eng664 computer analytical tools b -...

Finite Element Assignment

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ENG664 Computer Analytical Tools B

Amador Calvo, Koldo Aparicio, Rodrigo Folgueira

Lecturer: Dr. Richard Grant

Date: 19 November 2010


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INDEX

1. - Summary 3

2. - Tools 6

2.1. - Abaqus

2.2. - MDSolid 3.5

2.3. - HP Calculator

2.4. - Excel

3. - Material 8

4. - Theory 9

4.1.- Theory about beams

4.2.- Theory about “Finite Element Method”

5. - Methodology 10

6. - Results 11

6.1. - Results with Abacus

6.1.1.- C3D8 results

6.1.2.- C3D8I results

6.1.3.- C3D8R results

6.1.4.- C3D20 results

6.1.5.- C3D20R results

6.2. - Results with MDSolid 32

6.3. - Results with HP Calculator

7. - Conclusions 38

8. - References 39


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8.1. - Internet

8.2. - Books


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1. - SUMMARY

The present report consists in the analysis of a simply supported beam using

the finite element method.

The purpose of this project is to design a supported beam with concrete

specifications like a concentrated load and uniformly distributed loads. The beam is

designed in Abaqus 6.10-1. Finally the aim of the project is to study how the forces

acts in the beam that we are studying and compare the results that we obtain in 3D

with the Abaqus between that we obtain in 2D. For obtain this results in 2D we are

going to use the MD Solid 3.5 program.

For being completely sure that the solution is correct we have used another

tool to calculate the structure, an HP Calculator.

It is also necessary to try different element types like the solid elements to

obtain the most close reality results, for it we will use different kinds of noded bricks,

noded quadrilaterals, and using also different density meshes for being approach to

the most efficient result in our case.

Finally the most important part of the project is to see the different results that

we have obtain and to make some conclusions about them, some graphics made on

Microsoft Excel program have helped us in this proposition because the results are

more visual and clear in this graphics.


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2. - Tools

Some tools have been used for making this report; most of them for calculate

the forces of the beam, and other ones to compare the final results. Here there is a

small summary of them:

2.1) Abaqus

For making this project it has been used

Abaqus that is a suite of software applications for

finite element analysis and computer-aided

engineering. Concretely the 6.10-1 version.

Abaqus contains an extensive library of

elements that can model virtually any geometry. It

has an equally extensive list of material models

that can simulate the behavior of most typical

engineering materials including metals, rubber,

polymers, composites, reinforced concrete, crushable and resilient foams, and

geotechnical materials. Abaqus can be used to study more than just structural

problems; it can simulate problems in such diverse areas as heat transfer, mass

diffusion, thermal management of electrical components and piezoelectric analysis.

Abaqus is simple to use even though it offers the user a wide range of

capabilities. The most complicated problems can be modeled easily. For example,

problems with multiple components are modeled by associating the geometry

defining each component with the appropriate material models.


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2.2) MDSolid 3.5

MDSolid 3.5 is a program for elasticity

and resistance of materials of easy managing, It

possesses a simple and intuitive interface,

between whose possibilities are twist, Mohr's

circles, calculation of reactions in beams, Laws of

Efforts, etc.

2.3) Graphical calculator HP 50g

HP Calculators the last thing in graphical

calculators for professionals and students of

topography, engineering, mathematics and

sciences is even better. The new graphical

calculator has HP Solve, structured

programming and algebraic system for computer

(CAS).

2.4) Excel

Microsoft Excel is a spreadsheet

application. It features calculation, graphing tools,

pivot tables and a macro programming language

called Visual Basic for Applications. It has been a

very widely applied spreadsheet for these

platforms.

http://en.wikipedia.org/wiki/Spreadsheet


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3.- Material

The beam in study is an Aluminum alloy 7075 beam; this alloy is one of the

alloy with more raised characteristics inside the aluminiums. The development of this

alloy has made possible his utilization fields till now reserved to the stils.

It is applied in armament, aeronautics and in the industry of the car.

Construction of machinery, molds for injection of plastic, molds of blown, molds for

extrusion of plastic, supports for all kinds of molds, screws, molds for injected of

rubbers, etc.

