text:
University of Florida/Eml4500/f08.qwiki.MATLAB

Sections 1-4
1. Accessing MATLAB.

Almost all versions of MATLAB include an icon on the desktop in which you can open MATLAB. Also there is the option of going through the start menu. While in MATLAB, to exit the command is quit or exit.

2. Entering matrices.

In MATLAB, all entries are treated as a matrix. They can be only a 1x1 matrix, up to nxn matrices, where n is an integer. Scalars are considered 1x1 matrices and vectors are either 1 row or 1 column matrices.

There are multiple ways to write matrices in MATLAB. The following are two different codes for producing matrix A.

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

A =

1    2     3

4    5     6

7    8     9

EDU>> A= [

1 2 3

4 5 6

7 8 9]

A =

1    2     3

4    5     6

7    8     9

Complex numbers also have multiple ways of entering them into MATLAB. The following code demonstrates two approaches.

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

A =

1.0000 + 5.0000i  2.0000 + 6.0000i

3.0000 + 7.0000i  4.0000 + 8.0000i

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

A =

1.0000 + 5.0000i  2.0000 + 6.0000i

3.0000 + 7.0000i  4.0000 + 8.0000i

The variables i or j are considered imaginary numbers but can be overwritten as regular variables. If this does happen then one can create a new name for the imaginary number by defining it in code. (jj = sqrt(-1) ).

Rectangular arrays consisting of numbers can be viewed from a file using MATLAB’s code:

EDU>> load filename.ext.

The matrix may then be called up on MATLAB using its filename.

A special rule to note when involving exponential form is to avoid spaces else they will be viewed as two different objects (1.23e-10).

There are several already programmed functions to create matrices. One of which is rand, which creates a randomly numbered matrix. The function rand(n) will create a square matrix of n2 entries. However, if a rectangular matrix is desired, rand(m,n) produces a matrix m x n.

EDU>> rand(3)

ans =

0.9501   0.4860    0.4565

0.2311   0.8913    0.0185

0.6068   0.7621    0.8214

EDU>> rand(3,2)

ans =

0.4447   0.9218

0.6154   0.7382

0.7919   0.1763

Another built-in function that MATLAB contains is the magic command. magic(n) yields a matrix where no matter the size of the square matrix, all rows, columns, and diagonals will have the same sum.

EDU>> magic(5)

ans =

17   24     1     8    15

23    5     7    14    16

4    6    13    20    22

10   12    19    21     3

11   18    25     2     9

The final example of programmed functions in MATLAB is the Hilbert matrix. Typing hilb will yield a matrix with each element following the equation:.

EDU>> hilb(3)

ans =

1.0000   0.5000    0.3333

0.5000   0.3333    0.2500

0.3333   0.2500    0.2000

To call up a single entry from a matrix, one must use the format matrixname(i,j). The following is an example of this using the last version of matrix A.

EDU>> A(2,2)

ans =

4.0000 + 8.0000i

3. Matrix operations, array operations.

There are several operations in MATLAB that can be performed on matrices and scalars, including addition, subtraction, multiplication, power, conjugate transpose, and left and right division. The left and right division for matrices is shown below. Two rules to follow are:

If x = A\b then A*x=b (Using Gaussian elimination if A is square, else                                                    Householder orthogonalization.)

If x = b/A then x*A=b (Defined by b/A=(A'\b')'.)

where A is an invertible square matrix and b is a compatible column.

EDU>> A

A =

0    0     8

8    0     4

0   16     0

EDU>> b= [ 2 3 7; 5 4 3; 5 9 0]

b =

2    3     7

5    4     3

5    9     0

EDU>> x=A\b

x =

0.5000   0.3125   -0.0625

0.3125   0.5625         0

0.2500   0.3750    0.8750

EDU>> A*x

ans =

2    3     7

5    4     3

5    9     0

EDU>> x=b/A

x =

0.7500   0.2500    0.1875

0.0625   0.6250    0.2500

-0.3125   0.6250    0.5625

EDU>> x*A

ans =

2    3     7

5    4     3

5    9     0

EDU>> (A'\b')'

ans =

0.7500   0.2500    0.1875

0.0625   0.6250    0.2500

-0.3125   0.6250    0.5625

Array operations.

While the matrix operations mentioned above operate matrix-wise, they can be also used to operate entry-wise. If a period is added before the operation, then MATLAB will recognize this command as an entry by entry operation. Examples are shown below.

EDU>> [1,2,3,4].*[1,2,3,4]

ans =

1    4     9    16

EDU>> [1,2,3,4].^2

ans =

1    4     9    16

4. Statements, expression, and variables; saving a session.

When typing in commands or variables, MATLAB is case sensitive. Therefore, if variable A is assigned to a matrix then calling up a in the future will not bring up matrix A.

When entering a statement into MATLAB, it is stored into the memory either to the variable name assigned by the user or to ans which can be called back for future use. Statements usually end with the enter key, however if there is a comma or semicolon more than one statements may be made in the same line. Additionally instead of terminating the statement at the end of a line, a semicolon will continue until ended with the enter key without one.

To list all variables in the workspace, the command who or whos can be used. The variables can be reassigned or can be cleared from the memory. The command clear variable is used to clear the variable. Below are examples of the commands who and clear. clear alone deletes all user assigned variables. After logging out of a MATLAB session, all variable assignments are lost.

EDU>> who

Your variables are:

A   ans  b    p    x

EDU>> clear ans

If an operation is runaway or seems to never stop computing, pressing CTRL-C will break the computation in MATLAB.

Saving a session.

To save the workspace before exiting, the command save will write all variables to a diskfile. After restarting MATLAB, the last workspace saved can be called back to its previous state with the command load.

EDU>> save

Saving to: matlab.mat

Sections 5-7
% Matrix building functions % A few matrix building functions include % eye : identity matrix % zeros: matrix of zeros % ones : matrix of ones % diag : create or extract diagonals % 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 % zeros(m,n) produces an m-by-n matrix of zeros and zeros(n) produces an n-by-n one zeros(3)

ans =

0    0     0    0     0     0    0     0     0

zeros(2,3)

ans =

0    0     0    0     0     0

A = ans; zeros(size(A))

ans =

0    0     0    0     0     0

% diag(x) produces a matrix with the vector x down the diagonal; % if A is a square matrix, then diag(A) is a vector consisting of the diagonal of A x = [1 2 3 4]; diag(x)

ans =

1    0     0     0    0     2     0     0    0     0     3     0    0     0     0     4

A = [1 2; 3 4]

A =

1    2    3     4

diag(A)

ans =

1   4

diag(diag(A))

ans =

1    0    0     4

% matrices can be built from blocks; if A is a 3-by-3 matrix, then A = ones(3,3); B = [A, zeros(3,2); zeros(2,3), eye(2)]

B =

1    1     1     0     0    1     1     1     0     0    1     1     1     0     0    0     0     0     1     0    0     0     0     0     1

% Flow Controls % for : executes a block of code a specified number of times n=4; x=[]; for i=1:n, x=[x,i^2],end

x =

1

x =

1    4

x =

1    4     9

x =

1    4     9    16

m=3; for i=1:m, for j=1:n, H(i,j)=1/(i+j-1); end, end, H

H =

1.0000   0.5000    0.3333    0.2500   0.5000    0.3333    0.2500    0.2000   0.3333    0.2500    0.2000    0.1667

% the semicolon in the above code suppresses printing of H % while the last H displays the final result, a hilbert matrix % while loops allow their statements to be repeatedly executed as long as % the relation remains true a=5; n=0; while 2^n < a, n=n+1; end, n

n =

3

% the "if" statement executes the procedures inside the loop as long as the relation is true if n<0, parity = 0; elseif rem(n,2) == 0, parity = 2; else, parity = 1; end parity

parity =

1

% Here are the relation operators in MATLAB % < less than % > greater than % <= less than or equal % >= greater than or equal % == equal % ~= not equal % note: "=" is used for assignment while "==" is used for relation % relations can be connected or quantified by logical operators % "&" and, "|" or, "~" not % when applied to scalars, a relation is actually the scalar 1 for true % or 0 for false 3<5

ans =

1

3>5

ans =

0

3==5

ans =

0

3==3

ans =

1

a = rand(5)

a =

0.9355   0.3529    0.1987    0.7468    0.8462   0.9169    0.8132    0.6038    0.4451    0.5252   0.4103    0.0099    0.2722    0.9318    0.2026   0.8936    0.1389    0.1988    0.4660    0.6721   0.0579    0.2028    0.0153    0.4186    0.8381

b = triu(a)

b =

0.9355   0.3529    0.1987    0.7468    0.8462        0    0.8132    0.6038    0.4451    0.5252        0         0    0.2722    0.9318    0.2026        0         0         0    0.4660    0.6721        0         0         0         0    0.8381

a==b

ans =

1    1     1     1     1    0     1     1     1     1    0     0     1     1     1    0     0     0     1     1    0     0     0     0     1

% a relation between matrices is interpreted by "while" and "if" to be true if each % entry of the relation matrix is nonzero. for example if A and B are equal matrices, then % if A==B, "statement"; end % certain functions operate on scalars, but operate element-wise when applied to a matrix % common functions include: % sin   asin    exp    abs    round % cos   acos    log    sqrt   floor % tan   atan    rem    sign   ceil % see help files for examples of these

% Matrix building functions % A few matrix building functions include % eye : identity matrix % zeros: matrix of zeros % ones : matrix of ones % diag : create or extract diagonals % 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 % zeros(m,n) produces an m-by-n matrix of zeros and zeros(n) produces an n-by-n one zeros(3)

ans =

0    0     0    0     0     0    0     0     0

zeros(2,3)

ans =

0    0     0    0     0     0

A = ans; zeros(size(A))

ans =

0    0     0    0     0     0

% diag(x) produces a matrix with the vector x down the diagonal; % if A is a square matrix, then diag(A) is a vector consisting of the diagonal of A x = [1 2 3 4]; diag(x)

ans =

1    0     0     0    0     2     0     0    0     0     3     0    0     0     0     4

A = [1 2; 3 4]

A =

1    2    3     4

diag(A)

ans =

1   4

diag(diag(A))

ans =

1    0    0     4

% matrices can be built from blocks; if A is a 3-by-3 matrix, then A = ones(3,3); B = [A, zeros(3,2); zeros(2,3), eye(2)]

B =

1    1     1     0     0    1     1     1     0     0    1     1     1     0     0    0     0     0     1     0    0     0     0     0     1

% Flow Controls % for : executes a block of code a specified number of times n=4; x=[]; for i=1:n, x=[x,i^2],end

x =

1

x =

1    4

x =

1    4     9

x =

1    4     9    16

m=3; for i=1:m, for j=1:n, H(i,j)=1/(i+j-1); end, end, H

H =

1.0000   0.5000    0.3333    0.2500   0.5000    0.3333    0.2500    0.2000   0.3333    0.2500    0.2000    0.1667

