PSI - Issue 84

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

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Fig. 2. Principle of CSFM: (a) stresses in local coordinate directions; (b) principal stresses; (c) stress state according to CSFM

Fig. 2c illustrates the stress state of the same element analyzed using CSFM. In the direction of the principal compressive stress σ₂, the load is carried by compressed concrete, forming a compressive stress field with stress σ _c2 . For tensile stresses, the method adopts the fundamental and conservative Eurocode assumption that concrete does not carry tension. Consequently, the principal tensile stress σ₁ cannot be resisted by concrete, and a crack forms perpendicular to its direction. Tensile forces must therefore be transferred by reinforcement, which is explicitly included in the model. The CSFM analysis is based on nonlinear finite element analysis. In planar problems, concrete is modeled using two-dimensional continuum elements, while reinforcement is represented by one-dimensional bar elements. The analysis is incremental and nonlinear; loads are applied in increments, and the nonlinear equilibrium equations are solved using the Newton– Raphson method. Distributed fictitious cracks (with ε₁ representing the average crack strain) develop perpendicular to the principal tensile stress direction, which may change during the incremental loading process as cracking progresses. When applied continuously over the entire analyzed two-dimensional region, the CSFM yields continuous compressive stress fields in concrete and corresponding tensile and compressive stresses in the reinforcement. A simplified graphical representation of the stress field in a pier head or cross-beam is shown in Fig. 3.

Fig. 3. Visualization of stress fields: (Left) pier head; (Right) cross-beam (indirect support)

2.2. Extension to the third dimension – 3D CSFM The method described above has been extended to three dimensions (3D CSFM), enabling analysis of fully three dimensional stress states in concrete modeled using solid elements. As in planar CSFM, the 3D CSFM considers compressive principal stresses in concrete and tensile stresses in reinforcement at crack locations, while neglecting the tensile strength of concrete except for its tension-stiffening effect on reinforcement. For this reason, the method is not suitable for plain or weakly reinforced concrete; insufficient