introduction to finite element method - faculty...

Introduction to

Finite Element Method

Wong Foek Tjong, Ph.D.

Petra Christian University

Surabaya

Guest Lecture in Prodi Teknik Sipil

2 2014/6/7

Lecture Outline

1. Overview of the FEM

2. Computational steps of the FEM

3. Learning the direct stiffness method using ta29.petra.ac.id

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Behavior of a

Real Structure

Experiment

Replicate conditions of the

structure (possibly on a

smaller scale) and observe

the behavior of the model

Simulation

Simplifications and

assumptions of the

real structure

Mathematical

Model Physical

Model

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An example of the FEM

applications

Real experiment FE simulation It is often expensive or

dangerous

It replicates conditions of

the real experiment

Source: W.J. Barry (2003), “FEM Lecture Slides”, AIT Thailand

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Mathematical

Models

Analytical Solution

Techniques

Numerical Solution

Techniques

Closed-form

Solutions

Only possible for

simple geometries and

boundary conditions

•Finite difference methods

•Finite element methods

•Boundary element methods

•Mesh-free methods

•etc.

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It is a computational technique used to obtain approximate solutions of engineering problems.

The results are generally not exact.

However, the accuracy of the results can be improved either using finer mesh (h-refinement) or higher degree elements (p-refinement)

What is FEM?

Solution refinements in FEM

h-refinement

h=1/4

h=1/2

h=1

Solution refinements in FEM

(cont’d)

p-refinement

h=1 h=1/2 u=a+bx+cy+

dx 2+exy+fy 2

u=a+bx+cy

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In the context of structural analyzes, it may be regarded as a generalized matrix method of structural analysis.

It is originated as a method of structural analysis but is now widely used in various disciplines such as heat transfer, fluid flow, seepage, electricity and magnetism, and others.

Finite element method (1)

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Modern FEM were first developed and applied by

aeronautical engineers, i.e. M.J. Turner et al., at Boeing

company in the period 1950s.

1956: The first engineering FEM paper


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The name “finite element method” was

coined by R.W. Clough in 1960. It is

called “finite” in order to distinguish with

“infinitesimal element” in Calculus.

1967: First FEM book by O.C.

Zienkiewicz


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For General purposes:

NASTRAN, ANSYS, ADINA, ABAQUS, etc.

For structural analysis, particularly in Civil Engineering:

SANS, SAP2000, STAAD, GT STRUDL, MIDAS, DIANA, STRAND 7, etc.

For building structures:

ETABS, BATS etc.

Examples of FEM software

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Example of applications in

structural engineering

Source: MidasGen project apllications

http://en.midasuser.com/products/gallery.asp

Beijing National Stadium in China

(Bird’s Nest Stadium)

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Example of applications in

structural engineering (cont’d)

Source: http://gid.cimne.upc.es/gidinpractice/gp01.html

The structural analysis

of an F-16 aircraft The analysis of the Cathedral of

Barcelona using 3D solid elements.

(courtesy of Barcelona Cathedral)

Simulation of Wave Propagation

Passing a Solid Barrier

15 7-Jun-14

Source: G. B. Wijaya (2005), a Petra-AIT Thesis

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Lecture Outline




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Fundamental concept is discretization, i.e.

dividing a continuum (continuous body,

structural system) into a finite number of smaller

and simple elements whose union approximates

the geometry of the continuum.

Discretization

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Some basic element shapes

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Examples of discretization (3)

Cooling Tower– Nuclear Power Plant (taken from a FEM Course Project of Doddy and Andre, Dec 2008)

150 m

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Structural Model and Its Example

of the Analysis Results

The structure is divided into smaller parts called “element”

Membrane force contour in the circumferential direction

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The FE Model with

a Finer Mesh

The structure is modeled with a finer mesh

The result is now better

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Discretize the structure (problem domain)

Divide the structure or continuum into finite elements

Once the structure has been discretized, the

computational steps faithfully follow the steps in the

direct stiffness method.

The direct stiffness method:

The global stiffness matrix of the discrete structure are obtained

by superimposing (assembling) the stiffness matrices of the

element in a direct manner.

Computational steps of the FEM-

the direct stiffness method

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Generate element stiffness matrix and element

force matrix for each element.

Assemble the element matrices to obtain the

global stiffness equation of the structure.

Apply the known nodal loads.

Specify how the structure is supported:

Set several nodal displacements to known values.

Computational steps… (cont’d)

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Solve simultaneous linear algebraic

equation.

The nodal parameters (displacements)

are obtained.

Calculate element stresses or stress

resultants (internal forces).

General steps of the FEM (cont’d)

x2

D1

D2

D3

CONTINUOUS MODEL

DISCRETE MODEL

p1

p2

p3 Joint

= Free DOF

{D} = Joint Degree-of-freedom

= Locked DOF

Discretization of Plane Frame Structures

D4

D5

D6

D7

D8

D9

D10

D11

D12

Source: ACECOMS, AIT-Thailand

Plane Frame Element

The stiffness equation for the plane frame

element can be obtained by superposing

the bar and beam equations.

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1

2

3

4

5

6

x

y

z Local

coordinate

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1

2

3

4

5

6

The discretized equation:

kd=f

𝐝 =

𝑑1

𝑑2

𝑑3

𝑑4

𝑑5

𝑑6

=

𝑢1

𝑣1

𝜃1𝑢2

𝑣2

𝜃2

; 𝐟 =

𝑓1

𝑓2

𝑓3

𝑓4

𝑓5

𝑓6

=

𝑁1

𝑉1

𝑀1

𝑁2

𝑉2

𝑀2

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1

2

3

4

5

6

The stiffness matrix:

𝐶1 =𝐸𝐴

𝐿 ; 𝐶𝟐 =

𝐸𝐼𝑧

𝐿3 (1+𝜑𝑦) ; 𝜑𝑦 =

12𝐸𝐼𝑧

𝐴s𝐺𝐿2

𝐤 =

𝐶1 0 0 −𝐶1 0 00 12𝐶2 6𝐶2𝐿 0 −12𝐶2 6𝐶2𝐿

0 6𝐶2𝐿 (4 + 𝜑)𝐶2𝐿2 0 −6𝐶2𝐿 2 − 𝜑 𝐶2𝐿2

−𝐶1 0 0 𝐶1 0 00 −12𝐶2 −6𝐶2𝐿 0 12𝐶2 −6𝐶2𝐿

0 6𝐶2𝐿 (2 − 𝜑)𝐶2𝐿2 0 −6𝐶2𝐿 (4 + 𝜑)𝐶2𝐿2

1

2

3

4

5

6

7

8

9 10

11

12

=

1 2 3 4 5 6 7 8

9 10 11 12

Source: ACECOMS, AIT-Thailand

1

2

3

4

5

6

7

8

9 10

11

12

r2 = r2

=

1 2 3 4 5 6 7 8

9 10 11 12

-

1

2

3

4

5

6

7

8

9 10

11

12

r2 = r2

=

1 2 3 4 5 6 7 8

9 10 11 12

-

Final System

35 2014/6/7

Lecture Outline




Example (SAP2000 Verification

Manual, 2007)

Find the internal forces, support reactions and the

deflection at the middle of the beam (node 5)

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introduction to finite element method - faculty...

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