I'm trying to develop a finite conical shell element with six dof at each node. I want to express the rotations $\phi$, $\theta$ and $\psi$ in terms of the displacements $u$, $v$ and $w$. The displacements are functions of $x$ and $y$.
Using the geometric relations and small angle approximation, I have found $\phi\approx \tan(\phi)=\frac{dw}{dv}$, $\theta\approx \tan(\theta)=\frac{dw}{du}$ and $\psi\approx \tan(\psi)=\frac{dv}{du}$. But I'm not yet convinced that this is the way to go.
Do you know any other general approach to express the rotations in terms of the displacements?
Edit: I have attached a picture of the geometry of the cone and its coordinate system. $x$ is the meriodional coordinate along the generator of the cone which is presumed to be zero at the apex of the cone. $y$ is the circumferential coordinate.