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First Semester 
Teaching style
Lingua Insegnamento

Informazioni aggiuntive

Course Curriculum CFU Length(h)
[70/85]  MECHANICAL ENGINEERING [85/00 - Ord. 2019]  PERCORSO COMUNE 6 60


A) Knowledge and understanding.
The student will acquire basic knowledge of finite element methods for mechanical design. Emphasis will be placed on the applicative aspects of the method, in particular on the possibility of solving structures in the linear elastic field. There are some references to methods for solving nonlinear problems.
B) Applying knowledge and understanding
The student will acquire the knowledge that will allow him to choose the types of elements and the most suitable algorithms for the solution of the specific problem under examination. The student will be able to correctly interpret the results.
C) Making judgments.
The student will acquire the ability to identify and propose appropriate procedures for calculating the stress and deformation state of the mechanical components.
D) Communication skills.
The student will develop the ability to communicate requirements and performances expressed in terms of stresses, deformations, and displacements.
E) Learning skills
The student will acquire the ability to deal with typical problems related to the study of mechanical structures.


Undergraduate courses


1) Introduction to Finite Element Method: application to static stress analysis. Context and history.
2) Truss frame example: division into elements, selection of variables and shape functions, stiffness derivation, assembly of stiffness equations, application of boundary conditions, solution for displacements, computation of element stresses and strains.
3) The constant strain triangle: Continuum model and role of continuum elements; Geometry of constant strain triangle (CST), nodal variables and shape functions, stiffness derivation (plane stress and plane strain), element stress and strain computation.
4) Element formulation: the need for more advanced and generalized procedures; element stiffness by virtual work; general procedure applied to CST; detailed derivation of linear rectangular element (Gauss quadrature, location of element integration points etc.); quadratic rectangular element, axisymmetric elements, 3-dimensional solid elements, truss and beam elements, membrane, plate and shell elements. Isoparametric elements, Jacobian mapping for arbitrarily shaped elements. Element performance: Stiffness and accuracy considerations. Nonconforming elements, reduced integration.
5) Element libraries: elements offered by commercial programs (shapes, nodes, degrees of freedom, allowable load types, etc.). Materials, loads, supports and solution: analysis procedure for modern commercial programs (definition of structure and loads, supports and other constraints, solution, post-processing); material property definitions and matrices; geometric properties (thicknesses, cross-sectional areas etc.); load types (point forces and moments, pressure, body forces, thermal) and internal conversion to nodal loads; supports, prescribed displacements, rigid links; symmetric and antisymmetric boundary conditions and their application to reduce model size; stiffness transformations to model supports or loads at arbitrary angels; stiffness matrix assembly and solution, bandwidth and its minimisation.
6) Guide to good modelling: Identification of appropriate domain of solution (2-/3-dimensional, axisymmetry, beams/shells etc.). Selection of elements, degrees of freedom, stress assumptions etc. Creation of mesh (refinement, shape, aspect ratios, curvature); Definition of material and geometric properties; application of loads and supports; pre-analysis checks; post-processing results - typical options; importance of verification, development of checking strategies; sources of inaccuracies and errors.

Teaching Methods

Duration: 10 weeks.
Lectures: 4h/ week.
Tutorials: 2h/ week.

The course consists of 20 frontal lessons two hours each, presented in a traditional way on the blackboard, sometimes with the aid of projected images and the use of a commercial FEM program. Some theoretical explanations are followed by the description of their implementation in a FEM code written by the teacher in MATLAB.
Students are then required to perform some exercises (with a technical report) to be delivered to the teacher. The theme of the exercises is as follows:
1) Calculation of a plane or spatial frame made up rods;
2) Calculation of a plane or spatial frame consisting of beams;
3) Calculation of a plane structure under stress or plane deformation;
4) Calculation of an axial symmetry structure;
5) Calculation of a plane shell in bending.

To perform the calculations, students must use a FEM elastic linear code illustrated in the classroom. They have to compare the results obtained by discretizing the structure with mesh of different density and with different types of elements; the exercises end with a verification of results with respect to analytical solutions.

Verification of learning

The delivery of the exercises (and their acceptance by the teacher) is mandatory for admission to the oral examination during which the candidate must explain the problems faced and justify his choices.
The student will have to demonstrate that he is able to choose, within a large library of finite elements, the types best suited to solving a specific static problem. It will have to be able to choose the appropriate density of the mesh and decide to update it.
It will have to be able to evaluate the quality of the results in the light of an analytical reference analysis. It will have to know how the FEM codes calculate the stiffness matrices of some elements and the equivalent nodal forces, how the boundary conditions are introduced, calculated the displacements, and then the stress and deformations.
The quality of the tutorials delivered is 50% on the final grade.
In evaluating exercises and examining the final vote, determination takes into account the following elements:
1) the logic followed by the student in the resolution of the question;
2) the correctness of the procedure identified for the solution of the question;
3) the adequacy of the proposed solution in the light of the theoretical knowledge developed in the course
4) the use of adequate Language.
In order to pass the exam, you must give a vote of not less than 18/30, the student must demonstrate how to choose the finite elements suitable for certain stress distribution, know how to determine the boundary conditions, and be able to judge the final results.
To achieve a score of 30/30 and praise, the student must demonstrate that he has gained an excellent knowledge of all the topics dealt with during the course.


[1] A.Gugliotta: Elementi finiti, OTTO editore, 2002.
[2] R.D.Cook: Concepts and applications of finite element analysis, John Wiley & Sons. 1974.
[3] O.C.Zienkiewicz, R.L.Taylor: The Finite Element Method: Basic Formulation and Linear Problems. McGraw-Hill, 1989.
[4] R.W. Clough, J. Penzien "Dynamics of Structures", Mc Graw-Hill , 1975.

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