The course has the scope of providing the students with all the conceptual elements and tools necessary to understand and manage brittle and ductile collapse mechanism of structures. The brittle behaviour assessment of next-generation reinforced concrete structures (Hybrid RC, FRP-bar RC, and High-strength PC) is studied in detail. In the case of metallic structures subjected to cyclic loading, the fatigue assessment is illustrated for both resistance and subcritical crack growth.

Fundamental notions are required from the basic mathematical courses (Calculus, Linear Algebra, Numerical Methods) and engineering courses (Structural Mechanics and Structural Engineering; Advanced Structural Mechanics; Reinforced Concrete structures).

The program of the course is subdivided into the following topics: (1) Plasticity: Prandtl diagram; elastic-plastic bending; plastic incremental analysis of beam systems; theorems of plastic limit analysis; plastic collapse load of beam systems. Collapse load for plates in bending; strip method. (2) Plane elasticity. Plane stress and plane strain conditions: Airy stress function in Cartesian and polar coordinates; Flamant solution. Application to Brazilian Disk (BD); Thick-walled cylinder; Circular hole in a plate subjected to tension (3) LEFM 1: Fracture energy and Griffith energy criterion; crack driving force and strain energy release rate; application to the Double Cantilever Beam Test. Stress-singularity at a crack and a V-notch tip: Williams method. Stress-intensity factors; asymptotic stress and displacement fields. (4) LEFM 2: Irwin fundamental theorem. Fracture toughness. Stable and unstable crack growth. Paris equation. Application to the Three Point Bending (TPB) test of notched concrete specimen. Extension of the plastic zone at the crack tip; Ductile-to-brittle size-scale transition; Brittleness number, Mixed-mode loadings. (5) Bridged crack model and the problem of minimum reinforcement; Applications to fibre-reinforced materials (6) Computational fracture mechanics. Cohesive crack model and snap-back instability; Applications to plain concrete. Overlapping crack model and the problem of maximum reinforcement; Applications to steel-bar reinforced and to prestressed concrete (7) Fatigue resistance and Woehler curve; Fatigue crack growth and Paris law; Fatigue limit and fatigue threshold

The theoretical classes are followed by practical exercises. Numerical methods are applied for the solution of the problems already treated from an analytical point of view during the theoretical classes. Commercial finite element codes as well as software developed ad-hoc within the research activity are provided by the teachers. A couple of lectures are given at the Laboratory of Materials and Structures, where students, divided in small groups can, attend to a test on a BD and a notched TPB concrete specimen.

Reference text books: A. Carpinteri, “Advanced Structural Mechanics”, Chaps. 11 and 12, CRC Press, New York, 2017. A. Carpinteri, “Structural Mechanics: A Unified Approach”, Chaps. 19 and 20, Chapman & Hall, London, 1997. A. Carpinteri, “Fracture and Complexity”, Springer Nature, Berlin, 2021.

Lecture slides; Video lectures (previous years);

Exam: Written test; Compulsory oral exam;

The exam is composed by a written test (taking place at the official date of the exam session) where the evaluation of the plastic collapse load of a frame is required. Time for the written test is 1 hour and 15 minutes. The test of the written test is out of 30; the minimum to get access to the oral test is 18. The weight of the written test over the whole is 1/3. In case of a written test score higher or equal to 18, an oral interview follows (within the academic year, at student’s choice), during which the student replies to scientific and technical questions related to the course topics (mostly the part about fracture mechanics). Students must prepare also reports about experimental activities done during the course and can be asked about them. The weight of the oral test over the whole is 2/3. In this way, the level of preparation of the student is carefully assessed on both theoretical and practical aspects.

In addition to the message sent by the online system, students with disabilities or Specific Learning Disorders (SLD) are invited to directly inform the professor in charge of the course about the special arrangements for the exam that have been agreed with the Special Needs Unit. The professor has to be informed at least one week before the beginning of the examination session in order to provide students with the most suitable arrangements for each specific type of exam.