mechanical and electromagnetic units (Hoque, 2011).

A recent MCEER report compares several alternative solutions for seismically isolated electrical transformers. In particular, the performance of non-isolated solutions, isolated horizontally and then three-dimensionally are analyzed considering pendulum devices (FP) in the horizontal direction and viscous dampers in the vertical one (Kitayama, 2016).

Structural base insulation against vertical vibrations in parallel with standard horizontal base insulation remains a challenge in both research and application practice.

Summary of the invention

An object of the present invention is to address the reduction of the structural and non-structural dynamic effects associated with the vertical component of the earthquake shaking. The present invention overcomes the present limitation of standard base isolation systems which only mitigate horizontal seismic forces. The concepts of tensegrity and negative stiffness are implemented to provide effective vertical vibration absorption in constrained bistable structures.

Another object of the present invention is to face the great challenge of how to realize a 3D base isolation system to protect a wide range of structures and precious goods, overcoming the drawbacks and disadvantages of the prior art, providing a device that uses the principle of negative stiffness for vertical seismic isolation. Furthermore, the device of the present invention can be of the modular type so that it can be easily applied to protect structures having different masses simply by adopting an appropriate number of modules. The modularity of the device makes its mass production and diffusion easy and relatively cheap. Another object of the invention is to provide a device which is simple, economical, reliable, easy to maintain even without the use of specialized personnel or customized tools.

A particular aspect of the device consists in the weakening of its vertical behavior, which therefore simulates the failure of the global system, without modifying the real stiffness of the structure. This allows the structural response to be removed from possible dangerous phenomena linked to the amplification of the response. Therefore, the mechanism decouples the vertical shaking of the ground from the superstructure, thus generating vertical isolation at the base.

Taking advantage of its constitutive advantage, a modular device is described that uses negative stiffness for vertical seismic isolation. The modular device includes an element of the upper body to be fixed for use to the structure to be isolated from vibrations. The device also includes an element of the lower body which is able to slide vertically with respect to the element of the upper body, and which is adapted to slide horizontally substantially without friction with respect to the foundation of the structure. At least one prestressed spring is included between the structure of the upper body and the structure of the lower body so as to transfer substantially zero forces from the foundation to the structure. The ability of the lower body element to slide substantially frictionlessly relative to the base, while at the same time being able to slide vertically relative to the upper body structure, provides the desired decoupling of the vertical seismic isolation from the horizontal one, and this then makes the device particularly advantageous and effective.

According to another aspect, the device uses negative stiffness for vertical seismic isolation. The device can be easily reproduced in a modular way, preferably using a cylindrical shape that can be composed of different modular sectors, as if they were slices of a cake. The device can therefore be easily used to protect structures with different masses simply by adopting an appropriate number of modules or sectors.

According to a particular aspect, the alternative idea is considered of inserting a negative stiffness device (NSD) only in the vertical direction, inside a conventional horizontal base isolation system, thus generating a system of 3D isolation.

In one of its embodiments, the mechanism allows to isolate the superstructure from the vertical component of the earthquake and can comprise a mechanical assembly of passive components (i.e. springs, bars, pins). The concept of tensegrity (assembly of tense and compressed elements - Fuller 1961) guides the concept together with that of negative stiffness (Molyneaux, 1957). Vibration control can be accomplished by changing the structure's self-stress level to shift natural frequencies away from excitation (Bel Hadj Ali and Smith 2010). Recent research has shown that constrained bistable structures can provide an effective absorption capacity of horizontal vibrations (e.g. Pasala et al., 2015; Sarlis et al., 2016).

According to a particular aspect, the mechanism for the vertical base insulation described here adopts consolidated mechanical solutions. Unlike current approaches (Suhara et al., 2003; Inoue et al., 2004; Morishita et al., 2004; Zhou et al., 2016; Kitayama et al., 2017) which are based on linear elastic springs (helical and air spring), the device described here is capable of exhibiting non-linear weakening behavior which is important for limiting external forces that could be transmitted directly to the superstructure.

The concept of negative stiffness was first introduced in Molyneaux's pioneering publication (Molyneaux, 1957) in several proposals for vibration isolation systems. Subsequently Platus (Platus, 1991; Platus, 2007) proposed an effective passive isolator with negative mechanical stiffness only for the isolation of low frequency vibrations. Vertical movement isolation is provided by a rigid spring that supports a load (weight), combined with a negative stiffness mechanism. The limitation of this device is that it requires preload forces of the order of the supported weight. Therefore, it is usually used for light special equipment.

So far, therefore, the application of NSD devices has been limited to vibration isolation of small and highly sensitive equipment, or of seats in automobiles (Lee et al., 2007), since large forces are required for massive structures to develop negative stiffness (typically the same order as the weight of the building).

