PSI - Issue 78

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

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failure surface. In addition, the proposed model simultaneously considers different modes of failure that can involve the reduced- or the full-depth section of the dapped-end beam. The model considers the effects of horizontal, vertical and diagonal reinforcements as well as the effects of the transverse compressive stresses originating from hangers, stirrups and loading plates on the compressed uncracked concrete. Moreover, a local check is carried out on the compressed concrete close to the horizontal legs of the transverse reinforcement on the top of the beam in order to limit the diagonal compressive stress of concrete in the case of beams with low strength concrete and high strength transverse rebars. The shear capacity is obtained by means of a nonlinear constrained optimization problem that is built and solved by means of the program Excel by Microsoft. A similar approach, i.e. based on the solution of a nonlinear constrained optimization problem, has already been successively applied by the authors to predict the shear strength of RC rectangular or circular columns (Rossi 2013, Rossi and Recupero 2013, Rossi and Spinella 2023). The model is applied here to predict the shear capacity of 96 laboratory dapped-end beams that were tested in the past by other researchers. 2. The proposed method The considered dapped-end beam is simply supported and characterized by reduced-depth and full-depth sections (Fig. 1). The beam is subjected to concentrated forces F , which are resisted by reaction forces R at the supports. Owing to the non-uniform geometry of the beam and to the arrangement of longitudinal and transverse reinforcements, the method considers multiple possible failure surfaces that involve the reduced-depth or the full-depth section of the beam.

Fig. 1 Examined failure surfaces of the dapped-end beam

2.1. Failure surfaces In the present proposal four different failure surfaces, later named S 1 -S 4 , are considered. In particular, the first failure surface (S 1 ) intersects the top fiber just before the first hanger; the second (S 2 ) intersects the top fiber just before the first stirrup of the full-depth section; the third (S 3 ) intersects the top fiber just before the second stirrup of the full-depth section whereas the fourth (S 4 ) intersects the top fiber just before the loading plate. In the upper part of the member, the assumed failure surface is plane and orthogonal to the longitudinal axis of the beam. In the lower part, instead, the failure surface is assumed to be inclined and parallel to the principal compression direction. Even if no specific assumption is necessary regarding the plain, concave or convex aspect of the lower part of the failure surface, the gross knowledge of the overall inclination of the failure surface is necessary to identify the steel rebars that are intersected by the failure surface. To this latter end, the lower part of the failure surface is simplistically assumed to be plain and characterized by the angle  S with respect to the horizontal axis. Once the value of the angle  S is known, the rebars that are cut off by the failure surface S 4 are found by means of geometric considerations. This angle is not required to be equal for all the assumed failure surfaces and, with regard to surfaces S 1 to S 3 , is not specifically assigned here. However, based on the proximity of the failure surface to the support the authors assume that it is such that the longitudinal rebars of the nib and the transverse rebars from the support to the re-entrant corner are cut off by surface S 1 , the longitudinal rebars of the nib and the hangers on the left of the failure surface are cut off