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