Elastic Postbuckling Response of Bilaterally Constrained Non-prismatic Columns Suihan Liu1 , Rigoberto Burgueño2

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.