PSI - Issue 44

Fabio Mazza et al. / Procedia Structural Integrity 44 (2023) 1172–1179 Fabio Mazza / Structural Integrity Procedia 00 (2022) 000–000

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A nonlinear static analysis of the original (fixed-base) structure is preliminarily carried out with reference to invariant lateral forces along the in-plan X and Y directions, increasing proportionally to “uniform” (i.e. proportional to the floor masses) and “modal” (i.e. proportional to the fundamental vibration mode) vertical distributions. Then, each capacity curve is converted to that of an equivalent single-degree-of-freedom system and bilinearized in an elastic-perfectly plastic force-displacement law using an equal energy criterion. Finally, the fundamental vibration period ensuring elastic behaviour of the retrofitted (base-isolated) structure (T BI,1H ) is determined on the reduced elastic response spectrum of acceleration corresponding to the equivalent viscous damping ratio ( x e,H ), starting from the acceleration at the yield point of the modal capacity curve, characterized by the lowest shear strength, as well as satisfying the condition T BI,1H ≥max(3∙T FB,1X , 3∙T FB,1Y ). Note that different values of x e,H generally correspond to different design approaches, while x e,V =5% is assumed in the vertical direction. The design of the HDRBs is carried out at the collapse prevention (CP) limit state, considering the horizontal seismic loads acting in combination with the vertical ones and assuming a high-risk seismic zone (PGA H =0.510g and PGA V =0.476g). A shear modulus G=0.5MPa and a volumetric compression modulus E b =2000MPa are assumed for the elastomer. Specifically, HDRBs fulfil provisions regarding the design shear strains (UNI EN 15129 (2009)): i.e. g tot = g s + g c + g a ≤7 and g s ≤2.5, where g tot represents the total shear strain, while g s , g c and g a represent the shear strains of the elastomer due, respectively, to seismic displacement, axial compression and angular rotation. Moreover, the maximum compression axial load does not exceed the critical load (P cr ) divided by a safety coefficient equal to 2.0, with d ≤0.7 (i.e. d =d dC /D l , being d dC the horizontal displacement at the CP limit state and D l the diameter of the internal steel reinforcing plates). Finally, ultimate tensile stress of all isolators equal to 1 MPa and maximum compressive stress of steel plates lower than f yk (=275 MPa) are assumed. The following data are also reported in Table 1: i.e. primary (S 1 ) and secondary (S 2 ) shape factors; total thickness of the elastomeric layers (t e ); horizontal (K e,H1 ) and vertical (K e,V1 ) secant stiffnesses and equivalent damping coefficients (C H1 and C V1 ). Table 1. Design properties of a HDRB for the BI.H and BI.HV base-isolation systems (units in kN, m and s). UNI EN 15129 T 1H T 1V d dC S 1 S 2 D l t e P cr x e,H x e,V K e,H1 K e,V1 C H1 C V1 Nominal 2.62 0.053 0.415 34.2 3.15 0.656 0.208 25176 15.0% 5% 862 2.07×10 6 108 959 Upper bound 2.26 0.045 0.354 34.2 4.54 0.645 0.143 28132 13.7% 5% 1225 2.94×10 6 114 1056 Lower bound 2.98 0.060 0.466 33.9 2.34 0.710 0.303 17350 15.0% 5% 690 1.66×10 6 95 884 Mixed 2.37 0.047 0.428 34.1 3.93 0.686 0.175 27191 9.24% 5% 1120 2.69×10 6 73 1028

In the combination of horizontal and vertical base-isolation, the total vertical stiffness resulting from the in-series arrangement of HDRB and HDRL is evaluated as follows

) 1 -

(

e,Vtot K 1 K 1 K = + e,V1

K

(3)

=

×a

e,V2

e,H1 K,e

where the vertical stiffness of the HDRL is evaluated as

2 HDRL c HDRL D E 4 t × ×

p×

(4)

K

HDRL HDRB , D D 15cm = -

=

e,V2

being D HDRL the diameter and t HDRL the thickness, while

(

) 2

(5)

c 1 E E 1 2 k S , E=5.12 MPa and k=0.56 = × + × ×

represents the effective compressive modulus. It should be noted that the total horizontal stiffness is the same as the HDRB, because HDRL does not affect the horizontal shear deformation of the base-isolation system (Pourmasoud et al. (2020)). Design properties of the HDRL corresponding to the four design approaches are reported in Table 2.