PSI - Issue 64

Luigi Granata et al. / Procedia Structural Integrity 64 (2024) 1073–1080 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

1075

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a

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t

b

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a)

b)

c)

Fig. 1. a) FRP-RC cross-section; b) 3D modelling of a quarter reinforcement; c) front and top views.

The static scheme adopted to model the phenomenon is shown in Fig. 2. It consists of a plane-curved continuous beam on an elastic foundation (modelled by a continuous distribution of radial and axial springs). In detail, the radial springs simulate the contact between the stirrup and the longitudinal CFRP bar, while the axial springs simulate the friction between the beam and the surrounding concrete. Finally, the load condition is represented by a force applied at one of the two ends of the beam (F b ). The reference system is represented in polar coordinates ( , ) with 0≤ψ≤ ⁄2 as depicted in Fig. 2. 2.1. Mathematical model The elastic problem relative to the static scheme of Fig. 2 is represented by the following set of equations:

− + =0 + + =0 − + = 0 + + = − = ( − ) = − 1 ( − )

(1a-f)

{

Fig. 2. Static Scheme Eqs (1a-c) represent the equilibrium equations, while Eqs (1d-f) are the compatibility equations coupled with the constitutive equations. All variables are functions of ψ : ( ) , ( ) and ( ) are the axial force, the shear force and the bending moment, respectively; ( ) and ( ) , are the tangential and radial displacement; ( ) is the rotation about the axis perpendicular to the plane. Furthermore, and , are the distributed longitudinal and radial loads, which are related to the corresponding stiffness of the springs by the following equations: