Issue 37

P. Bernardi et alii, Frattura ed Integrità Strutturale, 37 (2016) 15-21; DOI: 10.3221/IGF-ESIS.37.03 17 that occurs after cracking, as explained in [10, 12]. Moreover, since in the cracked stage the hypothesis of perfect bond is no longer valid (and consequently the two strain vectors { ε c } and { ε s } cannot be set equal to each other), the steel strain { ε s } is assumed coincident with the total average strain { ε }, with a negligible approximation in excess. By inverting the equilibrium condition in RC between cracks, concrete strains { ε c } can be then expressed as:               sh s 1 c c D D         . (4) The strain vector { ε cr 1 } can be in turn determined by inverting the equilibrium equation at crack location:         1 1cr 1cr D   , (5) being [ D cr1 ] the crack stiffness matrix, whose expression – which is not modified in presence of shrinkage – can be still found in [10]. By substituting Eqs 4 and 5 into the compatibility Eq. 3, the total stress { σ } vector in case of shrinkage can be then expressed as follows:                       sh s 1 c 1 1 1cr 1 c D D I D D           , (6) being [ I ] the identity matrix. M ODEL V ALIDATION he effectiveness of the above described procedure, as well as of its correct implementation into a commercial finite element (FE) code (ABAQUS), are verified herein through comparisons with detailed test data on shrunk RC beams subjected to short-term bending. The first considered experimental program (carried out by Gribniak, [13]) mainly focuses on the effects of concrete shrinkage on beam deflection and first cracking moment in case of different amounts of top reinforcement, while the second one (by Sato et al. [14]) compares the flexural behavior of RC beams subjected or not to shrinkage prior to loading. Numerical vs. experimental results for RC beams with different amounts of top reinforcement [13] Four RC beams tested by Gribniak [13] – respectively named S1, S1R, S2, S2R – and subjected to four-point bending are first analyzed. The considered specimens were characterized by the same geometry, with a rectangular cross-section (300 mm deep and 280 mm wide) and a total length equal to 3280 mm, with a net span of 3000 mm. Series 1 and 2 were obtained from different concrete batches, showing slightly different compressive strengths f c , as reported in Tab. 1. Sample Steel Concrete A s,bottom [mm 2 ] A s,top [mm 2 ] E s [GPa] f sy [MPa] f c [MPa] ε sh (10 -6 ) φ c S1 309.0 56.6 212 566 47.3 -194.6 1.6 S1R 309.0 749 212 566 47.3 -188.2 1.6 S2 309.0 56.6 212 566 48.7 -152.6 1.4 S2R 309.0 749 212 566 48.2 -155.7 1.4 Table 1 : Material properties of RC beams tested by Gribniak [13]. All the specimens contained the same amount of lower tensile reinforcement (denoted as A s,bottom in Tab. 1, corresponding to 4 ϕ 10 mm bars), as well as of transverse reinforcement ( ϕ 6 mm / 100 mm spaced). The two specimens with designation “R” had an higher amount of top reinforcement ( A s,top , Tab. 1), constituted by 3 ϕ 18 mm bars, instead of 2 ϕ 6 mm bars. The main mechanical properties of steel reinforcement are summarized in Tab. 1 for reading convenience, together with T