A numerical approach for linear analysis in solid mechanics based on the finite element methods is presented. Continuum finite elements as well as special structural elements are derived. The advantage of the numerical approaches in comparison with classical analytical methods is demonstrated. The efficiency and accuracy of the finite element approach are demonstrated by numerical examples.

Student responsibilities:

Grading and evaluation of student work over the course of instruction and at a final exam:

Evaluation of the project assignments (20%). Evaluation of the results achieved at the colloquia or oral and written exams (80%).

Methods of monitoring quality that ensure acquisition of exit competences:

The verification of the student attendance at the classes and evaluation of the personal commitment of the students at the laboratory exercises. Evaluation of the results, theoretical understanding and the quality of the presentation of the project assignments. Evaluation of the results achieved at the colloquia or exams. Survey of the output data of the official university questionnaires completed by the students at the end of the course.

Upon successful completion of the course, students will be able to (learning outcomes):

Understand and derive the variational formulation of the finite element method. Understand and derive the global finite element equation. Understand and discuss the convergence of the solution and the organization of computational programs. Understand and derive one-dimensional finite elements. Understand and derive two-dimensional finite elements. Understand and derive three-dimensional finite elements. Understand and derive plate and shell finite elements. To be capable to apply finite elements in structural analysis and to assess the computational errors.

Lectures

1. Variational formulation of the finite element method. Finite element equation for static and dynamic analysis.

2. Global finite element formulation.

3. Convergence of the solution. Finite element program description.

4. Natural coordinates and interpolation polynomials.

5. One-dimensional finite elements.

6. Two-dimensional finite elements: basic triangular element and high-order triangular elements.

7. Two-dimensional finite elements: basic rectangular element and high-order rectangular elements.

8. Three-dimensional finite elements.

9. Axisymmetric finite elements, solution of axisymmetric problems.

10. Axisymmetric finite elements, non-symmetrical loading.

11. Isoparametric finite elements, two-dimensional and three-dimensional elements.

12. Isoparametric finite elements, solution convergence and numerical integration.

13. Finite elements for plate bending analysis based on the Kirchhoff-Love plate theory.

14. Finite elements for shell analysis, flat elements, axisymmetric shell elements.

15. Mixed and hybrid finite element formulations.

Exercises

1. On finite element method. Outline of its application.

2. An example of the global finite element formulation.

3. Presentation of finite element programs.

4. On application of natural coordinates and interpolation polynomials.

5. Application of the one-dimensional finite elements to the strength analysis of framed structures.

6. Application of triangular elements in solid mechanics.

7. Application of rectangular elements in solid mechanics.

8. Application of three-dimensional finite elements.

9. Application of axisymmetric finite elements.

10. Application of axisymmetric finite elements.

11. Application of isoparametric finite elements.

12. Error measures and finite element formulation.

13. Solution of the plate bending problems by means of finite element methods.

14. Solution of shell problems by means of finite element methods.

15. Application of mixed and hybrid formulation. Comparison with the displacement formulation.

Compulsory literature:

Sorić, J.: Finite element methods. Golden marketing, Zagreb, 2004.

K-J. Bathe: Finite Element Procedures, Prentice Hall, New Jersey 1996.

Zienkiewicz, O.C., Taylor, R.L., The Finite Element Method, Fifth edition, Vol. 1, Vol. 2, Butterworth-Heinemann, Oxford 2000.

Recommended literature: