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Elastic instability, long considered mainly as a failure limit state or a safety guard against ultimate failure is gaining increased interest due to the development of active and controllable structures, and the growth in computational power. Mode jumping, or snap-through, during the postbuckling response leads to sudden and high-rate deformations...

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... with the dissipated energy from the equilibrium path transitions. Research by [20] has shown that the geometry of a cylindrical shell dictates their buckling response, and that obtaining multiple local buckling patterns requires cylinders with a small length to radius ratio ( L / R ) and a large radius to thickness ( R / t ) ratio. Thus, the base cylinder used for this study had an effective length of 203 mm and internal diameter of 203 mm. The thickness varied from 1.32 mm to 0.28 mm depending on the material design as described in the following sections. Anisotropic coupling effects in composite laminates are known to reduce the buckling capacity of cylindrical shells. Coupling terms are thus generally avoided through special laminate designs. However, anisotropic coupling can provide useful response characteristics. York [21] provides a summary on benchmark configurations of composite laminates that indicate how new and emerging applications may require coupled laminates. Thus, coupling terms may help induce local mode jumps and trigger a desirable equilibrium path in the postbuckling response. Eight 4-ply CFRP cylinders with laminate stacking sequences exhibiting different coupling behavior, see Table 1, were considered. The cylinders were assumed to be made from unidirectional carbon/epoxy tape with a ply thickness of 0.1397 mm and ply properties of: E 11 =144.8 GPa; E 22 =9.655 GPa; G 12 =5.862 GPa; ν 12 = 0.25. The first buckling mode for Cases 1, 3, 5 and 6 (see Table 1) shown in Figure 4 illustrates the influence of laminate stacking sequence on buckling response. Figure 5 shows the axial load vs. end shortening response of the same designs to illustrate their predicted postbuckling behavior. Table 1 compares the postbuckling response for all cases in terms of the first buckling load ( P1 ), the first load jump (Δ P1 ), the end shortening gap between the first two mode jumps ( δ1 ), the dissipated energy ( A ), and the number of mode transitions ( k ). A pilot study to evaluate the effects of locally varying material stiffness on the postbuckling response of cylindrical shells is presented here. Typically, local variation of geometry for cylindrical shells refers to stiffeners/ribs along the surface or holes/bumps on the surface; which have a significant effect on buckling load. As noted on the discussion of imperfection sensitivity, the postbuckling response is driven by the extent of imperfections and their amplitude. The aim here is to induce a similar effect through artificial changes on the material properties and their distribution on the cylinder such that dominant buckling modes can be triggered. A set of eight cylinders (Table 2) with ad-hoc patterned material distributions were considered. The patterns were empirically chosen upon evaluation of the eigenshapes observed from linear and nonlinear buckling studies on isotropic and laminated cylinders. The approach to the material layout distributions consisted in providing regions of locally varying (softer) material stiffness. The patterned material distributions were grouped in four categories as shown in Figure 6. For this study the material properties of the base cylinder were assumed to be isotropic with E =14 GPa and = 0.25, while the locally varying material patches (red zones) were assumed to have half the stiffness of the base material. The local material variation allowed the nonlinear postbuckling numerical simulations to be conducted without the need of seeding imperfections. Further, numerical convergence was not a problem for these cases and thus there was no need to use numerical ...

## Citations

... They used an analytical approach in their study and observed that peak tensile hoop stresses and radial expansion of the cylinder inner walls could be substantially reduced. Burgueño et al. [2] carried out a numerical study on the post-buckling response of cylindrical shells under axial compression for application in smart structures. Damanpack et al. [3] studied the axisymmetric thermo-mechanical behavior of Epoxy/ Aluminum matrix composite cylinders reinforced with SMA fibers under internal pressure, axial, torsional, and thermal loading based on the micro-macro method. ...

... The SMA's Young modulus is assumed to be based on the Reuss definition [15] as: (2) where and are the Young modulus of SMA in austenite phase and martensite phase, respectively. The tensile recovery stress of SMA during phase transformation can be determined by using the simple Brinson's model [16] as: ...