Wrought aluminium alloy composition limits (% weight)

Alloy Si Fe Cu Mn Mg Cr Zn V Ti Bi Ga Pb ZR

Limits

Al Each Total

7075 0.40 0.50 1.2-2.0 0.30 2.1-2.9 0.18-0.28 5.1-6.1 0.20 0.05 0.15 remainder


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4.- Theory

4.1.- Theory about beams:

A beam is a structural element capable of supporting the load principally by

resisting bending.

It is called the force of bending in the material of the beam to all those

external loads, as the own weigh, the span and external reactions to these loads is

called “bending moment”.

Beams generally are vertical gravitational forces but can also be used

horizontal loads.

The beams are characterized by their profile, their length and their material.

Beams generally carry vertical gravitational forces but can also be used to

carry horizontal loads. The loads carried by a beam are transferred to columns, walls,

or girders, which then transfer the force to adjacent structural compression members.

In light frame construction the joists rest on the beam.

http://en.wikipedia.org/wiki/Vertical_direction

http://en.wikipedia.org/wiki/Gravitational

http://en.wikipedia.org/wiki/Force

http://en.wikipedia.org/wiki/Horizontal_plane

http://en.wikipedia.org/wiki/Column

http://en.wikipedia.org/wiki/Wall

http://en.wikipedia.org/wiki/Girder

http://en.wikipedia.org/wiki/Compression_member

http://en.wikipedia.org/wiki/Light_frame_construction

http://en.wikipedia.org/wiki/Joist


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The primary tool for structural analysis of beams is the Euler–Bernoulli beam

equation. Other mathematical methods for determining the deflection of beams

include "method of virtual work" and the "slope deflection method". Engineers are

interested in determining deflections because the beam may be in direct contact with

a brittle material such as glass. Beam deflections are also minimized for aesthetic

reasons. A visibly sagging beam, even if structurally safe, is unsightly and to be

avoided. A stiffer beam (high modulus of elasticity and high second moment of area)

produces less deflection.

Mathematical methods for determining the beam forces (internal forces of the

beam and the forces that are imposed on the beam support) include the "moment

distribution method", the force or flexibility method and the direct stiffness method.

4.2.- Theory about “Finite Element Method”:

The finite element method (FEM) (its practical application often known as

finite element analysis (FEA)) is a numerical technique for finding approximate

solutions of partial differential equations (PDE) as well as of integral equations. The

solution approach is based either on eliminating the differential equation completely

(steady state problems), or rendering the PDE into an approximating system of

ordinary differential equations, which are then numerically integrated using standard

techniques such as Euler's method, Runge-Kutta, etc.

In solving partial differential equations, the primary challenge is to create an

equation that approximates the equation to be studied, but is numerically stable,

meaning that errors in the input and intermediate calculations do not accumulate and

cause the resulting output to be meaningless. There are many ways of doing this, all

with advantages and disadvantages. The Finite Element Method is a good choice for

solving partial differential equations over complicated domains (like cars and oil

pipelines), when the domain changes (as during a solid state reaction with a moving

boundary), when the desired precision varies over the entire domain, or when the

solution lacks smoothness. For instance, in a frontal crash simulation it is possible to

http://en.wikipedia.org/wiki/Structural_analysis

http://en.wikipedia.org/wiki/Euler%E2%80%93Bernoulli_beam_equation

http://en.wikipedia.org/wiki/Euler%E2%80%93Bernoulli_beam_equation

http://en.wikipedia.org/wiki/Deflection_%28engineering%29

http://en.wikipedia.org/wiki/Virtual_work

http://en.wikipedia.org/wiki/Brittle

http://en.wikipedia.org/wiki/Glass

http://en.wikipedia.org/wiki/Stiffness

http://en.wikipedia.org/wiki/Modulus_of_elasticity

http://en.wikipedia.org/wiki/Second_moment_of_area

http://en.wikipedia.org/wiki/Moment_distribution_method

http://en.wikipedia.org/wiki/Moment_distribution_method

http://en.wikipedia.org/wiki/Flexibility_method

http://en.wikipedia.org/wiki/Direct_stiffness_method

http://en.wikipedia.org/wiki/Numerical_analysis

http://en.wikipedia.org/wiki/Partial_differential_equation

http://en.wikipedia.org/wiki/Integral_equation

http://en.wikipedia.org/wiki/Ordinary_differential_equation

http://en.wikipedia.org/wiki/Euler%27s_method

http://en.wikipedia.org/wiki/Runge-Kutta

http://en.wikipedia.org/wiki/Partial_differential_equation

http://en.wikipedia.org/wiki/Numerically_stable


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increase prediction accuracy in "important" areas like the front of the car and reduce

