18.6 1D Truss Element

18.6.1 Natural coordinate system

For any element, we can define a natural coordinate system. For example: \[\zeta\left(x\right)=\frac{2\,\left(x-x_1\right)}{x_2-x_1}-1\]

Thus, we can introduce interpolation functions: \[N_1\left(\zeta\right)=\frac{1-\zeta}{2}\] \[N_2\left(\zeta\right)=\frac{\zeta+1}{2}\]

Which allow us to write a displacement interpolation.

The unknown displacement field within an element will be interpolated by a linear distribution. \[u\left(x\right)=q_2\,N_2\left(\zeta\left(x\right)\right)+q_1\,N_1\left(\zeta\left(x\right)\right)\]

Thus, we define a set of linear shape functions:

Note also that: \[x\left(x\right)=x_2\,N_2\left(\zeta\right)+x_1\,N_1\left(\zeta\right)\]

Since this is a Rayleigh-Ritz approxation

• The first derivative of the shape function must be finite

• The displacements must be continous across the element boundary