# COMPUTATIONAL MECHANICS

## 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.

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.

## Prerequisites

It is necessary to have at least attended to the following courses: structural mechanics and Advanced Structural Mechanics.

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.

Variational methods.

Residual methods.

Basic concepts of the finite element method.

Isoparametric formulation.

Structural discretisation with finite elements.

Use of finite elements in non linear problems.

Some advanced aspects about the finite element method.

The topics treated in the course are listed below:

Basic concepts in computational mechanics.

Modeling of structures.

Variational methods.

Residual methods.

Basic concepts of the finite element method.

Isoparametric formulation.

Structural discretisation with finite elements.

Use of finite elements in non linear problems.

Some advanced aspects about the finite element method.

## Course contents

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 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.

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

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.

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

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.

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.

## Recommended readings

Reference books:

- R. Brighenti, Analisi numerica dei solidi e delle strutture: fondamenti del Metodo degli Elementi Finiti. Esculapio Editore (Bologna), 2014 (ISBN: 978-88-7488-798-9).

- 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 Graw-Hill, 1995.

Teaching stuff:

- Stuff provided by the teacher (see the teacher’s website: http://www2.unipr.it/~brigh/index.htm) or from the Elly website of the Univ. of Parma.

Reference books:

- 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

Graw-Hill, 1995.

Teaching stuff:

- 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

school.

## Teaching methods

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.

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 an oral test.

It is facultative the execution of a project consisting in the development of a simple finite element program 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).

The evaluation of the final exam will be as follows:

- Only oral test

Oral test (theoretical questions 50%, exercises 40%) (knowledge).

Clarity of presentation (Communication skills, 10%).

- Oral test + project

Project development (Applying knowledge, 40%).

Oral test (theoretical questions 30%, exercises 20%) (knowledge).

Clarity of presentation (Communication skills, 10%).

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%)

(knowledge).

- Clarity of presentation (Communication skills, 10%)