Dynamics of structures/Computational Mechanics (Computational Mechanics)

The course provides a comprehensive introduction of the methods and theory of computational mechanics for solids and structures. The course aims to provide concepts, theories and methodologies at the base of the techniques currently used for the numerical analysis of materials and structures, with particular regard to the Finite Element Method (FEM), which is the most used computational method in practical applications. The theoretical contents of the course are oriented to provide a solid conceptual basis and a deep understanding of the potential and limitations of different methods. Topics include problem formulation, discretization and approximation, the finite element method for linear and nonlinear analyses and transient dynamics. The course also involves the use of a commercially available finite element software to gain experience and insight on the course concepts.

Dynamics of structures/Computational Mechanics (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/Computational Mechanics (Computational Mechanics)

The course aims at giving the student: - an insight into the use of computational techniques in applied mechanics - an insight into how technical computations are used in the design process - a theoretical understanding of the Finite Element Method - the ability to independently build and solve a variety of mechanical problems, spanning from linear static, nonlinear, and dynamics problems, by using a commercial FE software - the ability to make a critical analysis of the obtained results

Dynamics of structures/Computational Mechanics (Dynamics of structures)

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

Dynamics of structures/Computational Mechanics (Computational Mechanics)

Fundamental notions from the bachelor mathematical courses (Mathematical Analysis 1 and 2, Geometry, Linear Algebra, Analytical Mechanics, Numerical Methods) and engineering courses (Structural Mechanics I, Structural Mechanics II) are required.

Dynamics of structures/Computational Mechanics (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/Computational Mechanics (Computational Mechanics)

(1) DISCRETIZATION METHODS FOR PHYSICAL PROBLEMS Physical problem statement; approximation; discretization. Formulation of the Finite Element Method. (2) LINEAR ELASTICITY Finite element formulation for bars, continuum (triangular, quadrilateral, and 3D solid elements), structural elements (beams, plates and shells). Calculation of the stiffness matrix; numerical integration; solution of equilibrium equations. Solution of example problems by commercial codes. (3) NONLINEAR ANALYSES Geometrically nonlinear problems: finite deformation. Nonlinear structural problems: large displacement, instability, cable structures Material nonlinearity: elastoplasticity. Iterative solution methods, solution of nonlinear equations. Solution of example problems by commercial codes. (4) LINEAR AND NONLINEAR DYNAMICS Solution of equilibrium equations in dynamic analysis: direct integration methods; mode superposition; stability and accuracy analysis. Solution of nonlinear equations in dynamic analysis (explicit and implicit integration). Solution of example problems will be proposed by means of commercial codes. (5) FUNDAMENTALS ON IMPLEMENTATION OF THE FINITE ELEMENT METHOD

Dynamics of structures/Computational Mechanics (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.

Dynamics of structures/Computational Mechanics (Computational Mechanics)

Dynamics of structures/Computational Mechanics (Dynamics of structures)

Dynamics of structures/Computational Mechanics (Computational Mechanics)

Approximately two third of the lectures are given in classroom (mainly at blackboard) whereas one third are held at the computer laboratory (LAIB) to learn the use of a finite element software. An optional part of computer programming of some simple problems and methods is also proposed.

Dynamics of structures/Computational Mechanics (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/Computational Mechanics (Computational Mechanics)

C.L. Dym, I.H. Shames, Solid Mechanics: A Variational Approach, augmented edition, Springer, 2013 J.N. Reddy, An Introduction to the Finite Element Method, Mc-Graw Hill Education, 2005 K.-J. Bathe, Finite Element Procedures, Prentice Hall, 1996 T. Belytschko et al., Nonlinear Finite Element for Continua and Structures, Wiley, 2nd edition, 2000 I.M. Smith, D.V. Griffiths, L. Margetts, Programming the Finite Element Method, Wiley, 2014 Y.W. Kwon, H. Bang, The Finite Element Method using MATLAB, CRC Press

Dynamics of structures/Computational Mechanics (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/Computational Mechanics (Computational Mechanics)

Modalità di esame: Prova scritta (in aula); Prova orale facoltativa; Elaborato scritto individuale;

Dynamics of structures/Computational Mechanics (Dynamics of structures)

Modalità di esame: Prova orale obbligatoria; Elaborato scritto individuale;

Dynamics of structures/Computational Mechanics (Computational Mechanics)

Exam: Written test; Optional oral exam; Individual essay;

Dynamics of structures/Computational Mechanics (Dynamics of structures)

Exam: Compulsory oral exam; Individual essay;

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Dynamics of structures/Computational Mechanics (Computational Mechanics)

Scope of the exam is to ascertain that the student has assimilated all topics presented and is able to apply the theories and methods for the solution of practical structural modeling problems. The final exam consists of a written examination on theory and the development of an analysis project on a commercial finite element method code (homework & class assisted). The written exam consists of open queries and exercises. It lasts about 2 hours. During the written test, students are not allowed to use notes, books and any other didactic material. A report concerning the analysis project has to be delivered by the day of the written test. The above two parts will give an evaluation that will not exceed 25/30. The results will be published on the Portale della didattica, together with the dates for the optional oral examination and for consultation. In order to get maximum marks an oral exam is required. During the first part of the oral examination, students have to be able to defend their projects by presenting and justifying the modeling choices, showing to have actively contributed to the project development. Then, the oral exam continues with the deepening of one of the topics discussed during the classes.

Dynamics of structures/Computational Mechanics (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.

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.