User:Eml4500.f08.delta 6.guzman/Lecture Notes 11/07

Stiffness Term


Note: Nodes do not have to be equidistant.

Plot


Note: Since you are drawing a straight line between nodes you are assuming the their displacement was linear.

Assume displacement $$ u\bigl(\begin{matrix}x\end{matrix}\bigr)$$ for $$ x_{i}\leq x\leq x_{i+1} $$ (i.e. $$x \epsilon \left [ x_i, x_{i+1} \right ]$$)

Motivation for linear interpolation of $$ u\bigl(\begin{matrix}x\end{matrix}\bigr)$$


In this plot can be easily observed that the deformed shape is a straight line confirming that there was an implicit assumption of the linear interpolation of the displacement between the two nodes. Consider the case where these are only axial displacements (i.e. 0 translational displacement).

Question: Express $$ u\bigl(\begin{matrix}x\end{matrix}\bigr)$$ in terms of $$d_{i}=u\bigl(\begin{matrix}x_{i}\end{matrix}\bigr)$$ and $$d_{i+1}=u\bigl(\begin{matrix}x_{i+1}\end{matrix}\bigr)$$ as a linear function in x (i.e. linear interpolation)

$$u\bigl(\begin{matrix}x_{i}\end{matrix}\bigr)=N_{i}(x)d_{i}+N_{i+1}(x)d_{i+1}$$

where $$N_{i}\bigl(\begin{matrix}x\end{matrix}\bigr)$$ and $$N_{i+1}\bigl(\begin{matrix}x\end{matrix}\bigr)$$ are linear functions of x.



Note: From this plot it can be easily observed that $$ N_{i+1}(x)=\frac{x-x_{i}}{x_{i+1}-x_{i}} $$.