Learning objectives
<br />To present concepts and tools for computational mechanics applied to generic solid structures.
Prerequisites
<br />Propedeuticities (suggested)<br />Analisi A-B, Analisi C, Geometria, Meccanica Razionale, Scienza delle Costruzioni A-B (Structural Mechanics A-B).
Course unit content
<br />Basic concepts in computational mechanics.<br />Introduction to the finite element method: displacement method for plane beam structures.<br />Variational methods.<br />Weak and strong form of a physical problem. Natural and essectial boundary conditions. Variational principles. Virtual work theorem. Approximate polinomial solution. Bubnov-Galerkin method. <br />General formulation of a problem by using finite elements: differential and integral forms.<br />Minimum potential energy principle. Displacement field approximation. Rayleigh-Ritz method.<br /><br />Residual methods.<br />Weighted residual method: subdomain method, collocation method, least square method, Galerkin method. The finite element method as a particular case of the Weighted residual method.<br />Basic concepts of the finite element method<br />Algebraic static and dynamic equilibrium equations of a structure discretized by finite elements. Stiffness matrix and nodal force vector . Stiffness matrix assembling. Treatment of boundary conditions and their classification: linear and non linear, single freedom constraints, multi freedoms constraints. Master-slave method, penalty method, Lagrange's multipliers method.<br /><br />Structural discretisation with finite elements.<br />Choice of the finite element and of the shape functions. Shape functions in the local reference system and their derivatives. Examples of linear shape functions. Isoparametric elements: convergence requirements. Lagrangian and Serendipidy elements. <br />Isoparametric elements in one, two and three dimensions.<br />Numerical integration methods. Variable transformation in 1D, 2D, 3D. Gauss rule. Accuracy of the numerical integration. Examples.<br />Use of finite elements in non linear problems<br />Eigen analysis: linear buckling problems (geometry stiffness matrix), vibration mode shapes of a structure (mass matrix). Material non linear problems in static and dynamic situations.<br /><br />Some more aspects about the finite element method<br />Flow-chart of a simple program for finite element analysis. Substructuring. Post-processing of the results. Accuracy of the solutions, reduced integration, hourglass modes, incompressible materials.
Full programme
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Bibliography
<br />Literature/Course material<br /><br /><br />Carpinteri: "Scienza delle Costruzioni", Vol. 1 e 2, Ed. Pitagora, Bologna.<br /><br /><br /><br />Corradi dell¿Acqua, L.: "Meccanica delle strutture", Vol. 1,2 e 3, Mc Graw-Hill, 1995.<br /><br /><br /><br />Cesari, F.: "Introduzione al metodo degli elementi finiti", Pitagora Ed., Bologna.<br /><br /><br />Cook, R.D., Malkus D.S., Plesha, M.E.: "Concept and application of finite element analysis", John Wiley & Sons.<br /><br /><br />Zienkiewicz, O.C.: "The finite element method", Mc Graw-Hill, 1986.<br /><br />Specialised books and Literature<br /><br /><br />Hughes, T.J.R.: "The finite element method. linear static and dynamic finite element analysis", Prentice Hall, 1987.<br /><br />Owen, D.R.J., Hinton, E.: "Finite elements in plasticity", Pineridge Press, 1980.
Teaching methods
<br />Form of teaching<br />Theory supported by exercises.<br />Assessment method<br />Oral examination
Assessment methods and criteria
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Other information
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