User:Eml4500.f08.a-team.melvin/Lecture 34

p.31-4: N1(1) $$ (\tilde{x}) $$ = HW6

N2(1) $$ (\tilde{x}) = \frac{\tilde{x}}{L^{(i)}} = \begin{cases} 0, & \mbox{at }\tilde{x}\mbox{ = 0} \\ 1, & \mbox{at }\tilde{x}\mbox{ = L} \end{cases} $$

shape function N1(i), N2(i)

(basis)

HW6 : book p.159

set E1 = E2 = E

Let A($$ \tilde{x} $$) be linear as on p.33-5

Obtain k (i) from previous problem (p.33-5) and compare to expression given in book.

$$ \frac{E}{L^{(i)}} \underbrace{\frac{(A_1 + A_2}{2}}_{Ave. \ Area} \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix} $$ = k (i)



Next, compare the general k (i) on p.33-5 to the stiffness matrix obtained by using $$ \frac{1}{2} (A_1 + A_2) $$. Note: $$ E_1 \ne E_2 $$

$$ \frac{(E_1 + E_2)(A_1 + A_2)}{4L^{(i)}} \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix} $$ = k (i)ave.

Find k (i) - k (i)ave.

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

MVT: $$ \int_{x=a}^{x=b} f(x)\, dx = f(\overline{x})[b-a] $$

for $$ \overline{x} \ \epsilon \ [b-a] $$

$$ \epsilon \ $$= "belongs to"

$$ a \le \overline{x} \le b $$

$$ \int_{A}^{} x\, dx = \overline{x} \int_{A}^{} \,dA = \overline{x} A $$

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

$$ a \le \overline{x} \le b $$

But $$ \ f(\overline{x}) \ \ne \ \underbrace{\frac{1}{b-a} \ \int_{a}^{b} f(x) \, dx}_{Ave. \ value \ of \ f} $$

In general

$$ \ g(\overline{x}) \ \ne \ \underbrace{\frac{1}{b-a} \ \int_{a}^{b} g(x) \, dx}_{Ave. \ value \ of \ g} $$

Modify 2-bar truss code to accommodate general k (i) on p.33-5