it in its rear (thus reducing cost of the simulation); another example would be the

simulation of the weather pattern on Earth, where it is more important to have

accurate predictions over land than over the wide-open sea.

Visualization of how a car deforms in an asymmetrical crash using finite element analysis

The Finite Element Method (FEM) allows more complex element behavior to

be modeled. The FEM was originally just an extension of matrix structural analysis,

developed by structural engineers. It has since been used in just about every field

where differential equations define the problem behavior. The result of the FEM is to

create a stiffness matrix and a set of loads.

After that, the solution process is identical to that covered in this text. There

are many excellent books covering the FEM, this section is intended only as an

introduction.

The basic idea of the finite element method is to break up a continuum into a

discrete number of smaller "elements". These elements can be modeled

mathematically by a stiffness matrix and are connected by nodes that have degrees

of freedom. This is identical to what we have done with bending and truss elements.


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However, beams and trusses have natural locations at which to define nodes.

In addition, the derivation of their stiffness matrices can be done on a physical basis.

Simple FEM Theory

More general finite elements require slightly more complicated procedures

than used for beams in order to derive the stiffness matrix. The basic procedure is to

assume a shape function that describes how the nodal displacements are distributed

throughout the element based. From the differential equation, we form an operator

matrix that will convert the displacements within the element into strains. Next the

internal and external virtual work can be formed and equated to develop the stiffness

matrix.

The last step is identical to that used for truss and bending elements.


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5. - Methodology

Before starting to define the beam, all of the units must be put in the same unit

because all input data is specified in consistent units. This table shows some of the

most common systems of consistent units and is that is has been used in this

project:

This image

represents the labeling

convention used for the

displacement and

rotational degrees of

freedom in Abaqus.


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This is the convention that has been used in this project for deciding what is

the best option to choose the correct supports of the beam.

In this case the best solution in the left (for de fixed support), has been to bear

in mind that U1 and U2 are equal zero and in the right (for the rolled support) only

the U2 is equal zero.

This is the beam that must be studied, with the specifications that would be

mentioned then:

P1 = 500 N Width = 10mm

W1 = W2 = 5 N/mm Deep = 20mm


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First it is indispensable to get all input data in consistent units, how we explain

before. It is an 3D program so it is necessary to transform the N/mm into N/mm2

(MPa).

The operation is that:

= 0, 25 MPa

When all is defined, is the time to start working with Abaqus. We think this

work must not be a catalogue of the individual commands you have used to set up

and solve the problem so we only include images or steps that we judge to

important. First the beam must be defined with the forces and the boundary

conditions.


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Then it is necessary to create the steps, the mesh, and running the job.

The second part consist in get the results and study them, for this is

necessary to obtain more results with other tools like MDSolid and in our case with a

calculator.

The last part is to compare these results and to obtain the conclusions about

the study of them, for this it is possible to use some graphics that have been made in

Excel and helps to understand better these results. An important step of any finite

element simulation is to discretize the geometry of the beam using finite elements.

The finite elements are joined by nodes. The compilation of finite elements and

nodes is the mesh. The number of elements used in a mesh is the mesh density. In

the next four pages, you can noticed number of elements (longitudinally and

laterally) and the mesh density used for the simple supported beam simulation.

No of elements:

16 32 64 112 160 1280 4752

16 elements


16

32 elements


17

64 elements

112 elements


18

160 elements

1280 elements


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4752 elements

At this point, we are ready to begin the symple supported beam simulation. We

present a few images of our simulation results. These images are the displacements

2-direction. Elements types used to create the mesh are as follows:

C3D8 An 8-node linear brick

C3D8I An 8-node linear brick, incompatible modes

C3D8R An 8-node linear brick, reduced integration, hourglass control.

