User:Eml4500.f08.bottle.loschak/HW2

HW 2
9/17 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.



9/19 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 room 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.