User:Eml4500.f08.ramrod.c/11/14/08

Meeting 34 November 14, 2008
$$N_1^{(i)}(\tilde{x}) = N_2^{(i)}(\tilde{x}) = \frac{\tilde{x}}{L^{(i)}} = \begin{cases} 0 \; at \; \tilde{x}=0\\ 1 \;at \; \tilde{x}=L^{(i)}\end{cases}$$

Shape Functions N1(i), N2(i)

Comparing Value in Notes with Value in Textbook
Set E1 = E2 = E

Let $$A(\tilde{x})$$ be linear. k(1) will be obtained from the problem above and will be compared to expression given in the textbook, which uses the average area.

From Above: $$\mathbf{K}^{(i)}(\tilde{x})=\frac{\left[N^{(i)}_1(\tilde{x})E_1+N^{(i)}_2(\tilde{x})E_2 \right]\left[N^{(i)}_1(\tilde{x})A_1+N^{(i)}_2(\tilde{x})A_2 \right]}{L^{(i)}}\begin{bmatrix} 1 &-1 \\ -1& 1 \end{bmatrix}$$ From Book: $$\frac{E}{L^{(i)}}\frac{(A_1+A_2)}{2}\begin{bmatrix} 1 &-1 \\ -1 & 1 \end{bmatrix} = \mathbf{k}^{(i)}$$

$$\frac{(A_1+A_2)}{2}$$ = Average Area



Comparing General Value with Average Value
Next, compare the general k(i) from above to the stiffness matrix obtained by using 1/2(A1+A2) and 1/2(E1+E2)
 * Note: $$ E_1 \neq E_2$$

General Value: $$\mathbf{k}^{(i)}=\frac{EA}{L^{(i)}}\begin{bmatrix} 1 &-1 \\ -1 & 1 \end{bmatrix}$$

Average Value: $$\mathbf{k}_{ave}^{(i)}=\frac{(E_1+E_2)(A_1+A_2)}{4L^{(i)}}\begin{bmatrix} 1 &-1 \\ -1 & 1 \end{bmatrix}$$ Find k(i) - k(i)ave $$\mathbf{k}^{(i)}-\mathbf{k}_{ave}^{(i)} = \frac{EA}{L^{(i)}}\begin{bmatrix} 1 &-1 \\ -1 & 1 \end{bmatrix} \; - \; \frac{(E_1+E_2)(A_1+A_2)}{4L^{(i)}}\begin{bmatrix} 1 &-1 \\ -1 & 1 \end{bmatrix} = \frac{4EA-(E_1+E_2)(A_1+A_2)}{4L^{(i)}}\begin{bmatrix} 1 &-1 \\ -1 & 1 \end{bmatrix}$$

Remark: Mean Value Theorem
Recall the Mean Value Theorem (MVT) and its relation to the centroid.

MVT: $$\int_{x=a}^{x=b}{f(x)dx} = f(\bar{x})[b-a]$$
 * for $$\bar{x} \;in \;interval \;[a,b]$$ (that is $$a \leq \bar{x}\leq b$$)

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


 * $$a\leq\bar{x}\leq b$$

But in general:


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


 * $$\frac{1}{b-a}\int_{a}^{b}{f(x)dx}$$  is the average value of f


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


 * $$\frac{1}{b-a}\int_{a}^{b}{g(x)dx} $$ is the average value of g



Modifying 2-Bar Truss Code
The 2-bar truss code is modified to accommodate for the general stiffness, k(i)