Elastic Beam Research Articles

Stability of a Multi-Span Current-Carrying Beam Resting on Periodic Supports and Exposed to an External Magnetic Field

Abstract The stability of a current-carrying beam in an external magnetic field is studied in this paper. The beam rests on periodic supports and is modeled as an Euler–Bernoulli elastic beam. Based on the linearized stability equation and propagator transfer matrix approach for a multi-span finite-length beam, the stability equation and solutions of the several boundary value problems are obtained. While the Floquet–Bloch theory is widely used in the dynamic problem of phononic and photonic structure conditions, its application to the solution of infinite length beam static stability problem is novel. The stability values of the periodic infinite beam model are in very good agreement with one of the finite-length multi-span beams.

An Improved Winkler Foundation Modulus for a Beam in a Full Space

The Winkler foundation modulus is key to evaluating the response of underground structures using the elastic foundation beam model. In this paper, an improved formula of the Winkler foundation modulus for a beam embedded in a full space is proposed to overcome the limitation of inconsistent assumptions in previous studies. To achieve this goal, the bending responses of the beam are obtained using the elastic foundation beam model and three-dimensional elastic continuum model, respectively, wherein a consistent assumption is proposed that tangential interactions at the beam–ground interface are ignored in the two models. In addition, as deformation of the site is an important source of the underground structure response, the beam is applied to standard soil displacement of the free field on the Winkler foundation to improve the accuracy of the Winkler modulus obtained by fitting solutions based on the concentrated force on the beam. The formula for the Winkler foundation is obtained by equating the first zero of the bending moment in the two models. The Winkler foundation modulus is verified by comparing the results with numerical solutions and previous studies.

Farklı Zemin Sınıfına Göre Elde Edilmiş Düşey Elastik Tasarım Spektrumlarının Yapı Davranışına Etkisi ve K.Maraş Deprem Verileri ile Karşılaştırılması

In practice and theory, many types of reinforced concrete slabs are used, such as beamed, beamless, and ribbed slabs. Beamed floors are connected to the columns by means of beams, and their vertical and horizontal loads are transferred to the columns through these beams. Therefore, beams are accepted as an important structural element by earthquake experts and many formulas and approaches have been developed in terms of horizontal and vertical load effects that occur in order to transfer the load transfer to the elements as desired. In this study, beam shear forces that occur under the influence of vertical earthquakes at the i and j ends of the beams of a fictional building whose floors are modeled according to the rigid diaphragm assumption in the ETABS were investigated. In this study, the shear forces that occur in the beams of a fictionally designed building under the effect of vertical earthquakes, and the floors carried by the beams are examined by designing with the assumption of a rigid diaphragm. The structure is designed with 3 openings in the X and Y directions and each opening is equal and 5 m. The 5-storey building has been designed to have equal floor heights of 3 m. The structure was obtained by using parameters belonging to 5 different soil classes as ZA, ZB, ZC, ZD, ZE, and 5 different vertical elastic design spectrums represented by the abbreviations EZ_ZA, EZ_ZB, EZ_ZC, EZ_ZD were activated in the ETABS. Seismic station coordinates numbered 4631 in K.Maras/Elbistan were used to obtain the parameters to be used in obtaining the vertical elastic design spectral of the structure. These coordinates were selected in order to compare the i and j end shear forces of the beam with the previously obtained beam i and j ends shear forces by influencing the Mw:7.6 Elbistan earthquake recorded by the seismic station to the building. Since the structure is symmetrical in both X and Y directions, the results obtained from beams B2, B3, B5, B19 represent the beam shear forces occurring at the i and j ends of all beams of the structure. As a result, it has been seen that the highest values of beam i and j end shear forces, obtained by affecting both the vertical elastic design spectral and the data of the vertical component of the Elbistan Mw:7.6 earthquake, were obtained on the 5th floor, which is the top floor of the building. The structure showed similar behavior among the different vertical elastic design spectral, and the highest value was obtained for each vertical elastic design spectrum at the 5th floor. Among these vertical elastic design spectral, the EZ_ZC spectrum gave the highest value at the 5th floor of the building. When the EZ_ZC vertical elastic design spectrum beam i and j end shear forces are compared with the beam i and j end shear forces of the vertical component data of the Elbistan Mw:7.6 earthquake, it is seen that the difference is not significant. It has been seen that the beam i and j end shear force value of the spectrum is higher than the shear force value.