% the semicolon in the above code suppresses printing of H % while the last H displays the final result, a hilbert matrix % while loops allow their statements to be repeatedly executed as long as % the relation remains true a=5; n=0; while 2^n < a, n=n+1; end, n

n =

3

% the "if" statement executes the procedures inside the loop as long as the relation is true if n<0, parity = 0; elseif rem(n,2) == 0, parity = 2; else, parity = 1; end parity

parity =

1

% Here are the relation operators in MATLAB % < less than % > greater than % <= less than or equal % >= greater than or equal % == equal % ~= not equal % note: "=" is used for assignment while "==" is used for relation % relations can be connected or quantified by logical operators % "&" and, "|" or, "~" not % when applied to scalars, a relation is actually the scalar 1 for true % or 0 for false 3<5

ans =

1

3>5

ans =

0

3==5

ans =

0

3==3

ans =

1

a = rand(5)

a =

0.9355   0.3529    0.1987    0.7468    0.8462   0.9169    0.8132    0.6038    0.4451    0.5252   0.4103    0.0099    0.2722    0.9318    0.2026   0.8936    0.1389    0.1988    0.4660    0.6721   0.0579    0.2028    0.0153    0.4186    0.8381

b = triu(a)

b =

0.9355   0.3529    0.1987    0.7468    0.8462        0    0.8132    0.6038    0.4451    0.5252        0         0    0.2722    0.9318    0.2026        0         0         0    0.4660    0.6721        0         0         0         0    0.8381

a==b

ans =

1    1     1     1     1    0     1     1     1     1    0     0     1     1     1    0     0     0     1     1    0     0     0     0     1

% a relation between matrices is interpreted by "while" and "if" to be true if each % entry of the relation matrix is nonzero. for example if A and B are equal matrices, then % if A==B, "statement"; end % certain functions operate on scalars, but operate element-wise when applied to a matrix % common functions include: % sin   asin    exp    abs    round % cos   acos    log    sqrt   floor % tan   atan    rem    sign   ceil % see help files for examples of these

Sections 17-19
17. Hardcopy

The Hardcopy function is used to save your work to a file on your computer. To use this feature, the user enters in “diary filename” where filename is the desired name of the file. If no name is given, the file will automatically be called diary. The file can be edited as desired and printed using the !-feature.

18. Graphics

To graph a planar plot in MATLAB, one can use the plot function. Once the x and y vectors and defined by the user, a graphic plot can be obtained by entering “plot(x,y).” Sample code is provided as an example.

x=-4:.01:4 y=sin(x) plot(x,y)

This will open up a window with the graph, called figure 1. To open a new blank figure without editing the old one, the user can type in “figure(2).” This new figure is now the active graph, and any code entered will only effect this window. The user can enter in “figure(1)” to make the old figure active when so desired.

Graphing over an interval of numbers can be done by placing a colon on either side of the values of x. For example,

x=-1.5:.01:1.5

will set x to the value of .01 only over the interval from -1.5 to 1.5.

Graphs can also be produced by using the fplot command. Typing in “fplot(‘functionname’)” will graph a function that has been previously saved as an M-file.

To make a parametric plot, simply enter in the values for x,y, and t. Sample code is provided.

t=0:.001:2*pi

x=cos(3*t)

y=sin(2*t)

plot(x,y)

Once a plot is made, the axis and title can be edited using the “title”, “xlabel”, and “ylabel” commands. “gtext” Will place a text box at a point specified by the user with the mouse. “text” will place a text box at the given coordinates. A samlple title code is given below.

title(‘Best Least Squares Fit’)

The axis can also be edited by using the “axis([xmin,xmax,ymin,ymax])”, “axis(axis)”, “axis auto”, “v=axis”, “axis square”, “axis equal”, “axis off”, and “axis on” commands.

To plot multiple y functions on a single graph, one can designate multiple y values. Sample code is provided.

x=0:.01:2*pi

y1=sin(x)

y2=sin(2*x)

y3=sin(4*x)

plot(x,y1,x,y2,x,y3)

This same feature can be done by creating a Y matrix and then plotting that matrix with respect to x.

x=0:.01:2*pi

Y=[sin(x)',sin(2*x)',sin(4*x)']

plot(x,Y)

Multiple plots on the same graph can also be achieved with the hold command. Entering “hold” will cause MATLAB to plot all subsequent graphs on the current active window.

It is also possible to change the color and type of the data line as well as the color and type of the data points. Sample code is provided.

x=0:.01:2*pi

y1=sin(x)

y2=sin(2*x)

y3=sin(4*x)

plot(x,y1,'--',x,y2,':',x,y3,'+')

plot(x,y,'r--')

That code will cause y1 to be plotted with a dashed line, y2 to be plotted with a dotted line, and y3 to have a plus sign an each data point. The r before the “—“ in the last line of code will cause the dashed line to be red.

The command subplot will place several smaller plots on the same figure.

To print a hardcopy of a graphic, one can use the print command. If the command is followed by the filename, MATLAB will save the graphic file to the computer. The graphic can be appended if edited by entering “print –append filename”.

MATLAB is also capable of producing 3-D plots. Sample code for producing a parametric 3-D plot is given below.

t-.01:.01:20*pi

x=cos(t)

y=sin(t)

z=t

z=t.^3

plot3(x,y,z)

The title, data points, and axis can be label and edited just as a 2-D plot could be.

MATLAB can also render various graphical effects, such as a mesh grid or surf plot. The “mesh” command will produce a mesh over the surface of the graph, while surf will produce a shaded surface over the graph. Sample code to produce a mesh graph is shown below.

xx=-2:.2:2

yy=xx

[x,y]=mesh

grid(xx,yy)

z=exp(-x.^2 - y.^2)

mesh(z)

Entering in “surf(z)” would turn this graph into a surf plot.

The shading schemes of the surface can be edit using the “faceted”, “interpolated”, and “flat” commands. The color schemes available in MATLAB are “hsv”, “hot”, “cool”, “jet”, “pink”, “copper”, “flag”, “gray”, and “bone”.

Parametric surface plots can be made using the sphere and cylinder commands. The following sample code is an example of how to graph a torus.

function [x,y,z] = torus(r,n,a)

if nargin < 3, a=1; end

if nargin < 3, a=1

if nargin < 2, n = 30

if nargin < 1, r = 0.5

theta = pi*(0:2:2*n)/n

phi = 2*pi*(0:2:n)'/n

xx = (a + r*cos(phi))*cos(theta)

yy = (a + r*cos(phi))*sin(theta)

zz = r*sin(phi)*ones(size(theta))

if nargout == 0

surf(xx,yy,zz)

ar = (a+r)/sqrt(2)

axis([-ar,ar,-ar,ar,-ar,ar])

else

x=xx

y=yy

z=zz

end

19. Sparse Matrix Computations

MATLAB can reduce a matrix to a “sparse” matrix by only saving the elements in the matrix that are non-zero. Therefore, it can perform calculations faster by storing a matrix as a sparse matrix rather than a full “dense” matrix. The commands “full” and “sparse” are used to change a matrix to either type. The following code will create a sparse matrix.

F=floor(10*rand(6))

F=triu(tril(F,1),-1)

S=sparse(F)

To return this matrix to full, enter the command “F=full(S)”. It is also possible to see if a matrix is full or sparse using the “issparse(F)” command.

To create a sparse banded matrix using the spdiags command, use the following code.

m=6

n=6

e=ones(n,1)

d=-2*e

T=spdiags([e,d,e],[-1,0,1],m,n)

Another way to enter in a sparse matrix is to only enter in the non-zero elements. Example code for this is shown below.

i=[1 2 3 4 4 4]

j = [1 2 3 1 2 3]

s = [5 6 7 8 9 10]

S=sparse(i,j,s,4,3), full(S)

Operations can by applied to any matrix regardless if it is sparse or full.


text:
University of Florida/Eml4500/f08.qwiki/Lecture 1

Lecture 1

 * Dr. Vu-Quoc Website
 * Course Website
 * Syllabus

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


text:
University of Florida/Eml4500/f08.qwiki/Lecture 2

Lecture 2

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text:
User:Eml4500.f08.qwiki.nobles

{| class="collapsible prettytable" !Lecture 6

Continued from Lecture 5
2. Element Picture

3. Global FD at element level

The global force relationship is given by the equation $$F=KD$$, where $$F$$ is the resultant force matrix, $$K$$ is the global stiffness matrix, and $$D$$ is the global displacement matrix.

In matrix format,

$$ k^e = k^e \begin{bmatrix} (l^e)^2 & (l^e m^e) & -(l^e)^2 & -(l^e m^e) \\ (l^e m^e) & (m^e)^2 & -(l^e m^e) & -(m^e)^2 \\ -(l^e)^2 & -(l^e m^e) & (l^e)^2 & (l^e m^e) \\ -(l^e m^e) & -(m^e)^2 & (l^e m^e) & (m^e)^2 \end{bmatrix} $$

Note: The superscript "e" represents the truss element under examination.

$$k^{e}=\frac{E^{e}A^{e}}{L^{e}}$$ This is the axial stiffness of bar element "e"

$$l^{e},m^{e}$$ are director cosines of $$\tilde{x}$$ axis (from local node 1 to local node 2) with respect to the global (x,y) coordinate system.

$$l^{e}=\vec{\tilde{i}}\cdot \vec{i}=cos\theta^{e}$$

$$m^{e}=\vec{\tilde{i}}\cdot \vec{j}=sin\theta^{e}$$

$$\vec{\tilde{i}}\cdot\vec{i}=(cos\theta^{e}\vec{i}+sin\theta^{e}\vec{j})\cdot\vec{i}$$

Distributing by unit vector "i" it is easily seen that the equation reduces to $$l^{e}$$ because a unit vector dotted with itself equals 1, whereas a unit vector dotted with its perpendicular counter part equals 0.

Note: the director cosines are the components of $$\vec{\tilde{i}}$$

$$\vec{\tilde{i}}=cos\theta ^{e}\vec{i}+sin\theta ^{e}\vec{j}$$

HW 2
Solve: $$\vec{\tilde{i}}\cdot\vec{j}$$

$$\vec{\tilde{i}}\cdot\vec{j}=(cos\theta^{e}\vec{i}+sin\theta^{e}\vec{j})\cdot\vec{j}$$

Distributing by unit vector "j" it is easily seen that the equation reduces to $$m^{e}$$ because a unit vector dotted with itself equals 1, whereas a unit vector dotted with its perpendicular counter part equals 0.

Solve the element stiffness matrices using the provided information from the problem at the end of Lecture 5.

