text:
User:Eml4500.f08.wiki1.aguilar/hw1

Section 1 How to Access MATLAB
After logging onto a computer with MATLAB installed, run the system command matlab or click on the desktop icon. To exit MATLAB, enter quit or exit in the MATLAB command window, or click on the File menu and select Exit MATLAB.

When the Editor in MATLAB is opened, it is convenient to have the Editor and Command windows opened separately. The specifics of the Editor are covered later on in this tutorial.

Section 2 Entering Matrices
MATLAB is based around operating with matrices. In fact, MATLAB is an abbreviation for Matrix Laboratory. The program can interpret these matrices as either scalars or vectors.

Matrices can be created by entering an explicit list of elements, generating built in statements and functions, creating a disk file with a local editor, or loading external data files or applications.

The following are all examples for creating matrix A, which is a 3x3 matrix. A = [1 2 3; 4 5 6; 7 8 9]

A = [1,2,3;4,5,6;7,8,9]

A = [ 1 2 3	4 5 6	7 8 9]

Notice that the elements in the rows of the matrix can be separated by commas or blank spaces.

The following is an example of how to enter a number in exponential form.

x = 2.34e-9

Notice when the number in entered, there are no spaces in between the e.

MATLAB is capable of doing computations with complex numbers. The following are examples of how to create complex matrices.

A = [1 2;3 4] + i*[5 6;7 8]

A = [1+5i 2+6i;3+7i 4+8i]

Notice when the complex numbers were listed inside the matrix, no blank spaces were used. It should also be noted that either i or j can be used as the imaginary unit. Also, if i or j are assigned as variables to represent a new value, they will not retain their value as an imaginary number. For example if the user enters the command

i = 5

then i will no longer be equal to 'sqrt(-1)', it will be equal to 5.

If a large matrix must be entered, creating an ASCII file is a good option because it can make it easier to correct errors. If an ASCII file is created and named data.ext, simply type load data.ext into the MATLAB command line and press enter. This command will read the file and save it as the variable data. MATLAB also has built in functions such as rand, magic, and hilb. If rand(n) is typed into the command line, MATLAB will create a n x n matrix with randomly generated numbers distributed uniformly between 0 and 1. If rand(m,n) is typed into the command line, MATLAB will create a m x n matrix with the same criteria as before. If magic(n) is typed into the command line, MATLAB will create an integral n x n matrix which is a magic square. This means all the rows, columns, and diagonals have a common sum. If hilb(n) is typed into the command line, MATLAB will create a n x n Hilbert matrix.

Individual entries in a matrix can be easily referenced. For example,

A(2,3)

selects the value in the second row, third column of matrix A. Also

X(3)

Would select the third coordinate of vector X.

Section 3 Matrix and Array Operations
The following matrix operations are available in MATLAB:

+	addition -	Subtraction ^	power ‘	conjugate transpose \	left division /	right division
 * multiplication

Since a scalar value entered into MATLAB is stored as a 1 x 1 matrix, these operations also apply to those values. If MATLAB is told to perform an operation that is not possible for a specific matrix, an error message will appear on the screen.

Matrix division can sometimes be confusing in MATLAB. The difference in right and left division is shown below.

x = A\b is the solution of A*x = b

x = b/A is the solution of x*A = b (another way of writing this could be x = (A’\b’)’ )

It is also important to note that addition and subtraction operate entry wise, but multiplication, power, and left and right division operate as matrix operations. These matrix operations can operate as entry wise operations by placing a period in front of them. An example is

[1,2,3,4].*[1,2,3,4]	or 	[1,2,3,4].^2

Both of these operations will give [1,4,9,16] as the solution.

Section 4 Statements, Expressions, and Variables and Saving Sessions
MATLAB uses expression language. This means when values are typed in, they are entered in the form of variable = expression or just as expression. For example,

x = 5*4

The evaluation of this expression is created as a matrix and displayed on the screen, such as

x = 20

If a variable and equal sign are not used, the result is automatically saved as the variable ans.

The user can also decide to continue the statement to the next line by using three or more periods (…) followed by enter, or several statements can be placed on a single line by separating them with commas (,) or semicolons (;).

If the variable is rather large, it may be useful to suppress the statement. By placing a semicolon (;) at the end of a statement, the assignment is carried out, but the result will not be printed on the screen.

It is also important to note that MATLAB is not case sensitive. For example, alpha is not the same as Alpha.

In order to see a list of variables that are currently being used in the workspace, type who (or whos) in the command line. If a variable needs to be cleared from the workspace, enter clear variablename'. If clear is entered into the command line, all nonpermanent variables will be cleared.

If the unit round off needs to be changed, type in eps (epsilon) to have MATLAB use a round off of 10^-16. This can be useful in specifying tolerances for convergence of iterative processes.

Sometimes a computer will enter a loop that does not stop or takes too long to calculate. These computations can be stopped in MATLAB by pressing CTRL-c.

When MATLAB is closed, all of the variables in the workspace are deleted. If these variables need to be saved, enter save in the command line before closing the program. These variables are written to a disk file named matlab.mat. When MATLAB is started again, enter the command load and the variables will be restored to the workspace.

Section 5 Matrix Building Functions
The following are convenient matrix building functions:

eye		identity matrix zeros 		matrix of zeros ones		matrix of ones diag		create or extract diagnols triu		upper triangular part of a matrix tril		lower triangular part of a matrix rand		randomly generated matrix hilb		Hilbert matrix magic		magic square toeplitz	see help toeplitz

For example, zeros(m,n) creates a m x n matix of zeros, and zeros(n) produces a n x n matrix of zeros. If A has already been defined as a matrix, then zeros(size(A)) will create a matrix of zeros that is the same size as matrix A.

If x has already been defined as a vector, diag(x) creates a diagonal matrix with x down the diagonal. If A has already been defined as a square matrix, then diag(A) creates a vector consisting of the diagonal of A.

Matrices can also be built from blocks consisting of other matrices. For example, if A is defined as a 3 x 3 matrix, then B = [A, zeros(3,2); zeros(2,3), eye(2)] will be 5 x 5 matrix.

Section 6 For, While, If, and Relations
As with other computer programming languages, MATLAB has the flow control statements For, While, If, and Relations.

For
For loops are started with the command. For example, one of the most simple examples of a for-loop is:

for j=1:4, j end Here, a loop is cycled through, setting the variable j to 1, then 2, then 3, and then 4, and showing each of these outputs in the command window.

Another example, for a given "n", the statement: x=[]; for i=1:n, x=[x,i^2], end which is the same as x=[]; for i=1:n x=[x,i^2] end will produce a certain "n" vector. The statement, x=[]; for i=n:-1:1, x=[x,i^2], end produces the same vector but in the reverse order.

The statements for i=1:m for j= 1:n H(i,j) = 1/(i+j-1); end end H generate an m-by-n Hilbert matrix.

Note: Using the semicolon in the code stops MATLAB from printing unwanted intermediate results.

MATLAB allows for any matrix to be used in place of 1:n. The variable just consecutive assumes the value of each column of the matrix.

While
While loops are done in the exact same way using the command. The general form of the  loop is:

relation

 * statements

As expected, the  loop will continue to be executed as long as the relation remains true.

A basic  loop example is given below. n=0; while n<5 n=n+1; n end This loop cycles through, increasing the variable n by 1 until the relation is no longer true.

If
The  command allows for various if-then statements to be executed. The general form of the  loop is:

 * relation

 * statements

As expected, the  command executes only when the relation is true.

A basic  command example is given below. n=0; if n==0 n=2; n end Here, we set n=0, so the relation is true and theprint statement exectues accordingly. If the code was repeated, n would now be equal to 2, so nothing would be printed.

Multiple branching is also possible. This is done with  and   commands, as shown below. n=x; if n>0 m='Positive'; elseif n<0 m='Negative'; else m='0'; end

We can see how the output changes depending on the value of x.

Relations
MATLAB uses typical programming relationship operators: <    less than >    greater than <=   less than or equal to     >=    greater than or equal to     ==    equal ~=   not equal Note: The assignment statement "=" is different from the "==" relationship operator.

You can also use these logical operators in conjunction with the relationship operators: &    and |    or     ~     not

When applied to scalars, the relation is the scalar 1 if the relation is true, and 0 if the relation is false. For example, executing 1<2 prints "ans = 1", however, if you enter 2<1, the result is "ans = 0".

These relationship operators can also be applied to matrices. If the are the same size, the output is a matrix of 0's and 1's giving the value of the relation between corresponding entries.

Note: if you wish for a statement to be executed if matrices A and B are equal, you could type:

if A == B  statement end

but the seemly obvious opposite below would not execute as expected:

if A ~= B  statement end This statement would only execute if each of the corresponding entries was different. So, alternatively, to execute a statement from two matrices that were even one entry different, you would use:

if A == B else statement end

Section 7 Scalar Functions
Many MATLAB functions are used mainly on scalars. A list of the most common include: sin            asin             exp                    abs               round cos            acos             log (natural log)      sqrt              floor tan            atan             rem (remainder)        sign              ceil

Note: The operate element-wise when applied to a matrix.

Section 8 Vector Functions
Other MATLAB functions operate essentially on a vector (row or column), but act on an m-by-n matrix (m$$>=$$2) in a column-by-column fashion to produce a row vector containing the results of their application to each column.

For example: m=[1,3;2,4]; mean(m) gives the result: ans =

1.5000   3.5000 As you can see, the mean of 1 and 2 is 1.5 and the mean of 3 and 4 is 3.5.

This means that in order to find the maximum entry in a matrix A, we use max(max(A)) rather than max(A).

Note: Row-by-row action can be obtained by using the transpose; for example, mean(B')'.

