Thin Elastic Beam Research Articles

Computation of dynamic deflection in thin elastic beam via symmetries

The deflection profiles governed by Euler Bernoulli's fourth-order equations under varied applied loads are investigated in this research. This study provides essential insights for engineers designing aircraft components, bridges, and similar structures, ensuring system safety and efficiency. The investigation emphasizes critical factors such as amplitude and frequency, load history, and material properties. Initially, conservation laws of the equations with applied loads are derived by expressing them in the Euler-Lagrange form, where the resultant conservation laws satisfy the divergence expression. The association between symmetries and conservation laws is demonstrated, followed by the application of double reduction theory, which reduces both the variables and the order of the equation. Graphical representations of the outcomes illustrate the impact of load variations on the beam's deflection profiles. These visual aids facilitate a deeper understanding of the influence of different loading conditions. A comparison between varying loads is presented, showcasing the impact of these variations on structural behavior. The findings are crucial for enhancing structural design and ensuring safety under varied loading conditions, showcasing the novelties in the analytical approach and the practical applications of the derived results.

End-rotation-driven kinetic elastica structures: Concept and morphology design

Euler’s elastica, a family of planar curves describing the equilibrium shapes of a thin elastic beam subjected to end forces and moments, demonstrates both geometric versatility and mechanical functionality, leading to its diverse applications across multiple disciplines. Despite its roots in ancient structural engineering practices, the kinetic potential of Euler’s elastica remains largely unexplored in this domain. To address this gap, this paper proposes the concept of end-rotation-driven kinetic elastica (ERKE) structures. This novel kinetic form consists of a sequence of discrete elastica arches transformed from initially straight members by elastic bending. The kinetic behavior arises from manipulating the end rotations of each elastica arch, prompting a continuous flow of motions resulting from adaptations in the arch shape. This adaptive quality opens up limitless possibilities for creating visually captivating dynamic patterns in the overall structure by coordinating the motions of individual arches. As the inaugural exploration of ERKE structures, this paper centers around their conceptualization, with an emphasis on morphology design, which refers to the creation of desirable dynamic patterns. To this end, the paper provides an exposition of the general design procedure and criteria, along with demonstrative examples. The paper also envisions potential application scenarios and identifies research needs, paving the way for future investigations.

Simulations of interaction between flow and a thin elastic beam using CIP-based model with adaptive mesh

The simulation of interaction between flow and the elastic structure is a complicated fluid-solid interaction problem, involving the structure's large deformation and fluid interface variation. In this paper, an adaptive CIP (constrained interpolation profile)-based numerical model has been presented to simulate this problem. In this CIP-based model, the constrained interpolation profile method, a high-order finite difference method, is employed as the base flow solver and THINC/SW (Tangent of Hyperbola for Interface Capturing with Slope Weighting) method, a volume of fluid method, is applied to capture the flow interface. IBM (immersed boundary method) method is used to treat the fluid-solid interface. The body deformation is calculated by coupled FDM-FEM (finite difference method and finite element method) method. Adaptive mesh is generated by Afivo (Adaptive Finite Volume quadtree/Octree mesh) framework. The cases of flow past a square and water entry of a symmetric solid wedge are carried out to validate the reliability of the flow solver in this present model. Cantilever beam deformation case is used to validate the accuracy of the solid solver in the present model. Then cases of a thin elastic beam in sloshing tank and dam-break flow are carried out by using this model. After that, the case of regular wave past a membrane breakwater is studied. Our numerical results are compared with previous experimental and numerical studies, and good agreements are obtained. It is demonstrated that this present numerical model has good applicability to simulate the interaction between flow and a thin elastic beam.

Экспериментальное и аналитическое исследование геометрически-нелинейного изгиба консоли под действием распределенной нагрузки гравитационного типа

This paper describes an approximate analytical solution for the geometrically nonlinear bending of a thin elastic cantilever beam under a uniformly distributed gravity load. The solution is based on the linearized Euler-Bernoulli equation of mechanics of materials. Traditionally, such a linear approach is used for small (geometrically linear) deflections. The authors have modified the original equation with an arc-length preservation condition. The modified solution allows one to obtain bending shapes, deflection, and axial displacement in the range of loads corresponding to geometrically nonlinear bending of a beam (large deflections). An experimental study is conducted to verify the proposed solution. A thin steel band bent by gravity is used as a sample. Changes in the length of the bent sample part allow one to obtain various dimensionless load parameters. The deflections and axial displacements averaged on experimental statistics are determined. Bending shapes are obtained by the least square method of 5th order. Experimental and theoretical data are shown to be in good agreement. This fact confirms that the approximate analytical solution can be applied to solve large deflection problems in a wider range of loads than normally considered in the original linear theory.

