{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,29]],"date-time":"2026-07-29T17:58:52Z","timestamp":1785347932097,"version":"3.55.0"},"reference-count":38,"publisher":"MDPI AG","issue":"15","license":[{"start":{"date-parts":[[2026,7,29]],"date-time":"2026-07-29T00:00:00Z","timestamp":1785283200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Materials"],"abstract":"<jats:p>Reinforced concrete (RC) structures are susceptible to crack initiation and propagation during service, making accurate numerical simulation of crack behavior essential for assessing structural durability and safety. Current numerical approaches for simulating concrete cracking include smeared\/continuum approaches, extended finite element method (XFEM), phase-field methods, and meso-mechanical models. In particular, smeared\/continuum approaches (e.g., smeared crack and plastic-damage models such as CDP) indirectly reflect cracking through diffusive damage fields without providing explicit geometric information on crack locations and propagation paths. The XFEM module in commercial software is further restricted to first-order elements and encounters difficulties in simulating multi-crack propagation. These limitations indicate that further development of complementary crack-simulation frameworks is warranted. To this end, this paper presents a cracking simulation framework for RC members within the theoretical framework of the Specified Stress Method, adopting an adaptive degree-of-freedom strategy to balance computational accuracy and efficiency. The method introduces inelastic strain as an additional unknown and establishes a variational principle and the corresponding virtual work equation. Concrete cracking is described by specifying the stress on the crack plane to zero, so that the crack-surface stress remains zero after cracking, thereby avoiding the issue of damage reversibility and improving computational convergence. The method requires neither a predefined crack path nor remeshing after cracking. Unlike smeared\/continuum approaches that rely on diffusive damage fields, the crack propagation paths, distribution characteristics, and evolution of multiple cracks are characterized through the spatial distribution of cracked integration points within the finite element mesh. In the present implementation, crack initiation is governed by the maximum tensile stress criterion, and a linear elastic constitutive model is adopted for concrete as a deliberate simplification to establish and verify the core computational mechanism of the framework. The proposed method was examined through three numerical examples. First, comparison with theoretical solutions confirmed the algorithm\u2019s correctness in simulating cracking in heterogeneous RC tension members. Second, comparison with experimental results demonstrated qualitatively consistent crack propagation trends and load\u2013displacement responses for RC beams under mixed-mode cracking; the calculated ultimate load of the plain concrete beam is lower than the experimental value, which is attributable to the use of the maximum tensile stress criterion without fracture energy considerations, and certain crack morphology deviations are observed due to the neglect of reinforcement\u2013concrete bond-slip. Third, a multi-crack simulation of an under-reinforced RC beam showed that, whereas the XFEM module in ABAQUS captures only a single dominant crack near the mid-span, the proposed algorithm predicts multiple distributed cracking zones on both sides of the mid-span, qualitatively consistent with the typical flexural cracking behavior of under-reinforced RC beams; the algorithm also supports second-order elements (e.g., C3D20R) unavailable in the ABAQUS XFEM implementation. While the method is still in an exploratory stage, these results confirm the feasibility and potential of the Specified Stress Method as a complementary framework for RC cracking simulation, providing a basis for further development.<\/jats:p>","DOI":"10.3390\/ma19153231","type":"journal-article","created":{"date-parts":[[2026,7,29]],"date-time":"2026-07-29T17:01:46Z","timestamp":1785344506000},"page":"3231","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Application of the Specified Stress Method to Crack Propagation Analysis in Reinforced Concrete Members"],"prefix":"10.3390","volume":"19","author":[{"given":"Xiaoqing","family":"Zhang","sequence":"first","affiliation":[{"name":"School of Civil Engineering, Chongqing Jiaotong University, Chongqing 400074, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jialin","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Civil Engineering, Chongqing Jiaotong University, Chongqing 400074, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhijian","family":"Yi","sequence":"additional","affiliation":[{"name":"School of Civil Engineering, Chongqing Jiaotong University, Chongqing 400074, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Tuo","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Civil Engineering, Chongqing Jiaotong University, Chongqing 400074, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,7,29]]},"reference":[{"key":"ref_1","first-page":"152","article-title":"Nonlinear analysis of reinforced concrete beams","volume":"64","author":"Ngo","year":"1967","journal-title":"J. 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