PSI - Issue 84

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

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• B- and D-regions identification , by applying the Saint Venant principle (Love, 1927). The extension of the D region is determined as in Fig. 2 (Desnerk et al., 2018; fib, 2008; Goodchild , 2014; Narayanan, 2007; Schlaich et al., 1987). Finally, internal forces are determined at the interface between B- and D-regions. • STM development , where struts, ties and nodes are arranged based on the reinforcement layout geometry. Multiple admissible models may be formulated for the same structural details. For half-joints the most commonly used configurations are the diagonal model, the orthogonal model or their combination (Fig. 2). In particular, ties are placed as close as possible to reinforcement, while struts could be put freely but respecting a minimum angle between them and ties. For instance, in ACI Committee 318 (2014) a minimum angle of 25° is suggested. • Design of STM members . Table 2 reports the equations needed to design struts and ties. As for the nodes, it is generally considered that a maximum of 3 STM members converge at the node, or that they can be converted into a series of 3-member nodes according to a modified hydrostatic approach. The possible configurations are CCC, TTT, CCT or CTT, the last two being the most common. The allowable node compressive stress is obtained by reducing the concrete compressive strength by a reduction factor. Also, in nodal zones, tension ties anchorage checks are made by considering a development length l d (Desnerk et al., 2018).

Table 2. Equations for the STM members design (derived from Desnerk et al., 2018). Struts Concrete compressive stress Allowable limit prescribed: reduced concrete compressive strength •

F n,st : bar force in the strut

 s,st ≤ f yd : compressive stress in the reinforcement, given the strut force A c,st : strut effective concrete area

•

, = , − , ⋅ , ,

, = ⋅ ′

• •

A s,st : provided compression reinforcement area

Ties Reinforcement inducted tensile stress

Allowable limit prescribed: reinforcement design yield stress

F n,tie : bar force in the tie A s,tie : provided tensile reinforcement area

• •

, = , ,

3. Assessment techniques analysis The two methods previously considered for design can also be used for the assessment of existing half-joints. However, historical design procedures may differ significantly from today’s procedures, making the application of modern safety-check concepts difficult for their assessment. It should be noted that as most half-joints were designed using the UBM rather than the LBM (as suggested from today’s Standards), the reinforcement layout is often poorly aligned with the required strut-and-tie mechanism. This may lead to not exploiting the full structural capacity. To this it should be added that to date poor instructions are offered by design codes for taking into account defects, such as insufficient anchorage length, reinforcement degradation (corrosion or pitting, i.e. the reinforcing bars section reduction), concrete cracking due principally to corrosion (other cracking causes are taken into account in the stress analyses), the latest also influencing the reinforcing bars bond behavior (Desnerk et al., 2018). Regarding the general procedure for the assessment of existing structures, in the Chapter 10 of Handbook 2 Reliability backgrounds (2005) general indications may be found for the assessment of existing structures, through two core requirements: (i) “currently valid codes for verification of structural reliability should be considered, historic codes valid in the period when the structure was designed should be used as guidance documents, only”; (ii) the obligation to consider the actual properties (materials, actions, geometry, and behavior) of the structure. 4. Application to a case study The LBM and UBM described in Section 3 are applied to a real case-study, an existing RCB having half-joints, in order to evaluate the Capacity/Demand ( C/D ) ratio. The 60-year-old RCB is located on a suburban road (SP01) in the Province of Pisa. It has a total length of 130 m and consists of three RC arch spans. The arches support half-joint