User:Eml4500.f08.team.guan/fixed1015

 HW Redo: The free body diagrams of the elements were added. Further explanation of the math and equations were provided.

Justifying the Global Stiffness Matrix
The assembly of the elemental stiffness matrices, k(e) into a global striffness matrix, K, can be justified by reviewing the two bar truss problem.

If we look back at the Two bar truss problem, and recall the element force-displacement relation $$ k^{(e)}d^{(e)} = f^{(e)}$$, we can use the Euler Cut Principle to give the equilibrium of the node relationship.

Then, if the free body diagrams of elements 1 and 2 are considered, and the global degrees of freedom for global node 2 are compared to the degrees of freedom for elements 1 and 2, the equilibrium of the node can be confirmed. Below are the free body diagrams of element 1, element 2, and the global node number 2. It is simple to see that the forces are the global node 2 in the elemental diagrams are directed in the opposite directions of the forces in the free body diagram of global node 2. This is the convention used in statics.

  

Looking at the equilbrium of node 2, the addition of k values is because the forces in the x and y direction sum up to equal zero. Using a statics approach and calculating the forces in each direction gives:

$$ \Sigma F_x = 0= -f^{(1)}_3 - f^{(2)}_1=0$$     (1) $$ \Sigma F_y = 0= P-f^{(1)}_4 - f^{(2)}_2=0$$     (2)

This next step is to involve the force displacement relationship. Replacing f with kd will get the middle of the 6 by 6 global stiffness matrix. Thus, statics and equilibrium evaluation of global node 2 has verified the middle of the global stiffness matrix.