User:EML4500.f08.A-team.VandenBerg/HW1

 Comment: This is HW1 located in the user namespace of Lukas Vandenberg --EML4500.f08.A-team.VandenBerg 20:12, 19 September 2008 (UTC)

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)

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

5-1 Meeting 5: Friday, 5 September 08, EML4500

Reading assignments:

 * Chapter 4: Trusses, Beams, and Frames
 * Chapter 1: By Picture
 * Section 1.1: Discretization
 * 1.1  Plane Truss Elem.
 * 1.2  Assembly of Elem. Eqs. (Ex. 4.1: Five-Bar truss)
 * 1.3  Elem. Solu. and Model Validity
 * 1.4  Plane Truss Elem.

Steps to solve simple truss system:
1. Global picture (description) -At structure level: *global d.o.f.’s (displ. d.o.f’s) --> unknowns in general *global forces -Actually --> displ. d.o.f.’s are partitioned into: *a known part: e.g. fixed d.o.f.’s, constraints *an unknown part: solved using FEM -Similarly for the global forces: *a known part: applied forces *an unknown part: reactions

2. Element Picture -Element d.o.f.’s and element forces:
 * either in global or local coordinate system

3. Global FB relation -Element stiffness matrices in global coordinates -Element force matrices in global coordinates -Assembly of element stiffness matrix and element force matrix into global FB    relation:  K * d = F  “free-free system” (unconstrained)  K=singular
 * K --> N x N
 * d --> N x 1
 * F --> N x 1

4. Elimination of known d.o.f.’s to reduce the global FB relation -Stiffness matrix non-singular --> invertible -K * d = F     *K --> M x M      *d --> M x 1 M < N      *F --> M x 1 -M = number of unknown displ. d.o.f.’s -N = number of both known & unknown d.o.f.’s -K is non-singular --> K(-1) exists *d = K(-1) * F

5. Compute element forces from now known d --> elem. stresses

6. Compute reactions (unknown forces)

Specific example to see how method work.

Data:

-L(1) = 1 -L(2) = 2

Cross-sectional Area:
-A(1) = 1 -A(2) = 2

Young’s Modulus:
-E(1) = 3 -E(2) = 5

-Global picture: Global d.o.f.’s:

Numbering the displ. d.o.f.’s:
-Follow order of global node number -For each node, follow the order of global coord. axes, number displ d.o.f.’s for that node -Global forces --> same thing

Getting Started
Quick example of how text is input and shown in Matlab

% Matlab

EDU>> a = [ 1 2; 2 1 ]

a =

1    2     2     1

EDU>> a*a

ans =

5    4     4     5

EDU>> quit

Matrices
This is a matrices tutorial showing a user how to input different matrices, computing different mathematical equations. Some of these equations are matrix multiplication, matrix addition, matrix division, matrix inverses, and determinates.

EDU>> %Matlab EDU>> a = [ 1 2; 3 4 ]

a =

1    2     3     4

EDU>> a

a =

1    2     3     4

EDU>> a * a

ans =

7   10    15    22

EDU>> b = [ 1 2; 0 1 ]

b =

1    2     0     1

EDU>> a*b

ans =

1    4     3    10

EDU>> b*a

ans =

7   10     3     4

EDU>> a + b

ans =

2    4     3     5

EDU>> s = a + b

s =

2    4     3     5

EDU>> inv (s)

ans =

-2.5000   2.0000    1.5000   -1.0000

EDU>> s * inv (s)

ans =

1.0000  -0.0000         0    1.0000

EDU>> s/s

ans =

1    0     0     1

EDU>> s\s

ans =

1    0     0     1

EDU>> inv (s) * s

ans =

1.0000        0         0    1.0000

EDU>> a/b

ans =

1    0     3    -2

EDU>> a\b

ans =

-2.0000  -3.0000    1.5000    2.5000

EDU>> c = [ 1 1; 1 1 ]

c =

1    1     1     1

EDU>> inv(c); Warning: Matrix is singular to working precision. EDU>> det(a)

ans =

-2

EDU>> det(c)

ans =

0

EDU>> quit

Vectors in MatLab
Quick way to show vector algebra in Matlab with matrices:

EDU>> a = [ 1 2; 3 4 ]

a =

1    2     3     4

EDU>> b = [1;0]

b =

1    0

EDU>> a\b

ans =

-2.0000   1.5000

EDU>> quit

Editing Systems of Equations
Linear equations can be easily manipulatedwith use of matlab and matrices.

If you have to equations: ax + by = p    cx + dy = q

These two equations can be better understood, written, and compacted using matlab. AX = B

Where the coeffient matrix is the A

A = [ a b; c d ]

with the unknown matrix with the x and y coefficients

Loops in MatLab
A loop is a programming tool in Matlab that is used to do repetitive work. It saves the user time. In the tutorial, matrix a represents the probabilities that two island populations will move or stay put on their side of the island. x is a column vector representing the initial populations.

a=    0.8  0.1 0.2 0.9

x=    1 0

Successive population states can be predicted by multiplying the two matrices. Using a loop, the operation can be repeated a defined number of times using the following form:

>> a = [ 0.8 0.1; 0.2 0.9 ] >> x = [ 1; 0 ] >> for i = 1:20, x = a*x, end

This operation will be repeated 20 times, and the populations of the island can be predicted 20 time units away.

Graphing in MatLab
Here are some images of the types of graphing that MatLab can perform, with the description of the code used to draw the graph directly beneath its corresponding image. Some of the nuances of Matlab graphing can be seen here, such as changing the color (function call "colormap" in images 5 and 6), removing mesh lines (function call "shading faceted" in image 5), and using the function name "sphere" to create a sphere with default dimensions.

IMAGE 1

EDU>> x = 0:.01:6; y = cos(x); plot(x,y)

IMAGE 2

EDU>> t = -3:.1:3; x = cos(1.5*t); y = sin(0.5*t); z = t.^2; plot3(x,y,z)

IMAGE 3

EDU>> surf(magic(10))

IMAGE 4

EDU>> [x,y] = meshgrid(-3:.01:3, -3:.01:3); z = (x.^3 - y.^3); mesh(z)

IMAGE 5

EDU>> surf(eye(10)/hilb(10)); colormap(cool); shading faceted

IMAGE 6

EDU>> sphere; colormap(flag); shading flat

 Mtg. 6; Monday 8, September 2008  EML 4500

2) Elem. Picture

3) Global Force Displacement (F.D.)

Contributions for HW1
Contributors:

Lukas Vandenberg --EML4500.f08.A-team.VandenBerg 20:12, 19 September 2008 (UTC)

Brian Morford --EML4500.f08.A-team.morford 20:12, 19 September 2008 (UTC)

Joseph Rieth --EML4500.f08.A-team.rieth 20:13, 19 September 2008 (UTC)

Joseph Melvin --EML4500.f08.A-team.melvin 20:23, 19 September 2008 (UTC)

Brian Kirley --EML4500.f08.A-team.kirley 20:32, 19 September 2008 (UTC)

Daniel Robinson --Eml4500.f08.a-team.robinson 20:32, 19 September 2008 (UTC)