A review on buckling and postbuckling of thin elastic beams

This paper provides a review of models and solutions for the buckling and postbuckling of beams available from the 50s of the last century to date. Beams with axially unrestrained (movable) ends and restrained (immovable) ends are covered. In each class, the formulation of the nonlinear buckling problem for the buckling loads and the postbuckling states is discussed and the underlying assumptions are highlighted. For relatively large-amplitude buckling of beams with movable ends, approximate analytical solutions up to the third order are presented and compared with the exact solutions expressed in terms of elliptic integrals. For beams with immovable ends, buckling involves midplane stretching that makes the nonlinear buckling problem takes the same form of the standard eigenvalue problem and, hence, exact solutions are affordable. This review combines the research outcomes on buckled beams from two scientific viewpoints: the structural dynamics and the nonlinear vibration viewpoints, respectively. Moreover, it presents in one place the formulation and the exact solutions for the buckling of beams to serve as an on-demand resource for researchers concerned with the buckling and postbuckling of beams.

A gradient reproducing kernel based stabilized collocation method for the static and dynamic problems of thin elastic beams and plates

A gradient reproducing kernel based stabilized collocation method for the static and dynamic problems of thin elastic beams and plates

Infinitely many periodic solutions for a semilinear Euler-Bernoulli beam equation with variable coefficients

We consider the periodic solutions for a semilinear Euler-Bernoulli beam equation with variable coefficients, which is used to describe the infinitesimal undamped transverse vibration of a thin straight elastic beam in a plane. The presence of variable coefficients leads to the destruction of spectral separability, which implies a loss of compactness on the range. By translating the spectrum, we construct a suitable function space which plays a crucial role in this paper. On this basis, we establish a theorem on the existence of infinitely many periodic solutions for the nonlinearity satisfying sublinear growth.

Exponential Stability for a Nonlinear Timoshenko System with Distributed Delay

This paper is concerned with a nonlinear Timoshenko system modeling clamped thin elastic beams with distributed delay time. The distributed delay is defined on feedback term associated to the equation for rotation angle. Under suitable assumptions on the data, we establish the exponential stability of the system under the usual equal wave speeds assumption.

LaCoDe: A Lagrangian two-dimensional thermo-mechanical code for large-strain compressible visco-elastic geodynamical modeling

We present LaCoDe (Lagrangian Compressible Deformation), a MATLAB solver for the Stokes equations for compressible non-Newtonian visco-elastic flow in two dimensions, based on a Lagrangian formulation of the Finite Element Method. The incompressible Boussinesq approximation is a widespread assumption in numerical models of lithospheric deformation, thus potentially masking a significant contribution of mechanisms linked to volumetric changes that occur in the asthenospheric mantle and the lithosphere. LaCoDe employs a compressible formulation of the Stokes equations designed to address such volume-changing processes. First, we provide a description of the equations governing the deformation of Earth rocks and detailed overview of the algorithm, its numerical implementation, treatment of the non-linearities rising from the compressible formulation and ineleastic deformation, and the remeshing algorithm that tracks and transfers the physical fields from a highly-distorted mesh to a high-quality one. LaCoDe is then benchmarked by comparing numerical results to analytical solutions for the bending of a thin elastic beam under a constant uniform load, flow around a rigid inclusion, Rayleigh-Taylor instability, stress build-up in a visco-elastic Maxwell body, and Couette flow with viscous heating. The Rayleigh-Taylor instability test is further used to demonstrate the accuracy of the remeshing algorithm. The importance of including volumetric strain for geodynamic processes is illustrated by two numerical experiments: i) volumetric-strain inducing phase changes in amagmatic slow-spreading ridges, and ii) subducting slabs.