Effects of printer variation on vibration response of fused deposition modeling-fabricated elastic beams with symmetric and asymmetric resonators: A round robin study

Additive manufacturing has expanded rapidly as a production tool due to its ease of use for rapidly producing parts. The most common polymer-based additive approach is fused deposition modeling (FDM) which uses plastic filament to build parts additively, one line at a time, to build a 3D shape. FDM printers are ubiquitous in university, government and industry settings, making them ideal for mass-production of shared designs across institutions. While it is commonly known that FDM finished products have variations due to differences in printer model and slicing and extrusion settings, little has been done to quantify the effects of these variations on elasto-dynamic response. In this study, a multi-institutional round robin approach is used to quantify the printer-to-printer variations of a structure comprised of a thin beam with attached resonators. The parameters of the round robin study involve printing the same geometry, with the same base material, on whatever FDM-type printers are available at each of six contributing institutions. All samples were then sent to a common location and tested on the same apparatus to limit experimental variability. This presentation will discuss the design of the study, test and modeling efforts, and a summary of results.

The Approximate Analysis of Higher-Order Frequencies of Nonlinear Vibrations of a Cantilever Beam With the Extended Galerkin Method

Abstract The nonlinear vibrations of elastic beams with large amplitudes are frequently treated as a typical problem of an elastica. As the continuation of the analysis of the deformation of an elastica, the nonlinear vibration equation of the elastic beam in the rotation angle of the cross section has been established. Using the deformation function, the nonlinear equation with the inertia effect has been solved by the newly proposed extended Galerkin method (EGM). The solution to the vibration problem of the elastica is compared with earlier approximate solutions including the frequencies and mode shapes obtained by other methods, and the rotation angle and energy of each mode at the high-order frequency are also calculated. This solution procedure provides an alternative technique to the elastica problem by the EGM with possible applications to other nonlinear problems in many fields of science and technology.

Vibration isolation by relative resonance perceived in combination of rigid bodies and elastic beams

This paper develops a novel low-frequency vibration isolation device based on rigid body dynamics called a Rigid Elastic Vibration Isolator (REVI). The REVI model is realized by coupling four elastic beams with two rigid bodies monolithically made using 3D printing. The system operates based on the vibration of the intermediate rigid bodies, which leads to the anti-resonance phenomenon at low frequencies. The dynamic analysis of the proposed REVI system has been meticulously investigated through analytical methods and real-world experimentation. The analytical method uses the Spectral element method to obtain the dynamic response, which is also validated by the experimental findings. Furthermore, the REVI transmittance sensitivity analysis was conducted by adjusting the rigid mass and system load and exploring methods for generating wide low-frequency bandgaps. The inclination angle of the REVI system is also varied, revealing the bandgap characteristics regarding negative transmittance level. The parametric study varying the geometric properties of the REVI system enhances our understanding of the bandgap and attenuation characteristics within the attenuation band. The REVI mechanism is practical and easily implemented, allowing for accurate and repeatable modeling. Moreover, the analytical observations assist in refining the shape of the REVI mechanism to achieve the necessary bandgap for the desired transmittance.

An Optimal Control Approach to the Problem of the Longest Self-Supporting Structure

The characterization of the self-supporting slender structure with the furthest length is of interest both from a mechanical and biological point of view. Indeed, from a mechanical perspective, this classical problem was developed and studied with different methods, for example using similarity solutions and stable manifolds. However, none of them led to a complete analytical solution. On the other hand, plant structures such as tree branches or searcher shoots in climbing plants can be considered elastic cantilevered beams. In this paper, we formulate the problem as a non-convex optimisation problem with mixed state constraints. The problem is solved by analysing the corresponding relaxation. With this method, it is possible to obtain an analytical characterization of the cross-section