Some of these vector functions are listed below: max         sum          median          any min         prod         mean            all sort                     std

Section 9 Matrix Functions
MATLAB is especially useful when using any of its various matrix functions. Some of the most useful include: eig      eigenvalues and eigenvectors chol     cholesky factorization svd      single value decomposition inv      inverse lu       LU factorization qr       QR factorization hess     hessenberg form schur    schur decomposition rref     reduced row echelon form expm     matrix exponential sqrtm    matrix square root poly     characteristic polynomial det      determinant size     size norm     1-norm, 2-norm, F-norm, infinite-norm cond     condition number in the 2-norm rank     rank

Note: Some of these functions may have single or multiple output arguments. While  produces a column vector containing the eigenvalues of A,   produces a matrix U whose columns are the eigenvectors of A and a diagonal matrix D with the eigenvalues of A on its diagonal.

For example, if m=[1,2,3;4,5,6;7,8,9], then [u,d] eig(m) outputs:

u =

-0.2320  -0.7858    0.4082        -0.5253   -0.0868   -0.8165        -0.8187    0.6123    0.4082

d =

16.1168        0         0              0   -1.1168         0              0         0   -0.0000

title: User:Eml4500.f08.wiki1.aguilar/hw1


text:
User:Eml4500.f08.bottle.loschak

EML 4500, Fall 2008 Finite Element Analysis Paul Loschak Loschak@ufl.edu

Homework
http://en.wikiversity.org/wiki/User:Eml4500.f08.bottle.loschak/HW1

http://en.wikiversity.org/wiki/User:Eml4500.f08.bottle.loschak/HW2

http://en.wikiversity.org/wiki/User:Eml4500.f08.bottle.loschak/HW3

http://en.wikiversity.org/wiki/User:Eml4500.f08.bottle.loschak/HW4

http://en.wikiversity.org/wiki/User:Eml4500.f08.bottle.loschak/HW5

http://en.wikiversity.org/wiki/User:Eml4500.f08.bottle.loschak/HW6

http://en.wikiversity.org/wiki/User:Eml4500.f08.bottle.loschak/HW7

Group Info
TA EMAILS: nvigroux@ufl.edu, zsunguk@ufl.edu PROFESSOR EMAIL: vu-quoc@ufl.edu TEAM EMAILS: jrbball3@ufl.edu, briz102@ufl.edu , loschak@ufl.edu , jchipps3@ufl.edu , bwb2005@ufl.edu, tvitello@ufl.edu

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: User:Eml4500.f08.bottle.loschak


text:
User talk:Eml4500.f08.bottle.loschak

 Hello Eml4500.f08.bottle.loschak, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

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And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! --JWSchmidt 23:26, 7 September 2008 (UTC)

title: User talk:Eml4500.f08.bottle.loschak


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User talk:Eml4500.f08.echo.sell.cm

 Hello Eml4500.f08.echo.sell.cm, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

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title: User talk:Eml4500.f08.echo.sell.cm


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User talk:Eas4200c.f08.carbon.clausen

 Hello Eas4200c.f08.carbon.clausen, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

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 -	Working Here: -	Message -	Eas4200c.f08.carbon.clausen 17:23, 26 September 2008 (UTC) -

title: User talk:Eas4200c.f08.carbon.clausen


text:
User talk:Eml4500.f08.group.clausen

 Hello Eml4500.f08.group.clausen, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

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title: User talk:Eml4500.f08.group.clausen


text:
User:Eml4500.f08.echo.bruce.jp

eml4500.f08.echo.bruce,jp/HW1

title: User:Eml4500.f08.echo.bruce.jp


text:
User:Eml4500.f08.team.foskey.ckf/HW1

Course Overview

The Fall 2008 Finite Element Analysis and Design (EML 4500) course is taught in FLG 270 at 3:00pm (8th period) by Dr. Loc Vu-Quoc. The prerequisites for the course include Computer Programming for Engineers (CGS 2425), Intermediate Engineering Analysis (EGM 4313), and Mechanics of Materials (EGM 3520).

The course website can be accessed at www.mae.ufl.edu/~vql

The course Wikiversity page can be found at http://en.wikiversity.org/wiki/User:Eml4500.f08

The current grade determination for the class is that each of the three exams will make up 23% of the student’s grade, and that the homework, projects, and class participation make up the remaining 31%. Team evaluation forms will need to be printed out prior to the exams, completed, and submitted with the exams. Grade penalties will result if a team evaluation form is not submitted.

The exam schedule for EML 4500 is as follows:

Exam 1: Wed, Sept. 24, 2008

Exam 2: Wed, Oct. 22, 2008

Exam 3: Wed, Dec. 3, 2008

Course Policy

Students are expected to arrive to class on time, and turn off cell phones, beepers and/or pagers. Students arriving late to class should use the doors at the back of the room. Talking during the lecture, as well as eating or drinking is not permitted. All emails submitted to the instructor of the TA’s should be written in formal letter format.

Wiki Homework

Creating an account & creating articles

Each student will be required to create a wikipedia account for the wikiversity portion of wikipedia.com. This is accomplished by logging on to the wikiversity page and clicking on log in/ create account. The student’s username should be in the form : course.semester.TeamName.LastName.initials.

For example: For John Q. Doe in team gamma for the fall 2007 EML 4500 course, John’s username would be eml4500.f08.gamma.doe.jqd.

To edit a page, the users log into wikiversity, and then click on their username. Links to other wiki pages are created by typing the links in the editing box. To create a link to a page for homework 1, the user would type User:eml4500.f08.gamma.doe.jqd/hw1 (in double square brackets) or User:eml4500.f08.gamma.doe.jqd/hw1| HW 1 (again, in double square brackets) The second of which will shorten the link to “HW 1”. Clicking on the link will take the user to a new page that the user can edit with the information pertaining to homework 1.

All information in the wiki article should be rewritten in the student’s words, to avoid plagiarism. Citations should be given for all reference materials. All figures should be redrawn by the individuals, not copied or scanned from textbooks.

Submitting Wiki Homework

To submit a Wiki Homework article, each member of the team will add a link to an ARCHIVED version of their contribution to the team coordinator’s page. Once the team coordinator has a completed assignment, the team coordinator will then email an ARCHIVED link of the page to the TA’s. The homework must be submitted prior to the due date time, as the homework will contain a time stamp indicating its submission time.

The top 3 reports will be selected by the TA’s and will be posted for the entire class to see. Those top 3 reports will receive an additional bonus of 10%, 5% and 3% respectively.

Guidelines for Homework

All homework should be written so that anyone not involved with the assignment would be able to understand it. The reports can be typed by word processor, or hand written if done neatly and legibly. The problem should be restated in simple terms. A detailed description of the methods of solving the problem should be given, and any computer code used should be explained. The input data should be given, and the end results should be given and discussed. An appendix should contain the computer code used, as well as the input and output data. Since the report is done as a group assignment, it is important for all group members to contribute, and to proofread the report prior to submission.

Additional Wiki Notes

To help reduce outside tampering to the articles, students should not use their class accounts to edit non class related wikipedia articles. There will be a grade penalty for any user that edits a non class related article with their class account. Additionally, to reduce the possibility of teams stealing information from other teams, the usernames for team members should only be known by fellow teammates. A safe method would be to wait until the assignment is completed to create the articles, compile them on the team coordinators page, and then submit them. In this way, the information is only viewable for a limited amount of time before it is turned in.

title: User:Eml4500.f08.team.foskey.ckf/HW1


text:
User:Eml4500.f08.team.foskey.ckf/teamHW1

 NOTE: The link below that is preceded by the bullet point was originally a link to a page created in the main space of Wikiversity. The page has since been moved to a user subspace, and the page in the main space has been deleted. As a result, the link in the originally submitted HW1 assignment is no longer functional. For that reason, the new link is dated for a date after the original HW1 assignment was due. For a comparison between the original submission, and the corrected version (but without this comment box), click here

Eml4500.f08.team.foskey.ckf 17:31, 1 November 2008 (UTC)

Team: "TEAM"

WIKI Homework Report 1

Summary of lecture notes from Aug. 25 - Sept. 5

[| What is Finite Element Analysis]

[| EML 4500 Overview]

[| The Matrix Method]

[| Two Bar Truss Problem Data]

[| Solving the Two Bar Truss Problem]

Matlab Matters The Following is a summary of the MATLAB Primer - Third Edition by Kermit Sigmon from the Department of Mathematics at the University of Florida

[| Matlab Summary (1/4)] - Sections 1-6

 * Matlab Summary (2/4)- Sections 7-12

[| Matlab Summary (3/4)] - Sections 13-18

[| Matlab Summary (4/4)] - Sections 19-20

Homework Contributions

Cory Foskey Eml4500.f08.team.foskey.ckf 02:48, 17 September 2008 (UTC) Jin Yu Guan Eml4500.f08.team.guan 03:06, 17 September 2008 (UTC) Joseph Figuerrez Eml4500.f08.team.figuerrez 12:37, 17 September 2008 (UTC) Owen Allen Eml4500.f08.team.allen.oca 13:54, 17 September 2008 (UTC) Timothy Bengtson Eml4500.f08.team.Bengtson 20:13, 17 September 2008 (UTC) James Dwyer Eml4500.f08.team.dwyer.jd 16:18, 18 September 2008 (UTC)


text:
User talk:Pohick2

 * http://en.wikiversity.org/wiki/Topic:Mechanical_engineering
 * http://en.wikiversity.org/wiki/Topic:Mechanical_engineering

title: User talk:Pohick2


title: User:Pohick2


text:
User:Eml4500.f08.team.figuerrez

hw1

hw2

hw3

hw4

hw5

hw6

Determining if the electric pylon is statically indeterminate

hw7

title: User:Eml4500.f08.team.figuerrez


text:
User:Eas4200C.f08.spars.mcgilvray

This is Thomas McGilvray's user page. Submitted HW1 Submitted HW2 Submitted HW3 HW4 HW5

title: User:Eas4200C.f08.spars.mcgilvray


text:
User talk:Eml4500.f08.team.figuerrez

 Hello Eml4500.f08.team.figuerrez, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random random project.
 * Browse everything on Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
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And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! --JWSchmidt 02:37, 8 September 2008 (UTC)

title: User talk:Eml4500.f08.team.figuerrez


text:
User talk:Eas4200C.f08.spars.mcgilvray

 Hello Eas4200C.f08.spars.mcgilvray, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random random project.
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 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! --JWSchmidt 02:38, 8 September 2008 (UTC)

title: User talk:Eas4200C.f08.spars.mcgilvray


text:
User talk:Tgretton

 Hello Tgretton, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

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 * Browse everything on Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
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 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
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 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! --JWSchmidt 02:39, 8 September 2008 (UTC)

title: User talk:Tgretton


text:
User:Eml4500.f08.team.figuerrez/HW1

MAT LAB Primer (3rd edition) Summary Page 1 - Page 6 (BEFORE "Scalar Functions")

title: User:Eml4500.f08.team.figuerrez/HW1


text:
User:EML4500.f08.Ateam.Miller

Homework Contributions
Homework 1

Homework 2

Homework 3

Homework 4

Homework 5

Homework 6

Homework 7

title: User:EML4500.f08.Ateam.Miller


text:
User:Eml4500.f08.ateam/hw 1

A-Team Teampage

Homework Report #1

This page is intended for Homework Report #1 of the A-Team.

