ME 457 Introduction to Finite Element Method

ME 457 Introduction to Finite Element Method

‘Engineering is an art of assumption’

Exact analytical solutions of engineering problems faced in nature are very limited, so
engineers need approximation techniques and certain assumptions for the solution of
problems

Finite element method is one of the discretization method that is widely used
It is a method that uses mesh structure
(There are also other meshless techniques)
Finite Element Method: FEM
Finite Element Analysis: FEA

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ME 457 Introduction to Finite Element Method

Exact analytical solution example

A plate is supported and loaded with distributed force p as shown in the figure.

Elastic modulus : E
Poisson ratio : 

The exact solution for this simple problem can be found easily as follows:
Displacement:

Strain:

Stress:

Exact (or analytical) solutions for simple problems are severe

Suppose there is a hole in the plate
These is no exact solution for that case
That is FEM or other approximation methods are needed
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ME 457 Introduction to Finite Element Method

Numerical Methods used in Engineering

(Approximation methods)

Finite Element Method

Boundary Element Method
Finite Difference Method
Finite Volume Method
Meshless Methods

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ME 457 Introduction to Finite Element Method

Finite Element Method: FEM

Finite element method (FEM), is a discretization technique in which the whole domain
is divided into several manageable (finite number of) pieces.
These pieces are called elements and the solution for these elements are previously
known (derived).
The solution for the whole domain is obtained after assembling the whole elements
and imposing the boundary conditions

Elements

9 Finite element model consists

of elements and nodes.
Nodes E1
9 E1: Element number
1 Element 4,5,9: Node numbers (connectivity)
Finite Element Model
The node numbering sequence is important
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ME 457 Introduction to Finite Element Method

k1 k2

- FEM divides a structure F

F into several elements
(pieces of the structure).
- Then reconnects elements kN
at “nodes”
- This process results in a set
of simultaneous algebraic
equations. kN is the local stiffness matrix for element N

K is the global stiffness matrix for the whole body

f=kx for each element

Combine (assemble) all stiffness matrices of
elements within the domain and obtain K
F=kx
F=KX for the whole body
x
x=F/k 𝐾 𝑋 = 𝐹
F
−1
𝑋 = 𝐾 𝐹
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ME 457 Introduction to Finite Element Method

Fundemental Concepts

FEM in Structural Analysis

Procedure:
· Divide structure into pieces (elements with nodes)
· Describe the behavior of the physical quantities on each element
. Calculate stiffness behavior of each element
· Connect (assemble) the elements by considering the connectivity (proper node order)
property of the elements to form approximate system of equations for the whole
structure
· Solve the system of equations involving unknown quantities at the nodes
(displacements for structural analysis)
· Calculate desired quantities (e.g., strains and stresses) at selected elements

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ME 457 Introduction to Finite Element Method

Objectives of this FEM Course

· Understand the fundamental ideas of the FEM

· Know the behavior and usage of each type of elements covered in this course
· Be able to prepare a suitable FE model for given problems
· Be able to use Matlab for the solution of simple FE problems
· Be able to use Ansys Workbench for linear problems
· Be able to interpret and evaluate the quality of the results (know the physics of the
problems)
· Be aware of the limitations of the FEM (don’t misuse the FEM. It is a numerical tool)

If eneyone has the exact analytical solution and if it is easy to apply it, it is nonsense to
search for a solution by FEM.

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ME 457 Introduction to Finite Element Method

Linear Static Analysis

Many of structural engineering problems can be threated as linear static problem with
the acceptance of the following assumptions.

1. Small deformations (loading pattern do not change with deformation)

2. Materials are elastic
3. Loads are static or applied slowly and uniformly (quasistatic)

Linear static analysis gives an idea about the behavior of the structure and can be
accepted as a good approach for many of the analyses.
Also for many cases static analysis is the base for nonlinear analysis. Before going into
details of nonlinear analysis a simple static analysis should be performed to gain an idea
about system behavior.

Within this course only static analysis subjects will be covered.

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ME 457 Introduction to Finite Element Method

Computational Mechanics

Computational Mechanics branch can be separated according to problem’s physical

content.

Nano and Micromechanics

Computational
Mechanics Continium Mechanics:
Solids and Stuctures,
Fluids,
Multiphysics

Systems

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ME 457 Introduction to Finite Element Method

Computational Solid and Structural Mechanics

Statics (Linear)

Computational
Solid and Structural
Mechanics

Dynamics (Nonlinear)

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ME 457 Introduction to Finite Element Method

Formulations and Solution Methods used in Finite Element Method

Displacement
Equilibrium
FEM Model Formulation Mixed
Hybrid

Stiffness
FEM Model Solution Flexibility
Mixed

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ME 457 Introduction to Finite Element Method

As a summary:

This Course Covers

Computational structural mechanics

Linear static problems

Spatially discretized by displacement-formulated FEM

Solved by the stiffness method

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ME 457 Introduction to Finite Element Method

Reasons of Using Finite Element Method

▪ Design Analysis: Reduces the number of experiments and hand calculations and
enables computer
▪ FEM/FEA is the commonly worldwide used analysis and simulation method
▪ Can work with CAD/CAM applications in an integrated manner
▪ ...

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ME 457 Introduction to Finite Element Method

Engineering Applications of Finite Element Method

▪ Mechanics/Space/Aeronautics/Civil/Automotive
▪ Structural Analysis (statics/dynamics, linear/nonlinear)
▪ Heat/Fluid
▪ Electromagnetics
▪ Geomechanics
▪ Biomechanics
▪ ...

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ME 457 Introduction to Finite Element Method

Finite Element Types

1-D Line Element (spring, truss, beam, pipe, ...,etc.)

2-D Plane Element (membrane, plate, shell, ...,etc.)

3-D Solid Element (3-D fields - temperature, displacement, stress, flow velocity, ...,etc.)

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ME 457 Introduction to Finite Element Method

Brief History of FEM

Initial studies began in 1943

finite element term was first used by Clough in 1960.

In 1960s, engineers used FE for approximate solutions of problems

in stress analysis, fluid flow, heat transfer, and other areas

The first book on the FEM was published in 1967 (by Zienkiewicz and Chung)

FEM was applied to a wide variety of engineering problems in late 1960s and early
1970s

Most commercial FEM software packages originated in the 1970s.

(Abaqus, Ansys, etc.)

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ME 457 Introduction to Finite Element Method

References

• Lee H.H., Finite Element Simulations with Ansys Workbench 2019, SDC Publications, 2019
• Kattan P.I., Matlab Guide to Finite Elements An Interactive Approach, Springer, 2002
• Cook D.C., Finite Element Modeling for Stress Analysis, John Wiley and Sons, 1995
• Liu Y., Lecture Notes: Introduction to Finite Element Method, University of Cincinnati, 2003
• KWon Y.W. and The Finite Element Method Using MATLAB, CRC Press, 1997
• Ansys Worknech Traing Notes, ANSYS Inc., 2020

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