Numerical Methods for Structural Analysis (Metodi Numerici per l’Analisi Strutturale) ( 9 CFU )
Prof. Roberto Brighenti
Tel. 0521 905910 - Fax. 0521 905924           E-mail. brigh@unipr.it
Prof. Elena Michelini
Tel. +39 0521 905709 - Fax. +39 0521 905924           E-mail. elena.michelini@nemo.unipr.it

Finalità
To present concepts and tools for computational mechanics applied to generic solid structures.

Programma
Basic concepts in computational mechanics.
Introduction to the finite element method: displacement method for plane beam structures.

Variational methods.
Weak and strong form of a physical problem. Natural and essential boundary conditions. Variational principles. Virtual work theorem. Approximate polynomial solution. Bubnov-Galerkin method.
General formulation of a problem by using finite elements: differential and integral forms.
Minimum potential energy principle. Displacement field approximation. Rayleigh-Ritz method.

Residual methods.
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.

Basic concepts of the finite element method
Algebraic static and dynamic equilibrium equations of a structure discretized by finite elements. Stiffness matrix K and nodal force vector f . 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.

Structural discretisation with finite elements.
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.
Isoparametric elements in one, two and three dimensions.
Numerical integration methods. Variable transformation in 1D, 2D, 3D. Gauss rule. Accuracy of the numerical integration. Examples.

Use of finite elements in non linear problems
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.

Some more aspects about the finite element method
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.

Modalità d'esame
Development of a project related to FE implementation followed by and oral examination

Propedeuticità
Analisi A-B, Analisi C, Geometria, Meccanica Razionale, Scienza delle Costruzioni A-B (Structural Mechanics A-B).

Testi consigliati
1. Stuff provided by the teachers.
2. Cook, R.D., Malkus D.S., Plesha, M.E.: “Concept and application of finite element analysis”, 4th edition, John Wiley & Sons, 2002.
3. Zienkiewicz, O.C.: “The finite element method”, Mc Graw-Hill, 2000.
4. Corradi dell’Acqua, L.: "Meccanica delle strutture", Vol. 1,2 e 3, Mc Graw-Hill, 1995.

Testi d'approfondimento
1. Hughes, T.J.R.: “The finite element method. linear static and dynamic finite element analysis”, Prentice Hall, 1987.
2. Owen, D.R.J., Hinton, E.: “Finite elements in plasticity”, Pineridge Press, 1980.
3. Bathe, K.J., “Finite element procedures”, Prentice Hall, 1996.