User:Eml4500.f08.delta 6.castrillon/HW 1

Matrix Method
The matrix method provides basic steps that are use as a recipe while solving truss systems helping solve problems more efficiently. This is achieved because a common labeling system for the diagrams is used.

Notation

Consider a truss system with two elastic bars, meaning that they are deformable since no materials are perfectly rigid, and both supported ends fixed and constrained to zero displacement in the x and y, as well as the x̃ and ỹ directions. Additionally, a force P loads the truss at the vertex.



Notice the global picture of the above truss system. It is composed of two bar elements that are joined and fixed at both ends. This calls for the need to identify three global nodes, and their reactions; it has been chosen from this point forth to circle the global node numbers for easy notation. Developing a free-body diagram (fbd) will result in the analysis above for the whole truss system (global picture).



We will analyze this system by choosing to use the y-x coordinate system. This global fbd yields 4 unknowns and 3 equations, making this system statically indeterminate. However, we can separate each bar element and build its own fbd.



New notation had to be used to number the bar element number with numbers inside a triangle. The diagram above is, thus, for bar element number 1. Also, local notation for element nodes have also been identified with a square around its node number, each corresponding to a global node (circled). Forces and reactions at each node have also been shown. The x and y directions of the reactions at the supports have also been separated, These have been labeled with an R with a subscript and a superscript that reflect the degree of freedom and the element number, respectively. The labeling of these forces is governed by these rules, in order: The directions of displacement of each node can also be represented by the red arrows in the diagram above, and both represent the degrees of freedom of the element. To save time, it has been decided to also label these arrows as being the direction of movement, each serving a dual purpose. The distances are represented by a d with the labels being the same for its force counterpart. We have done the same for the second bar element as for element 1, shown in the figure below.
 * Follow the order of the global node number
 * For each global node number, label the x-direction reaction first, then the y-direction reaction.



Notice that the number in the triangle has been changed to 2 to represent the element number. Also, the rules described above to label nodes has been followed. Global node 2 (circled) is now local node 1 (in square) and the global node 3 is now the local node 2. There is also another choice of local node numbering, where global node 2 can be local node 2 and the global node 3 is the local node 1. Both modes of numbering will yield the same outcome once the problem is solved, and the second is illustrated below.



Force-Displacement Relation

Recall a 1-D spring:



The above model is for a 1-D spring element with one end fixed. The following force-displacement (FD) relationship for this element is described below.


 * $$f=k*d$$

Where $$f$$ is the force on the free end, $$d$$ the distance the spring is deformed because of the force, and $$k$$ the stiffness of the spring and it depends on the material the spring is made out of.

If we were to free the left end of the above spring then the spring would be affects by the FD relationship as seen below.



The FD relation for this element is more complex as the displacements depend on each other, and each depends on both forces. We model this relationship with the aid of matrices, where each term is represented by a matrix.


 * $$\begin{pmatrix}

f_1\\ f_2\end{pmatrix}$$ = $$\begin{bmatrix} k & -k\\ -k & k\end{bmatrix}$$ $$\begin{pmatrix} d_1\\ d_2\end{pmatrix}$$

The first is a 2x1 column matrix that represents the forces on the spring, the second is a 2x2 matrix that represents the stiffness of the spring, and the third is also a 2x1 column matrix that represents the displacements of both free ends or nodes. This matrix equation is product of two cases that superimpose, each representing the behavior of both nodes as the spring stretches or compresses.


 * Case 1: The observer sits on node 1 and watches how the spring stretches out, represented by:
 * $$f_2=k(d_2-d_1)$$


 * Case 2: The observer sits on node 2 and watches how the spring stretches out, represented by:
 * $$f_1=k(d_1-d_2)=-f_2=-k(d_2-d_1)$$