Shape memory alloys (SMA) are often implemented
in smart structures as the active components. Their ability to recover
large displacements has been used in many applications, including
structural stability/response enhancement and active structural
acoustic control. SMA wires or fibers can be embedded with
composite cylinders to increase their critical buckling load, improve
their load-deflection behavior, and reduce the radial deflections under
various thermo-mechanical loadings. This paper presents a semianalytical investigation on the non-linear load-deflection response of
SMA-reinforced composite circular cylindrical shells. The cylinder
shells are under uniform external pressure load. Based on first-order
shear deformation shell theory (FSDT), the equilibrium equations of
the structure are derived. One-dimensional simplified Brinson’s
model is used for determining the SMA recovery force due to its
simplicity and accuracy. Airy stress function and Galerkin technique
are used to obtain non-linear load-deflection curves. The results are
verified by comparing them with those in the literature. Several
parametric studies are conducted in order to investigate the effect of
SMA volume fraction, SMA pre-strain value, and SMA activation
temperature on the response of the structure. It is shown that suitable
usage of SMA wires results in a considerable enhancement in the
load-deflection response of the shell due to the generation of the
SMA tensile recovery force.

... In these configurations, the ambient oscillations excite the beam laterally and the axial load or the axial displacement is constant. The axial load is directly used as a source of excitation in [21][22][23][24]. In this configuration, the successive buckling of the piezoelectric beam can result in a frequency up-conversion mechanism. ...

... The next step is to check if the weight is more than the first critical buckling load of the piezoelectric beam. In order to calculate the first critical buckling force, the first vibration mode shape is considered as f(x) in the equation (23). Similarly, for calculating the nth buckling critical force, f(x) in equation (23) is replaced by the nth mode shape. ...

... First and second critical buckling forces are shown for different thicknesses and lengths of piezoelectric layers in figure 6. For a uniform simply supported beam, the critical buckling force is calculated using equation (23). As the equation shows, the second critical buckling force in a uniform beam is four times the first critical buckling force. ...

A piezoelectric vibration energy harvester is presented that can generate electricity from the weight of passing cars or crowds. The energy harvester consists of a piezoelectric beam, which buckles when the device is stepped on. The energy harvester can have a horizontal or vertical configuration. In the vertical (direct) configuration, the piezoelectric beam is vertical and directly sustains the weight of the vehicles or people. In the horizontal (indirect) configuration, the vertical weight is transferred to a horizontal axial force through a scissor-like mechanism. Buckling of the beam results in significant stresses and, thus, large power production. However, if the beam's buckling is not controlled, the beam will fracture. To prevent this, the axial deformation is constrained to limit the deformations of the beam. In this paper, the energy harvester is analytically modeled. The considered piezoelectric beam is a general non-uniform beam. The natural frequencies, mode shapes, and the critical buckling force corresponding to each mode shape are calculated. The electro-mechanical coupling and the geometric nonlinearities are included in the model. The design criteria for the device are discussed. It is demonstrated that a device, realized with commonly used piezoelectric patches, can generate tens of milliwatts of power from passing car traffic. The proposed device could also be implemented in the sidewalks or integrated in shoe soles for energy generation. One of the key features of the device is its frequency up-conversion characteristics. The piezoelectric beam undergoes free vibrations each time the weight is applied to or removed from the energy harvester. The frequency of the free vibrations is orders of magnitude larger than the frequency of the load. The device is, thus, both efficient and insensitive to the frequency of the force excitations.

... Subsequently, the shell repeated consecutive snap-through buckling (secondary buckling) as the number of waves decreased one by one in the circumferential direction as the compression progressed [5]. Mode jumping or snap-through and snap-back during the post buckling response leads to sudden and high-rate deformations due to generally smaller changes in the controlling load ordisplacement input to the system [12]. ...

Present paper observed the postbuckling behavior of welded box section bridge compression members using static and response analysis. Parametric study using beam and shell varied in width-thickness and slenderness ratio compared with full truss bridge beam model were used. Results indicated that for design based purposes, beam element disregarding initial imperfection factors is still suitable. But for analysis based purposes, which need the capability to perform real structure behavior and to explore the postbuckling regime, shell is the best choice as it can perform more detail compression members behavior and has more severe strength reduction in postbuckling regime, especially.

... The design of the mechanical energy concentrator follows the test setup reported in [21]–[23] in which a polycarbonateTable I) with fixed end supports is placed between rigid continuous bilateral plexiglass frame as shown inFig. 5(top). ...