Due date: Friday, 12 September 08, 5 pm EDT = 17:00 EDT (or 21:00 UTC).

Class Overview
EML4500 is an upper level engineering course in the field of Finite Element Analysis and Design. The course, taught by Dr. L. Vu-Quoc at the University of Florida is an overview of the Finite Element Method and its applications. Throughout the course of this semester we will be introduced to the different aspects of the FEM and be applying these to solve different engineering problems. The curriculum will be presented to us through lectures from the professor and homework assignments that will assist in our understanding of the material. To ease in the understanding of the material we will work in small groups that will collaborate on weekly homework reports to be submitted using Wikipedia.

Wiki Homework Reports

As stated previously, we will be using Wikipedia (MediaWiki, WikiVersity,...) to submit our weekly HW reports. Working in small teams, we will be collaborating on the current HW questions and lecture notes to form a weekly HW report. These reports will consist of two parts: the HW problem statement/solution and a summary of that week's lecture notes. For the problem statement and solution we will work with our team to come up with a solution to the problem to the best of our ability and form this into the report. After proof-reading and, if needed, editing each of our team members contribution to the report, our team leader will submit a final copy using our Wiki page. The second half of our weekly reports will consist of a summary of the prevous week's lecture notes. For this section, we will take the lecture notes, rewrite them in our own words and also add any comments or suggestions to clarify and topics that we might not fully understand. In addition to the notes we will also redraw any figures or examples that would assist in our learning of the material.

Pros/Cons of Using Wikipedia

Using Wikipedia to work together and submit our HW reports has its pros and cons. Working in our teams will help our understanding of the material, but also presents some difficulties. This is where using Wiki will help. With this, we will be able to collaborate no matter where we are. As Dr. Vu-Quoc stated, "Even if a group member is in Australia, they will still be able to collaborate with the team." This will be beneficial further in the semester when our schedules keep the entire team from being able to meet in person. As with most good things there is the bad part. For our situation, that problem is the "vandalizing" of Wikipedia pages. Wikipedia is a free online encyclopedia that can be edited by anyone on the internet. This is an issue for us when it comes to submitting our weekly reports. Anyone with malicious intent could go to our page and edit our work to provide incorrect results or even just vandalize it with inappropriate content as seen by an example in class. To counteract this issue, we will be submitting the archived version of our report. Once a page on Wikipedia has been archived, that version can not be changed. The following versions might have been edited, but the archived copy will remain the same.

Trusses, Beams, Frames
Trusses are structural systems consisting of long slender members. The members are arranged so that they are loaded axially and do not experience bending or shear forces. For more information on trusses, please refer to the Truss article on Wikipedia.

Global Picture
Before trusses are analyzed with the Matrix Method or any other method, it is important to develop a standardized system for labeling and drawing the systems. First, the truss system is drawn and the components are labeled. Global nodes are labeled with a number circumscribed in a circle and elements are labeled with a number circumscribed in a triangle.

Global Free Body Diagram
The next step in the process is to draw the global free body diagram (FBD). The global FBD is a diagram of the whole structure that shows the applied forces, as well as the reaction forces. The FBD includes known and reactionary forces represented by arrows. The reaction forces are labeled Rna, where a denotes the direction of the force in reference to the global coordinate system and n represents the global node number. For example, R1X represents the reaction force R applied along the X axis at node 1.

Element Free Body Diagrams
After the global FBD is created, an element FBD must be created. The element FBD is a diagram of each element of the whole structure that includes known, reaction, and internal forces applied to that element. The elemental diagram is labeled with the element number as stated above. Each node of the element is labeled with its global node number as stated above, as well as a local node number that is a number circumscribed in a square. Each internal force of the element is labeled fi(e), where i is the force number and e is the element number. Force numbers are assigned first by local node number and then orientation related to the coordinate system. For example, a force f1(1) is the first degree of freedom (DOF) in element one.

Force-Displacement Relation
The next "big step" in applying the matrix method to a truss problem is to develop a relation between the applied force and the resulting displacement. The deformation of an elastic truss member can be modeled as the deformation of a spring. Therefore, mass-spring relations will be examined.

Spring with One End Constrained
First, consider a one dimensional spring with one end fixed. The force-displacement relation for this system is represented by the equation $$f=kd$$, where f is the force, d is the displacement, and k is the spring constant that it a property of the spring. This relation shows that there is linear relationship between the applied load and the resulting displacement.

Spring with Both Ends Free
Another spring system to consider is a one dimensional spring with both ends free. This system has two degrees of freedom so there is a system of two equations that govern the relationship between the applied force and the resulting displacement. Consider the spring system shown in the figure. Force 1, F1, acts at node 1, causing a displacement, d1 and force 2 acts at node 2, resulting in a displacement d2. The equations of motion for this system are developed for two cases. For the first case, consider an observer sitting on node 1. The resulting equation of motion is $$f_2 = k(d_2-d_1)$$. The second case is when an observer sits on node 2, resulting in the equation $$f_1 = k(d_1-d_2)$$. Therefore, the resulting force-displacement relation for this system is represented by the system of equations below.

$$\begin{bmatrix} F_1 \\ F_2 \\ \end{bmatrix} $$ = $$\begin{bmatrix} k & -k \\ -k & k \\ \end{bmatrix} $$ $$\begin{bmatrix} d_1 \\ d_2 \\ \end{bmatrix} $$

Steps to solve simple truss system described previously
1. Global picture

The global picture consists of a diagram of the entire structure detailing all of the nodes and nodal forces relevant to the problem. A global coordinate system is selected. At the structure level there are two components to be concerned about. These are the global degrees of freedom, or displacement degrees of freedom, and the global forces. Both are divided into known parts and unknown parts. In the previous truss example, the known parts consist of the applied force and the fixed degrees of freedom for nodes 1 and 3. These two nodes are fixed, therefore the degrees of freedom for each is constrained to zero. The unknown part is solved using the Finite Element Method (FEM). FEM is a numerical method used to obtain solutions to ordinary and partial differential equations. In the previous truss system, the reactions forces are unknown.

2. Element picture

An element picture is made of each individual part of the structure. These elements are considered "free" and a free-body diagram with a local coordinate system is made for each. Included in the element picture are both the element degrees of freedom and the element forces. These can be described either in the global coordinate system or the local coordinate system.
 * Element degrees of freedom: $$d_1,\ d_2,\ d_3,\ d_4,$$
 * Element forces: $$f_1,\ f_2,\ f_3,\ f_4$$

3. Global force-displacement relationship

The global force-displacement relationship is shown through matrices. These matrices consist of:
 * Element stiffness matrix in global coordinates ($$k$$) . The dimensions of this matrix are $$n$$ x $$n$$ where $$n$$ is the number of both known and unknown displacement degrees of freedom.
 * Element displacement matrix in global coordinates ( $$d$$) . The dimensions of this matrix are $$n$$ x $$1$$.
 * Element force matrix in global coordinates ($$f$$) . The dimensions of this matrix are $$n$$ x $$1$$.

To assemble the element stiffness matrices, the element displacement matrices, and the element force matrices into global force-displacement relationship matrix multiplication is applied in the form of:

 * $$kd= f$$

This is considered a free-free system where $$k$$ is singular, meaning $$k$$ is not invertible, or $$k^{-1}$$ does not exist.

4. Elimination of known degrees of freedom

In order to make the method more manageable, the global force-displacement relationship is reduced by eliminating the known degrees of freedom. This makes the stiffness matrix non-singular, or invertible. The form is:

 * $$KD= F$$

$$K$$ is the new stiffness matrix with dimensions $$m$$ x $$m$$, where $$m$$ is the number of unknown displacement degrees of freedom. Therefore, $$m$$ < $$n$$. Similarly, $$D$$ is the new displacement matrix with dimensions $$m$$ x $$1$$ and $$F$$ is the new force matrix with dimensions $$m$$ x $$1$$.

To compute the displacement, the equation is rearranged:

 * $$D = K^{-1}F$$

5. Compute element forces

Solving for $$D$$ above, the element stresses can now be determined.

6. Compute reactions

From the element stresses, the reaction forces can be determined.

Example: Truss problem

 Data

Element length:

L(1) = 4

L(2) = 2

Young's Modulus:

E(1) = 3

E(2) = 5

Cross section area:

A(1) = 1

A(2) = 2

Inclination angle:

θ(1) = 30°

θ(2) = -45°

1. Global picture

Global degrees of freedom: $$d_1,\ d_2,\ d_3,\ d_4,\ d_5,\ d_6$$

n = global node number

Note: When numbering the displacement degrees of freedom, it is useful to follow the order of the global node number. For each individual node, number the displacement degrees of freedom in the order of the global coordinate axes.