Element 1:

$$l^{1}=\frac{\sqrt{3}}{2}$$   $$m^{1}=\frac{1}{2}$$     $$k^{1}=\frac{3}{4}$$

$$\textbf{K}^{1}=\frac{3}{4}\begin{bmatrix} \frac{3}{4} & \frac{\sqrt{3}}{4} & -\frac{3}{4} & -\frac{\sqrt{3}}{4}\\ \frac{\sqrt{3}}{4} & \frac{1}{4} & -\frac{\sqrt{3}}{4} & -\frac{1}{4}\\ -\frac{3}{4} & -\frac{\sqrt{3}}{4} & \frac{3}{4} & \frac{\sqrt{3}}{4}\\ -\frac{\sqrt{3}}{4} & -\frac{1}{4} & \frac{\sqrt{3}}{4} & \frac{1}{4} \end{bmatrix}=\begin{bmatrix} \frac{9}{16} & \frac{3\sqrt{3}}{16} & -\frac{9}{16} & -\frac{3\sqrt{3}}{16}\\ \frac{3\sqrt{3}}{16} & \frac{3}{16} & -\frac{3\sqrt{3}}{16} & -\frac{3}{16}\\ -\frac{9}{16}& -\frac{3\sqrt{3}}{16} & \frac{9}{16} & \frac{3\sqrt{3}}{16}\\ -\frac{3\sqrt{3}}{16} & -\frac{3}{16} & \frac{3\sqrt{3}}{16} & \frac{3}{16} \end{bmatrix}$$

$$l^{2}=\frac{\sqrt{2}}{2}$$   $$m^{2}=-\frac{\sqrt{2}}{2}$$  $$k^{2}=5$$

$$\textbf{K}^{2}=5\begin{bmatrix} \frac{1}{2} & -\frac{1}{2} & -\frac{1}{2} & \frac{1}{2}\\ -\frac{1}{2} & \frac{1}{2} & \frac{1}{2} & -\frac{1}{2}\\ -\frac{1}{2} & \frac{1}{2} & \frac{1}{2} & -\frac{1}{2}\\ \frac{1}{2} & -\frac{1}{2} & -\frac{1}{2} & \frac{1}{2} \end{bmatrix}=\begin{bmatrix} \frac{5}{2} & -\frac{5}{2} & -\frac{5}{2} & \frac{5}{2}\\ -\frac{5}{2} & \frac{5}{2} & \frac{5}{2} & -\frac{5}{2}\\ -\frac{5}{2} & \frac{5}{2} & \frac{5}{2} & -\frac{5}{2}\\ \frac{5}{2} & -\frac{5}{2} & -\frac{5}{2} & \frac{5}{2} \end{bmatrix}$$

Calculate the axial forces for the statically determinate case:

 * }

title: User:Eml4500.f08.qwiki.nobles


title: User:Eas4200c.f08.zyx.lee


text:
User:Eml4500.f08.qwiki.berry/Eml4500.f08.qwiki/Lecture 1-3

=Lectures 1-3=
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 * Team names 6 characters long
 * Team info
 * Member names
 * UFID
 * Phone number
 * E-mail
 * Create a Wikipedia account
 * How do I create a wikipedia account, and then do a bunch of other stuff that I need to do to set up my junk so I can get credit on homework?...go here!
 * Wikipedia Security
 * The homework lockdown hacker-proof password is "fall08.wiki"
 * Submit archived copy of homework before deadline
 * How do I submit successfully and securely?...check this out and scroll down
 * How do I know what the homework grading criteria are?
 * Great question, find your answer to this question and others about life here.


text:
User:Saint Fink

Yo, daddy-o. I'm The Saint Fink

title: User:Saint Fink


text:
User talk:Eml4500.f08.qwiki.bishop

 Hello Eml4500.f08.qwiki.bishop, 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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 * 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 15:49, 11 September 2008 (UTC)

title: User talk:Eml4500.f08.qwiki.bishop


text:
Category:Wikiversity studies

ar:تصنيف:دراسات

title: Category:Wikiversity studies


text:
User talk:Eas4200c.f08.zyx.lee

 Hello Eas4200c.f08.zyx.lee, 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 15:55, 11 September 2008 (UTC)

title: User talk:Eas4200c.f08.zyx.lee


text:
User talk:Eml4500.f08.qwiki.nobles

 Hello Eml4500.f08.qwiki.nobles, 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 15:56, 11 September 2008 (UTC)

title: User talk:Eml4500.f08.qwiki.nobles


text: File:Mat Fig1.jpg

title: File:Mat Fig1.jpg


text:
User talk:Eml4500.f08.wiki1.brannon

 Hello Eml4500.f08.wiki1.brannon, 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 15:56, 11 September 2008 (UTC)

title: User talk:Eml4500.f08.wiki1.brannon


text:
User talk:Eas4200c.f08.ZYX

 Hello Eas4200c.f08.ZYX, 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 15:57, 11 September 2008 (UTC)

title: User talk:Eas4200c.f08.ZYX


text: File:Mat fig2.jpg

title: File:Mat fig2.jpg


title: User:Eas4200c.f08.ZYX.barnard


text:
User:Eml4500.f08.qwiki.bishop

=Course Information= EML 4500 - Finite Elemental Analysis (FEA)

Course wiki website: http://en.wikiversity.org/wiki/User:Eml4500.f08

Course website: http://clesm.mae.ufl.edu/~vql/courses/fead/2008.fall/

Professor : Dr. L. Vu-Quoc

Office: NEB 135

Office hours: Monday, Wednesday, Friday 4:05pm - 4:55pm

=Team "qwiki"=
 * Mike User:Eml4500.f08.qwiki.berry
 * Chris User:Eml4500.f08.qwiki.bishop
 * Andrea User:Eml4500.f08.qwiki.booher
 * Eric User:Eml4500.f08.qwiki.gatch
 * Lee User:Eml4500.f08.qwiki.harris
 * David User:Eml4500.f08.qwiki.nobles

=Other Assignments=
 * MATLAB Primer

=Lecture Notes=
 * Lecture 1-3
 * Lecture 4
 * Lecture 5
 * Lecture 6
 * Lecture 7
 * Lecture 8
 * Lecture 9
 * Lecture 10
 * Lecture 11
 * Lecture 12
 * Lecture 13
 * Lecture 16
 * Lecture 20
 * Lecture 22
 * Lecture 33

=Homework Assignments=
 * HW Assignment 1
 * HW Assignment 2
 * HW Assignment 3
 * HW Assignment 4
 * HW Assignment 5
 * HW Assignment 6
 * HW Assignment 7

title: User:Eml4500.f08.qwiki.bishop


text: File:Mat fig3.jpg

title: File:Mat fig3.jpg


title: File:Element2.jpg


text:
User:Eml4500.f08.qwiki.harris

= MATLAB Primer Notes =

Link to MATLAB Primer: http://apollo.mae.ufl.edu/~vql/courses/feem/matlab_primer_3rd.pdf

Managing M-Files
When using Matlab, it is sometimes necessary to move to another workspace (text editor, graphic viewer, etc.) while MATLAB is still active. The !- feature can be used to prevent exiting MATLAB while performing operations in a separate program. For instance, one can edit the file EML4500.m in a local text editor by typing

When the user is finished editing the file, they will be returned to an active MATLAB window.

M-files should be stored in a dedicated directory so that they can be easily accessed by MATLAB. The default active directory of MATLAB is C:\Program Files\MATLAB\R2006\work, but the user may choose to create another active work directory. To change the active directory or to create an additional MATLAB search path, navigate to File -> set Path... and perform any necessary changes. Note that MATLAB does include subdirectories in its search path, which allows the user to save M-files in subfolders and still be able to execute them in a MATLAB session.

The table below lists commands that will be useful when working with M-files:

Comparing efficiency of algorithms: Flops, tic and toc
When creating computer-intensive MATLAB code, it is often wise to create an algorithm that is efficient as possible. Two way to asses the efficiency of an algorithm are the amount of floating point operations (flops) performed and the elapsed time.

The MATLAB function records the total number of flops performed. The command  resets the flops to zero and is normally entered prior to running a an algorithm. Entering the  command after executing the algorithm gives the flops count for the algorithm. For example, the number of flops required to solve the equation c = a + b can be obtained with:

NOTE: With MATLAB ver. 7 and newer, the  command can no longer be used. MATLAB will display the following error:

While the flops command may not work for all users, the elapsed time can still be obtained. The time required to solve an algorithm can be found by using the stopwatch timers  and. For example, to find the total time needed to solve the equation, use the following command;

It should be noted, however, that the elapsed time may not be the best measure of algorithm efficiency because the rate of execution depends on how busy the processor is.

Output format
Even though all computations in MATLAB are performed in double precision, the format of the output display can be altered by the following commands:

Once introduced, the chosen format will remain in effect until changed.

The command  will suppress blank line allowing more information to be seen on the screen.

The command  returns MATLAB to the non-compact format.

=Introduction to Finite Element Analysis and Design=

=MATLAB Vision=

title: User:Eml4500.f08.qwiki.harris


text:
User talk:Xzapro4

 Hello Xzapro4, 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! --mikeu talk 16:24, 11 September 2008 (UTC)

Language
Would love to have your participation at the French Mentoring project. If you are willing, please sign up at the list there.

Your language learning idea sounds interesting, and like a good idea. I can see how it would be helpful to approach things like that on a number of levels. Your inclusion of embedded audio, pronunciation, and pictures in your example makes things that much more helpful, and the German explanations provide further helpful comments. I'll keep your method in mind for potential language courses here in the future; it may even be worth mentioning this as an effective way to organize language over at the Colloquium or at the School of Language and Literature. The Jade Knight (d'viser) 11:07, 4 November 2008 (UTC)

Mentoring
Would you be willing to Mentor Luca Saiu? He is a PhD candidate in Computer Science, living in Paris. He can communicate in French, but speaks Italian natively, and is also quite proficient in English. He is most interested in help with grammar, lexicon, or pronunciation. The Jade Knight (d'viser) 05:00, 24 November 2008 (UTC)

 * Interest in the Mentoring Project has increased! Would you be willing to take on additional students?  If so, how many?  The Jade Knight (d'viser) 21:09, 30 August 2009 (UTC)

title: User talk:Xzapro4


text:
User:EML4500.f08.A-team.rieth

Syllabus Overview:
Course website:

Then Click on:
 * teaching
 * my course website
 * Fall 2008 – Finite Element Analysis and Design

Office:		NEB 135

Office Hours:	MWF 9th Period

Exam Schedule:
 * Exam 1: September 24th
 * Exam 2: October 22nd
 * Exam 3: December 3

Book:	(Blackish in color) Finite Element Analysis with Mathematica and Matlab Computations and Practical Applications: Fundamental Concepts

Grading:	       HW + Project + Class Participation + Bonuses = 31%

Peer Evaluations:	wikiHW:fall08.wiki

Team Structure Assignment:
Due:	Wednesday, 3 September 08

Requirements:
 * 5-6 people
 * Team name (4-6 characters)
 * Member name + UF ID number
 * Wikipedia class username

Note:	When submitting documents on Wikipedia, send the archived version.

