User:EML4500.F08.RAMROD.B/Homework 6

=Continuous vs Discrete=

The following chart compares the continuous and discrete cases. Both cases are formulated using the principal of virtual work.

For the continuous case, the equations are valid for all $$\displaystyle w(x)$$ at $$\displaystyle w(0)=0$$ which is the boundary equation for the elastic bar. For the discrete case, the equations are valid for all $$\displaystyle \mathbf{w}$$. Here the boundary conditions are eliminated here.

Stiffness Term
the below picture depicts an elastic bar with all nodes. 

This photo depicts the element displacement more closely. 

Here we will assume the displacement $$\displaystyle u(x) $$ for $$\displaystyle x_i\leq x\leq x_{i+1}$$. This means that $$\displaystyle x \epsilon [x_i,x_{i+1}]$$ where the character $$\displaystyle \epsilon $$ means 'belongs to'.

Motivation For Linear Interpolation of $$\displaystyle u(x)$$ For The Two Bar Truss
The function $$\displaystyle u(x) $$ must be interpolated because the deformed shape is a straight line which means there must have been an assumption of displacement between two nodes. This allows a connection between the two cases where there are only axial displacements, or zero transverse displacements. The below picture shows this.