Global forces: $$f_1,\ f_2,\ f_3,\ f_4,\ f_5,\ f_6$$

Global force matrix

 * $$\begin{Bmatrix} F_1 \\ F_2 \\ F_3 \\ F_4 \\ F_5 \\ F_6 \end{Bmatrix}$$

Global stiffness matrix

 * $$\begin{bmatrix}

K_{11}&\cdots&\cdots&\cdots&\cdots&\cdots \\ \cdots&\cdots&\cdots&\cdots&\cdots&\cdots \\ \cdots&\cdots&\cdots&\cdots&\cdots&\cdots \\ \cdots&\cdots&\cdots&\cdots&\cdots&\cdots \\ \cdots&\cdots&\cdots&\cdots&\cdots&\cdots \\ \cdots&\cdots&\cdots&\cdots&\cdots&K_{66} \\ \end{bmatrix} $$

Global displacement matrix

 * $$\begin{Bmatrix} d_1 \\ d_2 \\ d_3 \\ d_4 \\ d_5 \\ d_6 \end{Bmatrix}$$

To solve for the unknowns, the matrices are arranged according to the equation above.

 * $$\begin{Bmatrix} F_1 \\ F_2 \\ F_3 \\ F_4 \\ F_5 \\ F_6 \end{Bmatrix} = \begin{bmatrix}

K_{11}&\cdots&\cdots&\cdots&\cdots&\cdots \\ \cdots&\cdots&\cdots&\cdots&\cdots&\cdots \\ \cdots&\cdots&\cdots&\cdots&\cdots&\cdots \\ \cdots&\cdots&\cdots&\cdots&\cdots&\cdots \\ \cdots&\cdots&\cdots&\cdots&\cdots&\cdots \\ \cdots&\cdots&\cdots&\cdots&\cdots&K_{66} \\ \end{bmatrix} \begin{Bmatrix} d_1 \\ d_2 \\ d_3 \\ d_4 \\ d_5 \\ d_6 \end{Bmatrix}$$

Matlab Tutorial
Matlab Primer, University of Florida, 3rd Edition by Kermit Sigmon

MatLab is developed by the MathWorks Company. Below is their link for further information outside the scope of this document.

http://www.mathworks.com/

About MatLab
MatLab is a matrix-based enginering computation program that is user-interactive. By using matrices to enter and solve complex equations, MatLab can do in a fraction of the time it would take older programming languages to do. When dealing with finite element analysis, depending on the number of nodes, there can be numerous partial differential equations that need to be solved.

Starting Matlab
MatLab can be started 2 ways. The first is using the command window and typing run/matlab. The second is through the start menu >> programs >> matlab.

Matrices
MatLab works solely with matrices to handle all computations. These matrices can range in size from 1 by 1 all the way up to thousands of elements. Entering a matrix into matlab is easy and is done with the following notation: A = [1 2 3; 4 5 6; 7 8 9] This creates a 3 by 3 matrix with the values shown and a new row started after each semi-colon. Additionally, matrices do not have to be entered with just real numbers. Complex numbers (i) can also be entered into matrices. For example: A = [1+3i; 2-4i; 3]

When it comes to large matrices, it may be very time consuming to enter in all the matrix values by hand. In such a case, the values can be read in from an ASCII file. As long as the file has a rectangular matrix in it, simply type: >> load numbers.ext This will load the matrix in the file numbers into the workspace.

Matrix Operations:
MatLab uses the same basic mathematical operators plus a few additional ones. Below are the operators: * + addition * - subtraction * * muntiplication * / right division * \ left division * ^ power * ` conjugate The left division operator is special in that it allows for division of matrices that are not square. If you use the right division operator, MatLab will give an error saying that the matrices are not square.

Statements, Expressions, and Variables
Matlab is an expression language meaning the user types an expression, usually composed of variables, functions, and operators, and Matlab interprets it and performs the expression. After the expression is evaluated, the output is put into a matrix. If no matrix name is specified by the user, matlab puts the result in a matrix called ans. Normally after each expression is evaluated, Matlab outputs the result of the expression. To supress(hold) the output, put a semi-colon at the end of the expression. Variables can have virtually any name you choose, as long as it is not a reserved variable or function name. Matlab is case sensitive but has no restriction on variable name length. Some Helpful Commands: * whos - displays a list of variables currently in the workspace Also, there are come commands that can be used to help the user build matrices Matrix Building Commands: * eye - identity matrix * zeros - matrix of zeros * ones - matrix of ones Each of the commands needs to be followed by the following notation: (# of rows, # on columns) Example: zeros(3,3) yeilds a square 3 by 3 matrix of zeros.

Loops
Loops are used to repeat a series of operations until a certain condition is met. Below are the two types of loops that are used frequently in Matlab.

For
For loops execute continually once a condition is met up until the condition is no longer met.

function problem_1_37

% This function will plot 3 conditional functions

dx = 0.1; x = [-2:dx:6]; for k = 1:length(x) if x(k) >= 5 y(k) = 10*(x(k)-5) + 1; elseif x(k) >= -1 y(k) = 2 + cos(pi*x(k)); else y(k) = exp(x(k)+1); end end

plot(x,y), xlabel('Time x (seconds)'), ylabel('Height y (kilometers)')

While
While loops are similar to for loops except for while loops are guaranteed to execute at least once. It will execute the statement, then check it against the while statement. If it is true, it will run back through the loop. If it is false, it will exit the loop.

function e = macheps

% This function determines the smallest number that can be added to 1 in % Matlab and still get a number greater than 1.

e = 1; while (1+e >1) e = (e/2); end

e = e*2;

'''In addition to loops, one can have an operation performed only IF a condition is met. That type of operator is an if statement. Below is an example of an IF statement.'''

function y = problem_1_36(x) % This function will check conditional statements to evaluate a function given different input values. if x >= 5 y = 10*(x-5)+1; elseif x >= -1 y = 2+cos(pi*x); else y = exp(x+1); end

NOTE: All code was taken from previous homework assignments in Numerical Methods

Relations
The relational operators are used to compare the value of a variable to another variable or to a constant value. Below are some common relational operators: * == equal * ~= not equal * < less than * > greater than * >= greater than or equal to    * <= less than or equal to     * & and * | or    * ~ not

Relational operators are always used in for, while,and if statements to see when to execute, or stop executing expressions within the loop.

Scalar Functions
Certain matlab functions operate on scalar numbers and when applied to a matrix, they act on 1 matrix element at a time. Some scalar Matlab functions are listed below: sin, cos, tan, asin, acos, atan exp, log, rem, abs, sqrt, sin, round, floor, ceil

Vector Functions
Vector functions act on a vector matrix column by column to produce a row matrix as the solution. These functions can only be used when the matrix is larger than 1 by 1. Below are some examples of vector functions: max, min, sort, sum, prod, median, mean, std, any, all

Command Line Editing
The command line in Matlab is denoted by 2 arrows (>>). The command line is the line that the user uses to type in commands. To keep the user from having to re-enter previously used commands, Matlab allows the user to scroll through the old commands used during the session by hitting the up-arrow and down-arrow keys.

Also, it is important to remember that instead of entering a series of commands by hand every time, create an m-file to execute. This will save loads of time and needless typing.

M-Files
M-files are special Matlab files that have a .m extension. These files contain a series of statements that can be exectued at one time, thus saving the user time from having to exectue expressions 1 at a time. There are two types of m-files, script files and function files.

Script files contain a series of normal Matlab scripts. They are traditionally used to enter large amounts of data into matrices. Any variable or matrix is global, meaning the parameter can be used in any other portion of the workspace.

Function files are a type of m-file that allow users to essentially create new Matlab functions. The functions can be specific to a particuar problem that the user is working on.

Text Strings
You can assign text strings to variables by using single quotes around the text to be entered into the variable. Additionally, text can be displayed to the screen using the disp function. The error function also outputs a message to the screen but also aborts the code in an m-file when it is executed. Below are examples of the 2 functions: disp('Hello World') error('Warning!!!')

Managing M-files
The !-feature allows for editing or printing to be done without having to exit MATLAB. Using the command !ed will let the named file be edited and then returned to MATLAB where you left off. Many commands control the working directory. Pwd returns the name of the present working directory. Cd changes the working directory. Dir or ls lists the contents of the working directory, but what lists only the M-files. Files named matlab are accessible from any directory. M-files have to be in a directory MATLAB can excess.

Comparing efficiency of an algorithms: flops, tic and toc
The efficiency of an algorithm can be measured by the number of floating point operations (flops) performed and the elapsed time. The total number of flops performed is kept track of by the function flops. This number can be reset by the command flops(0). Tic and toc are stopwatch times that can return the elapsed time. Tic starts the timer and toc returns the elapsed time.

Output format
The output on display is controlled by a variety of format commands, but the computations are done in double precision. Format compact suppresses most of the blank lines so more information can be added on the screen, while format loose returns the display back.

Hardcopy
The diary command causes the next code written to be written on the name diskfile until the command diary off is given. After this is done the file can be edited and printed on the local system.

Graphics
MATLAB can create planar plots of curves, 3-D plots of curves, 3-D mesh surface plots, and 3-D faceted surface plots.

Planar plots
The command plot produces a liner x-y plot. Plot(x,y) will draw an x-y plot if the vectors x and y are the same length. There can be many different graphic figures, but only one at a time will be call “current.” The command figure(1) will make figure 1 the current figure if it was not before, and gcf will return the number of which figure is current. An example of a linear x-y plot is shown below.

 * x = -1.5:.01:1.5; y = exp(-x.^2); plot(x,y)

Fplot is used to plot the graph of a function. Plots can be made of parametrically defined curves as well.

 * t=0:.001:2*pi; x=cos(3*t); y=sin(2*t); plotx,y)

All graphs can be given titles, labeled axes, and text within the graph.