Old method compared to new approach
Old methods:
 * E-learning
 * Texbooks
 * Individual work

New approach
 * Wikimedia, wikiversity, wikipedia
 * Team collaboration

WP polocies
 * username space is not the same as writing an article

Grade changes:
 * Old method; 10% homework, 30% for each test
 * New method; 31% homework, 23% for each test

More emphasis on team learning not on memorization.

Homework Assignments
Homework 1

Homework 2

Homework 3

Homework 4

Homework 5

Homework 6

title: User:EML4500.f08.A-team.rieth


text:
University of Florida/Eml4500/f08.qwiki/Lecture 4

Trusses, matrix method

Truss with 2 Elastic (deformable) Bars

Note: Points 1 and 3 are fixed (constrained) to zero in both the x and y directions

Global FBD

Note: There are 4 unknown reactions and three equations of equilibrium making this truss statically indeterminant

Separate FBD's of the bar elements

Bar Element 1

Note: The superscript 1 in parenthesis on each force variable denotes that the force is applied to element 1

Bar Element 2

Note: The superscript 2 in parenthesis on each force variable denotes that the force is applied to element 2

Next Big Step:Force displacement relation (FD)

Recall:

FD relation of 1-D spring element:

 * 1 end fixed: f = kd (see above figure)

f 2x1= k 2x2 d 2x1
 * 2 ends free:

Case 1: Observer sits on node 1
 * f2 = k(d2-d1)

Case 2: Observer sits on node 2
 * f1+f2=0 Therefore, f1 = k(d1-d2)


text:
User talk:Eas4200c.f08.spars.stoute

 Hello Eas4200c.f08.spars.stoute, 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 16:43, 11 September 2008 (UTC)

title: User talk:Eas4200c.f08.spars.stoute


text:
User talk:Eml4500.f08.qwiki.gatch

 Hello Eml4500.f08.qwiki.gatch, 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 16:43, 11 September 2008 (UTC)

title: User talk:Eml4500.f08.qwiki.gatch


text:
User talk:EML4500.f08.A-team.rieth

 Hello EML4500.f08.A-team.rieth, 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 16:44, 11 September 2008 (UTC)

Probably a bad idea
The page you created, Group Members:, can be viewed by anyone in the world and will be indexed by google. It's probably best to hide the email addresses (you can use the "email this user" button on the left hand column) and delete the cellphone numbers. Salmon of Doubt 16:46, 11 September 2008 (UTC)

Candidate for Speedy Deletion
Your page Group Members: has been tagged for speedy deletion. Would you like this deleted? Ottava Rima (talk) 02:38, 15 September 2008 (UTC)

pagename
9/8/08 is not the best way to name a page (not descriptive at all). Probably you wanted to make a subpage or something similar. If this is the case, please re-create/move it there. This will be deleted soon, I assume. If you need help with it, feel free to contact me. For the future, just browse through Naming conventions--Gbaor 06:05, 16 September 2008 (UTC)

Note:
Hi, I have delleted Note: page as you removed content, I found it useless. If you need it, we can restore it for you. Have a good day.--Juandev 12:44, 8 January 2009 (UTC)

title: User talk:EML4500.f08.A-team.rieth


title: File:Fig1.JPG


text:
User:Eas4200c.f08.radsam/Example: Problem 1.1

=Statement:= A beam (Figure 1) is designed to carry both a moment $$M\!$$ and torque $$T\!$$. If the perimeter of the cross section is $$L=2(a+b)\!$$ and is held constant, find the optimal $$b/a\!$$ ratio if $$M=T\!$$ and $$\sigma_{allowable}=2\tau_{allowable}\!$$. Also note: $$\tau=\frac{T}{2abt}\!$$.

=Proof:=  Error 1: Your diagram on shear flow is incomplete. You should also redraw the figure instead of scanning the hand-drawn figure and upload the scan. Eml4500.f08 19:13, 21 September 2008 (UTC)

Error corrected. Eas4200c.f08.radsam.a. 15:59, 23 September 2008 (UTC)

The figure looks better, but still incomplete; review your classnotes and check with your teammates. Eml4500.f08 13:15, 24 September 2008 (UTC)

Corrected again: if this is wrong I'll eat my Wikiversity page. ;) Eas4200c.f08.radsam.a. 00:38, 4 October 2008 (UTC)

 * {| class="toccolours collapsible collapsed" width="100%" style="text-align:left"

!Problem Explanation and Proof of Shear Stress The closed-walled beam under scrutiny is one of negligible thickness, thus one can assume that the shear stress distribution is of a uniform nature...

Therefore adding respectively the torque on each side of the box...

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

...where torque equals the lever arm times the shear stress by the wall thickness... $$T_{AB}=\frac{b}{2}(va) = \frac{1}{2}\tau(abt)$$

$$T_{BC}=\frac{a}{2}(vb) = \frac{1}{2}\tau(abt)$$

$$T_{CD}=\frac{b}{2}(va) = \frac{1}{2}\tau(abt)$$

$$T_{DA}=\frac{a}{2}(vb) = \frac{1}{2}\tau(abt)$$

and recalling...

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

$$T=2\tau(abt)$$

or

$$\tau=\frac{T}{2abt}$$

Contribution by radsam.z
 * }

 * {| class="toccolours collapsible collapsed" width="100%" style="text-align:left"

! Allowable versus Yield Stress

As explained in Section 1.2 in "Aircraft Structures and Materials," yield stress is the stress point where a material begins plastic deformation. The stress/strain relation prior to this yield point is relatively linear for most materials; this means that the strain is proportional to the stress and that the material will go back to its original state after the applied stress is released. However, if the applied stress is greater than the yield stress, the material will deform permanently. The allowable stress refers to a point which is lower than the yielding limit of the material. This stress is calculated to be less than the yield stress for safety reasons and due to uncertainties in the expected properties of the material. When designing a structure, engineers let stresses reach the allowable limit because the structure would still withstand the applied forces, but it is crucial for stresses not to reach the yield stress since this will cause permanent deformation, and ultimately, failure of the structure.

Contribution by radsam.d
 * }

=Solution:= Due in Homework #2


title: File:Fig2.JPG


title: File:Fig3.JPG


title: File:Fig4.JPG


title: File:Fig5.JPG


title: File:Fig6.JPG


text:
University of Florida/Eml4500/8/25/08

EML4500 – Finite Element Analysis and Design

(First Day of Class)

Syllabus Overview:

Course website:

Then Click on:
 * teaching
 * my course website
 * Fall 2008 – Finite Element Analysis and Design

Office:		NEB 135

Office Hours:	MWF 9th Period

Exam Schedule:

Book:	(Blackish in color) Finite Element Analysis with Mathematica and Matlab Computations and Practical Applications: Fundamental Concepts

Grading:	       HW + Project + Class Participation + Bonuses = 31%

Peer Evaluations:	wikiHW:fall08.wiki

title: University of Florida/Eml4500/8/25/08


text:
University of Florida/Eml4500/8/27/08

Team Structure Assignment:

Due:	Wednesday, 3 September 08

Requirements:
 * 5-6 people
 * Team name (4-6 characters)
 * Member name + UF ID number
 * Wikipedia class username

Note:	When submitting documents on Wikipedia, send the archived version.

title: University of Florida/Eml4500/8/27/08


text:
University of Florida/Eml4500/8/29/08

Old method compared to new approach

Old methods:
 * E-learning
 * Textbooks
 * Individual work

New approach
 * Wikimedia, wikiversity, wikipedia
 * Team collaboration

WP polocies
 * username space is not the same as writing an article

Grade changes:
 * Old method; 10% homework, 30% for each test
 * New method; 31% homework, 23% for each test

More emphasis on team learning not on memorization.

title: University of Florida/Eml4500/8/29/08


text:
User:EML4500.F08.RAMROD.B

=Homework Contributions=

/Homework 1

/Homework 2

/Homework 3

/Homework 4

/Homework 5

/Homework 5.1

/Homework 6

/Homework 7

title: User:EML4500.F08.RAMROD.B


text:
User:EML4500.f08.A-team.rieth/9/3/08

Today is Wednesday, September 3, 2008

Things to be covered in class:
 * Trusses
 * Matrix Method (Located in Chapter 4 of the Book)
 * MIT’s Open Course Ware (Located on the wiki page)User:Eml4500.f08

Truss with elastic (deformable) bars:
Figure 1

 * There are four unknowns in this example with only three equations. This is then defined as a statistically indeterminate example.

Global (whole structure) FBD:
FBD= Free Body Diagram
 * Everything needs to be in equilibrium. Supports are removed and replaced with reaction forces that are not known.

Figure 2

 * Global refers to the whole structure.
 * Local refers to each element separately.
 * A number with a circle around is known as a global node, while a number with a triangle around it is a bar element. The number one with a triangle around it is bar element 1 from global node one to two, while the number 2 with a triangle around it is bar element two from global node two to three.
 * “P” is the known force, and all the “R” terms are the four unknown reaction forces.

Two Free Body Diagrams of Two Bar Elements:
Free Body Diagram of Body One:

Figure 3

 * The subscript on the "f" also stands for the degree of freedom (d.o.f.).
 * So, instead of always writing "degree of freedom" in long-hand every time, it can just be written as the d.o.f.
 * The subscript on the "f" is for the internal force on the element, which is represented by the number in parenthesis in the superscript. For this example the superscript would take on the values of one and two, while the subscript would take on the values of one, two, three, or four.
 * In this example the numbering could either be done the way it is, or done such that the local node numbers would be reversed.

Free Body Diagram of Body Two:

Figure 4

 * When labeling the forces with their degrees of freedom (also known as d.0.f.) you should always start with the horizontal direction and then proceeding on to the vertical direction.

You can also label bar element two in the following manner, with both approaches being correct

Figure 5

 * Always make sure to include parenthesis around the element number in superscript over the force (f) and displacement.

Force Displacement (also known as the F.D.) Relation Recall:
Figure 6

 * K=Stiffness
 * d=Distance Stretched
 * f=Force

The Force Displacement (F.D.) Relation of a one-dimensional spring element with one end fixed: f=kd The Force Displacement (F.D.) Relation of a one-dimensional spring element with two ends free:

Figure 7

 * fH, dH -internal displacement force

Figure 8

The spring in the picture has been stretched displacing the nodes.

