PSI - Issue 84

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

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1. Introduction During the 20th century, half-joint and notch supports were commonly implemented in Reinforced Concrete Bridges (RCBs) and Prestressed Concrete Bridges (PCBs). These elements, now identified as critical components by the Italian Guidelines for bridge management (MIT, 2020), were often designed without the guidance of standardized codes. Their structural behavior is strongly influenced by the original design approach, reinforcement layout, and degradation phenomena, which can lead to fragile and sudden failures. Moreover, degradation due to corrosion and material aging (Ranaldo et al., 2024) may compromise the residual strength and service life (Tatangelo et al., 2025). In this paper several methods for half-joints design and assessment are examined and applied. To this purpose, a case study is selected, i.e. an existing half-joint belonging to an existing RCB. 2. Design techniques analysis In the past, half-joints, also known as “dapped-end beams”, “Gerber saddles” or “Gerber beams”, were designed following general structural principles, as no specific standards or guidelines were available. Recently, due to the many sudden collapses observed, attention has shifted towards safety assessment, monitoring, and structural strengthening of these elements. Historically, the kinematic approach based on the upper-bound theorem of plasticity (or simply Upper-Bound Method – UBM) was adopted to design half-joints, by assuming a hypothetical failure mechanism for calculating the collapse load. Several experimental campaigns were conducted by Reynolds, (1969), Mattock and Chan (1979) and Liem (1983), where the kinematic approach was adopted to quantify the necessary reinforcing steel and to identify its suitable layout. However, as known, this approach tends to overestimate the half-joint collapse load. Today, design is typically performed using the Strut-and-Tie Model (STM, fib, 2008), based on the lower-bound theorem of plasticity (or Lower-Bound Method – LMB). This approach is based on the equilibrium between external loads and internal forces. The main challenge is to define a model compatible with the existing reinforcement layout: ties should follow the reinforcing bars, while struts can be freely positioned, provided that the angle between struts and ties complies with certain limits and the concrete remains intact (Desnerck et al., 2018). In other design approaches, short concrete cantilevers and deep beams were dimensioned using the elastic theory, based on the analysis of principal stress curves. This methodology assumed a rotation center, and reinforcement was consequently arranged along the tensile stress directions to resist the corresponding forces. Concurrently, the cross section width was defined to be adequate for carrying the compressive stresses (Leonhardt, 1979). a b c

Fig. 1. Failure mechanism assumed in the kinematic method: (a) crack status observed under service load and (b) on the verge of collapse; (c) typical reinforcement layout design for half-joints using the UBM (derived from Mattock and Chan, 1979).

2.1. Upper-Bound Method (UBM) According to the UBM, collapse is assumed to occur through a predefined failure mechanism, and the corresponding collapse load is obtained by equating internal and external work (Smith and Gilbert, 2022). Several experimental tests on RC half-joints adopted this approach to determine both the amount and layout of reinforcement.