# 330-0512/01 – FEM II (MKPII)

 Gurantor department Department of Applied Mechanics Credits 5 Subject guarantor prof. Ing. Petr Horyl, CSc., dr.h.c. Subject version guarantor prof. Ing. Petr Horyl, CSc., dr.h.c. Study level undergraduate or graduate Requirement Compulsory Year 1 Semester summer Study language Czech Year of introduction 2015/2016 Year of cancellation 2020/2021 Intended for the faculties FS Intended for study types Follow-up Master
Instruction secured by
FUS76 doc. Ing. Martin Fusek, Ph.D.
MAW007 Ing. Pavel Maršálek, Ph.D.
POD10 doc. Ing. Jiří Podešva, Ph.D.
Extent of instruction for forms of study
Form of studyWay of compl.Extent
Full-time Credit and Examination 2+2
Part-time Credit and Examination 10+10

### Subject aims expressed by acquired skills and competences

Outline principle of derivation mass matrix in slope-deflection variant of FEM Identify meaning solution of natural frequencies and mode shapes and classify differences for Bernoulli and Timoshenko beam Define methods for numerical computing of eigenvalues and eigenvectors for undamped systems Construct reduction method for easy problem of natural frequency Solve matrix equation of motion by modal method Classify direct integration method, compare implicit and explicit method Clarify solution principle of nonlinear static problems Relate methods for analysis contacts problem by FEM

Lectures
Tutorials
Project work

### Summary

1. Dynamics and FEM 2. Mass matrix 3. Equations of motion of elastic systems 4. Natural frequencies and mode shapes - properties and normalization of mode shapes - methods for computing eigenvalues and eigenvectors 5. Reduction of the number of DOf in dynamics 6. Response history: modal method - proportional damping matrix - vibration caused by initial conditions - harmonic response - general excitation 7. Response history: direct integration method (implicit and explicit methods) 8. Principles of solution nonlinear static problems, fundamental numeric solution of contacts in FEM 9. Newton-Raphson method, arc-length method

### Compulsory literature:

Cook R. D., Malkus D.S., Plesha M.E., Witt R.J. CONCEPTS AND APPLICATIONS OF FINITE ELEMENT ANALYSIS. 4th edition. J. Wiley & Sons, Inc. NY, 2002, p. 719, ISBN 0-471-35605-0 Examples for ANSYS solutions: http://www.mece.ualberta.ca/tutorials/ansys/ REDDY, J.N., An Introduction Nonlinear Finite Element Analysis, Oxford University Press, 2004, p. 463, ISBN 0-19-852529-X BHATTI, M. A., Fundamental Finite Element Analysis and Applications: with Mathematica and Matlab Computations, Wiley, 2005, p.590, ISBN 0-471-64808-6 HORYL, P. FEM Finite Element Method Introduction,2014,https://www.fs.vsb.cz/330/cs/MKP-II/

### Recommended literature:

BHATTI,M.A., Advanced Topics in Finite Element Analysis of Structures: with Mathematica and Matlab Computations, Wiley, 2006, p.590, ISBN-13 978-0-471- 64807-9

### Other requirements

Good knowledge of numerical mathematics, especially Voight notation of matrix calculus.

### Prerequisities

Subject has no prerequisities.

### Co-requisities

Subject has no co-requisities.

### Subject syllabus:

1.0 Comparison of static and dynamic problems solved by FEM 2.0 Finite element mass matrix 2.1 Bar element 2.2 Beam element 2.2.1 Bernoulli‘s beam, diagonal mass matrix 2.2.2 Thimoshenko‘s beam, the second spectrum of natural frequencies 2.3 Plane and space frame finite element 2.4 Selected types of finite elements 3.0 Natural frequencies and mode shapes of un-damped vibration 3.1 Matrix equation for solving natural frequencies and mode shapes (eigenvectors) 3.2 Normalization of the eigenvectors 3.3 Solution methods 3.3.1 Solutions using determinant, disadvantages 3.3.2 Power method (method of inverse iterations) 3.3.3 Subspace method (method of simultaneous iterations) 3.3.4 Lanczos method 4.0 Reduction method in dynamics 5.0 Damping matrix 6.0 Solving response of the linear mechanical system by using mode shapes (Modal analysis method) 6.1 Self-induced oscillations by changing the initial conditions 6.2 Harmonic excitation 6.3 General continuous excitation 7.0 The response solution of nonlinear dynamical systems 7.1 Implicit method 7.2 Explicit method 8.0 Buckling (collapse of structures shape) 8.1 Introduction 8.2 Geometric stiffness matrix of bar FE 8.3 Geometric stiffness matrix of plane frame FE (compression-tension + bending) 8.4 Practical examples 9.0 Solution of nonlinear static tasks 9.1 Introduction 9.2 Newton-Raphson method 10. Contacts in FEM 10.1. Introduction 10.2. Penalty method 10.3. Langrange multiplier method 10.4. Augmented Lagrange method 10.5. Partitioning (Semi-analytical) method 10.6. Example

### Conditions for subject completion

Part-time form (validity from: 2015/2016 Winter semester, validity until: 2020/2021 Summer semester)
Min. number of points
Credit and Examination Credit and Examination 100 (100) 51
Credit Credit 35  20
Examination Examination 65  16
Mandatory attendence parzicipation:

Show history

### Occurrence in study plans

Academic yearProgrammeField of studySpec.ZaměřeníFormStudy language Tut. centreYearWSType of duty
2020/2021 (N2301) Mechanical Engineering (3901T003) Applied Mechanics P Czech Ostrava 1 Compulsory study plan
2020/2021 (N2301) Mechanical Engineering (3901T003) Applied Mechanics K Czech Ostrava 1 Compulsory study plan
2019/2020 (N2301) Mechanical Engineering (3901T003) Applied Mechanics P Czech Ostrava 1 Compulsory study plan
2019/2020 (N2301) Mechanical Engineering (3901T003) Applied Mechanics K Czech Ostrava 1 Compulsory study plan
2018/2019 (N2301) Mechanical Engineering (3901T003) Applied Mechanics P Czech Ostrava 1 Compulsory study plan
2018/2019 (N2301) Mechanical Engineering (3901T003) Applied Mechanics K Czech Ostrava 1 Compulsory study plan
2017/2018 (N2301) Mechanical Engineering (3901T003) Applied Mechanics P Czech Ostrava 1 Compulsory study plan
2017/2018 (N2301) Mechanical Engineering (3901T003) Applied Mechanics K Czech Ostrava 1 Compulsory study plan
2016/2017 (N2301) Mechanical Engineering (3901T003) Applied Mechanics P Czech Ostrava 1 Compulsory study plan
2015/2016 (N2301) Mechanical Engineering (3901T003) Applied Mechanics P Czech Ostrava 1 Compulsory study plan

### Occurrence in special blocks

Block nameAcademic yearForm of studyStudy language YearWSType of blockBlock owner
ECTS - MechEng - Master Studies 2015/2016 Full-time English Choice-compulsory 301 - Study and International Office stu. block