For the following matrix, statics is used to fill it in:

$$ \begin{array}{cccc} \begin{Bmatrix} f_1 \\ f_2 \end{Bmatrix} & = & \begin{bmatrix} k & -k \\ -k & k \end{bmatrix} & \begin{Bmatrix} d_1 \\ d_2 \end{Bmatrix} \\ \text{2 x 1}~\text{Matrix} & & \text{2 x 2} ~ \text{Matrix} & \text{2 x 1} ~ \text{Matrix} \\ \text{row x column} \end{array} $$ Figure 9

title: User:EML4500.f08.A-team.rieth/9/3/08


text:
User talk:Eml4500.f08.ramrod.c

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) 17:13, 11 September 2008 (UTC)

 Hello Eml4500.f08.ramrod.c, 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.

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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! Erkan Yilmaz uses the Chat (try) 17:13, 11 September 2008 (UTC)

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title: User talk:Eml4500.f08.ramrod.c


text:
User:Eml4500.f08.ramrod.c

/HW

/9-12-08

/9-29-08

/10-8-08

/11-20-08

/11/14/08

/12-1-08

title: User:Eml4500.f08.ramrod.c


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text: Wikiversity:Student union/Tab1

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text:
User:Eas4200c.f08.gator.edards/Problem 1.1

Problem 1.1
 Team gator, you need to register user "Eas4200c.f08.gator.edards" so that this wiki page can be preserved. I also moved the figure up so that it does not block the link to "show" your first collapsible table. Eml4500.f08 12:52, 18 September 2008 (UTC)

Problem 1.1 is taken from the textbook Mechanics of Aircraft Structures by C.T.Sun, copyright 2006.

Find the optimum ratio of $$b/a$$ to maximize the load carrying ability of the beam. Assume $$M=T$$ and $$\displaystyle \sigma_{allowable}=2\displaystyle \tau_{allowable}$$.

The question is asking for the optimal cross section carrying the maximum bending moment and maximum torque.

Since the cross section walls are very thin, assume shear stress distribution in the thin walls of the beam are distributed uniformly. This results in the equation $$\displaystyle \tau=\frac{T}{2abt}$$.

Case 1: Assume $$\displaystyle \sigma$$ (bending moment stress) reaches $$\displaystyle \sigma_{allowable}$$ first. Verify $$\displaystyle \tau \le 2\displaystyle \tau$$.

From Mechanics of Materials, the normal stress caused by a bending moment is $$\displaystyle \sigma=\frac{Mz}{I}$$


text:
Wikiversity:Student union/TabsTop

{| width="100%" cellspacing="0" cellpadding="0" valign="top" border="0" |

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text:
User:Guettarda

w:User:Guettarda

 * Teaching_and_Learning_Online

title: User:Guettarda


text:
Wikiversity:Student Union/Page2/About Wikiversity

For information about Wikiversity, please view the following:

 * Introduction
 * What is Wikiversity?
 * Presentations about Wikiversity
 * FAQ
 * culture
 * Research guidelines
 * Research community
 * History of Wikiversity


text:
User:Eml4500.f08.RAMROD.F

TEAM RAMROD
[Amory T.] [Chris P.] [Jeff  H.] [Dan   O.] [Kevin S.] [Chris F.]

Team Homeworks
/HW1 /HW2

/HW3

/HW4

/HW5

/HW6

/HW7

title: User:Eml4500.f08.RAMROD.F


text:
User:Eml4500.f08.jamama.justin

HW1

MatlabTutorial

HW2

HW3

HW4

HW5

HW6

HW7

HW7b

title: User:Eml4500.f08.jamama.justin


text:
User talk:Eml4500.f08.RAMROD.F

 Hello Eml4500.f08.RAMROD.F, 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.
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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 19:18, 11 September 2008 (UTC)

title: User talk:Eml4500.f08.RAMROD.F


text:
User talk:Eas4200c.f08.carbon.orear

 Hello Eas4200c.f08.carbon.orear, 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.

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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 19:19, 11 September 2008 (UTC)

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


text:
User talk:Eas4200c.f08.gator.mueller

 Hello Eas4200c.f08.gator.mueller, 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.
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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.
 * 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
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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 19:21, 11 September 2008 (UTC)

title: User talk:Eas4200c.f08.gator.mueller


text:
User talk:EML4500.F08.RAMROD.B

 Hello EML4500.F08.RAMROD.B, 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.
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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 19:21, 11 September 2008 (UTC)

title: User talk:EML4500.F08.RAMROD.B


text:
Learning from conflict and incivility/Cormaggio

''This is my (initial) "narrative" for the purposes of the Learning from conflict and incivility project. Cormaggio talk 19:21, 11 September 2008 (UTC)''

I became aware of the "ethics" project in a thread on foundation-l, which was followed up a month later by another. I didn't get involved for quite a while - SB_Johnny was doing a fine job of keeping track of it all - though I did look in on occasion to see how it was progressing. My general feeling was a mixture of curiosity, nervousness, and hope - I have always thought it would be a great opportunity for Wikiversity to provide a platform to learn about practices in Wikimedia projects (and others, of course), though I wasn't sure if this was the right set-up to do so. (I wasn't sure because I really didn't know - I was vaguely aware that some of the participants were banned from Wikipedia - but I just didn't have the time to dig through the mountains of text that it very quickly generated.)

I became directly involved in the project on 25th August 2008 when I noticed an edit war between Moulton and Salmon of Doubt - and I left a note on the associated talk page about "endless reverting" (now archived). Some of the relevant discussions around that time took place in that talk page, and threads on Colloquium, Moulton's talk page, and the request custodian action page (following various previous threads on the same page). I was disturbed by what I observed - neither Moulton nor Salmon_of_Doubt seemed prepared to genuinely engage with each other (ie openly, honestly, self-reflectively), and each seemed intent on characterising the other as the wrong-doer in the eyes of others. Also, neither seemed prepared to acknowledge any responsibility for their role in the escalation of a conflict situation. I saw much of it as completely childish behaviour on both of their parts - and WAS_4.250 seemingly agreed :

The above is part of a drama enacted, unwittingly perhaps, by Salmon_of_Doubt and Moulton, playing themselves and cast to type, entitled Conflict resolution via punishment vs. tit-for tat. Take your bows gentlemen for that learning resource. I think you have both demonstrated the poor qualities of both strategies for the purpose of conflict resolution. WAS 4.250 21:42, 27 August 2008 (UTC)

What especially bothered me was that the project that gave rise to all this "drama" was supposedly about ethics - and yet, Moulton and Salmon_of_Doubt seemed to be behaving in an entirely unethical manner towards each other (and where one of them, Moulton, was a founder of this project). Furthermore, whenever I put this to them, it was continually ignored. In one of several requests to genuinely engage with each other, Moulton indicated that he saw their engagement as "a little game of chess" :

We are engaged, Cormaggio. Salmon of Doubt and I are playing a little game of chess. This is Salmon of Doubt's preferred way to engage with me. It's a curious way to learn, but there are some people who prefer to learn by engaging in games, dramas, or similar kinds of structured asymmetrical competitions. It's not my favorite way, either, but sometimes one has to speak the other person's language. This is one of those times. There is no doubt that my participation in the English Wikipedia was entirely unproductive. I utterly failed to produce the outcome I sought, namely accuracy, excellence, and ethics in those dozen or so atrocious BLPs and related articles and non-article space pages.

Salmon of Doubt has not characterized my comments as "insulting" (even though some might characterize them that way). The problem we are jointly working on, Cormaggio, is how to play a high quality Two-Person Non-Zero Sum Game of Narcissistic Wounding, which is a scaled down version of the Massive Multiplayer Game of Narcissistic Wounding and Mugging, which was the game IDCab played with me last August and September. [...]

(I think this is a very revealing comment. However, I will leave analysis to another space, and continue with my narrative - which I don't want to go on forever. :-))

Since becoming involved, I have attempted to get both Moulton and Salmon_of_Doubt to engage with each other, and to use their conflicting viewpoints as an opportunity for some genuine learning (ie open, honest, self-reflective - I think the reader will be able to gauge something of my perspective on learning from this narrative!). However, I have been pushing Moulton more than Salmon_of_Doubt - partly because I believe he has a certain special responsibility as the instigator of the "ethics" project; and partly because I could see that JWSchmidt seemed to take up my corresponding role with Salmon_of_Doubt, urging him to address comments and questions, and not ignore or remove them altogether, as he has been doing frequently since he began participating in Wikiversity. However, this relationship between JWSchmidt and Salmon_of_Doubt has turned out to be less than civil - particularly the Trout of Doubt "learning project" (which JWSchmidt has reflected on here). It is in the wake of such events of the past few weeks, having been away on holiday for the past week, that I have felt the urge to prevent the situation from becoming any more toxic than it has got, and to initiate this learning project (via my post to the Colloquium).

I will add as a meta-commentary on this project (and this narrative), that I view conflict as a potentially powerful means of learning. (I have been writing about conflict and its role in learning since 2005, eg my Wikimania paper for that year.) However, conflict clearly does not always lead to the kind of learning that I aspire to - which is mutually reflective, and in which people are forced to confront something about themselves - whether or not they continue to disagree with the other. I am hoping that this project will go some way towards this goal, and that I might be able to facilitate this in some way. Cormaggio talk 19:21, 11 September 2008 (UTC)


text:
User:EML4500.f08.jamama.megan

EML4500 - Finite Element Analysis and Design

Professor: Loc Vu Quoc

This is Megan's Wikiversity Page for EML4500.

 * I am the Team Coordinator for Team Jamama.

To view our work it is posted here ->

title: User:EML4500.f08.jamama.megan


text:
Portal talk:Social entrepreneurship/Issue Tracker

Fancy Rounded Boxes
Removed them - when editing the rounded box top tag is not included in the edit box. This is awkward if one wants to cut the whole issue and move it to the "Resolved" list.


text:
Template talk:Catspy

Do we have a template which actually reveals which category should be placed in what category - that maybe more useful I have tried it on the category which I created in the past but it couldn't detect which category it should have been in, so I had to do it manually. DarkMage 20:38, 11 September 2008 (UTC)

title: Template talk:Catspy


text: File:Problem 1 1.JPG

title: File:Problem 1 1.JPG


text:
Talk:Learning from conflict and incivility/Cormaggio

Is there somewhere we can ask you questions? Salmon of Doubt 20:52, 11 September 2008 (UTC)

 * Absolutely! Here's fine. :-) Cormaggio talk 22:40, 11 September 2008 (UTC)


text:
User talk:Mac31

 Hello Mac31, 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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 * 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! --Salmon of Doubt 20:55, 11 September 2008 (UTC)

Just a quick question - are you User:Eml4500.f08.jamama.justin? You appear to be editing one of his homework pages. Salmon of Doubt 20:55, 11 September 2008 (UTC)

title: User talk:Mac31


text:
Talk:Learning from conflict and incivility/JWSchmidt

Is there somewhere we can ask you questions? Salmon of Doubt 20:56, 11 September 2008 (UTC)
 * The blog is an open forum. I have a hard time finding conversations that take place on pages like this one. I have thousands of pages on my watch list...just lucky I saw this. --JWSchmidt 23:42, 11 September 2008 (UTC)


text:
User:EML4500.f08.A-team.VandenBerg

Syllabus Overview:
Course website:

Then Click on:
 * teaching
 * my course website
 * Fall 2008 – Finite Element Analysis and Design

Office:		NEB 135

Office Hours:	MWF 9th Period

Exam Schedule:
 * Exam 1: September 24th
 * Exam 2: October 22nd
 * Exam 3: December 3

Book:	(Blackish in color) Finite Element Analysis with Mathematica and Matlab Computations and Practical Applications: Fundamental Concepts

Grading:	       HW + Project + Class Participation + Bonuses = 31%

Peer Evaluations:	wikiHW:fall08.wiki

Team Structure Assignment:
Due:	Wednesday, 3 September 08

Requirements:
 * 5-6 people
 * Team name (4-6 characters)
 * Member name + UF ID number
 * Wikipedia class username

Note:	When submitting documents on Wikipedia, send the archived version.

