2 Experimental Path Following of Unstable Static Equilibria for Snap-Through Buckling 19 Fig. 2.2 Load frame with beam in unloaded configuration side control points. Once convergence was achieved, the loads and displacements at the three control points were saved. By measuring the tangent stiffness experimentally, imperfections in the setup were accounted for. Despite the power of this control system, it has its limitations. As the tangent stiffness becomes singular at limit and bifurcation points, the Newton-Raphson scheme breaks down, limiting where equilibrium points can be found on both the stable and unstable paths. In addition, by monotonically increasing the displacement of the center actuator, reversals along the force-displacement curve will be lost. Another numerical method commonly used in models for snap-through buckling— the arc length method—was investigated but was not suitable. It is able to follow reversals and traverse singular points, but it relies on being able to control load independently of displacement. With the current load frame, the two cannot be decoupled, so an alternative numerical method is needed. 2.3 Results In Fig. 2.4, the force-displacement curve of a snap-through test is shown. The primary branch was obtained by displacement control of only the midspan control point, while the higher-order branch was obtained using the control system described above with three control points. One limit point for the higher-order branch occurs around ymid D 9 mm. Since the control system was not able to converge near this point, the data here is missing. Experimentally following the higher-order branch stopped when the beam displaced into an asymmetric configuration around ymid D4 mm. Theoretically, the higher-order branch would continue downwards and loop around into other higher-order paths, but that would require more than three control points to stabilize. A reflection of the higher-order branch could be followed starting at ymid D8 mm and decreasing in displacement, but it was not followed in this experiment. In Fig. 2.5, the displaced shapes at ymid D 0.75 mm on the primary path and the higher-order symmetric path are shown. A second stable configuration coexists with a mirror-image displaced shape. From Fig. 2.4, the unstable configuration requires a higher load than the stable configuration. The curvature in the symmetric shape is higher, resulting in higher internal moments. Both of these things put the symmetric configuration at a higher energy state. In addition, any perturbation of the symmetric shape would result in convergence to an asymmetric shape if the control points were not present.
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