C3D20 A 20-node quadratic brick.

C3D20R A 20-node quadratic brick, reduced integration.

The simply supported beam simulation is; each element type is tested in each mesh

density (No of elements). Therefore, we have thirty-five different jobs. 5 elements

types multiplied by 7 mesh density.


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6. - Results

6.1. - Abaqus Results:

6.1.1.- C3D8 results:

16 elements

32 elements


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64 elements

112 elements


22

160 elements

1280 elements


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4752 elements

The table shows displacements in 2-direction. Element type C3D8.


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6.1.2.- C3D8I results:

16 elements

32 elements

64 elements


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112 elements

160 elements


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1280 elements

4752 elements


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Displacements in 2-direction are shown in this table. Element type C3D8I


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6.1.3.- C3D8R results:

16 elements

32 elements


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64 elements

112 elements


30

160 elements

1280 elements


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4752 elements

The table, reproduced below, shows the displacements in 2-direction. Element type C3D8R.


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6.1.4.- C3D20 results :

16 elements

32 elements


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64 elements

112 elements


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160 elements

1280 elements


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4752 elements

This table shows the displacement in 2-direction. Element type C3D20


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6.1.5.- C3D20R results :

16 elements

32 elements


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64 elements

112 elements


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160 elements

1280 elements


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4752 elements


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6.2. - Results with MDSolid 32


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The diagram, shows below, are indicated the theory value of 180.382 mm in 2-direction (U2).


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6.3. - Results with HP Calculator:

Maximum deflection


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7.- Conclusions


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45


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Different finite element meshes have been used in simply supported beam

simulation. We have used in our work linear and quadratic elements, fully and

reduced integration and incompatible modes. The correct choice of element is very

important if you want to obtain accurate results at a low cost. It is necessary that we

use a sufficiently refined mesh to ensure correct results in our simulation. Little fine

mesh (not many elements) can lead to poor results. When you increase the mesh

density your numerical solution will go toward a single value, however your computer

needs much time for running a job and consequently your work goes down. These

ideas are suggested in the following book; Getting started with Abaqus/Standar,

Interactive version, chapter 4, page 37. We are going to check these ideas are

according our work and select one element type for simply supported beam

simulation.

We round up the theory value of 180.382 mm to 180.4 mm since the error produced

is ridiculous and are sure our results are more comfortable to understand with such

rounding.

With these considerations we are ready to discuss our results. First, we put the

results in tables and graphs and then discuss them,

The table, reproduced below, shows maximum displacements in 2-direction.

No of elements

Element type 16 32 64 112 160 1280 4752

C3D8 -4,6 -17,9 -63,4 -147,3 -218,9 -177,5 -179,8

C3D8I -176,4 -179,4 -180,2 -180,4 -180,4 -180,4 -180,4

C3D8R -16.230,50 -16.493,80 -16.559,70 -16.574,50 -16.578,20 -239,7 -202,7

C3D20 -178 -180 -180,4 -180,4 -180,4 -180,4 -180,4

C3D20R -180,4 -180,4 -180,4 -180,4 -180,4 -180,4 -180,4


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The ratios of the maximum displacements for the simply supported beam simulation to the theory value of 180.4

mm are shown below.

No of elements

Element type 16 32 64 112 160 1280 4752

C3D8 0,02549889

1 0,09922394

7 0,351441242 0,81651884

7 1,21341463

4 0,98392461

2 0,99667405

8

C3D8I 0,97782705

1 0,99445676

3 0,998891353 1 1 1 1

C3D8R 89,97 91,43 91,79 91,88 91,90 1,32871396

9 1,12361419

1

C3D20 0,98669623

1 0,99778270

5 1 1 1 1 1

C3D20R 1 1 1 1 1 1 1

Depends on the choice of your element type, the results vary greatly. First element

type, C3D8 in our conclusions, produce inaccurate results when the mesh density is

low until 160 elements. With 64 elements predicts a maximum displacement that is

only 35.1% of the theoretical value. With 160 elements we think of anything that we

can not explain; the value calculated is 21% more than theory value of 180.4 mm. At

the beginning, we thought our simulation was wrong but we made it again and the

result was the same. When the mesh density is increased the results are close to the

theory value. 98.39 % with 1280 elements and 99.66 % with 4752 elements.

