PSI - Issue 78

Pier Paolo Rossi et al. / Procedia Structural Integrity 78 (2026) 726–733

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2.6. Failure criteria for concrete and steel rebars The uniaxial response of the rebars is limited to the yield strength of steel. The stresses on the compressed uncracked concrete are limited so as to verify the equations proposed by Kupfer and Gerstle. To limit the contribution of very high strength transverse rebars in beams with concrete of low compressive strength, an additional check is carried out on the concrete close to the horizontal legs of the transverse rebars (hangers and stirrups) on the top part of the dapped-end beam. The inclined compressive stress of the concrete close to the transverse rebar is calculated by means of an equilibrium equation in the vertical direction assuming that the vertical component of the inclined stresses over the length of the horizontal leg of the transverse rebar is counterbalanced by the sum of the axial forces in vertical legs of the transverse rebar, i.e. where F s is the axial force in the single vertical leg of the transverse rebar,  s is the diameter of the horizontal leg of the transverse rebar, b 0 is the length of the horizontal leg of the transverse rebar and  r is the angle of inclination of the resultant of the compressive forces of the two struts that originate from the transverse rebar with respect to the horizontal axis. To limit the complexity of the constraint on the above stress of concrete, the transverse compressive stress of concrete on the top surface of the horizontal leg of the transverse rebar and the shear stresses on the upper and left surfaces of the horizontal leg have been neglected. The limit value of the inclined compressive stress of concrete, later named f cc , has been considered to be higher than the uniaxial cylindrical compressive strength because of the confinement effects given by the transverse rebars. Based on the values suggested by Eurocode 2 in the case of three-dimensional confinement and on the results of the proposed method with different values of the limit compressive strength, the value of f cc is assigned here equal to 2 times the uniaxial compressive strength. 2.7. Formulation of the proposed procedure Once the data that describe the geometry of the beam, the arrangement of the reinforcements and the mechanical properties of the materials have been collected, the variables and the constraints of the problem are defined. The constraints regard geometric and mechanical parameters. In particular, the depth of the failure surface S 1 is limited to the distance from the top of the beam to the tensile longitudinal rebars of the nib while the depth of the other failure surfaces, i.e. S 2 to S 4 , is limited to the distance from the top of the beam to the tensile longitudinal rebars of the full depth section. In addition, the cotangent of the angle  is forced to be not lower than the assigned lower limit, i.e. 0.4. Then, the state of stress of the rebars that are cut off by the considered failure surfaces and that of the concrete in the compressed part of the failure surface are checked. Finally, the compressive stress of the concrete that is close to the horizontal legs of all the transverse rebars is checked. The objective function of the nonlinear constrained optimization problem is the shear capacity of the beam and is calculated as the minimum of the shear resistances corresponding to all the considered failure surfaces. To obtain solutions of the above problem, random values are initially assigned to the variables of the problem. Then, at each failure surface, the shear resistance is obtained from the rotational equilibrium. The longitudinal stress in the uncracked concrete is determined from the equilibrium in the longitudinal direction and the shear stress from the translational equilibrium in the transverse direction. The transverse normal stress in the uncracked concrete is instead calculated by means of the procedure described in Section 2.5. The tentative values of the variables are automatically changed by the solver of the software Excel so as to obtain convergence over the maximum of the objective function, under the above-mentioned constraints. If convergence is reached, the procedure records the solution of the solver. Owing to the non-linearity of the mathematical relations, many runs of the programming problem are carried out by means of different sets of the starting values of the variables of the problem. Of the shear resistances corresponding to the different runs, the highest is considered as the shear capacity of the member. s    F * cd   0 s r sin b (7)