User:EML4500.f08.Ateam.Miller/homework2

Step 4: Elimination of Known Degrees of Freedom by Reducing the Global Force-Displacement Relationship
The next step in the Finite Element Method is to eliminate the known degrees of freedom by reducing the global force-displacement relationship. The displacement matrix from the previous two-truss system can be reduced by applying the fixed boundary conditions:

$$d_1=d_2=d_5=d_6=0$$

The resulting displacement matrix is:

$$d= \begin{bmatrix} 0 \\ 0 \\ d_3  \\ d_4 \\ 0 \\ 0 \end{bmatrix} $$

After applying these conditions, the corresponding columns in the global stiffness matrix can be deleted and the resulting matrix is shown below.

$$F=K \begin{bmatrix} 0 \\ 0 \\ d_3  \\ d_4 \\ 0 \\ 0 \end{bmatrix} = \begin{bmatrix} k_{13} & k_{14} \\ k_{23} & k_{24} \\ k_{33} & k_{34} \\ k_{43} & k_{44} \\ k_{53} & k_{54} \\ k_{63} & k_{64} \\ \end{bmatrix} \begin{bmatrix} d_3 \\ d_4 \\ \end{bmatrix} $$

The next step in reducing the force-displacement relationship is to apply the Principle of Virtual Work. This allows the corresponding rows of the global stiffness matrix and the global force matrix to be deleted. The resulting force-displacement relation is

$$ \begin{bmatrix} k_{33} & k_{34} \\ k_{43} & k_{44} \\ \end{bmatrix} \begin{bmatrix} d_3 \\ d_4 \\ \end{bmatrix}= \begin{bmatrix} F_3 \\ F_4 \\ \end{bmatrix} $$ or $$\mathbf{K}\cdot\mathbf{d} = \mathbf{F}$$

Based on the given information, the reduced force matrix is a known, as shown below.

$$\underline{\overline{F}}= \begin{bmatrix} F_3 \\ F_4 \\ \end{bmatrix}= \begin{bmatrix} 0 \\ P \\ \end{bmatrix}= \begin{bmatrix} 0 \\ 7 \\ \end{bmatrix} $$

Finally, the reduced global displacement matrix can be solved by multiplying the reduced force matrix by the inverse of the reduced global stiffness matrix.

$$ \begin{bmatrix} d_3 \\ d_4 \\ \end{bmatrix} =\begin{bmatrix} k_{33} & k_{34} \\ k_{43} & k_{44} \\ \end{bmatrix}^{-1} \begin{bmatrix} 0 \\ P \\ \end{bmatrix}= \begin{bmatrix} 4.352 \\ 6.127 \\ \end{bmatrix} $$

Step 5: Compute Reactions
After the displacements of Node 2 are computed, then the next step is to compute the reactions. There are two methods that can be used to compute the reactions. The first method is to use the element force-displacement relations

$$k^{(e)}d^{(e)}=f^{(e)}$$

for elements 1 and 2. The element displacement matrices can be determined from the global displacement matrix.

$$\underline{d}^{(1)}= \begin{bmatrix} 0 \\ 0 \\ 4.352 \\ 6.127 \\ \end{bmatrix}

\underline{d}^{(2)}= \begin{bmatrix} 4.352 \\ 6.127 \\ 0 \\ 0 \\ \end{bmatrix} $$