Dynamics of structures/Static and dynamic instability of structures (Dynamics of structures)

The course aims to provide the theoretical principles and practical tools to address the main topics of the dynamic analysis of structures. To this end, in addition to lectures, the course includes practical classes in the computer laboratory where the methodologies and tools illustrated in class are applied, together with some example experimental laboratory tests.

Dynamics of structures/Static and dynamic instability of structures (Static and dynamic instability of structures)

Purpose of the course is to present the fundamental concepts of static and dynamic stability of long-span and high-rise structures. In the introductory part of the course, the basic subjects of the buckling instability are presented for rectilinear elastic beams, plane frames, plates, arches and shells. The second part of the program is devoted to long-span roofing and bridge structures, with particular emphasis on the phenomena of snap-through and flutter, respectively. In the third and last part of the program, statics, dynamics, and stability of tall buildings are treated, with a particular emphasis on the structural behaviour of thin-walled open-section vertical elements (shear-walls, internal cores, external tubes). Vlasov theory is presented as a completion of Saint Venant torsion theory. The historical development of the different structural typologies (roofing structures, bridges, tall buildings) is also illustrated in the course with the related stability aspects.

Dynamics of structures/Static and dynamic instability of structures (Dynamics of structures)

Knowledge and understanding of structural dynamics analysis methodologies and their use for structural engineering applications.

Dynamics of structures/Static and dynamic instability of structures (Static and dynamic instability of structures)

The course has the scope of providing the students with all the conceptual elements and tools, which are necessary to face the stability problems, in both the static and dynamic regimes, and in the cases of slender, thin, or shallow structures.

Dynamics of structures/Static and dynamic instability of structures (Dynamics of structures)

Knowledge of the basics of mechanics and structural engineering, with specific reference to the following courses: - Structural Mechanics; - Advanced Structural Mechanics. Basic knowledge of the programming and numeric computing platform MATLAB is a plus.

Dynamics of structures/Static and dynamic instability of structures (Static and dynamic instability of structures)

Fundamental notions are required from the basic mathematical courses (Mathematical Analysis 1 and 2, Geometry, Rational Mechanics, Numerical Methods: Solution of ordinary and partial differential equations, eigenvalue/eigenvector problems; Statics, kinematics, and dynamics of rigid bodies) and engineering courses (Structural Mechanics, Advanced Structural Mechanics: Theory of Elasticity, Statically determinate and indeterminate beam systems; Automatic computation of frames; Static, kinematic, and constitutive equations for beams, arches, plates, shells; Finite Element Method).

Dynamics of structures/Static and dynamic instability of structures (Dynamics of structures)

Introduction to structural dynamics. Aspects of dynamic analysis. Types of dynamic loads. Displacements, velocities, accelerations: relationships between the three measures. Analytical solution of the equation of motion of a free undamped SDOF system. Analytical solution of the equation of motion of a free damped SDOF system (subcritical damping, critical damping, supercritical damping). Analytical solution of the equation of motion of a sinusoidally excited damped SDOF system. Analytical solution of the equation of motion of a damped SDOF excited by: -Periodic Excitation; -Step function excitation; -impulse excitation. Case of the generic excitation with homogeneus initial conditions. Duhamel's integral. Numerical solution of the equation of motion of a SDOF system excited by a generic force. Runge-Kutta methods. Physical significance of resonance, with and without damping. Equations of motion with a dynamic equilibrium approach (Newton's second law of motion) and with a variational approach (Hamilton's principle). Transversal vibration of a string: natural frequencies and mode shapes of a fixed-fixed string. Initial conditions. Example: the guitar string. Free axial and bending vibrations: Euler-Bernoulli model. Natural frequencies, mode shapes. Free bending vibrations: Timoshenko model. Comparison with results obtained with the Euler-Bernoulli model. Free bending vibrations: Euler-Bernoulli model for a cracked beam (equivalent rotational spring). Spring calibration according to fracture mechanics. Variation of natural frequencies as a function of crack size and position. Free vibration of the Kirchhoff plate. Review of dynamic analysis through finite element modelling. Mass matrix for the beam and the plate. Cracked Beam element. Comparison between the natural frequencies of a beam obtained analytically and numerically. Laboratory classes (experimental modal analysis). Introduction to non-linear dynamics. Free vibrations of a conservative single degree of freedom system. Free vibrations of a dissipative single degree of freedom system. Forced vibrations of a dissipative single degree of freedom system. Further types of damping and stiffness non linearities: a generalised power-law formulation.