Some application examples of a pseudo-negative stiffness system for civil engineering structures have also been developed. The "negative" hysteresis loops are first obtained using a fully active or semi-active hydraulic device (Iemura and Pradono, 2003). Hence, Iemura et al. (2006) proposed a variable damper with a combination of friction cycles with pseudo-negative stiffness ("negative" hysteresis cycles), to reduce the acceleration of the structure along with displacements. However, the dependence of the variable damper on the external power and the feedback signal to generate the negative hysteresis circuits limits its applications. The negative stiffness method is considered a promising control strategy for reducing absolute response, except that such control generally requires active or semi-active devices. In (Iemura, 2008) a structure is placed on top of convex pendulum devices, exactly the opposite of friction pendulum isolators which use a concave surface. Hence the negative stiffness is generated by the vertical loads of the structure applied to the convex surface, while the elastomeric isolators are arranged in parallel to provide positive stiffness.

The concept of NSD has some similarities with the protection system based on weakening and damping (WeD) of the structure (Cimellaro et al., 2009; Reinhorn et al., 2009; Viti et al., 2006). However, even if the WeD is able to reduce both the accelerations and the inter-plane drifts of the structural response, it generates permanent damage and deformation.

Nagarajaiah et al. (Nagarajaiah et al., 2010) proposed the concept of "apparent weakening" in which NSD is used to simulate global system failure (weakening) without changing the actual stiffness of the structure. Therefore, it allows you to move the system condition away from the resonant condition. In practice, a pseudo yield strength is created which is lower than the real one of the structure. The NSD is not designed to transfer forces to the structure as long as the displacement is greater than the gap displacement (displacement when pseudo-failure occurs). However, the concept thus proposed at this stage is impractical for large structures. In order to obtain negative stiffness more economically, a new structural control device that realizes negative stiffness in a passive way was initially proposed by Sarlis et al. (Sarlis et al., 2013) and tested on a vibrating table at the University of Buffalo (Pasala et al., 2013; Pasala et al., 2014).

The apparent weakening behavior and the negative performance of the stiffness device have been investigated more recently analytically and experimentally using vibratory table experiments (Attary et al., 2015; Pasala et al., 2015; Sarlis et al., 2016).

However, to the knowledge of the inventors, there are limited studies in the literature for the application in civil engineering of the concept of NSD to reduce vertical accelerations in base-isolated buildings, trying to achieve what can be called three-dimensional base insulation. For example, there is a theoretical study for 3-D base isolation using a negative stiffness mechanism (Mochida, 2015). There are also designs of solutions with negative and almost zero stiffness (P. Alabuzhev, 1989).

Certainly, the negligible performance in mitigating the vertical earthquake effect of conventional base isolation systems is well known. Reinforced rubber isolators, friction pendulum systems and rolling systems are included. Considering the most common class of elastomeric isolators, their intrinsic vertical stiffness does not allow isolating the seismic response excited by a vertical movement of the ground. Some attempts and innovative solutions have been proposed to overcome the isolation limits in the vertical direction. Examples include 3-D steel spring arrangements and fiber-reinforced elastomeric insulators (Harvey, 2016; Kelly and Van Engelen, 2016; Li et al., 2013).

In view of the above, according to one aspect a 3-D base isolation system is described to control both the horizontal and vertical components of ground movement. The system adopts a negative stiffness device which can be considered as an adaptive passive protection system, which therefore can apparently change the stiffness of the structure. The mitigation behavior of the negative stiffness device against strong earthquakes in the vertical direction was studied through numerical simulations. The base isolation system may comprise elastomeric isolators which act in both the horizontal and vertical directions and negative stiffness devices which act only in the vertical direction. In this way, a three-dimensional base insulation can be obtained, where it is assumed that the devices with negative stiffness affect only the vertical stiffness of the system. Numerical analyzes show that the presence of negative stiffness devices reduces vertical acceleration in the structure. However, according to passive control theory, relative displacements increase. Therefore, it seems advisable, although not necessary, that additional damping be adopted to mitigate this effect. Thanks to the presence of rubber insulators, it is possible to use their intrinsic damping without introducing specific shock absorbers in the vertical direction.

According to the above, a 3-D base isolation system is described which adopts an NSD device for the mitigation of vibrations in the vertical direction, combined with elastomeric devices acting on the horizontal plane. The main hypothesis is that the NSD is autonomous and therefore, once installed, affects only the vertical stiffness of the system, leaving the horizontal stiffness equal to the stiffness of the insulators. In other words, the NSD does not participate in the transfer of the horizontal actions between the structural base and the foundation but only of the vertical forces of interaction.