 * Title: graph title

 * xlabel: x-axis label

 * ylabel: y-axis label

 * gtext: place text on graph using mouse

 * text: place text at specified coordinates

Axes are auto-scaled, but this can be overwritten with the command axis. This command should always be given after the plot command. Multiple plots can also be displayed on a single graph.

 * x = 0: .01:2*pi; y1 = sin(x); y2 = sin(2*x); y3 = sin(4*x); plot(x,y1,x,y2,x,y3)

This can also be done with the hold command. It freezes the current on screen graphics so other plots can be put on the same graph, but the axes may become rescaled. Colors and line types can be added to a plot as well. The code plot(x,y,’r--‘) shows a red dashed line.

Graphics hardcopy
The command print is the easiest way to send a high-resolution copy of the current graphics figure to the default printer. The printopt M-file identifies the default setting used by the print command. Print filename saves the figure to the filename given in the default format. If there is no extension in the filename then the default file format will be assigned. A file will be overwritten if the filename already exists. The option –append will append the current graphics figure into the existing file so more than one figure can be saved in each file. The command code is print –append filename. This command can be overwritten with the option –f3.

3-D line plots
Plot3 makes a curve in three dimensions. If x, y, and z are defined vectors of the same size, then plot3(x,y,z) will create a 3-D curve passing through the coordinates of the respective elements. A zlabel can still be added to this plot. The helix shown below was created using the code:

 * t = .01: .01:20*pi; x = cos(t); y = sin(t); z = t.^3; plot3(x,y,z)

3-D mesh and surface plots
A three dimensional wire mesh surface plot with the elements of matrix z is created with the command mesh(z). The function meshgrid allows for a graph to be drawn over a rectangle by allowing matrices to be created. Surface plots are created using surf(z).

z=e-x^2-y^2 can be drawn over the square [-2,2]x[-2,2] with the following codes:
 * xx = -2: .2:2;

 * yy = xx;

 * [x,y] = meshgrid(xx,yy);

 * z = exp(-x.^2 – y.^2);

 * mesh(z)

Surf(z) of the same code is shown as well.

There are three different settings for the color shading of surfaces: faceted, interpolated, and flat. The command shading should be done after the surf command. Different color profiles can be produced by the command colormap.

Handle Graphics
MATLAB contains many different graphic functions. The user is able to control almost all of the graphics allowing for detailed plots to be made.

Sparse Matrix Computations
MATLAB provides the ability to save storage space, so larger problems can be solved, by only storing the nonzero elements of the matrix. Matlab has the storage mode full, the default, and sparse. The sparse matrix is stored as a linear array of its nonzero elements. The output of a matrix in sparse mode is a list of nonzero entries and their row and column indices. A matrix is sparse if any constituent block is sparse.

Contributing Team Members
The following students contributed to this report:

Daniel DiDomenico Eml4500.f08.ateam.didomenico 20:13, 16 September 2008 (UTC)

Matthew Philbin Eml4500.f08.Ateam.philbin 24:37, 16 September 2008 (UTC)

Mark Barry Eml4500.f08.Ateam.barry 18:23, 18 September 2008 (UTC)

Sean Miller EML4500.f08.Ateam.Miller 18:43, 18 September 2008 (UTC)

Jil Paquette --Eml4500.f08.Ateam.paquette 00:12, 19 September 2008 (UTC)

Ian Slevinski Eml4500.f08.Ateam.Slevinski  00:53, 19 September 2008 (UTC)

title: User:Eml4500.f08.ateam/hw 1


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User talk:Eas4200c.f08.aero.barcia

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Trusses, Matrix Method

Trusses are structural systems consisting of long slender members. The members are arranged so that they are loaded axially and do not experience bending or shear forces. For more information on trusses, please refer to the Truss article on Wikipedia.

Before trusses are analyzed with the Matrix Method or any other method, it is important to develop a standardized system for labeling and drawing the systems. First, the truss system is drawn and the components are labeled. Global nodes are labeled with a number circumscribed in a circle and elements are labeled with a number circumscribed in a triangle.

The next step in the process is to draw the global free body diagram (FBD). The global FBD is a diagram of the whole structure that shows the applied forces, as well as the reaction forces. The FBD includes known and reactionary forces represented by arrows. The reaction forces are labeled Rna, where a denotes the direction of the force in reference to the global coordinate system and n represents the global node number. For example, R1X represents the reaction force R applied along the X axis at node 1.

After the global FBD is created, an element FBD must be created. The element FBD is a diagram of each element of the whole structure that includes known, reaction, and internal forces applied to that element. The elemental diagram is labeled with the element number as stated above. Each node of the element is labeled with its global node number as stated above, as well as a local node number that is a number circumscribed in a square. Each internal force of the element is labeled fi(e), where i is the force number and e is the element number. Force numbers are assigned first by local node number and then orientation related to the coordinate system. For example, a force f1(1) is the first degree of freedom (DOF) in element one. SEE ILLUSTRATION

The next "big step" in applying the matrix method to a truss problem is to develop a relation between the applied force and the resulting displacement. The deformation of an elastic truss member can be modeled as the deformation of a spring. Therefore, mass-spring relations will be examined.

First, consider a one dimensional spring with one end fixed. The force-displacement relation for this system is represented by the equation $$f=kd$$, where f is the force, d is the displacement, and k is the spring constant that it a property of the spring. This relation shows that there is linear relationship between the applied load and the resulting displacement. SEE ILLUSTRATION

Another spring system to consider is a one dimensional spring with both ends free. This system has two degrees of freedom so there is a system of two equations that govern the relationship between the applied force and the resulting displacement. Consider the spring system shown in the figure. Force 1, F1, acts at node 1, causing a displacement, d1 and force 2 acts at node 2, resulting in a displacement d2. The equations of motion for this system are developed for two cases. For the first case, consider an observer sitting on node 1. The resulting equation of motion is $$f_2 = k(d_2-d_1)$$. The second case is when an observer sits on node 2, resulting in the equation $$f_1 = k(d_1-d_2)$$. Therefore, the resulting force-displacement relation for this system is represented by the system of equations below.

$$\begin{bmatrix} F_1 \\ F_2 \\ \end{bmatrix} $$ = $$\begin{bmatrix} k & -k \\ -k & k \\ \end{bmatrix} $$ $$\begin{bmatrix} d_1 \\ d_2 \\ \end{bmatrix} $$

title: Trusses, Matrix Method


text:
User:Bele

Just interested in information & learning, being a lifelong learner myself. Don't know if I'll have enough time to contribute much stuff, but I'll try.

Bele 06:10, 8 September 2008 (UTC)

title: User:Bele


text:
User:Leag

I'm sysop on french wikipedia. You can see my user page here.

fr:Utilisateur:Leag

title: User:Leag


text:
User talk:Leag

You can contact me here.

title: User talk:Leag


text:
User:Crochet.david/Editcounter

Month-Graph:yes Namespacesum-Graph:yes Weekday-Graph:yes Hour-Graph:yes Usercontributions:yes

title: User:Crochet.david/Editcounter


text:
User:Eml4500.f08.bike.pataky

Mon
A frame element is defined as an element with a truss or bar element and a beam element. This gives the frame element both axial and transverse deformations.

The frame shown above has a rigid connection between its two elements. This means that the angle between the two elements will stay constant after deformation.

Next the frame will be broken down into its two elements. The free body diagrams for these elements are shown below.

In the free body diagram above, a new degree of freedom is introduced for each node. This new degree of freedom is the moment around the node and adds two degrees of freedom. This brings the total to six degrees of freedom for element 1.

As with element 1, element 2 has the two new degrees of freedom.

In general, the degrees of freedom shown above for each element would be the general forces previously used. However, the new rotational degrees of freedom, $$d_3^{(e)}$$ and $$d_6^{(e)}$$, are now the bending moments, $$f_3^{(e)}$$ and $$f_6^{(e)}$$ instead of forces.

The global diagram contains nine degrees of freedom now. All three nodes now contain bending moments just like the elements do.

The elemental stiffness matrices $$\textbf{k}^{(e)}_{6x6}$$ where e = 1,2 are used to make the global stiffness matrix $$\textbf{K}_{9x9}$$. For the current frame element, the global matrix would look like the picture shown below.

title: User:Eml4500.f08.bike.pataky


text:
User talk:Jum21

Thank you for experimenting with Wikiversity. Your test was recorded, and it has been reverted or removed. Please use the sandbox for further testing. You may wish to learn more about what Wikiversity is to help you contribute. Consider registering if you are keen on our project. --McCormack 15:03, 8 September 2008 (UTC)

title: User talk:Jum21


text:
User talk:Eml4500.f08.bike.pataky

 Hello Eml4500.f08.bike.pataky, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random random project.
 * Browse everything on Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! --JWSchmidt 15:15, 8 September 2008 (UTC)

title: User talk:Eml4500.f08.bike.pataky


text:
Category:University of Florida team account

Fall 2008 engineering project from University of Florida. Apparent project leader: User:Eml4500.f08.


text: Category:University of Florida

title: Category:University of Florida


text:
User:Eml4500.f08.ateam.shah/Week 1

Week 1: Brief summary of Lectures 1-3

 * The finite element analysis method was first developed in 1943 by R. Courant. Courant modified and utilized the Ritz method of numerical analysis and variational calculus to determine an approximate solution to partial differential equations as well as integral equations. The solution is achieved by elimination the differential equation or creating several ordinary differential equations from a partial differential equation, which can then be solved using either the Euler’s method or the Runge Kutta method. The FEA method can be used in a wide variety of applications such as structural analysis, vibrational analysis, etc.

 * The Finite Element Analysis and Design class is offered at the University of Florida on Monday, Wednesday and Friday at 3:00 PM. The class is taught by Professor Loc Vu-Quoc.