Curvature-based, time delayed feedback as a means for self-propelled swimming

The development of bio-inspired robotics has led to an increasing need to understand the strongly coupled fluid–structure and control problem presented by swimming. Usually, the mechanical forcing of muscles is modeled with an imposed distribution of bending moments along the swimmer’s body. A simple way to exploit this idea is to define a central pattern forcing for this active driving, but this approach is not completely satisfactory because locomotion results from the interaction of the organism and its surroundings. Gazzola et al. (2015) have proposed that a curvature-based feedback with a time delay can trigger self-propulsion for a swimmer without necessitating such a pre-defined forcing. In the present work, we implement this feedback within a numerical model. We represent the swimmer as a thin elastic beam using a finite element representation which is coupled to an unsteady boundary element method for the resolution of the fluid domain. The model is first benchmarked on a flexible foil in forced leading edge heave.To recover previous findings, an imposed traveling bending moment wave is then used to drive the swimmer which yields peaks in the mean forward velocity when the driving frequency corresponds to the natural frequencies of the elastic structure. Delayed, curvature-based feedback is then applied to the swimmer and produces peaks in the velocity for delays that differ from the natural periods, associated to its deformations modes. Finally, a simplified model is shown to qualitatively describe the origin of the peaks observed in the feedback swimmer.

Pull-in Effect for Micro Beam with Tensile Force

Recent time the development and achievement of micro- and nano-electromechanical systems (MEMS and NEMS) are appeal the great interest of physics, biologists, engineers-electricians. The designing of MEMS based on pull-in effect consists in interaction of electrostatic field with thin elastic conductive beam. This interaction leads to pull-in instability – the effect of collapse of two initially parallel conductive layers, which play the role of capacitor. The important significance of MEMS have been acquired [1, 2] such, for example, as micro-switches with forward or rotary movement. These devices may be membrane else cantilever or another type, also high speed rotational actuator – contactless micro-gyroscope.

The asymptotic analysis of a junction of two elastic beams

AbstractThis paper is devoted to describing the asymptotic behavior of a structure consisting of thin elastic planar beams coupled at flexible joints. As the thickness of the beams tends to zero, we establish classical 1‐D beam equations for each individual structural element and transmission conditions across the joints.

Nonlinear Resonances of Chains of Thin Elastic Beams With Intermittent Contact

This paper deals with the forced response analysis of chains of thin elastic beams that are subject to periodic external loading and frictionless intermittent contact between the beams. Our study shows that the beams show nonlinear resonances whose frequencies are the same as the linear resonant frequencies if all the beams have the same stiffness. Furthermore, it is also shown that small gaps between the beams and small deviation or mistuning in the stiffness of each beam can cause drastic changes in the nonlinear resonant frequencies of the system dynamics. The system is modeled as a semidiscrete system of piecewise-linear oscillators with multiple degrees-of-freedom (DOF) that are subject to unilateral constraints, which is derived from a finite element discretization of the beams. The resulting equations of motions are solved by a second-order numerical integration scheme, and steady-state solutions are sought for various driving frequencies. Results of parametric studies with respect to the gaps between the beams and the number of beams are presented to discuss how these parameters affect the resonant behavior of the system.

Experimental Analysis of Lateral Torsional Buckling of Beams with Selected Cross-Section Types

Currently used design methods for solution of lateral torsional buckling are based on critical moment approach. This solution is also implemented in design codes but it is limited for use of single symmetric cross section only. The analytical solution of mono symmetric cross-section by critical moment is patterned on the theory of thin walled elastic beams by Vlasov [1]. In spite of the fact that this theory is known for many decades, the practical elaborated approach for solution of asymmetrical cross section is still the actual problem to solve.Nowadays the FEM based numerical models are the powerful instrument for detailed analysis of such kind of structure. However, there are still problems with definition of all imperfections and boundary conditions. Also the modelling of load effects during the loading procedure remains usually difficult. The most valuable source of information is the experiment.The goal of research which is presented in this paper is the experimental and theoretical analysis of cross section which can be asymmetrical. Particularly there are experimental test of several beams of selected cross-section shapes described and analysed. The main idea of those experiments was to observe the differences in behaviour between single symmetric cross-section and the slightly asymmetric one. Results of those experiments will serve as a base for further theoretical part of the research.

Effective wave propagation along a rough thin-elastic beam

Two methods for computing the complex-valued effective wavenumber of a rough beam in the context of linear time-harmonic theory are presented. The roughness of the beam is modelled as a continuous random process of known characteristic length and root-mean-square amplitude for either the beam mass or the beam rigidity. The first method is based on a random sampling method, with the effective wave field calculated as the mean of a large ensemble of wave fields for individual realisations of the roughness. The individual wave fields are calculated using a step approximation, which is validated for a deterministic problem via comparison to results produced by an integral equation approach. The second method assumes a splitting of the length scale of the fluctuations and an observation scale, employing a multiple-scale approximation to derive analytical expressions for the effective attenuation rate and phase change. Numerical comparisons show agreement of the results of the random sampling method and the multiple-scale approximation for a wide range of parameters. It is shown that the effective wavenumbers only differ by a real constant between the cases of varying beam mass and rigidity.

