Application of the Εxtended KDamper to the Seismic Protection of Bridges: Design Optimization, Nonlinear Response, SSI and Pounding Effects

ABSTRACT

The paper investigates the seismic performance of a novel passive vibration isolation and damping device termed Extended KDamper (EKD). The concept is applied to a representative two-span highway bridge, initially designed on conventional seismic isolation (CSI) bearings. An optimization process is developed and executed to design the EKDs, underscoring the importance of accounting for seismic motion variability. Compared to CSI, the incorporation of EKDs leads to a 40% to 70% reduction in deck drifts. In contrast to the CSI bridge, which may sustain excessive bearing shear strains when subjected to the most adverse seismic motions within the examined set, the bearings of the EKD bridge never exceed the 200% threshold. Through the use of nonlinear 3D time-history analyses, it is demonstrated that the nonlinearity of the EKD elements may result in residual deck drifts. The nonlinear EKDs exhibit a variation in maximum drifts and accelerations on the order of ±20% compared to the preliminary (linear elastic) design for the examined set of spectrum-compatible motions. The increased accelerations result from the stiffening of the negative stiffness elements (NSEs), being more pronounced for seismic motions that entail large displacement demands. With the aid of a fully nonlinear 3D model of the entire soil – foundation – structure system, the effects of soil-structure interaction (SSI) are explored and shown to significantly influence the seismic response of the system. Deck collision with the abutments restricts the movement of the deck and pier; however, it compromises the performance of the EKDs and leads to a substantial increase in deck accelerations. Overall, EKDs may facilitate a more economical design and enhanced seismic performance, particularly for displacement-sensitive structures like rail bridges.

KEYWORDS:

• Seismic isolation
• bridges
• KDamper
• negative stiffness
• earthquakes
• vibration control

Acknowledgments

This work was financially supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No INSPIRE-813424. The first author would like to thank Dr Athanasios Agalianos (ETH Zurich, Switzerland) and Mr Antonios Mantakas (NTUA, Greece) for the fruitful discussions and constructive criticism over the course of this research.

Disclosure Statement

No potential conflict of interest was reported by the author(s).

Data Availability Statement

The data that support the findings of this study are available from the corresponding author, Ioanni Anastasopoulo, upon reasonable request.