Highly sensitive light equipment or sculptures and artistic masterpieces could be effectively protected by the proposed insulation system.

The numerical results of a singledegree-of-freedom (SDOF) system isolated at the base with an NSD installed in parallel at the level of the traditional insulation system consisting of rubber insulators are reported. Three different systems are compared: (i) with a fixed base, (ii) isolated in the horizontal direction (traditional) and (iii) isolated with the addition of an NSD device considering a series of reference earthquakes. The numerical results show that adding the NSD in the vertical direction can reduce the vertical seismic forces in the structure.

Brief description of the drawings

Figure 1 is a schematic force-displacement behavior of the NSD implemented in the structure where (a) are the force components, (b) the structure is the NSD, (c) is the effect of the damper in parallel with the structure NSD.

Figure 2 (a) is a schematic of an isolated structure with NSD.

Figure 2 (b) shows the total vertical F∆ law at the level of the NSD isolator.

Figure 2 (c) shows the vertical stiffness: KA for fixed base structure, KB considering the columns and isolation devices (rubber isolator) in series. If NSD is added in parallel to the rubber insulators, KC is the total pre-yield stiffness and KD is the post-yield stiffness.

Figure 2 (d) shows a 3D schematic of the NSD device.

Figure 2 (e) shows a section of the device of Figure 2 (d) with the detail of the radial module for vertical insulation.

Figure 2 (f) shows a detail of the radial section of the GSA. Figure 2 (g) shows a deformed shape and free-body-like pattern of the NSD mechanism.

Figure 3 shows the horizontal acceleration response spectrum for (a) ξ = 5%; (b) ξ = 15%; and the displacement response spectrum for (c) ξ = 5%; and (d) ξ = 15%.

Figure 4 shows the vertical acceleration response spectrum for (a) ξ = 5%; (b) ξ = 15%; and the displacement response spectrum for (c) ξ = 5%; (d) ξ = 15%.

Figure 5 shows the horizontal time history of the Cape Mendocino earthquake: (a) accelerations; (b) speed; (c) travel. Vertical time history of the Cape Mendocino earthquake: (d) acceleration; (e) speed; (f) displacements.

Figure 6 shows a comparison of the vertical accelerations for traditional base insulation compared to the one that includes the NSD for different vertical stiffnesses. Cape Mendocino (a), mean values of all earthquake logs (b).

Figure 7 shows a comparison of the maximum vertical displacement at the isolator level for traditional base insulation compared to that which includes the NSD for different vertical stiffnesses. Cape Mendocino (a), mean values of all earthquake logs (b). Maximum lift: 24 mm in the structure with NSD.

Figure 8 shows a comparison of the minimum vertical displacement at the isolator level for traditional base insulation compared to that which includes the NSD for different vertical stiffnesses. Cape Mendocino (a), mean values of all earthquake logs (b). Minimum displacement: 56 mm in the structure with NSD.

Figure 9 shows a comparison of vertical accelerations of the proposed retrofit strategy versus the traditional base isolation strategy for different earthquake events.

Figure 10 shows a comparison of the vertical displacement of the insulator of the proposed retrofit strategy with respect to the traditional isolated structure at the base for different seismic events. Maximum vertical displacement of the insulator (a), minimum vertical displacement of the insulator (b).

Figure 11 shows a comparison of the vertical drift of the proposed retrofit strategy versus the traditional isolated base structure for different events. Maximum vertical drift (a), minimum vertical drift (b).

Figure 12 shows the total negative stiffness force required and normalized to the weight of the building.

Figure 13 shows the vertical input energy for different earthquake events and retrofit strategies

Detailed description

Characteristics of negative stiffness devices

When NSD is applied in the horizontal direction, it exhibits true negative stiffness and is capable of generating a force that pushes the isolated structure in the same direction as the relative horizontal displacement induced by seismic motion, between the base of the structure and the foundation (not in the opposite as in the case of positive stiffness). True negative stiffness for structural applications is a concept that was first introduced by Nagarajaiah et al. (S Nagarajaiah, 2010) and can be achieved without any external power supply, contrary to the pseudo-negative stiffness of active and semi-active devices.

The NSD can be classified as a passive seismic protection system but belongs to a more sophisticated type, the adaptive one. It means that it is able to change its characteristics when it deforms. Furthermore, even though it is a self-contained and inherently unstable device, the NSD is capable of exhibiting stable behavior when implemented in a facility; in other words, it is the structure itself that stabilizes the device.

Negative stiffness is generated by a pre-compressed spring within a mechanism, designed to contain the high compressive force of the spring. As the NSD deforms, the spring rotates and extends giving a force that assists the structure.

The NSD simulates an apparent weakening without changing the real stiffness of the structure, in order to move the system appropriately away from the resonant condition. In other words, a pseudo yield strength is created, lower than the actual yield strength of the structure.

The NSD is designed to transfer near zero forces to the structure as long as the displacement is greater than the gap displacement (displacement at which pseudo-yielding occurs). A so-called Gap Spring Assembly (GSA) is implemented to generate this behavior. It consists of a pair of mechanical springs, capable of generating a bilinear elastic positive stiffness. When the displacement is less than the gap, the GSA is able to cancel out the negative stiffness. Conversely, when the displacement exceeds the gap threshold, the stiffness of the GSA is close to zero, so the negative stiffness of the device is transferred to the structure.

NSD was originally used in parallel with passive shock absorbers. Indeed, the apparent reduction in stiffness produces an increase in displacement which must be controlled and, therefore, a small amount of damping is required.

Details on the NSD can be found in Sarlis et al. (Sarlis et al., 2013), where the force displacement law for the NSD without GSA, the GSA and the result of their combination is presented. Figure 1a illustrates the displacement force law for the main (linear spring) structure, a linear passive damper, and the NSD with GSA. Figure 1b shows the bilinear behavior coming from the parallel structure with NSD and GSA without damper. The rigidity of the NSD structure assembly is reduced to Ka = Ks-KNSD beyond the displacement y1. If F2 and y2 are the maximum force and displacement for the linear system, F3 and y3 are the same variables relating to the assembly. The maximum displacement y3 is increased with respect to the linear stiffness y2 (y3> y2). Therefore, the introduction of a damper in parallel with the NSD and the spring allows for an improved response in terms of displacement y'3 (Figure 1c).

A further positive feature of the NSD, GSA and isolated structural system is the possibility of re-centering. Therefore, residual deformations are restored at the level of the base insulation system, unless the main structure has developed plastic deformations due to yielding.

In this description, for simplicity, the NSD function is adopted with a bilinear model and acts only in the vertical direction. In the horizontal plane the decoupled insulation function coming from the rubber insulators is implemented. Since rubber insulators already have intrinsic damping capacity, as already mentioned, the latter can be useful for limiting relative vertical displacements. Description of the model adopted

The effectiveness of the proposed NSD for the mitigation of vertical vibrations was evaluated using a standard structure. It consists of a 2-D frame, a floor and a span with generic dimensions and characteristics. The case study is not related to a specific application, but is useful for evaluating the effectiveness of the proposed NSD in the vertical direction. It includes two columns and a rigid horizontal beam. The span measures 9.15m and the height is 3.96m. The mass distributed on the beam is 160000 kg. The frame has a total horizontal stiffness of 77000 kN / m, so the corresponding vibration period is TH = 0.28 s. The total vertical stiffness of the frame is 5.7 × 106 kN / m, so the vertical period is TV = 0.033 s. The damping for this structure is assumed to be ξ = 5% and has been included in the equation of motion using the Rayleigh damping matrix.

The performance of three structural configurations are compared: (i) a fixed base structure, (ii) a horizontally insulated structure with rubber insulators, and (iii) a horizontally insulated structure with rubber insulators and with NSD in the vertical direction (horizontally decoupled ). The fixed base structure (i) and the isolated versions (ii and iii) correspond to the frame with the parameters described at the beginning of this section. However, the three types studied are described in detail in the following: 1) Fixed base structure:

The horizontal beam is assumed to be rigid, the columns deformable and fixed to the base, therefore the frame has three degrees of freedom: the horizontal displacement, the vertical displacement and the angle of rotation in the vertical plane (the displacements refer to the center of gravity of the beam) . The stiffness matrix is:

where T, N, M are the internal forces in the columns: shear force, axial force, bending moment respectively. E is the Young modulus, h the total length, A and I the cross-sectional area and the moment of inertia, L the distance between the columns (see Figure 2a). The term K11 in the matrix (first row-first column) is 77000 kN / m, while the term in the second row and second column K22 is 5.7 × 106 kN / m.

The dynamic equations governing the process are presented in Equation (2), where the damping matrix is omitted. The mass matrix is diagonal and the term corresponding to the degree of freedom of rotation is the moment of inertia of the beam. The second member of the equation reports the forces transmitted by the movement of the ground. Obviously, the rotational part is negligible.

With reference to Equation 2, m is the mass, x, y, θ the coordinates

the acceleration components of the movement of the

ground in the horizontal and vertical direction respectively. The dot above the variables represents the derivative with respect to time. It is evident that the vertical movement is disjoint from the other two.

