PSI - Issue 44

Marco Gallo et al. / Procedia Structural Integrity 44 (2023) 618–625 Marco Gallo et al. / Structural Integrity Procedia 00 (2022) 000–000

622

5

response of the structure (inertial interaction effect) constrained via several base springs which simulate the soil foundation interaction. In scientific literature several solutions for the calculation of the impedances of rigid circular and rectangular shallow foundations are available such as those provided by Pais and Kausel (1988), Gazetas (1991), Mylonakis et al. (2006). In these formulas, the base spring stiffness is basically a function of the foundation dimension and G and v modulus of the soil. The following allows to evaluate the stiffness for the vertical translation to be assigned to the base springs: ! = (% " & # ') #3.1 ' # ) ( *.,- + 1.6+ (1) The soil-structure interaction (SSI) can be modelled in FEM models by assuming a different base constraint. Usually a fixed constraint is used, however a set of springs can be assigned to the base joints to take into account the soil mechanical properties. The springs stiffness should be evaluated by considering on the one hand the soil mechanical properties in terms of E , G and V s , on the other hand that must be dependent on the foundations system. Moreover, the foundation system also includes two piles whose effect in terms of SSI may be relevant. The piles effect can be equally considered by adding a set of springs that simulate their stiffness for displacements and rotation. The first type is obtainable from the product of a static stiffness, which depend on the Young’s modulus and diameter, and a dynamic factor. Regarding the rotational stiffness it is typically neglected as it is usually obtained from the vertical stiffness of the group of piles. However, for the case study purpose, three translational springs were assigned at the base of the two piers, also including the rotational one needed to make possible the equilibrium out of plane. 4.2. Effect of SSI on dynamic response of the bridge The SSI modelling affect the dynamic response of the bridge. As it has been previously shown the analysis carried out, static equivalent, provides for the application of horizontal forces which simulates the seismic action on the bridge. These forces depend on the permanent load applied on the portal (pier and pulvinus), as the y factor for bridge is equal to zero. For the linear analysis the seismic actions are modeled via response spectrum which allow to evaluate the acceleration as a function of the vibration period of the structure. The SSI particularly affect the period as the springs used to restrain the structure, replacing the fixed joints, make the flexibility of the structure increasing while the load applied are not modified in this model. As a result of that the vibration period of the structure are significantly higher in SSI case as shown in table 7.

Table 6. Vibration periods comparison.

FB

SSI

D [%]

T 1 [s] T 2 [s] T 3 [s]

0.40 0.37 0.29

0.84 0.72 0.39

112

97 34

Neglecting the first part of the response spectrum, the acceleration always decreases for vibration period greater than that leads to smaller seismic action. For this reason, the stresses which are considered in the verification are typically smaller so a reduction of the demand-to-capacity ration reduction is directly obtainable with the SSI modelling. The Figure 4 shows the comparing between the fixed and SSI case.

4.00

Fixed base transversal direction Fixed base longitudinal direction

2.00

D/C

0.00

bending moment

Shear

Fig. 4. Fixed base VS SSI: Demand-to-capacity ratios comparison.