


Politecnico di Torino  
Academic Year 2009/10  
01FOLDS Linear finite elements 

Master of sciencelevel of the Bologna process in Mechanical Engineering  Vercelli 





Objectives of the course
This course introduces students to theoretical and practical aspects of finite element method, regarding the static structural analysis in linear filed. Main finite element codes structures is introduced. Focusing on Modelling techniques and post processing of results.

Expected skills
This course introduces students to theoretical and practical aspects of Finite Element Method, regarding the static structural analysis in linear filed.

Prerequisites
The student must have frequented the courses of 'Machine Design'

Syllabus
' Theory of Matrix Structural Analysis
Matrices proprieties and algebra. Basic principles of structural finite element Matrix Structural Analysis. Stiffness matrix and properties. Spring element. Truss element. Coordinate rotation matrix. Matrix triangulariation of the stiffness matrix and Gauss method. Assembling the structural stiffness matrix. Methods of static solution. Bandwidth and frontal solutions. Equivalent nodal loads for truss and beams. Constitutive material matrices. ' Finite Element Method Methods to calculate stiffness matrix. Principle of virtual work. Truss element, bi dimensional beam element. Equivalent nodal loads shear and torsional factors. Bidimensional elements. Plane stress, plane strain. Generalized coordinates. Natural coordinates. Isoparametric elements. Jacobian evaluation and applications for isoparametric elements. Integration numerical methods : Newton Cotes and Gauss'Legendre methods; selection of integration points. Modelling techniques. Convergence criteria. Complete and compatible shape functions and elements. Axial symmetric elements. Plate and shell elements. di Kirchoff's shell formulation. Mindlin's shell formulation. Locking behaviour in Timoshenko and Mindlin elements. Torsional stiffness of shell elements. 
Laboratories and/or exercises
The practical lab will be done on the basic problems studied in the theoretical part of the course. Student groups have to prepare two reports. Practical lab consists in beamlike static analysis with ANSYS for the first report and the second report regarding the simulation of 2d and 3d structure modelling with plane or shell / solid elements.

Bibliography
' Gugliotta Elementi Finiti ' Otto Editore
' Aurelio Somà  Esercitazioni di laboratorio ' R.D. Cook, Concepts and applications of finite element analysis, Wiley. ' K.J. Bathe, E.L. Wilson, Numerical methods in finite element analysis, Prentice Hall. 
Revisions / Exam
Exams consist is oral. Examination starts from the discussion of the exercises regarding the practical lab and on the theoretical aspects presented in the classes.