 * The first lecture of the class was held on August 25, 2008. The lecture introduced the concept of the Finite Element Method and introduced student to the structure of the class. The class was instructed to divide into groups of 5-6. These groups acted as a single entity and represented those student efforts on HW, presentations and assignments. The class was also introduced to Wikipedia, a free website that can be used by students as a portal for posting assignments, comments, questions and problems. The students were instructed to create accounts on Wikipedia, with the following format: eml4500.f08.teamname.lastname. The accounts are created so that students would be able to communicate easily and also provide a forum to post HW assigned by the professor.

 * The following two lectures were a brief introduction into the workings of Wikipedia. The professor demonstrated the working of Wikipedia and his reason for using Wikipedia instead of university provided software such as E-learning. The professor provided a brief commentary on how to post articles and HW on the website and how to communicate with your fellow group members. MATLAB was also instructed to be one of the most important software programs to be used through out the class. The professor required a basic familiarity with MATLAB for all students. HW was assigned which included a brief tutorial on MATLAB and its uses.

 * The course website along with the syllabus were mentioned during these lectures. The syllabus listed the exam dates and the break up of the grading for the class. Strict policy rules were explicitly mentioned which provided for no make up exams and late HW.

title: User:Eml4500.f08.ateam.shah/Week 1


text:
User:Eml4500.f08.ateam.nobrega/week 1

8/25/08
Dr. Vu-Quoc reviewed the class syllabus in detail. The topics that will be learned in the eml4500 include matrix algebra and its application to finite element solution, linear stability analysis of beam/truss structures, and construction of a FE code. He also intoduced the concept of collaborative learning; we will be working in teams throughout the semester to complete homework reports. Each team is responsible for summarizing each leacture and identifing confusing or ambiguous points discussed. All of the teams' input will be used by Dr. Vo-Quoc as feedback and address any underliying issues if necessary. Each student is reponsible for creating their own wikipedia account to complete their homework. I am confused about how to submit or post information on my wikipage. I am unfamiliar with the interface and controls.

8/27/08
Wikipedia is a free open-content encyclopedia. The articles are written by users and can be edited by anyone with a wikipedia account. For this reason, vandalism is a concern. However, if a students wokr is vandalized the previous version of the article is available in the history and can be restored quite easily. Also, an archived copy of the final collective assignment must be created and submitted to the professor and teachong assistants as to avoid problems with vandalism. Groups working on similar research can collaborate across the globe using the various wikipedia applications.

title: User:Eml4500.f08.ateam.nobrega/week 1


text:
User:EAS4200C.F08.AERO.CLK04D

11/7

Example of $$\overrightarrow{t}$$ in aircraft

$$\displaystyle \overrightarrow{n} = n_y\overrightarrow{j} + n_z\overrightarrow{k}$$

$$\displaystyle ||n||= 1 $$

$$\overrightarrow{n} =$$ unit normal vector

$$ \displaystyle dz=ds cos \theta | n_y=cos\theta$$

$$\displaystyle dy=ds sin \theta | n_y=cos\theta$$

$$\displaystyle \sum F_y=0= -\sigma_{yy}*(dz*1)$$ (Where 1 is the unit depth along the x-dir)

 * $$\displaystyle -\sigma_{yz}*(dz*1) + t_y(ds*1)$$

$$\displaystyle 0 = -\sigma_{yy}*ds*n_y - \sigma_{yz}*ds*n_y + t_y*ds$$

Canceling ds we arrive at the following equation:

$$\displaystyle \sigma_{yy} n_y + \sigma_{yz}n_z = t_y $$ (1)

Note: $$[t_y] = \frac{F}{L^3}$$ => Units of force/area

$$\overrightarrow{t} =$$ traction vector (distributed surface force)

$$\displaystyle [t_y] = [\sigma] $$

$$\displaystyle t_z = \sigma_{yz}*n_y + \sigma_{zz}*n_z $$ (2)

Combining (1) and (2)

$$\displaystyle \begin{bmatrix} t_y \\ t_z \end{bmatrix}\, = \begin{bmatrix} \sigma_{yy} & \sigma_{yz} \\ \sigma_{zy} & \sigma_{zz} \end{bmatrix}\, \begin{bmatrix} n_y \\ n_z \end{bmatrix}\,$$ (3)

(3) can be generalized to the 3 dimensional case:

$$\begin{bmatrix} t_1 \\ t_2 \\ t_3 \end{bmatrix}\, = \begin{bmatrix} \sigma_{11} & \sigma_{12} & \sigma_{13} \\ \sigma_{21} & \sigma_{22} & \sigma_{23} \\ \sigma_{31} & \sigma_{32} & \sigma_{33} \end{bmatrix}\, \begin{bmatrix} n_1 \\ n_2 \\ n_3 \end{bmatrix}\,$$ (4)

$$\displaystyle t_i = \sum_{j=1}^{3} \sigma_{ij}n_{j} $$

11/10

For Homework Watch Ruth Goldberg Video Section 7.7 (Plate buckling)

Torsional Analysis Continued:

Case 1: Thin walled cross section (closed) e.g. NACA airfoil

Compare to fig 3.11

Case 2: Solid cross section

Now we consider the case of uniform bar with a solid, circular cross section. Refer to road map, expression for T and J in terms of $$\displaystyle \phi$$

$$ T = 2\int\int\limits_A\phi dA$$

$$J = \frac{-4}{\nabla^2\phi}\int\int\limits_A \phi dl $$

$$\phi(y,z) = C(\frac{y^2}{a^2} + \frac{z^2}{b^2} - 1)$$

(Note: b = 0 for a circle)

Revisiting the road map,

Kinematic assumption (Compare to pg. 70)

4 - Stain components (Compare to pg. 70)

4 zero stress components (Compare to pg. 70)

$$ \epsilon - \sigma $$ relationship (2 forms: tensorial or engineering)

Rewrite $$ \epsilon - \sigma $$ relation

$$ \varepsilon_{ij} = \begin{bmatrix} {\underline A}_{3x3} & {0}_{3x3}\\ {0}_{3x3} & {\underline B}_{3x3}\\ \end{bmatrix} \sigma_{ij} $$

Let $$ \begin{bmatrix} {\underline A}_{3x3} & {0}_{3x3}\\ {0}_{3x3} & {\underline B}_{3x3}\\ \end{bmatrix} = \underline C $$

An advantage of using this form is that $$\underline C$$ can be easily inverted.

$$ \sigma_{ij} = \begin{bmatrix} {\underline A}^{-1} & \underline 0\\ \underline 0 & {\underline B}^{-1}\\ \end{bmatrix} \varepsilon_{ij} $$

$$ \begin{bmatrix} {\underline A}^{-1} & \underline 0\\ \underline 0 & {\underline B}^{-1}\\ \end{bmatrix} = \underline {C}^{-1} $$

When the two matrices are dotted into each other we get the identity matrix.

$$\underline C \underline {C}^{-1} = \underline I$$

Goal: prove 0 stress components.

$$\sigma _{xx}, \sigma _{yy} , \sigma _{zz}, \tau_{yz} = 0$$

$$\epsilon _{xx}, \varepsilon_{yy} , \varepsilon_{zz}, \gamma_{yz} = 0$$

This implies,

$$ \sigma_{ij} = \begin{bmatrix} {\underline A}^{-1} & \underline 0\\ \underline 0 & {\underline B}^{-1}\\ \end{bmatrix} \begin{bmatrix} 0\\ 0\\ 0\\ 0\\ \varepsilon_{31}\\ \varepsilon_{12}\\ \end{bmatrix} $$

(Note, using numerical notation as introduced in MIT OCW assigned reading)

$$\sigma _{11}, \sigma _{22} , \sigma _{33} = 0$$

$$\sigma _{23} = 2G\varepsilon_{23} = \frac{1}{2}\gamma_{23} = \frac{1}{2}\gamma_{yz}$$

$$\sigma _{31} = 2G\varepsilon_{31}$$

Roadmap step C:

Implicit reading: Section 2.4 in book (Non-Uniform stress field)

Consider 1-D case as a model:

Multi Cell Airfoil (Sec. 3.6)

Recall:

$$ T = 2q\bar A $$

$$\theta = \frac{1}{G\bar A} \oint\frac{{q}_{i}}{{t}_{i}}ds $$

Specific example first (generalization later)

Compare this example with page 94 in textbook.

$$ T = 2\sum_{i=1}^1 {q}_{i} \bar {A}_{i}  $$

Find theta as a function of T and J(torsional constant)

$$T = {T}_{1} + {T}_{2} = 2{q}_{1}\bar {A}_{1} + 2{q}_{2} \bar {A}_{2} $$   (1)

$$\bar {A}_{1} = ac, \bar {A}_{1} = bc$$

$${\theta}_{1} = \frac{1}{G\bar {A}_{1}} \oint\frac{{q}_{1}}{{t}_{1}}ds $$

$$\oint\frac{{q}_{1}}{{t}_{1}}ds$$ q and t can be thought of as functions of (s), their curvilinear coordinates

therefore we can replace by discrete summation,

$${\theta}_{1} = \frac{1}{G\bar {A}_{1}} 2\frac{{q}_{1}a}{{t}_{1}} + \frac{{q}_{1}c}{{t}_{1}} + \frac{({q}_{1} - {q}_{2})c}{{t}_{12}}$$  (3)

$${\theta}_{2} = \frac{1}{G\bar {A}_{2}} 2\frac{{q}_{2}a}{{t}_{2}} + \frac{{q}_{2}c}{{t}_{2}} + \frac{({q}_{2} - {q}_{1})c}{{t}_{12}}$$  (4)

Since they are rigidly connect cells 1 and 2 must have the same rate of twist angle.

