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

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elastic field, in Liguori and Madeo (2021) for the analysis of slender structures and in Bilotta and Casciaro (2007); Bilotta et al. (2016) for 2D and 3D applications involving standard elastoplastic materials, i. e. von Mises-like models. The analysis of shell structures constituted by concrete, a material requiring more challenging elastoplastic mod elling choices, in presence of steel reinforcement bars, see for example Hofstetter and Mang (1996); Haussler (2014), constitutes a new context still not explored. The kinematic description is based on a first order shear deformable shell model whose primary kinematic fields are interpolated on the basis of a four node plane element, Madeo et al. (2021), whose mixed formulation contains also the discrete representation of the components of generalised stresses of the shell. The control of stress history and of the related evolution of plastic strain is however performed at 3D-level thanks to a numerical integration along the thickness of the shell. In this way is possible to accomodate a layered description of the materials composing the shell where concrete behaviour is modelled through a confinement-sensitive plasticity yield surface depending on all the three stress invariants, see Papanikolaou and Kappos (2007), and expressed in terms of only the uniaxial compressive strength of the concrete. At negligible computational costs the presence of steel rebars is computed by defining additional material layers contributing through a von Mises one-dimensional material response. It is worth to observe how the proposed approach can be readily used for the modelling of additional material layers as in the case of retrofitting of existing structures. Computational e ffi ciency and accuracy are assessed by comparing the proposed shell model with the solution obtained by standard displacement-based FEM formulations and the solutions provided by the commercial code ABAQUS.

2. Layer-wise shell formulation of the MISS-4 finite element

The MISS-4 shell finite element proposed in Madeo et al. (2021) is now reformulated in order to describe the di ff erent layers of materials composing the shell. The MISS-4 element interpolates both stress and displacement fields by choosing them in order to achieve high accuracy on very coarse meshes. In particular the assumed interpolation for the components of the generalised stresses are selected in order to a-priori satisfy the internal equilibrium equations. As a consequence the displacement field needs to be interpolated only along the element boundary. All other details, the assumed interpolations etc., can be found in Madeo et al. (2021).

2.1. Shell element response

gc -th IP

x g y g z c

e 1 e 2 e 3

z s

shell mid-plane

i s

a s

s -th reinforcement layer

Fig. 1: Reinforced concrete shell.

Figure 1 shows the generic material composition of the shell finite element. In particular, it is made of concrete and a certain number of reinforcement layers. Each reinforcement layer, identified by the the subscript s , has orientation