2) Insulated structure:

For this configuration, another slab is added to the insulation floor level which is supported by two insulators. The floor is assumed to be rigid and has the same mass m as the one above the deformable columns. The new degrees of freedom are the horizontal and vertical displacements of the center of mass (x1, y1) at the isolation level. Rotation at this level is not considered: in other words, rotational motion (such as rocking) is ignored at the isolation level. Ultimately, five degrees of freedom are considered in the structure.

Each insulator has a damping ratio set at 15% and their behavior is assumed to be linear elastic. This value is typical for high damping rubber insulators, or lead-core rubber insulators, with the largest hysteresis loops (Abe et al., 2004; Perotti et al., 2013), in correspondence with which the resulting benefits from the base isolation system, in terms of dissipation and decoupling of the system from the ground movement, are more evident. Equation 3 summarizes the stiffness matrix of the basic isolated structure:

where KSH and KSV are the total horizontal and vertical stiffness of the insulators, respectively. If a 2DOF system is assumed for basic isolated structures in the horizontal direction (x1, x2), the damping matrix is determined by the corresponding damping coefficients at the level of the superstructure c2 and at the level of the insulator c1. In particular, c2 is determined by assuming that the superstructure is an SDOF system with a damping ratio ξ = 0.05, while c1 is determined by assuming that the isolated base structure is an SDOF system with a damping ratio ξ = 0.15 .

Finally, the equations of motion are presented as follows:

Hence, the vertical problem remains decoupled from the others.

3) Insulated structure with NSD (Figure 2a):

The structure is the same as in the isolated case, but the proposed NSD device is introduced at the base. The vertical insulation relating to the NSD is designed to be decoupled from the horizontal one (relating to the standard rubber insulators) and the NSD is connected to the structure at the base center of gravity. Therefore, it acts only in the vertical direction and does not provide any force in the horizontal direction.

Therefore, in the horizontal direction, the rubber insulators ensure the insulation at the base, while the vertical force-displacement law at the level of the insulation plane is equal to the sum of both the proposed NSD and the contributions of the rubber insulators acting in parallel. . In practice, the proposed solution is designed to obtain a specific force-displacement law as illustrated in Figure 2b. This is non-linear elastic and the parameters that characterize the behavior are the following:

• The stiffness of the first branch, equal to the total stiffness of the KNSD vertical isolators.

• The stiffness of the second branch, equal to 10% of KNSD.

• The gap-displacement δ when the proposed NSD solution is used.

When the absolute value of the displacement is less than δ, the GSA provides an opposite force to the NSD, so that the structure behaves as if the NSD is not present. In this parametric study, five different gaps are considered: 1-2-3-4-5 mm. The dynamic equations are the same as in the isolated case, with the exception of the second equation, which is equal to:

As an extension of the previous structural configuration, the damping ratio of the superstructure is assumed to be 5%, while it is assumed to be 15% for rubber insulators, while the proposed NSD solution does not provide for damping.

Equations (4) and (5) of the structural model isolated at the base with NSD outline how the results of the vertical response are decoupled from the others. This quantitative aspect highlights the independent nature of the 3D insulation system, in which the vertical insulation function is autonomous from the other components. Figure 2c summarizes the force-displacement characteristics in the vertical direction for the three structural configurations considered. All numerical simulations were performed using MATLAB software (Matlab, 2015).

Figure 2d illustrates the 3D shape of the device 10 which is also schematized in Figure 2a between the superstructure and the base foundation. It has a cylindrical shape, rigidly connected to the superstructure by the top surface 11 and sliding without friction to the base foundation through the bottom surface 12. Thus, the decoupled behavior between the horizontal and vertical reaction forces in the insulation system is realized.

The radial section A-A in Figure 2e describes the internal (at rest) mechanism that characterizes the typical radial module, such as a slice or sector. The main dimensions of the NSD components are reported and must be identified during the design process, considering the specific conditions of the application. The same radial modulus is replicated symmetrically in the proposed solution, e.g. for 4-6-8 times according to the fixed dimensions of the device. The proposed system employs the highly prestressed spring 13, preferably a prestressed horizontal helical spring. The spring 13 belongs to an amplification mechanism which is realized through the lever 14, preferably an internal lever, which can rotate around the pin C. The lever 14 is connected to the upper component 11 by means of a pin or the like. Therefore, the extension of the spring 13 is amplified by the lever 14.

It is worth noting that, as the lever 14 tilts due to vertical displacements (Figure 2e), the distance Rp1 (from the center to the end of the lever 14) decreases due to fixed Rp2. Therefore, to avoid blocking the mechanism, a horizontal groove (slot) 16 is introduced at the rightmost pivot point of the lever 14, where the pin 15 is located.

The prestressed spring 13 remains at rest until the vibrations induced by the vertical component of the earthquake change the alignment between the lever 14 and the spring 13. Then the vertical component of the force of the prestressed spring 13 generates a bending moment, on the left in the figure 2e, around the pin C, in equilibrium with the bending moment generated to the right of the pin C. Since the length of the arm on the right of the pin C is smaller than that on the left, the force acting on the superstructure will be greater than the vertical component of the force of the prestressed spring 13. In this way the reduction of the vertical stiffness of the system is generated (weakening), which has the effect of amplifying the displacement induced by the vertical component of the earthquake. This mechanism decouples the vertical motion of the ground from the superstructure, generating isolation at the vertical base. Preliminary results highlight how the vertical isolation mechanism is effective in reducing the vertical acceleration of the superstructure and promising for further development and implementation on a wide range of structures.

Figure 2e also reports the position of the GSA mechanism inside the NSD, while its detailed drawing (radial section at rest) is provided by Figure 2f. The GSA mechanism is characterized by the same working principles described in (Sarlis et al., 2013). Therefore the displacement of the gap δ, also called NSD engagement displacement, is defined by Eq. (17) in (Sarlis et al., 2013).

To define the analytical model of the negative stiffness mechanism, the deformed configuration and the free body diagram must be considered as in (Sarlis et al., 2013). They are shown in Figure 2g without the contribution of GSA, where Pin and Kin are the pre-compression force and the spring stiffness, respectively. The vertical displacement u of the bottom surface 12 of the device is equal to the displacements of points B and E (non-deformable elements are assumed).

The length of the spring 13 in the deformed configuration of the mechanism is calculated as

Where Lp is the length of the spring 13 when u = 0 (undeformed configuration) and the increase ∆Lp is due to the rotation of the lever 14 around the pivot point C, as indicated in Figure 2g. The latter is

defined as

Fs is the force in the deformed configuration of the spring 13 and its component can be derived as

By writing the balance of the lever 14 around the pivot point C, the force exerted at point B can be calculated as

Ground movements

Civil engineering structures are subject to three-dimensional seismic movements, however, although both the horizontal and vertical components have been studied and considered with regard to structural design issues, the vertical component of ground movement has frequently been underestimated and neglected. (Ghaffarzadeh and Nazeri, 2015; Nagarajaiah et al., 2013; Shakib and Fuladgar, 2003). Some regulations and design codes, eg. NEHRP (1994) and UBC (1997) also assume that the vertical component of the ground movement is a fraction of the horizontal component. However, in destructive earthquakes such as 1989 Loma Prieta, 1994 Northridge, 1995 Kobe and 1999 Chi-Chi, it was found that vertical ground motion can match or even exceed horizontal ground motion locally.

There are several methods in the literature for the selection of the ground motion to be used for the analysis of the temporal history of the structural response in the horizontal direction. A summary of these existing methods can be found in (NIST, 2011). However, the main focus remains on the horizontal component only, while there are not as many criteria and approaches that can be used for the vertical components.

The principles adopted here for the selection of ground motion consist in finding records of earthquakes with a high vertical component of the ground motion, for example by observing the peak acceleration (PGA) of the vertical component and analyzing the displacement time history by identifying the records with pulses. This condition is usually common in near-field earthquakes where pulses are usually present in both the horizontal and vertical directions. The database adopted for the selection of earthquakes is the PEER database, while the selected set is shown in Table 1. The OPENSIGNAL software (Cimellaro and Marasco, 2014) was used in particular for the selection of earthquakes. The two-dimensional analyzes were performed considering both the vertical and the maximum horizontal component. The average response spectra of the selected series in terms of displacements and accelerations in the vertical direction are shown in Figures 3 and 4 where the period of the analyzed structure in the vertical direction is also shown.

For example, a selected earthquake record is shown in Figure 5, where both the horizontal and vertical components of ground motion are shown in terms of accelerations, velocities, and displacements. As can be seen, the vertical displacements of the ground motion are twice the horizontal displacements.

Design of the isolation system and devices

In order to evaluate the characteristics of the insulation system with the possible benefits deriving from the use of the proposed NSD solution, the three types described above are considered: fixed base frame, insulated and insulated with NSD. By focusing on laminated rubber insulator-like devices, assuming an initial horizontal stiffness range for the insulators, the goal is to find the stiffness at which the proposed NSD solution is most efficient in mitigating the vertical acceleration of the superstructure. To obtain a reasonable interval, the first hypothesis is to consider a stiffness interval corresponding to a horizontal vibration period between 1 s and 4 s. Therefore, to identify the total vertical stiffness of the insulators, it is assumed to have a ratio of vertical to horizontal reactions equal to 1000. The tests were performed in the vertical direction using the earthquakes listed in Table 1. For each recording and for each stiffness , all 5 gapdislocation values δ are assumed.

