Servizi per la didattica

PORTALE DELLA DIDATTICA

2020/21

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.

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 analyses and transient dynamics. The course also involves the use of the finite element softwares Ansys and Lusas, to gain experience and insight on the course concepts.

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

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

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.

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.

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

Computational Mechanics

(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.

Computational Mechanics

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.

Computational Mechanics

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.

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

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

Computational Mechanics

**Modalità di esame:** Elaborato scritto individuale; Prova scritta su carta con videosorveglianza dei docenti;

Computational Mechanics

Aim of the exam is to ascertain that the student has assimilated all the presented topics and is able to apply the theories and methods for the solution of practical structural modeling problems. The exam is composed by the development of intermediate homework and a final written test. The intermediate homework is intended to ascertain the acquisition of practical skills in the use of the finite element method for structural analysis. A report including all the assigned homework has to be delivered by the day of the written test. It contributes to the definition of the final mark with a weight of 10/30. The final written test consists of a paper-based written exam with open queries and exercises, with video surveillance by professors. It is intended to evaluate the comprehension of the background theory and the issues concerning the implementation/application of the FE method. It lasts about 2 hours. It contributes to the definition of the final mark with a weight of 20/30.

Computational Mechanics

**Exam:** Individual essay; Paper-based written test with video surveillance of the teaching staff;

Computational Mechanics

Aim of the exam is to ascertain that the student has assimilated all the presented topics and is able to apply the theories and methods for the solution of practical structural modeling problems. The exam is composed by the development of intermediate homework and a final written test. The intermediate homework is intended to ascertain the acquisition of practical skills in the use of the finite element method for structural analysis. A report including all the assigned homework has to be delivered by the day of the written test. It contributes to the definition of the final mark with a weight of 10/30. The final written test consists of a paper-based written exam with open queries and exercises, with video surveillance by professors. It is intended to evaluate the comprehension of the background theory and the issues concerning the implementation/application of the FE method. It lasts about 2 hours. It contributes to the definition of the final mark with a weight of 20/30.

Computational Mechanics

**Modalità di esame:** Prova scritta (in aula); Elaborato scritto individuale; Prova scritta su carta con videosorveglianza dei docenti;

Computational Mechanics

Aim of the exam is to ascertain that the student has assimilated all the presented topics and is able to apply the theories and methods for the solution of practical structural modeling problems. The final exam is composed by the development of intermediate homework and a final written test. The intermediate homework is intended to ascertain the acquisition of practical skills in the use of the finite element method for structural analysis. A report including all the assigned homework has to be delivered by the day of the written test. It contributes to the definition of the final mark with a weight of 10/30. The onsite written examination consists of open queries and exercises. The online written test consists of a paper-based written exam with open queries and exercises, with video surveillance by professors. The written test, in both modalities, is intended to evaluate the comprehension of the background theory and the issues concerning the implementation of the FE method. It lasts about 2 hours. It contributes to the definition of the final mark with a weight of 20/30.

Computational Mechanics

**Exam:** Written test; Individual essay; Paper-based written test with video surveillance of the teaching staff;

Computational Mechanics

Aim of the exam is to ascertain that the student has assimilated all the presented topics and is able to apply the theories and methods for the solution of practical structural modeling problems. The final exam is composed by the development of intermediate homework and a final written test. The intermediate homework is intended to ascertain the acquisition of practical skills in the use of the finite element method for structural analysis. A report including all the assigned homework has to be delivered by the day of the written test. It contributes to the definition of the final mark with a weight of 10/30. The onsite written examination consists of open queries and exercises. The online written test consists of a paper-based written exam with open queries and exercises, with video surveillance by professors. The written test, in both modalities, is intended to evaluate the comprehension of the background theory and the issues concerning the implementation of the FE method. It lasts about 2 hours. It contributes to the definition of the final mark with a weight of 20/30.

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Corso Duca degli Abruzzi, 24 - 10129 Torino, ITALY

Corso Duca degli Abruzzi, 24 - 10129 Torino, ITALY