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

547

4

invariants I 1 and J 2 and the Lode angle θ as follows

m √ 3 σ c 

3 σ 2 c

m 3 σ c

f c ≡

J 2 r ( θ, e ) +

I 1 − 1 .

(5)

J 2 +

The exact meaning of the material parameters, m and e , is defined in the already cited paper Papanikolaou and Kappos (2007) where their dependence from σ c , the compressive strength of concrete, is formulated and discussed. In this way an apparently three-parameter hydrostatic-pressure-sensitive loading surface, capable to describe parabolic meridians and a variable shape on the deviatoric plane, is simply defined on the basis of σ c only. The Newton-Raphson solution of Equation (3) provides the required stress σ gc but also the consistent tangent operator F − 1 tc relative to the concrete IP formulated as

f c ∂ σ ⊗ Ξ ∂ f c ∂ σ · Ξ

∂ f c ∂ σ

Ξ ∂

∂ 2 f c ∂ σ 2

F − 1

Ξ − 1 = F

(6)

tc = Ξ −

,

c + µ gc

,

∂ f c ∂ σ

and ready to be used in the assembly of the global tangent sti ff ness matrix.

2.3. Steel reinforcement layer mechanical response

Problem (3) is also solved for the generic steel reinforcement layer whose mono-axial mechanical response is now simply defined by the Young modulus E s , the yield stress σ s and the vector components c = cos α s and s = sin α s defined by rebar orientation, α s , with respect to the axes describing the shell mid-surface. In particular, in the elastic phase, the mechanical response is defined by

    .

    c 4 c 2 s 2 c 3 s 0 0 c 2 s 2 s 4 s 3 c 0 0 c 3 s s 3 c c 2 s 2 0 0 0 0 0 0 0 0 0 0 0 0

( n ) gs + F −

1 s ∆ ε gs , F − 1

s = E s

(7)

σ gs = σ

In the plastic phase the mechanical response is given by

σ gs = ± σ s  c

2 s 2 − cs 0 0  T

, F − 1

ts = 0 5 × 5 .

(8)

3. Numerical results

E ffi ciency and accuracy of the proposed FE strategy have been tested with respect to some numerical tests consist ing in the step-by-step nonlinear analysis of structures subjected to a system of loads amplified through a multiplier λ used to describe the loading history. The results provided by the layer-wise MISS-4 element are compared with those provided by the commercial software Abaqus. In particular, all the analyses are also performed by using the Abaqus S4 shell element and describing the nonlinear behaviour of concrete through the Concrete Damage Plasticity (CDP) model parameterized in order to obtain a perfectly plastic behaviour. The response of MISS-4 element is evaluated by using a 2 × 2 IP grid over the finite element mid-surface and 12 IPs through the shell thickness. The response of the steel reinforcement bar layers contribute with a single IP for each steel layer.