... This translates into a calibrated system that records only these transitions. The difference in voltage output between transitions is due to the levels of stored strain energy in the system before the transition events and can be tuned and calibrated as shown experimentally in [21]–[23].Fig. 15 shows the recorded voltage variations at the linear injector. ...

Changes in physical processes like ambient temperature or pressure variations occur at frequencies that are significantly lower than 1 Hz. This poses a challenge for designing self-powered sensors that monitor these quasi-static physical processes and at the same time scavenge operational energy for sensing, computation, and storage from the signal being monitored. In this paper, we present a novel paradigm for designing a self-powered sensor/data logger that exploits the physics of negative-stiffness mechanical energy concentrators with the physics of our previously reported piezoelectricity driven impact ionized hot-electron injection (p-IHEI)-based sensors. The operational principle is based on the sudden transitions from unstable mode branch switching during the elastic postbuckling response of slender columns, which are used to generate high-frequency deformations as an input to the p-IHEI-based sensor. The experimental results demonstrate that the proposed self-powered sensor based on an integrated circuit fabricated in a 0.5-μm CMOS technology can count and record the number of quasi-static input events with frequencies spanning less than 1 Hz.

... A pilot experimental study [40] showed that the postbuckling mode shapes were always asymmetric due to the high sensitivity to localized imperfections on the shell surface. Thus, the biased mesh seed method [25] was used to generate a non-uniform (size) mesh in the finite element model such that asymmetric buckling mode shapes would be obtained from the eigenvalue analysis. ...

... The 2-ply design was chosen to obtain a shell with relatively flexible axial stiffness in order to obtain the desired postbuckling response. The laminate sequence was chosen based on numerical studies that evaluated the effects of coupling stiffness terms in laminated composites on triggering multiple mode jumps in the postbuckling response of cylindrical shells [40]. The presence of extension-twist coupling terms (B 16 , B 26 ) was found to be desirable for the sought-after behavior. ...

... The design of the hybrid carbon/E-glass/epoxy cylindrical shells was inspired by the use of stiffeners for improving the loadcarrying capacity of shells. Linear and nonlinear buckling analyses on isotropic cylinders [40] showed that mode jumps in the postbuckling regime were more likely to occur when the seeded imperfection mode shapes had a relatively small number of waves in the longitudinal direction but a large number of waves in the circumferential direction. Thus, a hybrid cylinder concept was developed by using a chopped E-glass fiber mat with spaced unidirectional carbon/epoxy strips. ...

Buckling is a major structural instability of composite material that can lead to failure of large component without initial failure. The Finite Element Simulation is performed on CFRP, GFRP and KFRP composite material using COMSOL Multiphysics® and the buckling factor is compared with these composite materials. It is performed with a linear analysis of a composite cylinder under compressive loading and fixed end conditions. The composite materials are made up of eight layer (plies) of a composite. An Equivalent Single Layer (ESL) based approach is used for this analysis. The stacking sequence of composite materials are Layered 1[0/0/45/−45], Layered 2 [90/90/45/−45], Layered 3[90/0/90/0] and Layered 4 [45/45/45/45]. The critical buckling load factor is compared among the three materials with 4 layered sequence. Based on the Finite Element Simulation Result. The Kevlar Fiber Reinforced Polymer has considerable buckling load stability factor compared with other material. It is proposed that the KFRP material is best suitable for larger applications.

In this paper, an analytical investigation on linear mechanical buckling
of shape memory alloy (SMA) composite cylindrical shell reinforced
with eccentrically stiffeners subjected to axial compression is presented.
The equilibrium equations of the shell are derived based on Donnell's
shell theory. The effect of SMA recovery force is captured by using 1-D
simplified Brinson’s model due to its simplicity and accuracy. The
Lekhnitsky smeared stiffeners technique and Galerkin method are
implemented in order to obtain linear mechanical buckling load. The
obtained results are verified by comparing them with those in the
literature. Several parametric studies are conducted to examine the effect
of SMA volume fraction, SMA pre-strain value, and activation
temperature on the buckling response of the composite cylindrical shell.
Due to the generation of tensile recovery force, significant enhancement
in buckling load is observed by increasing each of the SMA fiber volume
fraction, SMA pre-strain value, and activation temperature.