User:Eml4500.f08.bike.viale/homework report 6

For the sixth homework, the class was asked to evaluate the function:

$$N_i(\tilde{x})=$$

Which equals:

$$N_i(\tilde{x})=\frac{\tilde{x}-\tilde{x}_{i+1}}{\tilde{x}_{i}-\tilde{x}_{i+1}}$$

Where:

$$\tilde{x} = x - x_{i}$$

Using the linear interpolation method described in an earlier lecture and defining $$\tilde{x}$$ as the length from the beginning of the interpolation to the value that is to be found.

A description of the shape function $$N_{2}^{(i)}$$ using the $$\tilde{x}$$ notation is as follows:

$$N_{2}^{(i)}(\tilde{x}) = \frac{\tilde{x}}{L^{(i)}} = \begin{cases} & \text{0 if } \tilde{x}= 0\\ & \text{1 if } \tilde{x}= L \end{cases}$$

Where the boundary conditions are provided, at $$\tilde{x} = 0$$ and $$\tilde{x} = L$$ respectively.

The class was also asked to complete the example provided in the book on page 159 while setting $$E_{1} = E_{2} = E$$ and to let $$A(\tilde{x})$$ be linear as it was described in the previous lecture. $$\mathbf{k}^{(i)}$$ is found from the problem given in the previous lecture and compared to the expression given in the book, which is:

$$\frac{E(A_{1}+A_{2})}{L^{(i)}2}\begin{bmatrix} 1 & -1\\ -1 & 1 \end{bmatrix} = \mathbf{k}^{(i)}$$

As seen from the previous lecture, this result for the stiffness matrix is identically the same.



Next, the general $$\mathbf{k}^{(i)}$$ matrix from the previous lecture is to be compared with the matrix obtained by using $$\frac{(A_{1}+A_{2})}{2}$$ and $$\frac{(E_{1}+E_{2})}{2}$$ while noting that in this case $$E_{1}\neq E_{2}$$. This produces the stiffness matrix:

$$\frac{(E_{1}+E_{2})(A_{1}+A_{2})}{4L^{(i)}}\begin{bmatrix} 1 & -1\\ -1 & 1 \end{bmatrix} = \mathbf{k}_{ave}^{(i)}$$

The average stiffness matrix is then subtracted from the general stiffness matrix to find that:

$$\frac{(2E - E_{1}+E_{2})(A_{1}+A_{2})}{4L^{(i)}}\begin{bmatrix} 1 & -1\\ -1 & 1 \end{bmatrix} = \mathbf{k}^{(i)} - \mathbf{k}_{ave}^{(i)}$$

In order to interpret this, it is useful to review the Mean Value Theorem from calculus.