PSI - Issue 44

Gaspar Auad et al. / Procedia Structural Integrity 44 (2023) 1474–1481 Gaspar Auad / Structural Integrity Procedia 00 (2022) 000–000

1476

3

! ( , , ) = " ! #$ ! % ! + ('()) ! ) ! − 1 = 0

(2)

2 a

z

0 5 10 z (cm) 50

50

2b

0

x

0

(a)

y (cm)

(b)

-50 -50

x (cm)

Fig. 1. (a) Ellipse with its geometric parameters; (b) Variable curvature sliding surface.

2.1. Local system of coordinates Fig. 2 shows illustrations of the physical model in undeformed and deformed configurations. Additionally, in this figure, the main geometric parameters are presented being: + and , , the vertical distances between the nodes I and J and the origin O in the undeformed configuration, respectively; and - , the lateral capacity of the isolator. The local system, ℑ . = { : } , is located on the sliding surface. The symbol S represents the instantaneous positions of the slider concerning the local coordinate system. The deformation of the device is described by the vector = > >> > = ? " , $ , ' A / . Note that, since the physical model considers the actual geometry of the sliding surface, a vertical constrain is imposed and needs to be satisfied at every time step: ' = − C . − ) ! % ! D ". + $. E (3) A gap element is included in the physical model. The uplifting and vertical impact of the isolator is considered due to the incapacity of the gap element to transmit tension forces. The kinematic constraints in the slider trajectory can be accounted for by adopting a large value in the axial stiffness of the gap element.

J-node

(J)

u z

(J)

u

x

(J)

r y

z

z

l J

gap

Z

S

x

x

O

local frame ℑ 2

O

ground frame ℑ 1

l I

(I)

u z

I-node

X

(I)

r y

L c

(I)

u x

Fig. 2. Representation of the physical model

2.2. Non-linear kinematics Since large deformations are accounted for in the approach used in the physical model, the relationship between the degrees of freedom of the structure and the deformation of the variable curvature isolator is nonlinear. The