Elastic Postbuckling Response of Bilaterally Constrained Non-prismatic Columns Suihan Liu 1 , Rigoberto Burgueño 2 1. Graduate Student Researcher 2. Professor Dept. of Civil and Environmental Engineering, Michigan State University Fig. 3. Schematic of the non-uniform design concept. Strips with three equal length segments with variations of thickness in the middle segment were considered. The thickness ti of the middle segment varied in proportion to t0 of a baseline strip design and was defined by the ratio α = ti/t0. When α is less than 1, the stiffness of the middle region is reduced; when α is greater than 1, the middle segment is stiffener compared to the reference geometry. Thus, two design groups were considered: (a) reduced stiffness group and (b) increased stiffness group. Numerical simulations were conducted using the finite element (FE) program ABAQUS. Experimental setup is shown in Fig. 4. All the test strips in this study were fabricated by a Connex350 3D printer using rigid photopolymer material. The strip was fully fixed at the bottom, and rotations and transverse translations were constrained at the top. The strip was subjected to a cycle of axial compressive load under displacement control to a target of 3.8 mm at a rate of 0.38 mm/s. Fig. 4. (a) View of a 3D printed strip test specimen (left) and (b) test setup APPROACH METHODS Fig. 1. Schematic of postbuckling response of uniform strip with/without continuous rigid constraints Axially loaded bilaterally constrained columns can attain multiple snap-through buckling events in their elastic postbuckling response for use as energy concentrators that transform external quasi-static displacement input to high-rate motions to excite vibration-based piezoelectric transducers. Fig. 2. Buckling-induced energy harvesting concept Regulation of the postbuckling behavior can lead to controlled acceleration input to the piezoelectric transducers and increased performance of the energy harvesting device. However, the geometries and material properties of the uniform column setup have limited control on the post-buckling response of the system at a given strain level. Therefore, it is of interest to develop the concept of using non-prismatic columns/strips with piece-wise variations in stiffness for controlling the buckling mode transitions in the elastic post-buckling regime. BACKGROUND MOTIVATION RESULTS • the FE analyses adequately captured the mode transitions and the initial and end response stiffness. The error is due to uncertainty in the modeling parameters, such as the actual imperfections, friction resistance between and strip and walls and the boundary conditions. • For reduced stiffness group, the end stiffness being lower for lower α values, and the number of mode jumps increased by one com-pared with the baseline. • For increased stiffness group, the change in stiffness did not significantly affect the overall stiffness, but the magnitude of the load drops increased considerably. Fig. 5. Numerical and experimental force-displacement response of case α = 0.7 (a) and case α = 1.5 (b). Force-displacement Responses Fig. 7. Buckling mode shapes of baseline and case α =0.7 Controlled Buckling Sequence & Location Fig. 6. Postbuckling responses of reduced stiffness group (a) and increased stiffness group (b). (a) (b) The mode shapes of the non- prismatic elastica is not uniform throughout the strip. The waves of the deformed shape are denser in the reduced stiffness (middle in this case) segment that superposed on the global buckling shape. RESULTS Through test observations for the non-prismatic strips, all higher order buckling modes were triggered at the stiffer regions of the strip except for the mode 1, which is invariant at the mid-span of the strip for any cases. Fig. 8. Post-buckling transition process of case α = 0.7 (a) and case α = 1.3 (b). Local Stress Wave Travelling Motions The traveling waves induced local motions largely increased the number of accelerations impulses and the maximum acceleration at the destination segment. Fig. 8. The number of acceleration impulses (a & c) and the maximum acceleration (b & d) of each segment along the non- prismatic strips compares with the baseline. SUMMARY By introducing non-uniform flexural stiffness regions: (1) The number of the global buckling mode transitions can be increased by one; (2) Localized buckling motions from traveling stress waves were generated. This significantly increased the number of valid acceleration impulses and the magnitude of accelerations; (3) A repeatable and controllable pattern of the location and sequence of the buckling events. The presented results confirm that non-prismatic columns are a viable way to control the elastic post-buckling response of these device elements. ACKNOWLEGMENTS The presented work was carried out with support from the U.S. National Science Foundation under grant number ECCS-1408506.