FEM I - lecture notes. INTRODUCTION 1 Page 1 of 7 ____________________________________________________________________________________________________________________

Course Title: FINITE ELEMENT METHOD I Type of course: undergraduate, graduate

Field of study, specialization Aerospace Engineering, Power Engineering, Robotics, Computer Aided Engineering Format (Teaching methods)

Lectures: 2hours/week

Laboratory: 1hour/week

Private study: 2hours/week

Lecturer: Pawe Borkowski

Objective: To supply the basic knowledge and skills required for understanding and simple practical applications of FEM

Contents (lectures program): Approximate methods in mechanics of structures - Finite Element Method, Finite Difference Method, Boundary Element Method/ Approximate solutions of 2D Poisson's equation/ FEM versus Ritz method - beams / Basic relations in analysis of truss structures/ Simple finite elements for 2D and 3D stress analysis/ Typical algorithms of FEM in static linear stress analysis, popular commercial FE software packages/ Accuracy of FE analysis. Computer lab: Introduction to practical problems of FE modeling in ANSYS/ 2D and 3D linear stress analysis/ Static analysis of simple shell structure/ Discretization error and adaptive meshing Abilities: After completing the course the students will be able to build simple FE models and will know the possible applications and limitations of the method in mechanics of structures. Assessment method: Assessment based on tests and results of computer lab work (reports). Practical work: Project/laboratory classes, where students will build and analyze the results of simple FE models of elastic structures Recommended texts (reading): [1] Huebner K. H., Dewhirst D. L., Smith D.E., Byrom T. G.: The finite element method for engineers, J. Wiley & Sons, Inc., 2001. [2] Zagrajek T., Krzesinski G., Marek P.: MES w mechanice konstrukcji. wiczenia z zastosowaniem programu ANSYS, Of.Wyd.PW 2005 [3] Bijak-ochowski M., Jaworski A., Krzesiski G., Zagrajek T.: Mechanika Materiaw i Konstrukcji, Tom 2, Warszawa, Oficyna Wydawnicza

PW, 2005 [4] Saeed Moaveni: Finite Element Analysis. Theory and Application with ANSYS, Paerson Ed. 2003 [5] Cook R. D.: Finite Element Modeling for Stress Analysis, John Wiley & Sons , 1995 [6] Zienkiewicz O.C., Taylor R.: The Finite Element Method.- different publishers and editions

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FINITE ELEMENT METHOD

The FEM is a numerical procedure that can be used to solve a large class of engineering problems including mechanics of structures, heat transfer, electromagnetism, fluid flow and coupled fields problems (e.g. electro-thermal). The simplest description. The method involves dividing the geometrical model of the analyzed structure into very small, simple pieces called finite elements, connected by nodes. The behaviour of the element is described by adequate physical laws. An unknown quantity (e.g. temperature, displacement vector, electrical potential) is interpolated over an element from the nodal values using specially defined polynomials (called shape functions). The procedure leads to the set of simultaneous algebraic equations with the nodal values being unknown. During the solution process the nodal values (DOF- degrees of freedom of the model) are found. Then all interesting quantities ( strains, stresses) are calculated within the elements. Finally the results may be presented in the required graphical form (the typical form of presentation is a contour map)

MPa

050100150200250300350400450

MPa

0-15-30-45-60-75-90-105-120-135

Finite Element Mesh Von Mises stress distribution Contact pressure between the shaft and the rotor disk

3D finite elements

u

w

uv

FEM was developed in 1950's for solving complex problems of stress analysis - mainly for aeronautical industry. The development of the method was connected with the progress in digital computers and numerical techniques. Today the method is considered as the most powerful analysis method for problems described by partial differential equations.

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FEM is one of the approximate methods for solving continuous problems of mathematics and physics.

Approximate methods flow chart

Real object

Methematical model

(continous)

Discrete

model

Real result wr

Exact solution of the mathematical model ws

Exact solution of the discrete model wd

geometry, boundary (initial) conditions material properties, laws pf physics

discretization approximation

numerical calculations

Numerical result

- wn

Partial differential equations

Boundary integral equations

Functionals minimalisation

FINITE DIFFERENCE M. BOUNDARY ELEMENT M. FINITE ELEMENT M. FDM BEM FEM

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Discretization of the continuous problem numerical estimation of the integral

The example wooden board Different models for the problem

a) model belki

b) model pyty

c) model trjwymiarowy bryy

q 0 N

m

p 0 m N

2

N m 0 3

number of DOF

one dimensional model - beam

three dimensional model solid volume

two- dimensional model - plate x

f(x)

x1 x2 bx =ah

0 x =a+ihih h h

f1

f2

f0

3f

f i

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BASIC STEPS IN THE FINITE ELEMENT METHOD (FE modeling)

Preprocessor (preprocessing phase)

In the preprocessing phase the mathematical problem is described and presented in the numerical, discrete form:

Steps: Description of : the analyzed domain (geometry of the analysis object) the material properties the boundary conditions (loads and constraints) the meshing (dividing the domain into the finite elements of the required density distribution)

FE model of the bolted joint of the high pressure vessel entire connection, representative part of the structure and its discretization, FE nodes with load symbols

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Processor (solution phase)

In this phase the user of the FE program defines the type of analysis (static, linear or nonlinear, dynamic, buckling ...... ) and other details describing the method of calculations and solution process. The FE program performs the calculations and writes the results in the adequate files.

Postprocessor

In this phase it is possible to present the interesting results in different forms: plots, graphs, animations, listings etc. The user can create contour maps, tables, graphs and generate the reports.

Von Mises equivalent stress distribution (MPa). Contour map

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The results of FE analysis

Deformed model compared to undeformed structure

Displacement vector ( ux, uy , uz ) Stress state components within the model {x , y , z ,xy , yz ,xz }

X

Y

Z

x

y

z

xy

xz

yz

xy

xzyz

Strain state components {x , y , z , xy , yz , xz }

Principal stresses: 1 , 2 , 3 and principal strains 1 , 2 , 3

Equivalent stress distribution according to an arbitrary criterion: eqv , int.

Any other entity defined by the user (ANSYS Ansys Parametric Design Language (APDL) commands)