PSI - Issue 84

Pavlina Lakatosova et al. / Procedia Structural Integrity 84 (2026) 1278–1285

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reinforcement may lead to unrealistic deformations and numerical divergence. Unlike planar analysis, spatial problems require consideration of the influence of triaxial stress states on concrete strength. In the 3D CSFM, concrete behavior under triaxial stress is described using a Mohr–Coulomb plasticity model. Although commonly associated with geomechanics, this model is also suitable for quasi-brittle materials such as concrete, which can be idealized as a cohesive material with an internal friction of approximately 35–40°. In the 3D CSFM, modified Mohr– Coulomb circles with an internal friction angle φ = 0° are used. This conservative assumption significantly simplifies the analysis while maintaining sufficient accuracy. A practical consequence is that the maximum difference between σ _c3 and σ _c1 remains constant regardless of the magnitude of σ _c3 . The resulting equivalent stress, defined by this difference, can therefore be directly compared with the uniaxial design compressive strength of concrete, f cd , as described in the Eurocodes. Fig. 4 shows the plasticity envelope in Mohr’s stress representation. The admissible stress domain is bounded by a horizontal line derived from the Mohr–Coulomb model and by the left-hand side of the Mohr circle corresponding to uniaxial compression.

Fig. 4. Modified Mohr–Coulomb circles for concrete used in the 3D CSFM

3. Practical applications At present, the 3D CSFM can be applied in engineering practice to assess a wide range of reinforced concrete anchorages. It is particularly advantageous in cases that cannot be reliably addressed by manual calculations according to EN 1992-4, such as anchor groups subjected to multiple load components, complex spatial arrangements, anchorages close to edges, or verification of existing reinforcement. In bridge engineering, the method can be applied to various anchorage details, including noise barrier posts, vehicle restraint systems and railings, anchorage of hangers in pedestrian bridges, or even suspension cable anchorages. Another important application is the assessment of locally loaded concrete edges, such as pier-caps subjected to concentrated loads during bearing replacement. Both applications are presented in detail below. 3.1. Noise barrier anchorage The following example illustrates the design and assessment of a noise barrier post and its anchorage into a bridge ledge, a common practical task. Noise barrier posts must be designed for both service wind loads and accidental vehicle impact loads. Common practice is to verify the plastic bending resistance of the post, which the anchorage must safely transfer.