Objectives and competences
At the end of this course, students will understand the connection between mathematics and mechanics of civil engineering structures, and consequently the necessary concepts and procedures for the modern numerical solving of differential equations of different orders
Content (Syllabus outline)
. Interpolation and approximation of functions.
2. Solving systems of linear equations.
3. Numerical integration (Gaussian integration).
4. Numerical solution of differential equations - boundary values problem.
5. Variational principle for solving ordinary differential equations, residual (collocation method, subdomain method, weighted integral: least squares method and Galerkin method).
6. Weak formulation of the problem.
7. Rayleigh-Ritz method for determining coefficients from weak form.
8. Solution of 2nd order ordinary differential equations by finite element method: "stiffness" matrix, "load" vector and vector of secondary variables.
9. Derivation of the stiffness matrix of the finite element and the load vector for the analysis of axial displacements of the straight line element.
10. Finite element method and solving of ordinary 4th order ordinary differential equations.
11. The stiffness matrix and the load vector of the Bernoulli-Euler finite element straight line element.
12. Stiffness matrices of elements with different boundary conditions.
13. The composition of the global stiffness matrix and the load vector of the structure, the inclusion of the prescribed boundary conditions, and calculation of the primary variables.
14. Modeling of beams on an elastic basis.
15. Computation of structures by substructure method and static condensation.
16. Computer applications (preparation of adequate numerical models and critical evaluation of results).
Learning and teaching methods
Lectures in the lecture room, supported by computer projection of material and simultaneous explanation of more complex details on the blackboard. Tutorial exercises by solving sample problems. Computer exercises by using computers to calculate structures. Individual preparation of seminar paper
Intended learning outcomes - knowledge and understanding
Upon completion of this course, the student will be able to:
• understand the need for numerical solution of coupled differential equations and to know the advantages of the finite element method and the principles of computational analysis of line structures,
• choose the correct computational model based on the concepts of the finite element method,
• be able to perform »manual« formation of stiffness matrices, load vectors of elements, and to compose corresponding matrix equations of structure,
• to calculate the static response of the structure in the form of discrete nodal displacements based on the solutions of the structure's equations,
• evaluate the relevance of the solutions obtained from different discretizations and select the best quality values,
• calculate the vectors of secondary variables / internal static quantities and reactions,
• calculate the functions of inner forces and displacements within the field of each finite element from the discrete nodal values of these quantities
Intended learning outcomes - transferable/key skills and other attributes
• Communication skills: written professional manner at written examination and oral professional manner at oral examination,
• Individual utilisation of acquired knowledge: numerical calculation of the structure with the help of software for the calculation of structures, understanding of the basic problem solving procedures,
• Team problem solving: ability to design and construct 1D civil enginering structures.
Readings
M. Skrinar, Metoda končnih elementov Zbirka rešenih primerov, UM Maribor, 2020
J.N. Reddy, An Introduction to the Finite Element Method, 2006 McGraw-Hill
K.J. Bathe, Finite Element procedures in Engineering Analysis, Prentice-Hall, 1982
Boris Lutar, Janez Duhovnik: Metoda končnih elementov za linijske konstrukcije, UM FG, Maribor 2004
Z. Bohte, Numerične metode, Ljubljana : Društvo matematikov, fizikov in astronomov SRS, 1985
M. Skrinar, Zbirka vaj iz vsebin MKE (v pripravi)
Dodatna literatura:
R. D. Cook: Finite Element Modeling for Stress Analysis, John Wiley & Sons, 1995
M. Petyt, Introduction to finite element vibration analysis, Cambridge University Press, 1990
Prerequisits
Recommended knowledge in mathematics and structural mechanics.
