PSI - Issue 48

Nemanja Ilić et al. / Procedia Structural Integrity 48 (2023) 318 – 325 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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3.2. Other methods

Several methods are used in literature to evaluate ultimate capacity of the hull girder. These methods are, for instance: first-yield, first-collapse, Caldwell, full plastic, Paik – Mansour, Paik – Mansour modified, nonlinear finite element methods (implicit, explicit, modified), various modified progressive collapse methods, ISUM, etc. However, a few simplified and “easy to use” methods are applied here for the purpose of comparison to PCA results. Those are: first-yield, first-collapse and Paik – Mansour method.

Fig. 3. Segment division, types of segments and general form of stress-strain curve shape.

First-yield method assumes that the capacity is reached when the one of the segments starts to yield. It could be bottom or deck segment, whichever delivers lower bending moment. Such bending moment is than considered as ultimate bending moment or ultimate capacity, see Eq. (5). Z D and Z B stand for hull section modulus of deck and bottom, respect ively. Yield stress is labelled as σ y . First-collapse method assumes that the capacity is reached when one of the segments starts to buckle. In sagging condition, the deck segments tend to buckle at first. In hogging condition, bottom segments would experience buckling. Ultimate bending moment is one that accounts for the first segment that collapsed, see Eq. (6). Buckling stresses in which the deck and bottom are intended to buckle, are labelled here as σ uD and σ uB, respectively. =min(σ › ǡ σ › ሻ (5) =min(σ — ǡ σ — ሻ (6) Therefore, both methods are based on beam theory within the elastic limits and are traditionally used in assessments. They are not accounting for nonlinearities or plastic behaviour of the material. Generally, in further structural assessment, such ultimate bending moment is reduced to account the safety margin, according to regulations. On the other hand, Paik – Mansour method considers the collapse of several segments along the height of the of the cross section. In its essence, this method is similar to PCA. However, it is simplified. It assumes that the outermost segment of the section in tension region experienced yielding. Therefore, the method is called presumed. Due to redistribution of the stresses the position of the neutral axis changes according to Eq. (1) and Eq. (2), meaning that axial force equilibrium must be achieved taking into account all segments within the section. Illustration of the Paik – Mansour collapse is presented in Fig. 4 for sagging condition. Starting from deck zone, buckling is then transferred to nearby segments along the height of the cross section, while the bottom segments are assumed to yield. Note that the diagram is inverted in case of hogging condition in which the bottom segments are in buckling.