C3D8I element, second here, predicts accurate results with 16, 32 and 64 elements

about 99% of theoretical value. From 112 elements the results are exact. Therefore,

C3D8I is a behavior that fits with the discussion in the first paragraph that is to say

when the mesh density increases your numerical solution will go toward a single

value.

C3D8R predicts the deflection so badly that the results are very poor. Until 160

elements the results are about 90 times more than in theory. From 1280 elements

the results improve a lot, however they are not accurate. 1280 elements 1.32 times

more and 4752 elements 1.12 times more. With this element type is demonstrated is

shown the number of elements is vital if you want to obtain accurate results.

C3D20 only underpredict the deflection 2 mm with 16 elements. With 32 elements

predicts a maximum displacement in 2-direction of 99.7%. From there the results are

exact.

Finally, C3D20R predicts accurate results from the 16 elements.


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No OF ELEMENTS vs NORMALIZED RESULTS (C3D8)

0

0,2

0,4

0,6

0,8

1

1,2

1,4

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

No of elements

No

rma

lize

d r

es

ult

No OF ELEMENTS vs NORMALIZED RESULTS (C3D8I)

0,975

0,98

0,985

0,99

0,995

1

1,005

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

No of elements

No

rma

lize

d r

es

ult

s


49

No OF ELEMENTS vs NORMALIZED RESULTS (C3D8I)

0,975

0,98

0,985

0,99

0,995

1

1,005

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

No of elements

No

rma

lize

d r

es

ult

s

No OF ELEMENTS vs NORMALIZED RESULTS (C3D20R)

0

0,2

0,4

0,6

0,8

1

1,2

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

No of elements

No

rma

lize

d r

es

ult

s


50

8.- References

8.1.- Internet

Wikipedia:

Technical information about the aluminum alloy:

http://en.wikipedia.org/wiki/7075_aluminium_alloy

http://www.electrocome.com/al7075.htm

Information about Excel program:

http://en.wikipedia.org/wiki/EXCEL

Theory about beams:

http://en.wikipedia.org/wiki/Beam_%28structure%29

Theory about Finite Element:

http://en.wikipedia.org/wiki/Finite_element_method

Description of the MDSolid 32:

http://www.el-hacker.com/md-solids-3-5-calculo-de-estructuras/

Information About the HP Calculator:

http://h41111.www4.hp.com/calculators/es/es/graphing/50g/index.html

Theory about Finite Element:

http://www.ce.ufl.edu/~mih/courses/CES4141/Notes%2062%20-

%20Basic%20Finite%20Element%20Theory.pdf

http://en.wikipedia.org/wiki/7075_aluminium_alloy

http://www.electrocome.com/al7075.htm

http://en.wikipedia.org/wiki/EXCEL

http://en.wikipedia.org/wiki/Beam_%28structure%29

http://en.wikipedia.org/wiki/Finite_element_method

http://www.el-hacker.com/md-solids-3-5-calculo-de-estructuras/

http://h41111.www4.hp.com/calculators/es/es/graphing/50g/index.html

http://www.ce.ufl.edu/~mih/courses/CES4141/Notes%2062%20-%20Basic%20Finite%20Element%20Theory.pdf

http://www.ce.ufl.edu/~mih/courses/CES4141/Notes%2062%20-%20Basic%20Finite%20Element%20Theory.pdf


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8.2.- Books

Getting Starter with ABAQUS / Standard (Interactive version)

Hibbitt, Karlson & Sorensen, Inc.


52

DAT

Datos numéricos del Abaqus

C3d8i

16 elements

32 elements

64 elements


53

112 elements

160 elements

1280 elements


54

4752 elements

C3d20

16 elements

32 elements


55

64 elements

112 elements


56

160 elements

1280 elements

4752 elements

C3d20r


57

16 elements

32 elements

64 elements

112 elements


58

160 elements

1280 elements

4752 elements


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