Dynamics of structures/Static and dynamic instability of structures (Static and dynamic instability of structures)

The program of the course is subdivided into the following topics: (1) Buckling of discrete mechanical systems: Rigid bars with elastic springs; Matrix of geometric stiffness (2) Buckling of rectilinear elastic beams: Euler problem; Different kinematic and static boundary conditions; Buckling of plane frames (3) Buckling of elastic plates and lateral-torsional instability (Prandtl problem) (4) Buckling of elastic rings and thin cylinders subjected to external pressure; Snap-through of shallow elastic arches subjected to vertical loads (5) Shell and space roofing structures: Historical and typological aspects; Shallowness and snap-through instability versus buckling (6) Long-span bridges: Arch, cable-stayed, suspension typologies; Historical aspects (7) Dynamics and stability: Conservative loads; Interaction between resonance and buckling (8) Dynamics and stability: Nonconservative follower loads; Aero-elastic instability (Flutter) (9) High-rise buildings: Historical and typological aspects; General Algorithm (10) Torsion of thin-walled open-section beams: Vlasov theory (11) Distribution of the external actions between the different opensection vertical elements: Capurso‘s method (12) Dynamics and stability of high-rise buildings

Dynamics of structures/Static and dynamic instability of structures (Dynamics of structures)

Dynamics of structures/Static and dynamic instability of structures (Static and dynamic instability of structures)

Dynamics of structures/Static and dynamic instability of structures (Dynamics of structures)

The course includes lectures, practical classes in the computer laboratory related to the topics covered in the course, and laboratory classes on experimental modal analysis. Students will also have to carry out individual assignments, targeted on the course topics that will contribute to the final grade.

Dynamics of structures/Static and dynamic instability of structures (Static and dynamic instability of structures)

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.

Dynamics of structures/Static and dynamic instability of structures (Dynamics of structures)

Notes will be provided during the course. For further consultation: • S.S. Rao Vibration of Continuous Systems John Wiley & Sons, Inc. 2007 • D. J. Ewins, Modal Testing: Theory and Practice. John Wiley & Sons Inc., 1995. • R. W. Clough J. Penzien Dynamics of Structures, McGraw-Hill, 1982. • Carpinteri. Dinamica delle strutture. Pitagora, 1998.

Dynamics of structures/Static and dynamic instability of structures (Static and dynamic instability of structures)

All the slides shown during the lectures are available on the course website for the students regularly registered. Reference text books: A. Carpinteri, “Advanced Structural Mechanics”, Chaps. 7-9, CRC Press, New York, 2017. A. Carpinteri, “Structural Mechanics: A Unified Approach”, Chapman & Hall, London, 1997. D.P. Billington, “The Tower and the Bridge: The New Art of Structural Engineering”, Princeton University Press, Princeton, 1985.

Dynamics of structures/Static and dynamic instability of structures (Dynamics of structures)

Dispense;

Dynamics of structures/Static and dynamic instability of structures (Static and dynamic instability of structures)

Slides; Dispense; Libro di testo; Libro di esercitazione; Esercizi; Esercizi risolti; Esercitazioni di laboratorio; Esercitazioni di laboratorio risolte; Video lezioni tratte da anni precedenti;

Dynamics of structures/Static and dynamic instability of structures (Dynamics of structures)

Modalità di esame: Prova orale obbligatoria;

Dynamics of structures/Static and dynamic instability of structures (Static and dynamic instability of structures)

Modalità di esame: Prova orale obbligatoria;

Dynamics of structures/Static and dynamic instability of structures (Dynamics of structures)

Exam: Compulsory oral exam;

Dynamics of structures/Static and dynamic instability of structures (Static and dynamic instability of structures)

Exam: Compulsory oral exam;

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Dynamics of structures/Static and dynamic instability of structures (Dynamics of structures)

The exam is aimed at ascertaining knowledge of the topics listed in the official course program and the ability to apply the theory and related calculation methods to determining the dynamic response of simple structures. The exam consists of an oral test with presentation and discussion of the assignments developed during the course and has the purpose of verifying the level of knowledge and understanding of the topics covered. The evaluations are expressed out of thirty and the exam is passed if the score reported is at least 18/30.

Dynamics of structures/Static and dynamic instability of structures (Static and dynamic instability of structures)

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 preparation level 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.