The second part of the analysis concerns the design of the insulators. After selecting the optimal vertical stiffness, the aim is to design an insulation system capable of withstanding the displacements and stresses induced by the seismic load. The design must take into account some requirements and verifications that appear in most codes: the control of the horizontal shear deformation, the reduced area, the instability and the stress of the materials.

Four geometric and mechanical parameters characterize a rubber insulator and for each a range of likely values is selected and all possible combinations are considered.

- the shear modulus of elastomer G (0.4 - 0.8 - 1.4 MPa)

- the diameter D (from 0.3 m to 2.5 m)

- the number of rubber layers n (from 3 to 60)

- the thickness of each layer t (from 3 mm to 50 mm)

The standard (AASHTO, 2010) makes it possible to identify suitable solutions within the preliminary design intervals that have been considered in the numerical simulations. The next section is dedicated to presenting the results of numerical simulations in terms of response mitigation by comparing the three different structural configurations. The last part of the study shows the comparison in terms of the amount of energy transmitted in the superstructure, as obtained from the integration of the equations of motion.

Numerical results

As described in the previous paragraph, the first simulations focus on finding the optimal vertical stiffness of the insulator with respect to which the proposed NSD solution is more effective in terms of reducing the vertical acceleration component in the structure, compared to the traditional isolated case (only horizontal ). Obviously, the limitation of displacements at the level of the insulators must be guaranteed. It turns out that the number of registrations that engage the NSD with displacement δ greater than 1 mm is negligible. However, with the value δ = 1mm the results are satisfactory in terms of vertical attenuation of the acceleration. Therefore, for the purposes of this preliminary study, all the analyzes presented below refer to this value.

Figure 6a shows the maximum vertical acceleration corresponding to each vertical stiffness for the Cape Mendocino earthquake. Figure 6b shows the mean values and standard deviations of all ground movements considered. With the same scheme, Figure 7 shows the response in terms of maximum vertical displacement at the level of the isolators, while Figure 8 shows the minimum displacement. These are measured from the absolute zero position of the insulators. It is important to separate the maximum and minimum displacement due to the different behavior of the rubber in traction versus compression. Traction is mainly due to the introduction of the NSD but also to the movement of the ground in the vertical direction and to the dynamics of the system. In fact, as shown in Figures 7 and 8, even the simple structure isolated by traditional rubber insulators experiences traction for a given stiffness ratio.

In Figure 7 and Figure 8, a dashed line represents static vertical displacement which is a step function because it depends on vertical stiffness.

Looking at the figures, we observe a transition zone in which the stiffness is greater and the behavior of the two types of structures is very similar, because the displacements are smaller. So the NSD is only engaged a few times. On the other hand, when the stiffness is small, a significant reduction in accelerations is obtained. However, for the same cases the higher tractive displacements are also highlighted. The optimal stiffness is identified within the highlighted transition area, where the most satisfactory acceleration reduction in the vertical direction is achieved, experiencing relative vertical displacements compatible with the structure isolated with rubber insulators.

A stiffness value of = 7.9 x 105 kN / m is adopted, which represents the optimal value in terms of acceleration reduction by limiting displacements.

The next step is to design a reference insulator capable of withstanding the demand in terms of displacements and stresses, starting with the preliminary analysis. After combining the independent parameters (G, D, n, t) described in the previous paragraph, the selected isolator has the characteristics in Table 2.

In this study, all the recordings used for the analyzes generate large vertical displacements in the isolators. To be compatible with these displacements, it is necessary to improve the deformability of the insulators by managing the number and thickness of the rubber layers. The characteristics of each isolator corresponding to each seismic record are shown in Table 3.

All regulatory conditions were met except (i) the instability condition in the Gazli earthquake and the (ii) allowable voltage limit for cavitation which is exceeded under the Northridge record. Regarding the first condition, the vertical load is less than the buckling load, but it is greater than half (4.34 MPa). This means that the stiffness of the insulators is affected and therefore a non-linear model must be used in the analysis. Regarding the second problem, even if the theoretical cavitation limit is exceeded for a short time at the peak of the structural response, the design of the isolation system for a real application should be re-evaluated. However, these questions are not in the scope of this work which represents a preliminary phase of the project, so the two cases described, relating to the Gazli earthquake and the Northridge earthquake, have been ignored.

