PSI - Issue 44

Francesco S. Liguori et al. / Procedia Structural Integrity 44 (2023) 544–549 F. S. Liguori, A. Corrado, A. Bilotta and A. Madeo / Structural Integrity Procedia 00 (2022) 000–000

548

5

The results compare the obtained equilibrium curves, i. e. the plot of the load multiplier λ versus a reference displacement component. For the sake of conciseness only one test is reported in the following.

3.1. Prestressed girder

The test regards a prestressed girder of length L = 30 [m] whose geometry, loads, reinforcement layout and boundary conditions are described in Fig. 2. The beam is simply-supported at the two end sections, while it is free to deform axially. The concrete compressive strength is σ c = 40 [MPa] from which the other parameters are derived. The yield stress of the steel reinforcement is σ s = 1500 [MPa] in the bottom plate and σ s = 450 [MPa] elsewhere, the Young modulus is E s = 2 . 1 × 10 5 [MPa].

l q z

f14/150

f14/300

30000

v,w=0

200

300 300

2000

f14/150

transversal rebars f14/150 500

3200 1200

Z Y X w=0

q 0

3D view

Cross-section

Fig. 2: Prestressed girder: geometry, loads and boundary conditions

The load is applied into two phases. In the first phase the girder is subjected to a compressive loading along the bottom plate q 0 = 4 . 62 10 3 [N / mm] to simulate the prestress load. A uniformly distributed pressure is subsequently applied to the top slab with magnitude q z = 0 . 04 [MPa] and ramped by the load multiplier λ . The first mesh used in the analyses is composed by 7 elements dividing the cross-section and 6 elements along the longitudinal direction. The other two meshes are an h -refinement of the initial one. Figure 3 reports the equilibrium curves, i. e. the load multiplier vs the vertical displacement of the midpoint of the beam. The proposed model provides accurate results if compared to the Abaqus solution. Furthermore, the equilibrium path evaluated using S4 and CDP coincides with that given by the proposed elastoplastic model. In the right-hand side of Fig. 3 the contour plots of the equivalent plastic strain at two di ff erent equilibrium points are shown. They indicate how plasticity concentrates in the beam midspan and collapse occurs in the areas undergoing tensile stresses. A mixed 4-nodes finite element has been formulated for the analysis of reinforced concrete shells. The element is based on the assumed interpolation of both displacement and stress fields. The interpolation of the stress field is chosen in order to a-priori satisfy the equilibrium equations inside the element domain. In this way e ffi ciency and accuracy are ensured. Moreover, the assumed interpolation for the displacement field can be defined only along the element’s boundary. The material mechanical response of the concrete is numerically integrated along the shell’s thickness assuming a plasticity yield surface depending on all the three stress invariants which allows the description of the confinement-sensitive behaviour. The steel reinforcement bars are modelled as additional layers, embedded in the shell thickness, with von Mises mechanical response. The accuracy and e ffi ciency of the proposed formulation are confirmed by the numerical tests performed. 4. Concluding remarks