DYNAMICS OF STRUCTURES WITH APPLICATION TO EARTHQUAKE ENGINEERING

The course Dynamics of Structures with Application to Earthquake Engineering aims to provide students with the theoretical background and practical tools required for the dynamic analysis of structures and for the understanding of structural response to earthquake loading.

Particular attention is devoted to the physical interpretation of dynamic phenomena, the formulation of mathematical models, and the application of the main tools used in modern earthquake engineering.

Knowledge and Understanding

At the end of the course, students will demonstrate:

• understanding of the fundamental principles of structural dynamics;

• understanding of the dynamic behaviour of single-degree-of-freedom and multi-degree-of-freedom systems;

• understanding of the role of mass, stiffness and damping in the dynamic response of structures;

• understanding of the formulation of the equations of motion;

• understanding of the main methods used in dynamic response analysis;

• understanding of the fundamental concepts of modal analysis;

• understanding of the principles governing the seismic response of structures;

• knowledge of response spectra and their applications in earthquake engineering;

• knowledge of the main numerical methods used for the evaluation of dynamic response.

Applying Knowledge and Understanding

At the end of the course, students will be able to:

• formulate the equations of motion of single-degree-of-freedom and multi-degree-of-freedom structural systems;

• determine natural frequencies, mode shapes, and dynamic properties of structures;

• evaluate the dynamic response of systems subjected to harmonic, impulsive, and arbitrary excitations;

• perform seismic response analyses of linear systems;

• use response spectra to estimate structural response;

• apply modal analysis procedures;

• use numerical methods for the integration of equations of motion;

• implement computational procedures and simple applications in Matlab;

• critically interpret the results of dynamic analyses.

Making Judgements

At the end of the course, students will be able to:

• critically evaluate the adequacy of a dynamic model for a given engineering problem;

• compare different structural response analysis strategies;

• identify the main assumptions and limitations of the adopted models;

• assess the physical plausibility of numerical results.

Communication Skills

At the end of the course, students will be able to:

• correctly use the technical language of structural dynamics and earthquake engineering;

• describe and discuss modelling and analysis procedures;

• interpret and comment on numerical results, graphical outputs, and response spectra;

• present numerical solutions and analyses in a clear and rigorous manner.

Learning Skills

At the end of the course, students will be able to:

• independently consult advanced textbooks in structural dynamics;

• further investigate analysis methodologies not fully covered during the course;

• use computational tools and bibliographic resources to broaden their expertise;

• pursue advanced studies in structural and earthquake engineering with an adequate degree of autonomy.

The course consists of theoretical lectures, practical exercises, and numerical laboratory activities.

Lectures focus on the presentation of the fundamental concepts of structural dynamics and earthquake engineering, while exercises and Matlab applications are aimed at developing the skills required for modelling, analysing, and interpreting the dynamic response of structures.

Tutoring activities and formative assessments are provided throughout the course to monitor students' progress.

Digital tools may also be used to support teaching and learning activities.

If the course is delivered in blended or remote mode, appropriate adjustments may be made to ensure consistency with the syllabus.

Students are expected to possess a solid background in Calculus, Linear Algebra, Mechanics, and Structural Analysis acquired during previous studies.

In particular, students should be familiar with:

• systems of equations, matrices, and eigenvalue problems;

• the fundamentals of particle and rigid-body dynamics;

• the fundamental principles of structural mechanics;

• scientific programming tools, preferably in the Matlab environment.

Attendance is not compulsory but is strongly recommended.

Due to the progressive nature of the topics and the extensive use of exercises, Matlab applications, and tutoring activities, regular participation in the course is highly beneficial.

• Fundamentals of Structural Dynamics: dynamic equilibrium, D'Alembert's principle, Hamilton's principle, formulation of the equations of motion.

• Single-Degree-of-Freedom Systems: free and forced vibrations, response to arbitrary excitations, Duhamel integral, seismic response, response spectra.

• Frequency-Domain Analysis: Fourier series, Fourier transform, transfer functions, frequency-domain response.

• Multi-Degree-of-Freedom Systems: matrix formulation, free vibration, modal analysis, damping, and reduction of degrees of freedom.

• Continuous Systems: natural frequencies, mode shapes, Rayleigh method, Rayleigh-Ritz method, and introduction to finite elements.

• Earthquake Analysis of Structures: Response History Analysis, Response Spectrum Analysis, modal combination, and response of linear and nonlinear systems.

• Numerical Methods for Structural Dynamics: direct integration of the equations of motion and major numerical algorithms.

• Matlab Applications: numerical implementation of dynamic and seismic analysis methods.

Main Reference Textbook

A. K. Chopra, Dynamics of Structures: Theory and Applications to Earthquake Engineering, Prentice Hall, Fourth Edition, 2012.

Supplementary Reference Textbook

R. W. Clough, J. Penzien, Dynamics of Structures, McGraw-Hill.

Additional Teaching Material

Lecture notes, solved exercises, Matlab scripts, collections of past examinations, and additional teaching material made available by the lecturer during the course.

Guidance on the Use of the Textbooks

Chopra's textbook is the primary reference for topics related to single-degree-of-freedom systems, multi-degree-of-freedom systems, modal analysis, and seismic response of structures. The textbook by Clough and Penzien is used for theoretical insights and for selected topics covered during the course, including variational formulations, Hamilton's principle, and applications of frequency-domain analysis.

Learning is assessed through written examinations, practical assignments, and oral examinations.

Students may choose between:

• a pathway consisting of an intermediate assessment at the end of the first semester and a final assessment focused on the topics developed during the second semester;

• a traditional pathway consisting of a single final examination covering the entire course syllabus.

Assignments carried out during the academic year may be discussed during the oral examination.

Assessment takes into account the correctness of the adopted models, mastery of theoretical and numerical tools, ability to interpret results, and the appropriate use of technical terminology.

Learning assessment may also be carried out online should circumstances require it.

To ensure equal opportunities and in compliance with current regulations, students may request an individual meeting in order to discuss appropriate compensatory and/or dispensatory measures according to their specific needs. Students may also contact the departmental CInAP representative (Centro per l'Integrazione Attiva e Partecipata – Services for Students with Disabilities and/or Specific Learning Disorders).

• Formulation of the equations of motion of single-degree-of-freedom and multi-degree-of-freedom systems.

• Determination of natural frequencies and mode shapes of discrete systems.

• Analysis of free and damped vibrations of single-degree-of-freedom systems.

• Determination of the response of single-degree-of-freedom systems subjected to harmonic, impulsive, or arbitrary excitations.

• Application of Duhamel's integral to determine dynamic response.

• Construction and interpretation of response spectra.

• Determination of the seismic response of linear single-degree-of-freedom and multi-degree-of-freedom systems.

• Application of modal analysis to the determination of structural dynamic response.

• Derivation of mass, stiffness, and damping matrices and formulation of the corresponding equations of motion.

• Computation of eigenvalues, eigenvectors, and modal participation factors.

• Application of numerical methods for the integration of equations of motion.

• Analysis of simple continuous systems using the Rayleigh and Rayleigh-Ritz methods.

• Discussion of the theoretical foundations of structural dynamics and earthquake engineering.

• Development and discussion of simple Matlab applications related to the course topics.

• Physical interpretation and critical assessment of the results obtained from dynamic and seismic analyses.

• Analysis of structural response in the time and frequency domains through Fourier series and Fourier transform.