Course detail

# Finite Element Method

Mathematical models and FEM, basic assumptions, linear 3D models, constitutive relations, design models for solving engineering problems (planar beam task models, bent plates, shells, tasks of heat flow), process solutions, variant of formulation of FEM, discretization, derivation matrix stiffness of the 2D element, equilibrium equations. Isoparametric elements, numerical integration to calculate the stiffness matrix and load vector elements for solving various problems, generation FE mesh and the influence on the accuracy of the solution, singularity, the possibility of nonlinear problems solving and problems of FEM stability, software based on FEM.

Language of instruction

Czech

Number of ECTS credits

3

Mode of study

Not applicable.

Department

Institute of Structural Mechanics (STM)

Entry knowledge

Static analysis of statically determinate and indeterminate planar beam structures with straight and curved centreline; calculation of deformations via unit forces method; force method; influence support relaxation and the influence of temperature changes; theory of strength and failure; stress and strain in point of the solid, the principal stresses.

Rules for evaluation and completion of the course

Extent and forms are specified by guarantor’s regulation updated for every academic year.

Aims

Not applicable.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Not applicable.

Not applicable.

Classification of course in study plans

• Programme N-P-C-SI Master's

branch S , 1 year of study, winter semester, compulsory

• Programme N-K-C-SI Master's

branch S , 1 year of study, winter semester, compulsory

#### Type of course unit

Lecture

26 hod., optionally

Teacher / Lecturer

Syllabus

1. Introduction to the Finite Element Method (FEM) of solids and structures. Mathematical models and FEM. Detail of models. The basic assumptions for solving problems of mechanics of structures. 2. Solution of beam structures. Linear 3D mathematical models. Deformation. Stress. Constitutive equations. Formulation of linear / non-linear tasks. 3. Mathematical models of structures for solving engineering problems (2D beam models, bent plates, shells, tasks of heat flow, other force fields). The principle of virtual work. 4. Procedure FEM. Formulation of 1D and 2D tasks. Discretization. Equilibrium equation. 5. Isoparametric elements. Basic considerations. Stiffness matrix and load vector of 1D and 2D element. Numerical integration to calculate the stiffness matrix and load vectors. 6. The finite elements (FE) for beams, plates and shells. 7. FEM modelling of structures. The combination of elements. Boundary conditions. Rigid connections. Spring. Singularity. 8. Generation of FE mesh. Check-shaped elements and softness meshes. The accuracy of the solution. 9. Potential solutions of nonlinear problems via FEM. Geometric, material nonlinearity and contact. 10. Identification of a critical load, the collapse of the structure. Matrix notation of stability task in FEM and its solution. 11. Software for solving FEM. Pre-processor, solver and post processors. 12. Solving problems with stress concentrators. 13. Introduction to Extended Finite Element Method.

Exercise

13 hod., compulsory

Teacher / Lecturer

Syllabus

1. Solving simple discrete problems of elasticity. 2. Analysis of the derivation of the element stiffness matrix for plane stress. Calculating deformations of simple wall. 3. Calculation of the matrix of elasticity constants of the different types of elements. 4. Analysis algorithm assembly stiffness matrix and load vector of the different types of elements. Approximate functions for various types of elements. 5. Stating stiffness matrix of isoparametric element. 6. Numerical integration – application examples. Entering the boundary conditions. Singularity and stress concentration. 7. Derivation of finite element of plates and shells. 8. Modelling of simple tasks of FEM. The combination of elements. Boundary conditions. Rigid connections. Spring. Joining elements. 9. Application software for solving stability – model creation. 10. Calculation of critical load and analysis of the results. 11. Analysis of modelling structures process. Definition of input data and selection of types of finite elements. 12. Analysis of results of the FEM solution of the selected model. 13. Stress concentrators. Credit.