It is worth remembering, with regard to the tensile forces in basic insulation systems, the studies of Roussis; for example (Roussis, 2009) where special connections between the devices and the structure are presented. Such solutions that remove the possibility of transferring the vertical load directly on the insulator causing traction in the material are also allowed by the standard (EN15129, 2009). However, the implementation of any lift prevention mechanism could change the behavior of the isolation system and should therefore be included in the analysis model.

The comparison of the response of the three types of structures adopted is shown in Figure 9 and Figure 10. The figures show how the proposed NSD solution is very effective in reducing vertical acceleration. For some earthquakes the reduction is significant, while for others it is less. However, the case with the proposed NSD solution is almost always the best option.

Figures 9 and 10 also show that the displacements for the two isolated cases (isolated structure and isolated structure with negative stiffness) are comparable in tension and compression, with the sole exception of the Northridge earthquake.

Figure 11 shows that the results in terms of vertical drifts in the superstructure are satisfactory. There are cases where the reduction is significant, while in other cases it does not occur. Furthermore, Figure 12 shows the forces required in negative-stiffness devices to achieve this performance, normalized with respect to the total weight of the building. The ability of the proposed NSD solution to limit the seismic energy transmitted to the structure during the earthquake is also investigated. Figure 13 shows the comparison between the structural types considered. Cape Mendocino and Northridge records are shown and, in both cases, when the proposed NSD solution is employed, the amount of energy entering the facility is reduced compared to the fixed base case and the simply insulated version.

In conclusion, the insertion of the proposed vertical isolation device in parallel with the rubber isolators in a building isolated from the base to control the 3D structural response was described. The structure is insulated horizontally with a system of elastomeric isolators and vertically with the proposed device in parallel with the rubber isolators. Both horizontal and vertical stiffness elements are implemented independently and thus a 3-D base isolation with decoupled reaction components is achieved.

In the analysis, a SDOF standard steel load-bearing frame was considered to test the effectiveness of the proposed configuration compared to the traditional fixed base and traditional insulated one. A series of near-field earthquake records with the characteristic pulse shape were selected to perform non-linear dynamic analyzes.

Numerical analyzes show that by implementing the proposed vertical device in the vertical direction, the vertical accelerations of the structural response are lower than in a simply isolated structure. However, consistent with basic insulation theory, there are increases in displacement at the level of the insulation system. Therefore, the proposed vertical isolation device, if properly designed, is able to reduce the effect of vertical seismic forces compared to traditional isolation systems at the base, without considerably increasing the absolute and relative displacements. The proposed vertical isolation device is also capable of reducing the input energy transferred to the superstructure.

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Table 1 - List of the records of the 10 earthquakes used for the analyzes.

Table 2 - Properties of rubber insulators.

Table 3 - Design verification of the insulator.

Claims (9)

CLAIMS 1. A modular device (10) that uses negative stiffness for vertical seismic isolation, comprising an upper body structure (11) to be fixed in use to a structure to be insulated from vibration and a lower body structure (12) which is able to slide vertically with respect to the upper body structure (11) and which is able to slide substantially without friction with respect to a base foundation of the structure, at least one prestressed spring (13) being included between the upper body structure (11 ) and the lower body structure (12) to transfer substantially zero forces to the structure. A modular device according to claim 1, having a generally cylindrical configuration composed of a number of equally spaced radial sections, each comprising at least one prestressed spring. A modular device according to claim 2, comprising four, six or eight radial sections. A modular device (10) according to any one of the preceding claims, wherein the prestressed spring (13) is comprised in a negative stiffness amplification mechanism. A modular device (10) according to any one of the preceding claims, further comprising a lever (14) articulated around a fulcrum (C), the lever (14) being connected to the upper body structure (11), the lever ( 14) being connected to the pre-compressed spring (13) to amplify its extension according to the ratio of the lever (14). A modular device (10) according to claim 5, wherein the lever (14) is connected to the upper body structure (11) by means of a pin (15) which interacts with a slot (16) on one end of the lever (14) to avoid blocking the mechanism. A modular device (10) according to any one of the preceding claims, comprising a gap spring assembly GSA which is capable of canceling the negative stiffness when the displacement is less than a fixed spacing, a greater displacement of spacing bringing the stiffness of the GSA close to zero so that the negative stiffness of the device is transferred to the superstructure. A modular device according to claim 7, wherein the spaced spring assembly (GSA) comprises a pair of mechanical springs which generate a bilinear elastic positive stiffness. A 3-D basement insulation system for controlling both the horizontal and vertical components of the earthquake shaking, comprising a modular device (10) according to any one of the preceding claims which provides only negative stiffness in the vertical direction.