Therefore,

$${\theta}_{1} = {\theta}_{2}$$  (2)

Think of q1 and q2 as 2 unknowns. Equations (1) and (2) are the 2 equations for the 2 unknowns (q1 and q2) with T being a variable. Use (1) and (2) to find an expression for q1 and q2 in terms of T.

$${q}_{1} = {\beta}_{1}T {q}_{2} = {\beta}_{2}T$$

Next use the experssion (3) or (4) to find the expression between theta and T

$$\theta = {\theta}_{1} = {\theta}_{2} = \frac{T}{GJ} $$

Finally from this equation the torsional constant J can be deduced.

Homework #3
NACA 4 digit airfoil series

For homework number four teams will develop Matlab code to plot and analyze a NACA airfoil. This homework assignment will be an ongoing project with the ultimate going of developing Matlab code to do structural analysis of a wing. For the first assignment teams should:

 * Plot the airfoil.
 * Compute location of the airfoil.
 * Plot the centroid of the airfoil with cross hair.
 * Compute A_bar (from any point in the plane).

The user will input the number of segments and the coordinate of the upper and lower surfaces of the airfoil.

Matlab code is provided to plot a straight line. Teams should modify the code to plot the entire airfoil. Given some wing thickness the code must then find the centroid of the airfoil segment. A summation of the individual segment centroids will yield the overall airfoil centroid.

For each small airfoil the perpendicular distance from the origin (or other point) will need to be calculated.

This distance is assigned the variable $$\rho$$ as shown in the illustration below for vector $$\overrightarrow{PQ}$$

Note,

$$ \overrightarrow{dT} = \overrightarrow{r} $$ $$ \mathbf{x}$$ $$\overrightarrow{dF} = q\overrightarrow{r}$$ $$ \mathbf{x}$$ $$\overrightarrow{PQ} = (2dA)\overrightarrow{i}$$

$$ T = \oint_C \,dT $$

$$= q \oint_C \rho \, dl $$

$$=2q \int\limits_A \, dA$$

END OF HOMEWORK 3

Torsion of Uniform, non-circular bars

When a torque is applied to a uniform bar it leads to warping along the longitudinal axis of the bar. The amount of warping depends on the applied torque.

Warping - axial displacement along longitudinal axis of a point on the deformed (rotated) cross section.

Virtual Displacement:

Uy = y - component of displacement vector $$\overrightarrow{PP'}$$

Uz = z - component of $$\overrightarrow{PP'}$$

Note: Horizontal Bar

Class Notes - 9/3/08
$$T_{max}= (2t\tau_{allow})(\frac{L}{4})^2 T_{max}$$

$$ = \frac{1}{8}tL^2\tau_{allow} = M_{max} $$

$$M_{max} = \frac{tL^2}{16} \sigma_{allow}$$

$$ => \sigma_{allow} = \frac{16{M}_{max}}{tL^2} $$

Recall: $$f(b) = \frac{I(b)}{b}$$

$$f(b) = \frac{I}{b} = \frac{3L - 4b}{12} $$

$$= \frac{tL^2}{24} $$ (Recall that b = $$\frac{L}{4}$$)

$$\sigma_{max} = \frac{{M}_{max}b}{2I} = {M}_{max} \frac{12}{tL^2}$$ = $$\frac{tL^2}{16} \sigma$$allow

Since, $$\sigma_{max} = \frac{12}{16} \sigma_{allowable} < \sigma_{allowable}$$

Case 2 is acceptible

Therefore, the optimal cross section for case two is a square.

=Wednesday=

Weight is a critical component in the design of aircraft structures. In order to minimize weight proper materials must be selected. Ideally materials used in aerospace applications have high stiffness and strength while still remaining light weight.

Stiffness is described by Young's Modulus ($$ \displaystyle E = \displaystyle \epsilon/ \displaystyle\sigma $$) the ratio of stress over strain. A higher Young's modulus corresponds with a higher material stiffness.

NOTE: Young's modulus is valid only in the elastic range (shown below).

The strength of a material is measured by the materials yield stress ($$\displaystyle \sigma$$y). Obviously a higher yield stress corresponds with a stronger material.

A materials toughness represents its ability to resist fracture. This is also know as fracture toughness.

Shown above is a graph of stress versus strain for a typical metal. Note that the graph is not exact and only for conceptual purposes. From the graph it is apparent that the ratio of stress and strain is linear inside the elastic range. Outside of this range the material deforms plasticly until the point of failure. Inside the plastic range a positive slope represents hardening of the material while a negative slope represents softening.

The Figure 1.1 to the right lists various categories of material properties and the materials which possess those properties.

The structure of an aircraft has two primary components:

 * 1) Geometry - Common aircraft geometries include monocoque shell like constructions and semimonocoque stiffened shell like structures.
 * 2) Material - Common aircraft materials include aluminum, titanium, and composite materials.

Below is a picture of the Boeing 787 Dreamliner which illustrates the aircraft's material composition. Composite materials make up fifty percent of the 787's total weight.

Boeing 787 $$Insert formula here$$

title: User:EAS4200C.F08.AERO.CLK04D


text:
User talk:Eas4200c.f08.aero.e

This is my page

 Hello Eas4200c.f08.aero.e, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random random project.
 * Browse everything on Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! --JWSchmidt 16:11, 8 September 2008 (UTC)

title: User talk:Eas4200c.f08.aero.e


text:
User talk:Eas4200c.f08.aero.arena

Hey look, I wrote something on your page. Ha Ha, Ha Ha.

title: User talk:Eas4200c.f08.aero.arena


text:
User talk:Eas4200c.f08.aero.r

Here is something.

title: User talk:Eas4200c.f08.aero.r


text:
User talk:Eml4500.f08.echo.mott/HW1draft

Discussion

 * The sections below the MATLAB section need to be filled with basic information and examples including the matlab input and output. Don't copy examples from the tutorial. Chris 16:11, 8 September 2008 (UTC)

 * We could start moving some stuff from Brad's page to this page. -Chris 20:47, 8 September 2008 (UTC)

 * I added the outline of items for 9/3 lecture. I will try and finish up the details and graphics after class. I will also start doing a few sections for the MATLAB tutorial. -Chris 18:38, 10 September 2008 (UTC)

 * Is anyone having trouble loading Dr. VQ's web page right now? --Charles 02:22, 12 September 2008 (UTC)

 * Yeah, it wasn't working for me. Chris 02:41, 12 September 2008 (UTC)

 * did we loose some pictures in the last update? --Eml4500.f08.echo.sell.cm 15:58, 12 September 2008 (UTC)

 * Check the link on the top of the page. I am leaving stuff that still needs to be done on the draft page and then putting completed stuff on the final page. Chris 16:55, 12 September 2008 (UTC)


text:
User:Eas4200c.f08.aero.E

 This HW report is not in the right format. There should be a separate wiki page for each HW report with a section with the signature of each contributing team member for that HW report. Here you have a section of contributing team members that is not clear to which HW report it applies. See Team Carbon as an example.

Also a team member did not register (create user account), and the work is threatened to be deleted; see the matlab code section for HW5.

Please fix the above problems immediately.

Do NOT remove this comment box; simply add to this comment box what you have done to fix the above problems.

Eml4500.f08 15:10, 11 November 2008 (UTC)

Solutions:

[Lecture: HW 6] --Eas4200c.f08.aero.E 03:14, 20 November 2008 (UTC)

We are also going to start putting a signature on the items we work one.

Team: Aero

EAS 4200C Aerospace Structures

Professor Vu-Quoc

=Book Preface= The book used for the class is Mechanics of Aircraft Structures by C. T. Sun. We will be using the second edition of the text. The text comes with two prefaces: one for the first edition and another one for the second edition (which is a complement for the first preface). On the preface of the first edition, the author explains the basic knowledge that the student must have in order to have a good understanding of the material covered during class. The book is intended for community of junior and senior level aerospace engineering students with prior knowledge of a mechanics of materials course. The material might be covered in a semester at a decent pace.

The author presents the topics of the book influenced by the recent developments in commercial finite codes which have been used lately in structural analysis. Therefore many ad hoc techniques that were used in the past have lost their usefulness in structural analysis. These recent changes have led to put more emphasis in understanding the mechanics of the structure rather than focus on the problem-solving drill. Another aspect that has influenced the writing of the book is the recent attention put on fracture mechanics, since this tool has became the most important tool in estimating aircraft structure damage tolerance and durability. The author believe that this knowledge (that traditionally was a subject directed to graduate students), is very important for the students to get exposed to. Finally, a third subject of influence for the book has been the recent development and study of advanced composite materials, which is a subject in which students must have contact to the characteristics and properties of such materials. The implementation of Finite Element Method brought with it the use of the consistency elasticity approach. Here, multidimensional stresses, strains, and stress-strain relations are emphasized. Also instead of using strain and stress to derive the equations for torsion and bending, displacement is used. The author uses the concept of Griffith stress energy release rate to bring the concepts of fracture mechanics. In which the strain energy release rate can be calculated for simple structures in a basic way. Composite materials are covered in Chapter 1, anisotropic stress-strain relations in Chapter 2, and in Chapter 8 analysis of symmetric laminates of composite materials is covered. Such coverage of composite materials is believed by the author of the book to be sufficient to give the student a basic background, however; the material covered should not be used as a substitute for a course based on entire composite materials. Elastic buckling is presented in Chapter 7, where sub-topics such as buckling of slender bars, the postbuckling concept and buckling of structures composed of thin sheets are briefly covered. Also, since Postbuckling strengths of bars or panels are so commonly used in aircraft structure, its briefly presentation is justified. Finally, the author appreciates Mrs. Marilyn Engel for typing the manuscript and also James Chou and R. Sergio Hasebe for making the drawings. On the second edition preface the author explains that the purpose of the second edition is to correct some typographical errors as well as to introduce new topics. The newly introduced topics are: primary warping, effects of boundary constraints, Saint-Venant's principle, the concept of shear lag, the Timoshenko beam theory, and briefly introduces the effect of plasticity on fracture. Chapters 1-2 remain unmodified. The modifications concentrate on Chapters 3, 4 and 6, and the number of pages are increased by 25 percent. Finally, and as in the first preface; the author expresses his gratitude to the students and colleagues for pointing out errors and making suggestions, and give special thanks to Dr. G. Huang for assisting him in making new drawings.