Old method compared to new approach
Old methods:
 * E-learning
 * Texbooks
 * Individual work

New approach
 * Wikimedia, wikiversity, wikipedia
 * Team collaboration

WP polocies
 * username space is not the same as writing an article

Grade changes:
 * Old method; 10% homework, 30% for each test
 * New method; 31% homework, 23% for each test

More emphasis on team learning not on memorization.

Homework Assignments
Homework 1

Homework 2

Homework 3

Homework 4

Homework 5

Homework 6

Homework 7

title: User:EML4500.f08.A-team.VandenBerg


text:
User:EML4500.f08.JAMAMA

EML4500 - Finite Element Analysis and Design

Team Name: JAMAMA

Professor: Loc Vu Quoc

Homepage: User:Eml4500.f08

title: User:EML4500.f08.JAMAMA


text: File:Spring Sys.jpg

title: File:Spring Sys.jpg


text: File:Free spring system.jpg

title: File:Free spring system.jpg


text:
User:Eas4200c.f08.blue.a/Lecture 6

Continuation of Problem 1.1

Because the shear stress is greatest at the top and bottom of the beam section (where z = b/2), the equation of shear stress can be rewritten as follows: $$\sigma = \frac{Mz}{I}  \Rightarrow   M=\frac{2I\sigma\ _{allowable}}{b} $$

Thus, it is necessary to find the value of I in terms of b (recalling that a can be written as a function of b: a=L/2-b). From the problem parameters we know that for this instance, Mmax = Tmax. Because Mmax has been determined, it must only be put into the relation:

$$\tau _{max}=\frac{T^{(1)}_{max}}{2a^{(1)}b^{(1)}t}=\frac{M^{(1)}_{max}}{2a^{(1)}b^{(1)}t}$$

Now check the values of $$\tau _{max}$$ with the value of $$\tau _{allowed}$$.If $$\tau _{max}$$ is larger than $$\tau _{allowable}$$ the optimal ratio of $$\frac{b^{(1)}}{a^{(1)}}$$would be acceptable. But $$\tau _{max}$$ is smaller than $$\tau _{allowable}$$, then the optimal ratio of $$\frac{b^{(1)}}{a^{(1)}}$$ would be unacceptable

Case 2
For this case we assume that the shear stress is eqaul to the maximum allowable shear stress in the section. Then we will test to see if the normal stress falls within acceptable tolerances (notably, less than $$\sigma _{max}$$)

By the equation given earlier in the problem the value of T becomes:

$$T=(2t\tau _{allowable})(ab)$$

Where the dimensions of the rectangle ab are equal to one another, hence a square. Therefore if the total perimeter of the square is L and all sides are equal, both a and b are equal to L/4. Plugging this quantity into the previous equation and using the given relation that M=T, we have:

$$M^{(2)}_{max}=\frac{1}{8}tL^{2}\tau _{allowable}$$

Using this relation and the previously defined equation for normal stress:

$$\sigma ^{(2)}_{max}=\frac{M^{(2)}_{max}b^{(2)}}{2I^{(2)}}$$

We can find $$\sigma ^{(2)}_{max}$$ and compare it with the allowable normal stress. If the maximum normal stress is less than the allowable shear stress, then the ratio of b to a for case 2 is acceptable. But, if it is greater, then that ratio is not correct.

Solving Case 1
First the moment of inertia, I, must be computed. This is done by summing the moments of inertia of each of the four parts of the section. In order to do this, the Parallel axis theorem must be applied. Solving for the horizontal axis yields:

$$I=\frac{tb^{2}}{6}(3a+b)$$

And with the relation a=L/2-b, the plot of f(b)=I/b is shown to the right.

Taking the derivative of the function f(b) yields the maximum value of the curve to be b(1)=3L/8. Thus the ratio of $$\frac{b^{(1)}}{a^{(1)}}$$ is equal to 3 and the value of a(1) is found to be L/8. Solving for $$\tau _{max}$$ with the values of a(1) and b(1) yield:

$$\tau _{max}=\frac{M^{(1)}_{max}}{2a^{(1)}b^{(1)}t}=\frac{32M^{(1)}_{max}}{3tL^{2}}$$

but because this larger than half of $$\tau _{allowable}$$ this case is not possible

title: User:Eas4200c.f08.blue.a/Lecture 6


text:
User:EML4500.f08.JAMAMA/Group Members

''' Cedric Adam

actex007@ufl.edu

eml4500.f08.jamama.adam

To visit Cedric's Wiki Page click here -> User:eml4500.f08.jamama.adam

Chris Alford

alford@ufl.edu

eml4500.f08.jamama.chris

To visit Chris's Wiki Page click here -> User:eml4500.f08.jamama.chris

Megan Alvarez (Team Coordinator)

meg1015@ufl.edu

eml4500.f08.jamama.megan

To visit Megan's Wiki Page click here ->User:EML4500.f08.jamama.megan

Arunas Janulevicius

arunas@ufl.edu

eml4500.f08.jamama.jan

To visit Arunas's Wiki Page click here ->User:Eml4500.f08.jamama.jan

Justin McIntire

mac31@ufl.edu

eml4500.f08.jamama.justin

To visit Justin's Wiki Page click here ->User:eml4500.f08.jamama.justin

William Mueller

willsm@ufl.edu

eml4500.f08.jamama.mueller

To visit Will's Wiki Page click here -> User:eml4500.f08.jamama.mueller

title: User:EML4500.f08.JAMAMA/Group Members


text:
User:Eml4500.f08.bottle.vitello/HW 2

Homework 2 describes the methods involved in solving truss analysis problems using methods of FEA.

Lecture Notes
The following is a diagram for the truss problem which will motivate the notes hereafter.

Note that all constants and physical constraints are specified in the subsequent notes.

When faced with a problem as presented above in Statics, the boundary conditions are removed and replaced with forces. When there are more unknowns then equations of equilibrium for a specific problem, the problem is considered statically indeterminate. The following steps are used to solve a statically indeterminate problem using matrices.

First, draw a global free body diagram. The axis for the FBD are chosen based on the direction of the reaction forces. The FBD is then labeled using the following notation:

1.    Nodes are numbered within circles 2.	Elements are then numbered within triangles 3.	Unknown reaction forces are labeled 4.	The three equations of quality are written.

An example of global free body diagram labeling is shown below.

Then, a FBD for each element is drawn. The following notation is used:

1.    Each node is labeled within a box 2.	Element is numbered within a triangle 3.	The reaction at each node is labeled in the following way: f (e) dof, d (e) dof begins with 1 on X axis and then moves to Y axis, then proceeds to next node, where (e) is the element number and dof is the degree of freedom number.

An example of local free body diagram labeling is shown below.

Once the elements are labeled the force displacement relationship is written. This relationship takes the form:
 * $$\textbf{k}^{(e)}\textbf{d}^{(e)}=\textbf{f}^{(e)}$$

where k(e) is the element's stiffness matrix. The stiffness matrix for element 1 would have the form:

 * $$ \textbf{k}^{(1)}= \left[ \begin{array}{cccc} k_{11}^{(1)} & k_{12}^{(1)} & k_{13}^{(1)} & k_{14}^{(1)} \\ k_{21}^{(1)} & k_{22}^{(1)} & k_{23}^{(1)} & k_{24}^{(1)} \\ k_{31}^{(1)} & k_{32}^{(1)} & k_{33}^{(1)} & k_{34}^{(1)} \\ k_{41}^{(1)} & k_{42}^{(1)} & k_{43}^{(1)} & k_{44}^{(1)} \end{array} \right]$$

The big picture: Steps to solve a simple truss problem: 1.	Global Picture (Description) a.	At structure level i.	Global degrees of freedom (displacement, dof’s) ii. Global forces 2.	Displacement degrees of freedom are partitioned into a.	A known part (e.g. fixed dof, constraint/applied force) b.	An unknown part (solve using FEM/resultant forces) 3.	Element picture a.	Element DOF’s b.	Element faces 4.	Global Force Displacement relationship a.	Element stiffness matrices in global coordinates b.	Element force matrices in global c.	Assembly of element stiffness matrix and element force matrix in global FD rel d.	Kd=F 5.	Elimination of known dof’s to reduce global relationship a.	Stiffness matrix is non singular 6.	Compute forces from known d= element stress 7.	Compute reactions

Specific example:

Element Length L(1)= 4 L(2)=2 Youngs modulus E(1)= 3 E(2)=5

θ(1) = 30° θ(2) = -45°

A(1) = 1 A(2) = 2

Where  K = (E A) / L

Matrices: {F}=[k]{d}

(Remember to use convenient coordinate system and apply the given load at only one node)

$$ \textbf{k}^{(1)}= \left[ \begin{array}{cccc} l_{s}^{2} & l_{s}m_{s} & -l_{s}^{2} & -l_{s}m_{s} \\ l_{s}m_{s} & m_{s}^{2} & -l_{s}m_{s} & -m_{s}^{2} \\ -l_{s}^{2} & -l_{s}m_{s} & l_{s}^{2} & l_{s}m_{s} \\ -l_{s}m_{s} & -m_{s}^{2} & l_{s}m_{s} & m_{s}^{2} \end{array} \right]$$

where l(e)= cosθ(e) and m(e)= sinθ(e)

We know that θ(1) = 30°. Therefore, l(1)= cos(θ(1))= cos 30° and m(1) = sinθ(1) = sin30°. Finally, K(1) = (3 * 1) / 4 = 0.75

Therefore, we can compute the stiffness matrix k(1), which is calculated in the following form:

 * $$ \begin{bmatrix} k_{11}^{(1)} & k_{12}^{(1)} & k_{13}^{(1)} & k_{14}^{(1)}

\\ k_{21}^{(1)} & k_{22}^{(1)} & k_{23}^{(1)} & k_{24}^{(1)} \\ k_{31}^{(1)} & k_{32}^{(1)} & k_{33}^{(1)} & k_{34}^{(1)} \\ k_{41}^{(1)} & k_{42}^{(1)} & k_{43}^{(1)} & k_{44}^{(1)} \end{bmatrix} $$

When looking at the stiffness matrix we are able to observe that only 3 values need to be calculated: l(e)m(e), m(e), and l(e). The absolute values of these numbers form the stiffness matrix with only the sign changing. We are also able to observe that the matrix is symmetric, or the transpose of k(e) is equal to k(e).