Asymptotic Analysis of an L-Shaped Junction of Two Elastic Beams

We propose a one-dimensional asymptotic model of an L-shaped junction of two thin two-dimensional elastic beams subject to boundary conditions of different type at external bean ends. The beams are described by standard systems of the Kirchhoff ordinary differential equations, whereas the transmission conditions on coincident (internal) nodes essentially depend on the type of boundary conditions on the external beam ends. The transmission conditions are obtained by analyzing the boundary layer near internal nodes. The asymptotics is justified with the help of a weighted anisotropic Korn inequality.

Global existence and exponential stability for a nonlinear Timoshenko system with delay

This paper is concerned with a nonlinear Timoshenko system modeling clamped thin elastic beams with time delay. The delay is defined on a feedback term associated to the equation for rotation angle. Under suitable assumptions on the data, we establish the well-posedness of the problem with respect to weak solutions. We also establish the exponential stability of the system under the usual equal wave speeds assumption.

Mechanics of relative and absolute displacements across normal faults, and implications for uplift and subsidence along the eastern escarpment of the Sierra Nevada, California

The magnitude of late Cenozoic rock uplift of the Sierra Nevada (California, USA) remains unresolved despite more than a century of investigation, with estimates ranging from essentially zero to ∼3 km of uplift at the range crest. Two sets of two-dimensional end-member mechanical models bracket how normal faulting along the eastern escarpment of the Sierra Nevada contributed to uplift of the range over a time span of millions of years. The short-term models are based on dislocations in an elastic half-space. The long-term models involve thin elastic beams resting on an inviscid fluid. Both sets of models predict that if the regional topography were entirely a response to faulting along the eastern escarpment, then the bedrock floors immediately east of the range should consistently lie thousands of meters below sea level, instead of thousands of meters above sea level as they generally do. Both sets of analyses indicate that although faulting would lift the range crest, it would drop the rock east of the range-front faults at least as much, and perhaps much more; model results suggest that ∼66%–85% of the current escarpment relief stems from subsidence of the grabens east of the Sierra Nevada, with only ∼15%–34% resulting from crestal uplift. Our results strongly indicate that range-front faulting in the last 3–10 m.y. uplifted rock at the Sierra Nevada crest by hundreds of meters to as much as 1 km, and that this uplift was superposed on high topography that predated the origin of the eastern escarpment. These conclusions are compatible with diverse geologic observations and measurements.

전자기기 냉각용 압전팬의 열전달 향상

Piezoelectric fans are thin elastic beams whose vibratory motion is actuated by means of a piezoelectric material bonded to the beam. These fans have found use as a means to enhance convective heat transfer while requiring only small amounts of power. This study presents new types of models with heat sink having air passage and investigates experimentally their heat transfer characteristics. From the comparison results for four models, the heat transfer coefficients of model 1 are approximately 44~66% higher than those of the reference model 0. The model 1 show the best overall performance about heat transfer and cooling capability. As shown in above results, it is necessary to design the heat sink with air pass for cooling of electronic devices, in order to increase the convective heat transfer coefficient of a piezoelectric fan for electronic cooling.

A structural gradient theory of torsion, the effects of pretwist, and the tension of pre-twisted DNA

A structural gradient theory of torsion of thin-walled beams is developed. A non-local estimate of the mean value of the angle of twist of the beam leads to a shear gradient that is energetically consistent with a bi-moment, in the spirit of the averaging theory of Vardoulakis and Giannakopoulos (2006). The geometric details of the cross section play the role of the microstructure of the beam, introducing a size effect in the torsion problem. The appropriate boundary conditions are derived from the variational formulation of the problem. The proposed gradient elasticity theory is identical to Vlasov’s torsion theory of thin walled elastic beams. The tension of pre-twisted DNA is analyzed at high axial loads, where enthalpic elasticity prevails. A size effect is naturally introduced, indicating that shorter DNA lengths lead to stiffer response in torsion. It is shown also that the complete unwinding of DNA triggers the debonding of its strands.