Issue 37

P. Bernardi et alii, Frattura ed Integrità Strutturale, 37 (2016) 15-21; DOI: 10.3221/IGF-ESIS.37.03 18 concrete compressive strength f c , shrinkage strains ε sh and creep coefficient φ c measured at the test date (see [13] for details). The experimental value of ε sh is used herein to compose the shrinkage strain vector that must be defined in the numerical model, as described in the previous Section. Since shrinkage-induced stresses develop gradually with time, the relief caused by creep should be included in numerical simulations, as suggested by many Authors in the literature (e.g., [1, 8]). The creep coefficient φ c reported in Tab. 1 is therefore adopted in the initial step of the analyses to correct the stresses in concrete, by applying the effective modulus method [1]. All tests were carried out under loading control. On the contrary, numerical analyses are performed under displacement control, in order to achieve a better numerical convergence. Taking advantage of the symmetry of the problem, only one half of each beam is simulated, by adopting a FE mesh constituted by quadratic, isoparametric 8-node membrane elements with reduced integration (4 Gauss integration points). 0 20 40 60 80 100 0 3 6 9 12 15 S1 Experimental [13] Numerical P [kN] δ [mm] (a) M cr,exp [kNm] 16.8 M cr,num [kNm] 14.9 δ i,num [mm] 0.13 0 20 40 60 80 100 0 3 6 9 12 15 S1R Experimental [13] Numerical P [kN] δ [mm] (b) M cr,exp [kNm] 19.8 M cr,num [kNm] 17.5 δ i,num [mm] -0.16 0 20 40 60 80 100 0 3 6 9 12 15 S2 Experimental [13] Numerical P [kN] δ [mm] (c) M cr,exp [kNm] 15.9 M cr,num [kNm] 15.4 δ i,num [mm] 0.09 0 20 40 60 80 100 0 3 6 9 12 15 S2R Experimental [13] Numerical P [kN] δ [mm] (d) M cr,exp [kNm] 17.9 M cr,num [kNm] 17.6 δ i,num [mm] -0.13 Figure 1 : Comparison between numerical and experimental [13] results in terms of total applied load P vs. midspan deflection δ for beams: (a) S1; (b) S1R; (c) S2; (d) S2R. A first comparison between numerical and experimental results is provided in Fig. 1, in terms of total applied load P vs. midspan deflection δ . Since the initial deflection due to shrinkage was not experimentally measured, numerical analyses are shifted to the origin along the horizontal axis, so as to match experimental results. The values of initial shrinkage deflection δ i , as obtained from numerical analyses before the application of external loading, is however reported in each graph of Fig. 1. A good agreement is found for all the considered specimens. Beams with a larger amount of top reinforcement (designation “R”, Fig. 1b,d) are characterized by an initial negative deflection δ i due to shrinkage. Moreover, they show an higher cracking resistance with respect to their twin specimens (Fig. 1a,c). This can be attributable to the increment of moment of inertia due to the presence of a heavier top reinforcement, but mainly to the difference of tensile stresses caused by shrinkage in the extreme bottom fiber of the cross-section. To better clarify this last aspect, the numerical variation within the depth of the section of the total strain ε , as well as of the stresses in concrete σ c and in steel σ s just before loading are reported in Fig. 2 for the twin beams S1 and S1R. The presence of a not symmetric reinforcement in the beam cross-section determines a not uniform restraint to free shrinkage, which in turn causes the element curvature