User:Eml4500.f08.ateam.carr/HW6

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

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

Homework:

Book p. 159 ; Set $$ E_{1}=E_{2}=E_{3}$$, let $$A(\tilde{x})$$ be linear as previously defined, and obtain $$\mathbf{k}^{(i)}$$ and compare to expression given in the book

Where:

$$A(\tilde{x})=N_{1}^{(i)}(\tilde{x})A_{1}+N_{2}^{(i)}(\tilde{x})A_{2}$$

and

$$E(\tilde{x})=N_{1}^{(i)}(\tilde{x})E_{1}+N_{2}^{(i)}(\tilde{x})E_{2}$$

Compare general $$\mathbf{k}^{(i)}$$ from before to $$\mathbf{k}_{avg}^{(i)}$$ where,

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



Homework: Find, $$\mathbf{k}^{(i)}-\mathbf{k}_{avg}^{(i)}$$

Solution:  

Remember: recall the Mean value Theorem (MVT) and its relations to the centroid:

Mean Value Theorem: for $$\bar{x}\epsilon [a,b]$$,     $$a\leq \bar{x}\leq b$$,

$$ \int_{x=a}^{x=b}{f(x)dx}=f(\tilde{x})[b-a] $$

Where the centroid is defined as:

$$\int_{A}^{}{xdA} = \bar{x} \int_{A}^{}{dA} = \bar{x}A$$

Therefore:

$$\int_{x=a}^{x=b}{f(x)g(x)dx} = f(\bar{x})g(\bar{x})[b-a]$$ where  $$ a\leq \bar{x}\leq b$$

In general:

$$f(\bar{x})\neq \frac{1}{b-a} \int_{a}^{b}{f(x)dx}$$

$$g(\bar{x})\neq \frac{1}{b-a} \int_{a}^{b}{g(x)dx} $$