Learning outcomes of the course unit
Knowledge and understanding: the course aims to present concepts and
tools for computational mechanics applied to generic solid structures.
Furthermore the course intends to provide to the students the basis to
perform numerical linear static or dynamic analyses of structures and
enables them to read and understand computational mechanics books
and to study autonomously the subject. Applying knowledge and under
standing: at the end of course the student should be able to correctly
develop a numerical model of structural elements or generic structures through the finite element technique; in particular the student should be able to choose the most suitable finite element kind to represent the structural problem under study, and to correctly introduce the boundary conditions and the mechanical properties of the materials. Making
judgments: at the end of course the student should be able to correctly
interpret the structural behavior of generic structures and to propose a
proper numerical modeling. Communication skills: at the end of course the student should have a proper use of the terminology of the
computational mechanics applied to structures and will be able to properly use it.
It is necessary to have at least attended to the following courses:
structural mechanics and Advanced Structural Mechanics.
Course contents summary
The topics treated in the course are listed below:
Basic concepts in computational mechanics.
Modeling of structures.
Basic concepts of the finite element method.
Structural discretisation with finite elements.
Use of finite elements in non linear problems.
Some advanced aspects about the finite element method.
1. Basic concepts of 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.
2. 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 applied to beams and plates. The finite element method as a subclass of the variational methods.
3. 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.
4. Basic concepts of the finite element method 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
5. 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.
Shape functions completeness.
6. Isoparametric elements in one, two and three dimensions.
Truss elements, beam bending elements (Bernoulli and Timoshenko
formulation). Finite elements for 2-D problems under plane stress, plane
strain and axisymmetric conditions (shells); 2-D bending plates elements
(Kirchhoff and Mindlin formulations). Finite elements for 3-D problems
with isotropic or orthotropic materials.
Numerical integration methods. Variable transformation in 1D, 2D, 3D.
Gauss rule. Accuracy of the numerical integration.
7. Convergence problems. Numerical errors and ill conditioning of a
matrix. Causes of ill conditioning. Matrix scaling. Scaling of a matrix.
Convergence requirements: completeness, compatibility, stability. The
patch test. The Babuška-Brezzi condition. Stiffness overestimation,
accuracy of the solution, reduced integration, hourglass modes.
8. Some more aspects about the finite element method. Flow-chart of a
simple program for finite element analysis. Substructuring. Post-
processing of the results. Basic concepts on FORTRAN programming;
development of simple FE programs for structural analyses.
9. Applications: numerical modeling of generic structures. Use of FE
software for the analysis of structures or generic structural elements.
Convergence tests. Analysis and interpretation of the results, assessment
of the solution accuracy.
- R. Brighenti, Analisi numerica dei solidi e delle strutture: fondamenti del metodo degli elementi finiti. Editrice Escupalpio, 2014.
- Cook, R.D., Malkus D.S., Plesha, M.E.: “Concept and application of finite
element analysis”, 4th edition, John Wiley & Sons, 2002.
-Zienkiewicz, O.C.: “The finite element method”, Mc Graw-Hill, 2000.
- Corradi dell’Acqua, L.: "Meccanica delle strutture", Vol. 1,2 e 3, Mc
- Stuff provided by the teacher (see the teacher’s website: http://www2.
unipr.it/~brigh/index.htm) or from the LEA website of the Univ. of Parma.
All the suggested textbooks are available in the library of the Engineering
The course is organized in theoretical and practical lessons (by making
use slides or other kind of presentations); the exercises are either
developed by the teacher and autonomously in class also by making use
of the computer and at home by the students.For every topic, the
practical activities are properly scheduled in order to provide the
students the ability to solve the proposed problems on the basis of the
previously explained theoretical concepts
Assessment methods and criteria
The final exam consists in the development of a project concerning the
solution of a structural problem by the finite element technique in groups of 1, 2 or 3 students (the teacher will decide on the number of students depending on the difficulty of the assigned work) and in an oral test.
The evaluation of the final exam will be as follows:
- Project development (Applying knowledge, 60%).
- Oral test (theoretical questions 20%, exercises 10%)
- Clarity of presentation (Communication skills, 10%)