Caricamento in corso...

01VKZVA, 01VKZMX

A.A. 2024/25

Course Language

Inglese

Degree programme(s)

Master of science-level of the Bologna process in Civil Engineering - Torino

Master of science-level of the Bologna process in Ingegneria Civile - Torino

Course structure

Teaching | Hours |
---|---|

Lezioni | 45 |

Esercitazioni in aula | 15 |

Tutoraggio | 30 |

Lecturers

Teacher | Status | SSD | h.Les | h.Ex | h.Lab | h.Tut | Years teaching |
---|---|---|---|---|---|---|---|

Cornetti Pietro | Professore Associato | CEAR-06/A | 30 | 0 | 0 | 0 | 2 |

Co-lectures

Espandi

Riduci

Riduci

Teacher | Status | SSD | h.Les | h.Ex | h.Lab | h.Tut |
---|---|---|---|---|---|---|

Corrado Mauro | Professore Ordinario | CEAR-06/A | 15 | 0 | 0 | 0 |

Context

SSD | CFU | Activities | Area context |
---|---|---|---|

ICAR/08 | 6 | B - Caratterizzanti | Ingegneria civile |

2024/25

The present course aims at introducing the fundamental concepts of Theory of Plasticity and Fracture Mechanics.
The basis of plasticity theory is set and applied to ductile collapse of structures. Then the fundamental concepts of Linear Elastic Fracture Mechanics (LEFM) and their applications in Civil Engineering are provided. The solution to elastic problems of cracked structural elements is presented, as well as both the stress-field and the energy-based fundamental approaches. The singular stress field in the crack tip vicinity is determined and the instability conditions for brittle crack propagation are defined. The size-scale effects and the related ductile-to-brittle transition are described using nonlinear crack models, and then applied to the assessment of the brittle behaviour in reinforced concrete structures.
Finally the problem of fatigue resistance and fatigue crack growth are introduced and some applications to metallic structures are proposed.

The present course aims at introducing the fundamental concepts of Theory of Plasticity and Fracture Mechanics.
The basis of plasticity theory is set and applied to ductile collapse of structures. Then the fundamental concepts of Linear Elastic Fracture Mechanics (LEFM) and their applications in Civil Engineering are provided. The solution to elastic problems of cracked structural elements is presented, as well as both the stress-field and the energy-based fundamental approaches. The singular stress field in the crack tip vicinity is determined and the instability conditions for brittle crack propagation are defined. The size-scale effects and the related ductile-to-brittle transition are described using nonlinear crack models, and then applied to the assessment of the brittle behaviour in reinforced concrete structures.
Finally the problem of fatigue resistance and fatigue crack growth are introduced and some applications to metallic structures are proposed.

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.

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 (Mathematical Analysis 1 and 2, Geometry, Rational Mechanics, Numerical Methods: Solution to ordinary and partial differential equations, eigenvalue/eigenvector problems; Statics, kinematics, and dynamics of rigid bodies) and engineering courses (Theory of Elasticity, Reinforced and prestressed concrete).

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: elastic-plastic bending; plastic incremental analysis of beam systems; theorems of plastic limit analysis; proportional loads; non-proportional loads; cyclic loads and plastic adaptation (shake-down), inflected flat plates, strip method.
(2) Plane elasticity: cartesian coordinates, polar coordinates
(3) Plane stress and plane strain conditions: Airy stress function in cartesian and polar coordinates; Thick-walled cylinder; Circular hole in a plate subjected to tension
(4) Introduction to Fracture Mechanics: Fracture energy; Griffith energy criterion
(5) Stress-singularity at the crack tip: Westergaard method (complex potentials); Mixed-mode conditions; Stress-intensity factors
(6) Stress-singularity at the vertex of a re-entrant corner: Williams method (series expansions); Irwin fundamental theorem
(7) Extension of the plastic zone at the crack tip; Ductile-to-brittle size-scale transition; Brittleness number
(8) Bridged crack model and the problem of minimum reinforcement; Applications to fibre-reinforced materials
(9) Cohesive crack model and snap-back instability; Applications to plain concrete
(10) Overlapping crack model and the problem of maximum reinforcement; Applications to steel-bar reinforced and to prestressed concrete
(11) Applications of Fractal Geometry to strength of materials and fracture mechanics; fractal cohesive crack model; Multi-fractal scaling laws
(12) Fatigue resistance and Woehler curve; Fatigue crack growth and Paris law; Fatigue limit and fatigue threshold

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 carried out in the Computational Laboratories. In such practical classes, 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 teacher.

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.

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.

Slides; Video lezioni tratte da anni precedenti;

Lecture slides; Video lectures (previous years);

...
The exam consists in an oral interview, during which the student replies to scientific and technical questions related to the course topics. In this way, the level of preparation of the student is carefully assessed on both theoretical and practical aspects.

Gli studenti e le studentesse con disabilità o con Disturbi Specifici di Apprendimento (DSA), oltre alla segnalazione tramite procedura informatizzata, sono invitati a comunicare anche direttamente al/la docente titolare dell'insegnamento, con un preavviso non inferiore ad una settimana dall'avvio della sessione d'esame, gli strumenti compensativi concordati con l'Unità Special Needs, al fine di permettere al/la docente la declinazione più idonea in riferimento alla specifica tipologia di esame.

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.