Learning objectives
Knowledge and understanding:
The course aims to present the fundamental concepts of computational mechanics, with specific reference to the methods of computational mechanics applied to the analysis of generic solid structures.
Furthermore, the course intends to provide the students with the basic knowledge to perform numerical linear static or dynamic analyses of structures and enable them to read and understand computational mechanics books, and to autonomously deepen the knowledge of the subject.
Skills:
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 type to represent the structural problem under study, and to correctly assign boundary conditions the mechanical properties of the materials.
Making judgments:
At the end of course the student should be able to correctly interpret the structural behaviour of generic structures and to propose a proper numerical modelling.
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.
Prerequisites
It is necessary to have attended the following courses: Structural Mechanics - Strength of Materials, and Mechanics of Structures.
Course unit content
The topics treated in the course are listed below:
• Basic concepts in computational mechanics.
• Modelling of structures.
• Variational methods.
• Residual methods.
• Basic concepts of the finite element method.
• Isoparametric formulation.
• Structural discretisation with finite elements.
• Introduction to finite elements for non-linear problems.
Full programme
1. Basic concepts of computational mechanics:
Concepts of modelling.
Foundations of variational methods.
Weak and strong form of a physical problem. Essential and natural boundary conditions.
2. Variational principles:
Virtual work principle. 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 (FE) 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 multipliers method.
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. Static condensation and substructuring.
6. Isoparametric elements in one, two, and three dimensions:
One-dimensional finite elements: truss elements, beam bending elements (Bernoulli and Timoshenko formulation). Two-dimensional finite elements: Finite elements for 2D problems under plane stress, plane strain and axisymmetric conditions (shells); 2D bending plates elements (Kirchhoff and Mindlin formulations). Three-dimensional finite elements: 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. Stiffness overestimation, accuracy of the solution, reduced integration, hourglass, incompressible materials.
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 FE programming; development of simple FE programs for structural analyses.
9. Applications - Numerical modelling of generic structures:
Use of FE commercial software for the analysis of structures and of generic structural elements.
Convergence of the solution (h- and p- convergence). Analysis and interpretation of the results, assessment of the solution accuracy.
10. Introduction to finite elements for non-linear problems:
Sources of non-linearity in structural problems. Concept of linearisation. Newton-Raphson method. Formulation of the virtual work principle for non-linear problems.
Bibliography
Reference books:
- Corradi dell’Acqua L.: "Meccanica delle strutture", Vol. 1,2 e 3, Mc Graw-Hill, 1995 (in Italian).
- Brighenti R.: “Analisi numerica dei solidi e delle strutture: fondamenti del metodo degli elementi finiti”. Esculapio Publisher, III Ed., 2019 (in Italian).
- Cook R.D., Malkus D.S., Plesha M.E.: “Concept and application of finite element analysis”, IV Ed., John Wiley & Sons, 2002.
- Zienkiewicz O.C.: “The finite element method”, Mc Graw-Hill, 2000.
Teaching material:
- Course slides, available to download from the Elly website of the University of Parma.
All the suggested textbooks are available for free consultation in the campus library.
Teaching methods
The course is organized in theoretical and practical lessons (by making use computer slides). Exercise sessions are either developed by the teacher or autonomously in class making use of the computer, and additionally at home.
For every topic, the practical activities are properly scheduled to provide the students with the ability to solve the proposed problems on the basis of the previously explained theoretical concepts.
Assessment methods and criteria
The assessment is based on the completion of a project, agreed upon with the lecturer, and on a written exam, which consists of questions related to the full programme of the course.
The project can be completed individually or in small groups (2-3 students max.) and is related to the development and verification of a simple finite element program for structural analysis (developed in FORTRAN language or MATLAB environment).
The evaluation of the final exam will be as follows:
- Project development (applying knowledge, 40%)
- Written test (theoretical questions 30%, exercises 30%)
Other information
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