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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α s with respect to the element local Cartesian axis e 1 , rebar area A s , spacing i s and depth z s with respect to the shell mid-surface. As shown in Fig. 1, a discrete number of IP is considered located in position x gc = [ x g , z c ], where x g is the position of a generic IP over the mid-surface of the element and z c indicates the depth of the generic IP along the thickness direction. The vector of the generalised shell stresses at the g -th mid-surface IP is evaluated as t g = t [ x g ] =  h / 2 − h / 2 E [ z ] T σ [ x g , z ] dz = n c  c = 1 E [ z c ] T σ gc w c + n c + n s  s = n c + 1 E [ z s ] T σ gs A s i s (1) where n c is the number of concrete IPs along the thickness, n s is the number of steel layers, σ gc are the concrete stresses at the c -th integration point along the thickness direction, w c is the corresponding weight and σ gs are the stresses of the s -th reinforcement layer. The stresses in both concrete and steel must satisfy the plastic admissibility in each IP, expressed by where with m we identify the generic IP and f m is the yield function. The stresses σ gc and σ gs are independently evaluated by solving the IP state determination scheme at level of material point   F m ( σ gm − σ ( n ) gm ) + µ gm ∂ f m [ σ gm ] ∂ σ gm − ∆ ε gm = 0 µ gm f m [ σ gm ] = 0 µ gm ≥ 0 f m [ σ gm ] ≤ 0 ∀ m = 1 , . . . , n c + n s . (3) where ∆ = ( · ) ( n + 1) − ( · ) ( n ) represents the di ff erence between quantities at step n + 1 and n , and µ gm are the positive plastic multipliers, ∆ ε gm = E [ z m ] ∆ ϱ g represents the strain on the m th IP, while the assigned ∆ ϱ g is constant for each m . The superscript ( n + 1) is omitted to simplify the notation. The stress parameters β e = β e [ β ( n ) e , ∆ d e ] for an assigned value of ∆ d e are obtained using the element state determi nation algorithm defined by the following additional equations   r g ≡ t g − N tg β e = 0 , ∀ g r e ≡ Q e ∆ d e −  g N T tg ∆ ϱ g w g = 0 . (4) where first one imposes that the generalised stresses t g coming from the solution of the IP state determination in Eq. (3) and evaluated using Eq. (1), are the same than those furnished by the assumed stress interpolation. The second equation imposes, in a weak form, the strain / displacement relation over the element. f m [ σ gm ] ≤ 0 , m = 1 , . . . , n c + n s (2)

2.2. Concrete mechanical response

The stress response of concrete, σ gc , relative to the generic IP along the thickness is evaluated by solving problem (3) on the basis of an elastic isotropic behaviour defined by the Young modulus, E c , Poisson ratio, ν c , of concrete and the yield function proposed in Papanikolaou and Kappos (2007) and here reformulated with respect to the stress