Notation

$C_b$ :	=	dashpot coefficient
$C_{\mathit{up}},C_{\mathit{down}}$ :	=	damping factors of the EKD device
$E_{\mathit{comp}}$ :	=	instantaneous compression modulus of the rubber-steel composite under specified level of vertical load in elastomeric bearings
$E_o$ :	=	small-strain elastic modulus of soil
$E_p$ :	=	pile elastic modulus
$E_s$ :	=	steel Young’s modulus
$F_{3\mathit{p}}$ :	=	ultimate abutment soil resistance
$F_{\mathit{base}}$ :	=	maximum base shear
$F_p$ :	=	mechanical spring precompression load
$K_{1\mathit{p}}$ :	=	initial abutment soil stiffness
$K_{\mathit{b},\mathit{H}}$ :	=	horizontal stiffness of elastomeric bearing
$K_{\mathit{b},\mathit{V}}$ :	=	vertical stiffness of elastomeric bearing
$K_c$ :	=	concrete damage plasticity model parameter
$K_s$ :	=	mechanical spring stiffness
$K_{\mathit{tot}}:$	=	total bridge stiffness
$L_p$ :	=	undeformed length of the precompressed mechanical spring
$S_u$ :	=	undrained shear strength
$T_E$ :	=	fundamental earthquake period
$T_n$ :	=	bridge fundamental natural period
$V_s$ :	=	soil shear wave velocity
$a_1$, $p_1$ :	=	coefficients employed in the calculation of abutment soil stiffness and ultimate resistance
$c_D$ :	=	damping factor of the K-Damper device
$c_s$ :	=	damping factor of a structural system
$d_s$ :	=	bearing drift
$f_{\mathit{ck}}$ :	=	characteristic concrete compressive strength
$f_t$ :	=	concrete tensile strength
$f_y$ :	=	steel yield strength
$k_1$ :	=	initial stiffness in the bilinear law characterizing elastomeric bearings behaviour
$k_2$ :	=	post-yielding stiffness in the bilinear law characterizing elastomeric bearings behaviour
$k_N$ :	=	negative stiffness element of the K-Damper (or EKD) device
$k_P$ :	=	positive stiffness element of the K-Damper (or EKD) device
$k_R$ :	=	positive stiffness element working in parallel with the EKD device
$k_b$ :	=	elastomeric bearing stiffness
$k_o$ :	=	tangential stiffness of the EKD device
$k_{\mathit{pier}}$ :	=	pier stiffness
$l_1$, $l_2$ :	=	lengths on the NSE device, used as input for the calculation of the NSE force-displacement relationship
$m_D$ :	=	additional mass of the K-Damper (or EKD) device
$m_{\mathit{deck}}$ :	=	deck mass
$m_{\mathit{pier}}$ :	=	pier mass
$m_s$ :	=	mass of bridge deck
$t_r$ :	=	height of elastomer layer
$u_s$ :	=	point of change from negative to positive stiffness for the precompressed spring (displacement value)
$u_y$ :	=	bearing yield displacement
$x_{\mathit{deck}},x_{\mathit{pier}},x_{\mathit{G},}{x}_{\mathit{D}1,}{x}_{\mathit{D}2}:$	=	horizontal displacements of the bridge deck, pier top, structure base, and of the additional EKD masses
${\mathrm{H}}_w$ :	=	abutment wall height
${\mathrm{K}}_{\mathit{eff}}$:	=	bearing effective stiffness
${\delta }_c$ :	=	clearance between the deck and the abutment stoppers
${\delta }_{\mathit{pier},\mathit{res}}$ :	=	residual pier drift
${\delta }_{\mathit{pier}}$ :	=	pier drift
${\epsilon }_N$, ${\epsilon }_P$ and ${\epsilon }_R$ :	=	variations in the positive and negative stiffness elements of the EKD devices
${\epsilon }_s$ :	=	seismic shear strain
${\xi }_{\mathit{eff}}$ :	=	effective damping ratio
${\sigma }_{\mathit{bo}}/{\sigma }_{\mathit{co}}$ :	=	concrete damage plasticity model parameter
$\mathit{A}$ :	=	Area
$\mathit{BW}$ :	=	bandwidth
$\mathit{CSI}$ :	=	Conventional seismic isolation
$\mathit{D}$ :	=	pier diameter
$\mathit{E}$ :	=	concrete Young’s modulus
$\mathit{G}$ :	=	shear modulus
$\mathit{H}$ :	=	height
$\mathit{H}$ :	=	pier height
$\mathit{HMCR}$ :	=	harmony consideration rate
$\mathit{HMS}$ :	=	harmony memory size
$\mathit{L}$ :	=	length
$\mathit{L}:$	=	pile length
$\mathit{MaxItr}$ :	=	maximum iteration number
$\mathit{PAR}$ :	=	pitch adjusting rate
$\mathit{PGA}$ :	=	Peak Ground Acceleration
$\mathit{Q}$ :	=	characteristic bearing strength
$\mathit{S}$ :	=	bearing shape factor
$\mathit{W}$ :	=	width
$\mathit{d}$ :	=	pile diameter
$\mathit{max}\left|a_{\mathit{Deck}}\right|$ :	=	maximum absolute deck acceleration
$\mathit{max}\left|{\delta }_{\mathit{Deck}}\right|$ :	=	maximum absolute deck drift
$\mathit{res}\left|{\delta }_{\mathit{Deck}}\right|$ :	=	residual (absolute) deck drift
$\mathit{s}$ :	=	distance between foundation piles
$\mathit{u}$ :	=	horizontal displacement
$\mathit{u},\dot{u},\ddot{u}$ :	=	relative displacement (drift), velocity & acceleration in the presented equations of motion
$\mathit{w}$ :	=	abutment wall width
$\mathit{\gamma }$ :	=	scalar coefficient that determines the rate of decrease of kinematic hardening with increasing plastic deformation in the employed soil constitutive model
$\mathit{\mu }$ :	=	friction coefficient
$\mathit{\nu }$ :	=	Poisson’s ratio
$\mathit{\xi }$ :	=	Rayleigh damping ratio
$\mathit{\rho }$ :	=	material density
$\mathit{\psi }$ :	=	concrete dilation angle
$\mathit{ϵ}$ :	=	flow potential eccentricity

Notes

1. https://peer.berkeley.edu/peer-strong-ground-motion-databases.

Additional information

Funding

This work was supported by the H2020 Marie Skłodowska-Curie Actions [INSPIRE-813424].