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# COMPUTATIONAL METHODS IN STRUCTURAL ANALYSIS

 Degree course Civil Engineering Curriculum PROGETTAZIONE STRUTTURALE, INFRASTRUTTURALE E GEOTECNICA Learnings Orientamento unico Academic Year 2014/2015 ECTS 6 Scientific Disciplinary Sector ICAR/08 Year Second year Time unit Second semester Class hours 48 Educational activity Formative educational activities

## Single group

 Professor ALBA SOFI Objectives N.D. Programme 1. Elastic problem and variational principlesReview: basic equations governing the elastic problem, force method, displacement method. Finite difference method. Elements of calculus of variations. Fundamental identity of Mechanics. Principle of virtual displacements. Principle of virtual forces. Principle of minimum total potential energy. Principle of minimum total complementary energy.2.Variational methods and matrix methodsWeighted-Integral formulation. Weak formulation. The Rayleigh-Ritz method. The Weighted Residuals method: Petrov-Galerkin method; Galerkin method; Least-Squares method; Collocation method; Sub-domains method.Matrix formulation of the displacement method. Element stiffness matrix. Equivalent nodal forces. Transformation from local to global coordinate system. Assembly procedure. Imposition of boundary conditions and solution. Post-processing of the solution: evaluation of stress and strain distributions. Applications to the analysis of truss structures and frame structures.3.The finite element methodIntroduction to the finite element method: the displacement approach. The discretization process. Formulation of the finite element model: weak form of the governing equations; derivation of the interpolation functions; derivation of the finite element equations and computation of the element properties (stiffness matrix and equivalent nodal forces). Assembly of element equations: interelement continuity and equilibrium conditions. Imposition of boundary conditions and solution. Post-processing of the solution. Mono-dimensional finite elements: derivation of interpolation functions and properties. Truss element. Beam element: Timoshenko beam; Euler-Bernoulli beam. Convergence conditions of the method: validity of finite elements, conformity and geometric isotropy. Two-dimensional finite elements: derivation of interpolation functions and properties. Triangular element. Rectangular element. Plate finite elements. Natural coordinates. Derivation of shape functions in natural coordinates. Families of rectangular finite elements: Lagrangian family and Serendipity family. Isoparametric finite elements. Numerical integration. Limitations of the displacement approach. Modelling, errors and accuracy. Hints on: mixed finite element methods, equilibrated and hybrid models. “Locking” phenomena. Structure and main features of a commercial finite element software (ABAQUS. Finite Element Software. Student Edition). Elaboration and discussion of illustrative examples during exercise classes. Books -K.J. Bathe, Finite Element Procedures, Prentice-Hall, Upper Saddle River, New Jersey, 2006.-L. Corradi Dell’Acqua, Meccanica delle strutture. Le teorie strutturali e il metodo degli elementi finiti. Volume 2. Second Edition. McGraw-Hill, Milano, 2010.-J.N. Reddy, An Introduction to the Finite Element Method, Third Edition, McGraw-Hill, Singapore, 2006.-O.C. Zienkiewicz, R.L. Taylor & J.Z. Zhu, Finite Element Method. Its Basis & Fundamentals, Sixth Edition, Elsevier, Oxford, 2005. Traditional teaching method Yes Distance teaching method No Mandatory attendance No Written examination evaluation No Oral examination evaluation Yes Aptitude test evaluation No Project evaluation Yes Internship evaluation No Evaluation in itinere No Practice Test No

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