KDamper Concept for Base Isolation and Damping of High-Rise Building Structures

Abstract

Throughout the past decades, seismic isolation of structures has attracted the attention of civil engineers and scientists. Research around this field has progressed tremendously, starting from the use of simple elastomeric bearings for the decoupling of the superstructure from the foundation and moving toward the invention of more complex devices (characteristic examples being the Tuned Mass Dampers—TMDs or the Quasi-Zero Stiffness oscillators—QZSs) aiming to enhance structural dynamic behavior. In this context, the implementation of novel passive seismic isolation devices incorporating negative stiffness elements to a high-rise building structure is introduced and proposed in this effort. The design of these devices follows the scope of general vibration isolation and damping concept, entitled KDamper concept based on Antoniadis et al. (2016). This paper considers the application of a KDamper system to 10-storey concrete building structure. A dynamic system consisting of a simplified flexible structure model and KDamper devices is considered and is subjected to artificial accelerograms designed to match the EC response spectra. The KDamper is designed to higher frequencies compared to the isolated system with seismic isolation bearings, exploiting the extraordinary damping properties it offers. A comparison with a base isolated structure using Lead Rubber Bearings designed to greatly increase the natural period of the system (2.0–2.5 s), confirms that KDamper base seismic absorption designs can provide great reduction to the inter-story drifts and absolute accelerations reducing at the same time the base displacement.

Appendix

In the limit cases of ζ_D = 0 or ζ_D $\to \infty$, H_AS of Eq. (10b) becomes

$$H_{AS} (0) = \left| {\frac{A}{C}} \right|;H_{AS} (\infty ) = \left| {\frac{B}{D}} \right|$$

(29)

The transfer function H_AS(q, ζ_D) of Eq. (10b) has two poles for two different values of q, and it presents two different maximal values (peaks) at these points. The optimal selection of the parameters of the KDamper requires that both these peaks are minimized and become equal to each other. This is ensured by the optimal design approach followed in Den Hartog [38], which will be also used in the current paper. The approach is based on the identification of a pair of frequencies q_L < 1 and q_R > 1, where the values H_AS(q_L) and H_AS(q_R) become independent of ζ_D. The first step for the optimization procedure is the requirement that the values of the transfer functions at these points are equal:

$$H_{AS} \left( {q_{L} } \right) \, = H_{AS} \left( {q_{R} } \right) \, = H_{ASI} = H_{AS} \left( \infty \right)$$

(30)

In order that a solution for such a pair of frequencies exists, two alternative conditions must be fulfilled as in Den Hartog [38]:

$${\text{Case I}}:AD = BC$$

(31a)

$${\text{Case II}}:AD = - BC$$

(31b)

As it can be verified, no solution of Eq. (31a) exists for a positive q², when the values κ, μ, and ρ are positive. Elaboration of Eq. (31b) results in

$$(A_{2} D_{2} + B_{0} )q^{4} + (A_{0} D_{2} + A_{2} D_{0} + B_{0} C_{2} )q^{2} + (A_{0} D_{0} + B_{0} C_{0} ) = 0$$

(32)

where

$$A = A_{2} q^{2} + A_{0} ;B = B_{0} \rho q;C = q^{4} + C_{2} q^{2} + C_{0} ;D = (D_{2} q^{2} + D_{0} )\rho q$$

(33a, b, c, d)

$$A_{2} = A_{2\rho } \rho^{2} + A_{20} ;A_{0} = A_{0\rho } \rho^{2} + A_{00} ;B_{0} = B_{0\rho } \rho^{2} + B_{00} ;C_{2} = C_{2\rho } \rho^{2} + C_{20}$$

(34a, b, c, d)

$$C_{0} = C_{0\rho } \rho^{2} + C_{00} ;D_{2} = D_{2\rho } \rho^{2} + D_{20} ;D_{0} = D_{0\rho } \rho^{2} + D_{00}$$

(34e, f, g)

$$A_{\rho } = (A_{0\rho } D_{2\rho } + A_{2\rho } D_{0\rho } + B_{0\rho } C_{2\rho } )D_{20} - 2(A_{2\rho } D_{20} + A_{20} D_{2\rho } + B_{0\rho } )D_{0\rho }$$

(35a)

$$B_{\rho A} = [(A_{0\rho } D_{20} + D_{2\rho } A_{00} ) + (A_{2\rho } D_{00} + D_{0\rho } A_{20} ) + (B_{0\rho } C_{20} + C_{2\rho } B_{00} )]D_{20}$$

(35b)

$$B_{\rho B} = - 2(A_{2\rho } D_{20} + A_{20} D_{2\rho } + B_{0\rho } )D_{00} - 2(A_{20} D_{20} + B_{00} )D_{0\rho }$$

(35c)

$$B_{\rho } = B_{\rho A} + B_{\rho B}$$

(35d)

$$C_{\rho } = (A_{00} D_{20} + A_{20} D_{00} + B_{00} C_{20} )D_{20} - 2(A_{20} D_{20} + B_{00} )D_{00}$$

(35e)

and the coefficients in the Eqs. (34a, b, c, d, 34e, f, g) are defined in Table 5.

Table 5 Coefficients in Eqs. 34a, b, c, d, 34e, f, g

As a result of Eq. (32), the pair of roots of Eq. (32) must satisfy:

$$q_{L}^{2} + q_{R}^{2} = - \frac{{(A_{0} D_{2} + A_{2} D_{0} + B_{0} C_{2} )}}{{(A_{2} D_{2} + B_{0} )}}$$

(36)

Additionally, both roots q_L and q_R must fulfill Eq. (29), which results in

$$\frac{{B_{0} }}{{D_{0} + D_{2} q_{L}^{2} }} = - \frac{{B_{0} }}{{D_{0} + D_{2} q_{R}^{2} }} \Rightarrow q_{L}^{2} + q_{R}^{2} = - \frac{{2D_{0} }}{{D_{2} }}$$

(37)

The combination of Eqs. (36) and (37) leads to an equation for the optimal value of the parameter ρ:

$$A_{\rho } \rho^{4} + {\rm B}_{\rho } \rho^{2} + C_{\rho } = 0$$

(38)

The optimal value of ρ is selected as the minimum positive value of the two roots of (38).