Curved Cantilever Beam
The curved cantilever in Figure 1{reference-type=“ref”
reference=“fig:bathe”} was first studied by [@bathe1979large], and has
become a staple in the literature on geometrically nonlinear rods. This
example is selected to demonstrate the path-independence of the Init
interpolation. The undeformed centerline of the cantilever follows a
$45^\circ$ arc with radius $R$ given by:
A point load $\boldsymbol{F} = 600 \, \mathbf{E}_2$ is applied at the tip, i.e. at $\xi = R$. There is no closed-form solution to the problem, and it is customary to present the final displacements at the tip:
$$\Delta x_i \triangleq \mathbf{E}_i \cdot \left(\boldsymbol{x}(L) - \boldsymbol{x}_0(L)\right).$$The following parameters are used for the simulations:
$$\begin{array}{lr} % Hockling R =& 100 \\ % ,& A &= 10 \\ E =& 1000 \\ % ,& I &= 0.0833 \\ G =& 500 \\ % ,& J &= 2.16 \\ \end{array} \qquad\qquad \begin{array}{ll} % Hockling A =& 10^4 \\ I =& 10^4/12 \\ J =& 10^4/6 \\ \end{array}$$ The analysis uses a discretization with 8 linear (2-node)
elements. The analysis is performed twice for each formulation under
consideration, first with 8 equal load increments and then with 10. The
results are presented in
Table \[tab:bathe\]{reference-type=“ref”
reference=“tab:bathe”}. Only the formulations with the Init
interpolation produce the same tip displacement in both load cases,
indicating an artificial path dependence for all other variants.
{#fig:bathe width=“60%”}