PSI - Issue 44

Laura Giovanna Guidi et al. / Procedia Structural Integrity 44 (2023) 1284–1291 Laura Giovanna Guidi et al. / Structural Integrity Procedia 00 (2022) 000–000

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Kostantinidis (2011), bearing is assumed as a continuous composite system, while plane sections, normal to the un deformed central axis, remain plane also after bending. This approach, modelling the isolator as a continuous beam, slightly modifies the linear elastic formulation of Euler buckling, by introducing shear stiffness contribution. Kelly provides a formula for deriving the critical load of elastomeric bearings, V crit,0 , in absence of lateral deformations: V cr,0 = (P E · P S ) 1/2 (1) where P E is the Euler load and P S is the shear stiffness per unit length. As argued in detail by the authors (De Luca et al., 2022), the value of critical load in absence of lateral deformations, V crit,0 can be expressed as function of both shape factors. For a circular bearing, having diameter ϕ , single rubber layer thickness t i , total rubber thickness t r , these dimensionless parameters can be defined in accordance to (2) and (3): (4) (5) To determine the critical load in view of the combined effects of lateral deformations, literature calls back the overlapping (or reduced area) formulation, firstly proposed by Buckle and Liu (1994), still used nowadays in practice and adopted by many design codes. When a horizontal displacement (d H ) occurs, the effective load-bearing area of is reduced to the only overlapped portion (between the upper and the lower base), i.e. the reduced area, A r , defined by literature, in accordance to (6), where R is the radius of the circular bearing and θ is the half-angle subtended at the center of the intersection of the top and bottom circles. For horizontal displacements of about 30-40% of device diameter, the reduced area can be estimated with good approximation through the simplified formula (7). Considering the effective loaded area of device when horizontal deformation occurs, the critical load is given by (8), as proposed by the authors (De Luca et al., 2022): V 0'1# = V 0'1#,3 ( 4 5 4 ) = 1,1 ∙ G ∙ A ' ∙ S : ∙ S ; (8) This formulation points out the main geometrical and mechanical parameters involved in buckling phenomenon: G related to material, A r related to the horizontal displacement, S 1 associated to local geometry, S 2 related to global geometry. Assuming V (= σ v ·A) as the vertical load that corresponds to the applied vertical stress, the corresponding buckling ratio is defined in (9). Theoretically, to prevent from buckling failure it is necessary that the buckling ratio is at least greater than 1. < =5$> < = G·S 1 ·S 2 · ) 4 5 4 . ∙ ? : @ (9) According to (9), the buckling ratio increases proportionally to the product of shape factors and reduces as the vertical stress grows. This theoretical tends to zero when the horizontal displacement is equal to the device width. This means that, theoretically, to define the stability limit under the combined effect of vertical load and lateral deformation, the widest horizontal displacement (d H ) cannot exceed the diameter of device itself, i.e. d H ≤ ϕ . As visible in (10), this parameter can be expressed also as function of the secondary shape factor: , - ! = , - # % ∙ # % ! = B A C (10) In terms of maximum horizontal deformation, the theoretical limit is d H = ϕ , while looking at the vertical stress, the ultimate condition is given by V=V crit , or equivalently in terms of the corresponding vertical stress. Opting for a V crit,0 = (1,11) · G · A · S 1 · S 2 σ crit,0 = (1,11) · G · S 1 · S 2 (6) A r = 2R ² ( θ - sin θ ·cos θ ) A ' = A )1 − , - ! . (7) (2) S 1 = " ! # $ S 2 = # ! % (3) As consequence, V crit,0 and the corresponding critical pressure, σ crit,0 can be defined as (4) (5):