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, in statics and dynamics, by using commercial FE softwares - the ability to make a critical analysis of the obtained results

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.

(1) INTRODUCTION Matrix mathematics; physical problem statement; approximation; discretization. (2) FORMULATION OF THE FINITE ELEMENT METHOD Virtual Work and Variational Principle; Galerkin method; Finite Element Method: displacement approach; interpolation of the displacement field; interpolation (shape) functions for general element formulation: polynomial series, Lagrange functions, Hermite functions; isoparametric formulation; stiffness matrix; numerical integration; boundary conditions; characteristics of the system of algebraic equations. (3) LINEAR ELASTICITY FEM for frame structures Stiffness of truss members; analysis of trusses; stiffness of beam members; analysis of continuous beams; plane frame analysis; analysis of grid and space frames. Solution of structural architecture example problems by commercial codes. FEM for two and three dimensional solids Constant strain triangle; linear strain triangle; rectangular elements; numerical evaluation of element stiffness; computation of stresses, geometric nonlinearity and static condensation; axisymmetric elements; finite element formulation for three dimensional elements. Solution of example problems by commercial codes. FEM for plates and shells Introduction to plate bending problems; finite element for the analysis of thin plates (Kirchhoff-Love theory); finite element for the analysis of thick plates (Reissner-Mindlin theory); finite element analysis of skew plates; shell elements. Modelling of long-span shell structures and bridge decks by commercial codes. (4) CONVERGENZE OF FEM AND ERROR ESTIMATES Convergence of analysis results: definition of convergence, properties of the finite element solution, rate of convergence; patch tests; definition of errors; error estimators. (5) FEM FOR LINEAR DYNAMIC ANALYSIS Solution of equilibrium equations in dynamic analysis: direct integration methods for the study of the transient dynamic response, mode superposition; stability and accuracy analysis; study of the structural response to an applied accelerogram; solution of seismic design example problems by commercial codes. (6) ADVANCED APPLICATIONS Presentation and discussion of advanced case studies, which include the study of mechanical and geometrical nonlinear problems and the modelling of real structural failure cases. Support will be provided to the students willing to replicate the FE models.

Approximately two third of the lectures are dedicated to the presentation of the theoretical basis of the finite element method and the practical issues related to its implementation, whereas one third is dedicated to learn the use of finite element softwares (Ansys and Lusas) and do practice by solving real engineering problems. The last two weeks of the course are devoted to the presentation and discussion of advanced applications.

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

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

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.

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.