We can use the same method described above to find the stiffness matrix for element 2. l(2)= cos(–45°)

m(2)= sin(-45°)

A brief description of the force-displacement relationship:

Elemental FD: k(e)d(e)= f(e)
 * $$ f^{(e)} = \begin{Bmatrix} f_1^{(e)} \\ f_2^{(e)} \\ f_3^{(e)} \\ f_4^{(e)} \end{Bmatrix}$$
 * $$ d^{(e)} = \begin{Bmatrix} d_1^{(e)} \\ d_2^{(e)} \\ d_3^{(e)} \\ d_4^{(e)} \end{Bmatrix}$$

Global FD:
 * $$ \begin{bmatrix} K_{11} & K_{12} & K_{13} & K_{14} & K_{15} & K_{16}

\\ K_{21} & K_{22} & K_{23} & K_{24} & K_{25} & K_{26} \\ K_{31} & K_{32} & K_{33} & K_{34} & K_{35} & K_{36} \\ K_{41} & K_{42} & K_{43} & K_{44} & K_{45} & K_{46} \\ K_{51} & K_{52} & K_{53} & K_{54} & K_{55} & K_{56} \\ K_{61} & K_{62} & K_{63} & K_{64} & K_{65} & K_{66} \end{bmatrix} \begin{Bmatrix} d_1 \\ d_2 \\ d_3 \\d_4 \\d_5 \\d_6 \\ \end{Bmatrix} = \begin{Bmatrix} F_1 \\F_2 \\ F_3 \\ F_4 \\ F_5 \\ F_6  \end{Bmatrix}$$

The global notation can be simplified into the form [Kij]{dj} = {Fi} where K= [Kij] is the global stiffness matrix, d= {dj} is the global displacement matrix and F= {Fi} is the global force matrix. The elemental notation can be simplified into the form k(e)d(e)=f (e), where k= [kij(e)] is the element stiffness matrix, d= {dj(e)} is the element displacement matrix and f= {fi(e)} is the element force matrix.

One can move from the element to the global matrices through an assembly process. The correspondence between the element degrees of freedom and the global displacement degrees of freedom can be seen below for element 1 and 2:

For node 1: d1 = d(1)1 and d2 = d(1)2

For node 2: d3 = d(1)3= d(2)1 and d4= d(1)4= d(2)2

For node 3: d5 = d(3)4 and d6 = d(2)4 When fully completed we can see that the topography of K resembles the figure below.

 * [[Image:K_Matrix.JPG]]

Therefore, for example 1, K6x6 =
 * $$ \begin{bmatrix} K_{11} & K_{12} & K_{13} & K_{14} & K_{15} & K_{16}

\\ K_{21} & K_{22} & K_{23} & K_{24} & K_{25} & K_{26} \\ K_{31} & K_{32} & K_{33} & K_{34} & K_{35} & K_{36} \\ K_{41} & K_{42} & K_{43} & K_{44} & K_{45} & K_{46} \\ K_{51} & K_{52} & K_{53} & K_{54} & K_{55} & K_{56} \\ K_{61} & K_{62} & K_{63} & K_{64} & K_{65} & K_{66} \end{bmatrix} $$

Where K11= k(1)11 = ;  K33= k(1)33 + k(2)11 = 3.0625; etc. At this step we can eliminate the known DOF’s. Because the truss example we are calculating is static, we know d1=d2=d5=d6= 0. When we apply these boundary conditions we can delete the corresponding columns in the stiffness matrix and by using the Principle of Virtual Work we can also eliminate the corresponding rows. We are left with the following equation:

 * $$ \begin{bmatrix} K_{33} & K_{34} \\ K_{43} & K_{44} \end{bmatrix} \begin{Bmatrix} d_3 \\ d_4 \end{Bmatrix} = \begin{Bmatrix} F_3 \\ F_4 \end{Bmatrix} $$

Once we have used the boundary conditions (global nodes 1 and 3 are fixed) to eliminate the 1st, 2nd, 5th, and 6th columns of the global K matrix, we are left with the matrix below.

 * $$\begin{bmatrix} K_{13}&K_{14}\\ K_{23}&K_{24}\\ K_{33}&K_{34}\\ K_{43}&K_{44}\\ K_{53}&K_{54}\\ K_{63}&K_{64} \end{bmatrix} = \begin{Bmatrix} d_3\\ d_4 \end{Bmatrix} =\overline{F}_{6x1}$$

We can also delete the 1st, 2nd, 5th, and 6th rows in the global K matrix and in the global F (force) matrix. The global F matrix will be a 2x1 matrix containing the values F3, F4. If we recall from the original problem statement that the force P acting on the truss system is directed upwards in the y-direction, we can say that F3=0 and F4=P.

Here are the resulting force displacement reactions.

Our problem is now reduced to a much more manageable matrix. Since we already know the values of F, we want to solve for the global displacement d3 and d4 terms. We will take the inverse of the 2x2 K matrix in order to do this.

The resulting global displacements d3=4.352 and d4=6.1271 are the displacements in the x- and y-direction of global node 2.

To compute the reaction forces in each individual element we solve the matrices k(e)d(e) = f(e) for e = 1,2 Recall that in element 1 the displacement terms d3 and d4 are in the 3rd and 4th rows. It is important here to note that the displacements terms are in different rows for each element. The local displacement matrices are below. Due to the fact that the local displacement matrix for element (1) has zeros in the first two rows, the first two columns of the local k matrix can be deleted. Plugging in numbers for the matrix k(1)d(1) = f(1) We get the local reaction forces for element (1) below. Since we know that element (1) is in equilibrium, the following three equations are true Element (2) can be represented by the following diagram with these forces acting. Taking the root sum squares of the local force components will help solve for the magnitude of forces P for both elements. There are two methods of statics to solve for the 2-bar truss system. These two principles describe the Euler Cut Principle.

Two-Bar Truss System
The following is a MATLAB example for a Two-Bar Truss System

% Two bar truss example clear all; e = [3 5]; A = [1 2]; P = 7; L=[4 2]; alpha = pi/3; beta = pi/4;

nodes = [0, 0; L(1)*cos(pi/2-alpha), L(1)*sin(pi/2-alpha); L(1)*cos(pi/2-alpha)+L(2)*sin(beta),L(1)*sin(pi/2-alpha)-L(2)*cos(beta)];

dof=2*length(nodes);

conn=[1,2; 2,3]; lmm = [1, 2, 3, 4; 3, 4, 5, 6]; elems=size(lmm,1); K=zeros(dof); R = zeros(dof,1); debc = [1, 2, 5, 6]; ebcVals = zeros(length(debc),1);

%load vector R = zeros(dof,1); R(4) = P;

% Assemble global stiffness matrix K=zeros(dof); for i=1:elems lm=lmm(i,:); con=conn(i,:); k_local=e(i)*A(i)/L(i)*[1 -1; -1 1] k=PlaneTrussElement(e(i), A(i), nodes(con,:)) K(lm, lm) = K(lm, lm) + k; end K R % Nodal solution and reactions [d, reactions] = NodalSoln(K, R, debc, ebcVals) results=[]; for i=1:elems results = [results; PlaneTrussResults(e, A, ... nodes(conn(i,:),:), d(lmm(i,:)))]; end format short g results

k_local =

0.75       -0.75        -0.75         0.75

k =

0.5625     0.32476      -0.5625     -0.32476      0.32476       0.1875     -0.32476      -0.1875      -0.5625     -0.32476       0.5625      0.32476     -0.32476      -0.1875      0.32476       0.1875

k_local =

5   -5    -5     5

k =

2.5        -2.5         -2.5          2.5         -2.5          2.5          2.5         -2.5         -2.5          2.5          2.5         -2.5          2.5         -2.5         -2.5          2.5

K =

0.5625     0.32476      -0.5625     -0.32476            0            0      0.32476       0.1875     -0.32476      -0.1875            0            0      -0.5625     -0.32476       3.0625      -2.1752         -2.5          2.5     -0.32476      -0.1875      -2.1752       2.6875          2.5         -2.5            0            0         -2.5          2.5          2.5         -2.5            0            0          2.5         -2.5         -2.5          2.5

R =

0    0     0     7     0     0

d =

0           0        4.352       6.1271            0            0

reactions =

-4.4378     -2.5622       4.4378      -4.4378

results =

1.7081      5.1244       8.5406       5.1244       17.081       0.6276       1.8828        3.138       1.8828        6.276

Homework Solutions
These problems supplement the material in the notes, much of which overlaps with or directly relates to the following assigned problems. As the class notes and the assigned problems are very much mutually integrated, it is essential to consider the class notes as a supplement and reference to the following homework solutions.

Derivation of $$\tilde{\textbf{i}}\cdot\textbf{j}$$: Given that m(e) = sin(θ(e)) for a particular element e it follows from the definition of director cosines that:

$$ m^{(e)}=\mathbf{\vec{i}}\cdot \vec{j}=\vec{j}\cdot (\cos \theta ^{(e)}\vec{i} + \sin \theta ^{(e)}\vec{j}=\cos \theta ^{(e)}\vec{j}\cdot \vec{i } + \sin \theta ^{(e)}\vec{j}\cdot \vec{j}=\cos \theta ^{(e)}\ (0) + \sin \theta ^{(e)}\ (1)$$

Such that $$m^{(e)}=0 +\sin \theta ^{(e)}$$

and hence $$m^{(e)}=\sin \theta ^{(e)}$$

Ergo, $$\tilde{\textbf{i}}\cdot\textbf{j}$$ is thus derived.

Find values for k(e) matrices from example truss problem

The example truss problem can be found in the lecture notes, above.

