User:Eml4500.f08.team.guan/hw61114

Evaluating and Comparing Stiffness Matrices with Constant and Varying Areas and Modulus of Elasticities
 Referring to the figure above which is the graphical representations of the shaping functions, x tilda is the coordinate transformed from the global coordinate x to be the local coordinate of the element i. This coordinate is used to describe the linearly varying area or modulus of elasticity or both. Using the figure above, the shape functions for the element i $$N_1^{(i)}(\tilde{x})$$ and $$N_2^{(i)}(\tilde{x})$$ can be derived in terms of the local coordinate and the length of the element. $$N_2^{(i)}(\tilde{x})=\frac{\tilde{x}}{L^{(i)}} \begin{cases} 0, \tilde{x}=0\\ 1, \tilde{x}=L^{(i)}\\ \end{cases} $$ $$N_1^{(i)}(\tilde{x})=\frac{-\tilde{x}+L^{(i)}}{L^{(i)}} \begin{cases} 1, \tilde{x}=0\\ 0, \tilde{x}=L^{(i)}\\ \end{cases} $$ For the case in which there is linearly varying area and linearly varying modulus of elasticity on the axial element i, the bar will have a stiffness varying a parabolic manner because multiplying the two linear functions together will yield a parabolic function. In the collapsible box below, the stiffness matrix for the this case is derived below. Now, we will set the modulus of elasticity to be constant and letting the area vary linearly with respect to the local coordinate. The stiffness matrix will be calculated and then compared with the one in the book.

Now, we will use the average modulus of elasticities and the areas to calculate the stiffness matrix then subtract this expression from the stiffness matrix of the variable area and modulus of elasticity case. This stiffness matrix will be denoted as the average stiffness matrix. Note: Recall that the mean value theorem (MVT) and its relation to the centroid of an area. MVT: $$\int_a^bf(x)dx=f(\bar{x})[b-a]$$ for $$\bar{x}\in[a,b]$$ or $$a \le \bar{x} \le b$$ The application to the centroid calculation is as follows: $$\int_AxdA=\bar{x}\int_AdA=\bar{x}A$$ Now, how is connected to the homework problem? Consider the following expressions. $$\int_{x=a}^{x=b}f(x)g(x)dx=f(\bar{x})g(\bar{x})[b-a]$$ for $$a \le \bar{x} \le b$$ but $$f(\bar{x}) \ne \frac{1}{b-a}\int_a^bf(x)dx$$ where then above expression is the average value of f. This is true for the function g(x) as well. $$g(\bar{x}) \ne \frac{1}{b-a}\int_a^bg(x)dx$$ What this essentially means is that the value resulting from the products of the mean value of the modulus of elasticity and the mean value of the area does not equal the value obtained by the integration of the parabolic stiffness function $$EA(\tilde{x})$$ along the length of the element.