=Class Notes=

 * Lectures: HW 1

 * Lectures: HW 2

 * Lectures: HW 3
 * Lectures: HW 4

 * Lectures: HW 5

 * [Lectures: HW 6]

 * [Lectures: HW 7]

=Matlab Projects=

 * Matlab 1: NACA Airfoil

 * Matlab 2: Convergence and Multi Cell

 * Matlab 3: Airfoil and Stringers

 * Matlab Projects assigned for HW 6 and 7 will be in their HW botton.

=Matlab-code certification=

I, the undersigned, certify that I can read, understand, and write matlab codes, and can thus contribute effectively to my team.

Eas4200c.f08.aero.r         --Eas4200c.f08.aero.r 15:38, 19 November 2008 (UTC) Eas4200c.f08.aero.arena     --Eas4200c.f08.aero.arena 05:14, 20 November 2008 (UTC) Eas4200c.f08.aero.gabo28   --Eas4200c.f08.aero.gabo28 20:08, 24 October 2008 (UTC) EAS4200C.F08.AERO.CLK04D   --EAS4200C.F08.AERO.CLK04D 03:47, 24 October 2008 (UTC) Eas4200c.f08.aero.E        --17:06, 23 October 2008 (UTC)Eas4200c.f08.aero.E EAS4200C.Fall08.AERO.Watlington.VG --    EAS4200C.Fall08.AERO.Watlington.VG 17:18, 23 October 2008 (UTC)

=Contributing Team Members=

The following students contributed to this report:

"Eas4200c.f08.aero.r"

"Eas4200c.f08.aero.arena"

"Eas4200c.f08.aero.gabo28"

"EAS4200C.F08.AERO.CLK04D"

"Eas4200c.f08.aero.E"

"EAS4200C.Fall08.AERO.Watlington.VG 17:18, 23 October 2008 (UTC)"

title: User:Eas4200c.f08.aero.E


text:
User:Eas4200c.f08.aero.arena

EAS 4200C Main page for Daniel Arena Class notes / Homework

Material Properties
Aircraft design is governed mainly by the limitations imposed by weight. Mass is the primary concern with respect to all areas including fuselage, wings, engines and everything else. Current design is a balance between performance and how much it will cost to lift off the ground.

One method of combating weight is to use as much light weight yet strong material as possible. Cutting edge composite development is a major part aerospace industry, composites can be used instead of metals and provide the same strength.

There are two main factors to consider with respect to aircraft structure and design, the strength and stiffness of the materials used. Material strength can be described and separated into deformations states. First there is elastic deformation then plastic deformation and permanent deformation. If the material is stressed beyond its maximum yield point it will reach rupture stress and fail.

The Modulus of elasticity also known as young’s modulus is the rate of change between strain and stress. Further informations can be found at

http://www.instron.us/wa/resourcecenter/glossaryterm.aspx?ID=99&ref=http://www.google.com/search.

title: User:Eas4200c.f08.aero.arena


text:
User:Eas4200c.f08.aero.r

thsi is the start

title: User:Eas4200c.f08.aero.r


text:
User talk:Eas4200c.f08.aero.gabo28

Testing.

title: User talk:Eas4200c.f08.aero.gabo28


text:
User:Eas4200c.f08.aero.gabo28

=Normal Stress= Is the averaged stress calculated over a cross sectional area of a member in which a force is applied in the axial direction. The figure below represents the force acting in the axial direction of a member.

title: User:Eas4200c.f08.aero.gabo28


text:
User talk:Hajveryprofessional

Hello, welcome around here. I hope we can help you succeed in your learning goals. Below more info about Wikiversity, Erkan Yilmaz uses the Chat (try) 16:39, 8 September 2008 (UTC)

 Hello Hajveryprofessional, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random random project.
 * Browse everything on Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! Erkan Yilmaz uses the Chat (try) 16:39, 8 September 2008 (UTC)

title: User talk:Hajveryprofessional


text:
User:Eas4200c.f08.aero.e/Week 2

=Wednesday=

Material Properties

Weight is a critical component in the design of aircraft structures. The weight in the fuselage, wings, and engines is the primary concern. Current design is a balance between performance and how much it will cost to lift off the ground. Ideally materials used in aerospace applications have high stiffness and strength while still remaining light weight.

Stiffness is described by Young's Modulus ($$ \displaystyle E = \displaystyle \epsilon/ \displaystyle\sigma $$) the ratio of stress over strain. A higher Young's modulus corresponds with a higher material stiffness.

NOTE: Young's modulus is valid only in the elastic range (shown below).

The strength of a material is measured by the materials yield stress ($$\displaystyle \sigma$$y). Obviously a higher yield stress corresponds with a stronger material.

A materials toughness represents its ability to resist fracture. This is also know as fracture toughness.

Shown above is a graph of stress versus strain for a typical metal. Note that the graph is not exact and only for conceptual purposes. From the graph it is apparent that the ratio of stress and strain is linear inside the elastic range. Then plastic deformation and permanent deformation. If the material is stressed beyond its maximum yield point it will reach rupture stress and fail. Inside the plastic range a positive slope represents hardening of the material while a negative slope represents softening.

The Modulus of elasticity also known as young’s modulus is the rate of change between strain and stress. Further informations can be found at

http://www.instron.us/wa/resourcecenter/glossaryterm.aspx?ID=99&ref=http://www.google.com/search.

The chart below lists various categories of material properties and the materials which possess those properties.

The structure of an aircraft has two primary components:

 * 1) Geometry - Common aircraft geometries include monocoque shell like constructions and semimonocoque stiffened shell like structures.
 * 2) Material - Common aircraft materials include aluminum, titanium, and composite materials.

Below is a picture of the Boeing 787 Dreamliner which illustrates the aircraft's material composition. Composite materials make up fifty percent of the 787's total weight.

Boeing 787

=Friday=

Problem 1.1

Problem 1.1in "Mechanics of Aircraft Structures" 2nd Ed. by C. T. Sun states:

The beam of a rectangular thin-walled section (ie., t is very small) is designed to carry both bending moment M and torque T. If the total wall contour length L = 2(a + b) is fixed, find the optimum b/a ratio to achieve the most efficient section if M = T and σallowable = 2τallowable. Note that for the closed thin-walled sections such as the one in Fig. 1.16, the shear stress due to torsion is $$ \tau = {T \over 2abt} $$

The problem asks to find the ratio of the height vs. the width of a shell beam. To find this, first some assumptions must be made.

 * 1) M = T
 * 2) σallowable = 2τallowable
 * 3) L = 2(a + b)
 * 4) We can also assume shear stress distribution uniform across the wall, because the cross-section walls are very thin.

 Methodology  The shear stress can be represented with the figure below.

$$ T = T_{AB} + T_{BC} + T_{CD} + T_{DA}$$

The force can be shown as a parabola, like water in a pipe, like Fig. A below. This will translate to the next force below to B and C.

$$ T_{AB} = {b \over 2} vb = {1 \over 2} \tau abt $$

 Case 1:

For case one, we assume σ (bending normal stress) reaches σallowable and then verify that τ less than or equal to τallowable

Recall:       $$ \sigma\  = {M_z \over I} $$

Where M is the bending moment, z is the of a point on axis parallel to the normal bending axis, and I is the 2nd moment of inertia. The equation for the moment of inertia is:

$$ I = \iint_{A} \,z^2 dy\, dz = {bh^3 \over 12}$$

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title: User:Eas4200c.f08.aero.e/Week 2


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User:Eas4200c.f08.aero.e/Week 1

=Monday= The class website is the following: http://clesm.mae.ufl.edu/~vql/courses/aero1/2008.fall/ The password is fall08.wiki

The finite element method (FEM) will be utilized in analyzing partial differential equations that represent aerospace structures. FEM is a tool used in many engineering applications such as locating underground oil reserves and applied mathematics.

Ad. hoc. methods of structural analysis are discouraged in this course. Emphasis will be place on FEM, mechanics, and on the ability to formulate problems.

=Wednesday= Form teams of 5 to 6 students by the third of September. Each team should pick a team name that does not exceed six characters (four characters are preferred). On the third of September, the group must turn in a piece of paper to the instructor which includes all information relevant to the group such as the names, UFID numbers, and user names of each group member. The name of the group should also be included.

To create a Wikipedia account one should navigate to Wikipedia, go to create an account, type in the user name as follows: eas4200c.f08.group_name.member_initial. Type in a personal password and click "create account". This will create your own account. A team leader is to be chosen. All homework/class notes are to be saved to the group leader's page.

=Friday= Reference material for this course includes the textbook entitled "Mechanics of Aircraft Structures" by C.T. Sun, MIT's open course ware, and Wikiversity. Confidentiality is emphasized in this this course. To keep group web pages within a confidential framework, students are to edit only the group's web page while signed in under the class account. Editing other web pages while signed in under the class account could attract other users from Wikipedia/Wikiversity to the group's page. Once the web page is discovered by other users, vandalism could take place.

Homework assignments and class notes are to be completed within the group, and submitted via email to the T.A. Under this "new approach" the assigned work will be weighted heavily. If the instructor feels that the "new approach" is not working well for the class, assignments will be done individually and will be weighted less. This approach is known as the "old approach".

The vision for this course includes collaboration on several levels. Naturally, groups are to work together to solve problems and to write out class notes. Collaboration within the entire class and with the instructor is also expected.

title: User:Eas4200c.f08.aero.e/Week 1


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User talk:Eas4200c.f08.aero6.chastain

 Hello Eas4200c.f08.aero6.chastain, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random random project.
 * Browse everything on Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! --JWSchmidt 17:28, 8 September 2008 (UTC)

title: User talk:Eas4200c.f08.aero6.chastain