Given that l(1)=cos(30o), m(1)=sin(30o), and k(1)=0.75, the k(1) matrix is then formed using the methods described above, resulting in:

$$\textbf{k}^{(1)}= k^{(1)} \left[ \begin{array}{cccc} l_{s}^{2} & l_{s}m_{s} & -l_{s}^{2} & -l_{s}m_{s} \\ l_{s}m_{s} & m_{s}^{2} & -l_{s}m_{s} & -m_{s}^{2} \\ -l_{s}^{2} & -l_{s}m_{s} & l_{s}^{2} & l_{s}m_{s} \\ -l_{s}m_{s} & -m_{s}^{2} & l_{s}m_{s} & m_{s}^{2} \end{array} \right]=\left[\begin{array}{cccc} 0.5625 & 0.3248 & -0.5625 & -0.3248 \\ 0.3248 & 0.1875 & -0.3248 & -0.1875 \\ -0.5625 & -0.3248 & 0.5625 & 0.3248 \\ -0.3248 & -0.1875 & 0.3248 & 0.1875 \end{array} \right]$$

Also, given that l(2)=cos(-45o), m(2)=sin(-45o), and k(2)=5, the k(2) matrix is then formed using the methods described above, resulting in:

$$\textbf{k}^{(2)}= k^{(2)} \left[ \begin{array}{cccc} l_{s}^{2} & l_{s}m_{s} & -l_{s}^{2} & -l_{s}m_{s} \\ l_{s}m_{s} & m_{s}^{2} & -l_{s}m_{s} & -m_{s}^{2} \\ -l_{s}^{2} & -l_{s}m_{s} & l_{s}^{2} & l_{s}m_{s} \\ -l_{s}m_{s} & -m_{s}^{2} & l_{s}m_{s} & m_{s}^{2} \end{array} \right]=\left[\begin{array}{cccc} 2.5 & -2.5 & -2.5 & 2.5 \\ -2.5 & 2.5 & 2.5 & -2.5 \\ -2.5 & 2.5 & 2.5 & -2.5 \\ 2.5 & -2.5 & -2.5 & 2.5 \end{array} \right]$$

Prove that -P1(1) is equal to P1(2)

Knowing that and given the root sum square method, it follows that the definitions of -P1(1) and P1(2) are

$$-P^{(1)}_{1}=\sqrt{(f_{1}^{(1)})^{2}+(f_{2}^{(1)})^{2}}$$

and

$$P_{1}^{(2)}=\sqrt{(f_{3}^{(1)})^{2}+(f_{4}^{(1)})^{2}}$$

where

$$(f_{1}^{(1)})=-4.4378$$, $$(f_{2}^{(1)})=-2.5622$$, $$(f_{3}^{(1)})= 4.4378$$, and $$(f_{4}^{(1)})= 2.5622$$

such that

$$-P_{1}^{(1)}=\sqrt{(-4.4378)^{2}+(-2.5622)^{2}}=5.1243$$

and

$$P_{1}^{(2)}=\sqrt{(4.4378)^{2}+(2.5622)^{2}}=5.1243$$

Thus -P1(1) is equal to P1(2)

Prove Equilibrium at Node 2 of Example Truss Problem

Using the value calculated above for P1(1) of -5.1243 and calculating P2(2) as 6.2760, it follows that:

$$\sum{F_{y}} = 7-5.1243\sin(30^\circ)-6.2760\sin(45^\circ)=0$$

and

$$\sum{F_{x}} = 5.1243\cos(30^\circ)-6.2760\cos(45^\circ)=0 $$

Hence equilibrium is satisfied.

Individual Contributions
Eml4500.f08.bottle.vitello 21:57, 24 September 2008 (UTC) User:Eml4500.f08.bottle.butler/HW 2 22:00, 24 September 2008 (UTC) Eml4500.f08.bottle.barnes 19:35, 25 September 2008 (UTC) Eml4500.f08.bottle.ranto 21:13, 25 September 2008 (UTC) Eml4500.f08.bottle.hipps 21:21, 25 September 2008 (UTC) Eml4500.f08.bottle.loschak 08:13, 26 September 2008 (UTC)

title: User:Eml4500.f08.bottle.vitello/HW 2


text:
Category:Homework Postings

 * Homework 1
 * Homework 2
 * Homework 3
 * Homework 4
 * Homework 5
 * Homework 6
 * Homework 7

title: Category:Homework Postings


text:
User:Eas4200c.f08.zyx.lee/test

 Lecture Summaries:  EAS 4200c - Group ZYX

Course Description
This course, EAS4200C Aerospace Structures, will help students to achieve a better comprehension of mechanics and advance their problem-formulating skills, in response to a recent movement away from ad hoc techniques and towards the finite element method. Topics covered in this course include stress, strain, torsion, bending, shear and failure. There will also be an introduction to composite materials, an important new category of materials in aircraft construction.

Textbook
C.T. Sun, Mechanics of Aircraft Structures Wiley, 2nd edition (April 28, 2006), ISBN-10: 0471699667.

title: User:Eas4200c.f08.zyx.lee/test


text:
University of Florida/Eml4500/Lecture Notes

Trusses, Matrix Method

The following image is of a truss with two elastic (deformable) bars.

Note: The displacement at points 1 and 3 is fixed (constrained) to zero, in both the x and the y direction.

Global Free Body Diagram:

The below image illustrates the 4 unknown reactions. This example is statically indeterminate because it has 3 equations and 4 unknown variables. The key to solving this example is to look at the individual bars' deformation.

2 Free body diagrams of separate bar elements

 * Bar Element 1:

The unknown solutions at the nodes are the nodal degrees of freedom. The choice of nodal degrees depend on the governing differential equation.

 * Bar Element 2:

Next step: Force Displacement relationship

Recall:

The force distance relationship of a 1 dimensional spring element (with one end fixed) is F=KD. (K is the spring constant)

The force displacement relationship of a 1D spring with 2 ends free is as shown above. To solve this more complex problem one needs to use a matrix to relate F=Kd. The matrix for this example is shown below.

Stiffness Matrix:

Case 1: Observer sits on node 1.
 * F2=K(d2-d1)

Case 2: Observer sits on node 2, or equilibrium f1+f2=0.
 * F1=-F2=-K(d2-d1)=K(d1-d2)

Steps to solve simple truss system.

1.Global Picture (description)
 * At structure level:
 * Global degrees of freedom (displacement unknowns)
 * Global forces

 * Actually, the displacement Dofs are partitioned into:
 * a known part, e.g., fixed Dof, contraints
 * an unknown part: solved by using the finite element method (FEM)

 * Similarly, for the forces:
 * known part: applied forces
 * an unknown part: reactions

2.Element Picture (draw)
 * Element Dof (displacements)
 * Element forces
 * Note: it can be either in global or local coordiante system

3.Global Force-Displacement (FD) relationships
 * Element stiffness matrix in global coordinates
 * Element force matrices in global coordinates
 * Assembley of element stiffness matrices and element force matrices into global FD relation: $$K*d$$ = $$F$$

4. Elimination of known Dof to eliminate/reduce the global FD relations (if stiffness matrix is non-singular >> invertable >> det(K) ≠ 0);

Km × m*dm × 1 =  F m × 1, where $$m < n $$ m = n 0 of unknown displacement Dof n = n 0 of known and unknown displacement Dof K is non-singular matix >> K -1 (invertable)>> d m × 1 =  K -1 m × m * F m × 1

5.Compute element forces form now known d >> element stresses

6.Compute the reactions (unknown forces)

Example:

Numbering of displacement Dof:
 * Follow the order of global node numbering. Assume x - axes are horizontal: If horizontal force was considered first in the first global node, then in the consequent nodes the horizontal force should be numbered first as well(vertical force follows the horizontal force in each node force/displacement numbering).


text:
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User talk:EML4500.f08.JAMAMA

 Hello EML4500.f08.JAMAMA, 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.JAMAMA


text: File:Shear flow.JPG

title: File:Shear flow.JPG


text:
User:Eml4500.f08.jamama.justin/HW1

Trusses, Matrix Method

The following image is of a truss with two elastic (deformable) bars.

Note: The displacement at points 1 and 3 is fixed (constrained) to zero, in both the x and the y direction.

Global Free Body Diagram:

The below image illustrates the 4 unknown reactions. This example is statically indeterminate because it has 3 equations and 4 unknown variables. The key to solving this example is to look at the individual bars' deformation.

2 Free body diagrams of separate bar elements

 * Bar Element 1:

The unknown solutions at the nodes are the nodal degrees of freedom. The choice of nodal degrees depend on the governing differential equation.

 * Bar Element 2:

Next step: Force Displacement relationship

Recall:

The force distance relationship of a 1 dimensional spring element (with one end fixed) is F=KD. (K is the spring constant)

The force displacement relationship of a 1D spring with 2 ends free is as shown above. To solve this more complex problem one needs to use a matrix to relate F=Kd. The matrix for this example is shown below.

Stiffness Matrix:

Case 1: Observer sits on node 1.
 * F2=K(d2-d1)

Case 2: Observer sits on node 2, or equilibrium f1+f2=0.
 * F1=-F2=-K(d2-d1)=K(d1-d2)

Steps to solve simple truss system.

1.Global Picture (description)
 * At structure level:
 * Global degrees of freedom (displacement unknowns)
 * Global forces

 * Actually, the displacement Dofs are partitioned into:
 * a known part, e.g., fixed Dof, contraints
 * an unknown part: solved by using the finite element method (FEM)

 * Similarly, for the forces:
 * known part: applied forces
 * an unknown part: reactions

2.Element Picture (draw)
 * Element Dof (displacements)
 * Element forces
 * Note: it can be either in global or local coordiante system

3.Global Force-Displacement (FD) relationships
 * Element stiffness matrix in global coordinates
 * Element force matrices in global coordinates.
 * Assembley of element stiffness matrices and element force matrices into global FD relation: $$K*d$$ = $$F$$

4. Elimination of known Dof to eliminate/reduce the global FD relations (if stiffness matrix is non-singular >> invertable >> det(K) ≠ 0);

Km × m*dm × 1 =  F m × 1, where $$m < n $$ m = n 0 of unknown displacement Dof n = n 0 of known and unknown displacement Dof K is non-singular matix >> K -1 (invertable)>> d m × 1 =  K -1 m × m * F m × 1

5.Compute element forces form now known d >> element stresses

6.Compute the reactions (unknown forces)

Example:

1) Globlal Picture:

Numbering of displacement Dof:
 * Follow the order of global node numbering. Assume x - axes are horizontal: If horizontal force was considered first in the first global node, then in the consequent nodes the horizontal force should be numbered first as well(vertical force follows the horizontal force in each node force/displacement numbering).

title: User:Eml4500.f08.jamama.justin/HW1


text: File:PauseReflect.png

title: File:PauseReflect.png


text:
User:Eml4500.f08.jamama.chris

Eml4500.f08/Matlab Summary

title: User:Eml4500.f08.jamama.chris


text: File:Hw1 fig1.jpg

title: File:Hw1 fig1.jpg