PSI - Issue 84

Antonella Ranaldo et al. / Procedia Structural Integrity 84 (2026) 821–828

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They observed a dominant crack initiating at the inner corner of the nib and propagating towards the beam neutral axis (Fig. 1a,b). Therefore, all the reinforcement intersecting this crack (Fig. 1c) was assumed to contribute to resisting the collapse mechanism (Reynolds, 1969; Mattock and Chan, 1979; Liem, 1983). Table 1 reports the equations needed for the design of half-joints. While the UBM proved to be accurate, it intrinsically relied on strong assumptions regarding location and type of collapse mechanism, associated cracks and neutral axis position. These assumptions often lead to an overestimation of the actual load-bearing capacity, where construction tolerances, non-uniform reinforcement layouts, and material degradation play a critical role (Asso et al., 2022). These limitations have progressively reduced the use of the UBM for assessment purposes, motivating the shift towards LBMs, such as STMs, which are better suited to capture the brittle and degradation-sensitive behavior of half-joints.

Table 1. Equations for the half-joints design according to the UBM (derived from Mattock and Chan, 1979). Ultimate stress shear Hanger reinforcement dimension

Flexural reinforcement dimension ( ρ max = 0.0252) ( ∕ ) < 1 = ⋅ + ⋅ ( − ) = ∕ ( ⋅ ⋅ 2 ⋅ ′ ) = ( ′ ∕ ) ⋅ (0.85 − √0.72 − 1.7 ) < = ⋅ ⋅ Horiz. force reinforc. and stirrups dimensions

= ⋅ ⋅ < 0.2 ⋅ ′ (with strength reduction factor η = 0.85) Shear force reinforcement dimension =( 0.8 ⋅ − 0.5 ⋅ ⋅ ) ⋅ 1 < 0.2 ⋅ ⋅

ℎ = ⋅

Nib main reinforcement 2 3 ⋅ < = + , ℎ = ℎ − +

= ∕ ( ⋅ ) ℎ = 0.5 ⋅ ( − )

2.2. Lower-Bound Method (LBM) The STM is a design tool used for RC regions where the standard beam theory according to the Bernoulli-Euler hypothesis no longer applies. These areas are known as D-regions (Discontinuity or Disturbed regions), differing from B-regions assessed simply by means Bernoulli-Euler theory (Desnerk et al., 2018). As previously mentioned, the STM is a LBM, implying that the structure is safe if any external loads is provided being in equilibrium with the internal forces being inferior or equal to the yield load (Braestrup, 2021). The STM is composed of three primary components: the struts , representing the compressed concrete; the ties , carried by the reinforcing steel; the nodes , as the intersections of struts and ties, where the internal forces are transferred. a b

Fig. 2. (a) Location of B- and D-regions on a half-joint; (b) examples of common STMs in RC half-joints (from Desnerk et al., 2018).

The phases in which the STM design process may be split are described in the following (Desnerk et al., 2018).