On matched asymptotic expansion solutions of a bucklewave propagation problem

AbstractIn the present paper, the method of matched asymptotic expansions is used to derive approximate analytical solutions to a model problem related to bucklewave propagation in undersea pipelines. The model problem, introduced by Chater, Hutchinson and Neale (in: Collapse – The Buckling of Structures in Theory and Practice (Thompson, J.M.T., Hunt, G.W., eds.) Cambridge Univ. Press, Cambridge, 1983), consists of a linear elastic beam resting on a nonlinear elastic foundation, with inclusion of axial tension and inertia, related to a propagating buckle. When tension dominates over inertia, it is found that the asymptotic solution describing a propagating buckle has the character of nonoscillatory motion. When tension and inertia forces are of equal magnitude, or when inertia forces dominate over tension, the solutions are found to be oscillatory, but still localized around the bucklewave front. Finally, it is shown that when the propagation speed of the buckle exceeds a critical value, a spatial instability sets in, with exponentially growing amplitude of the beam deflection.

Fiber Bragg grating-based flume tests for inversing the impact force coefficient of debris flow

In the piezoelectric sensor-based debris flow flume tests, the impact force coefficient (k) obtained from inversion has a significant dispersion range, which causes considerable uncertainty in the debris flow impact force. To reduce the dispersion range of k, this study designed a new measurement system by integrating the Fiber Bragg Grating (FBG) and the elastic cantilever beam as the core sensing unit. The impact of debris flows on the sensing unit causes the bending deformation of the cantilever beam, which subsequently causes the wavelength shift of the FBG fixed on the cantilever beam. The optical signals are converted into the dynamic strain signals of the cantilever beam using the photosensitive characteristic of the FBG. According to the mechanical characteristics of the cantilever beam and the Bingham model of debris flow, the method of inversing k using the strain signals was proposed. Based on the FBG sensor system, 45 tests were conducted under different working conditions. The inversion model obtained 38 sets of k by analyzing the test data. The standard deviation and dispersion coefficient of k were 0.4 and 0.25, respectively, making the k in the range of 0.9 to 2.8, which was superior to the k derived from the traditional measurement mode.

Modeling and design of a SMA self-actuating bistable beam

Bistable beams with implanted actuators show the self-actuating ability that further improve performance of the system. In this paper, a novel self-actuating bistable mechanism coupled by elastic straight beam and SMA curved beam was designed and tested. Snap-back behavior of the presented system was triggered by SMA curved beam actuator, which generated actuation moment and rotation angle at the end of flexible beam. The bistable mechanism was modeled through nonlinear governing equation of large deflection beam. A solution strategy that considered multiple inflection points was developed to solve the governing equation. Loading path, deflection curves, and energy landscape of bistable beam were illustrated and further analyzed. Moreover, parameters of SMA curved beam actuator were determined by a simplified model. A prototype of the self-actuating bistable beam was fabricated and the snap-back phenomenon was observed on the test bench. Accompanied by the snap-back, a 44% decrease of axial load could be reached. It was also found that the bistable beam could be reset by applying vertical concentrated force. Reciprocating snap-back tests were conducted to confirm that point. Possible applications of the presented bistable beam could be variable load mechanisms and novel flexible switches.

Elastic programmable properties and dynamic dissipation of gradient unstable structures

Metamaterials derive promising properties mainly from an internal micromechanical mechanism, rather than their chemical composition. Many interesting mechanical behaviors can be obtained by orderly assembling buckling elements. Elastic programmable properties, corresponding potential energy landscape and dynamic dissipation mechanism of elastic beam structures are studied in detail. To accomplish this work, a combination of theoretical models and extensive numerical calculations are applied to investigate a single element as well as homogeneous and gradient structures. This comprehensive investigation not only provides insights into the complex internal buckling behaviors and mechanisms of a specific unstable structure but also establishes conclusions that are applicable to other unstable structures having similar buckling behaviors. The potential energy of a serial combination of unstable elements is extremely wiggly, enabling multiple equilibrium configurations and resulting in multistable transformations and energy dissipation with elastic deformation and loading timescale independence. To the best of our knowledge, the most of programming properties investigated in this paper, the complex energy landscape, the deformation of internal elements, and programmable dynamic responses of gradient unstable structures have not been reported in previous papers.

Numerical Evaluation of the Elastic Moduli of AlN and GaN Nanosheets.

Two-dimensional (2D) nanostructures of aluminum nitride (AlN) and gallium nitride (GaN), called nanosheets, have a graphene-like atomic arrangement and represent novel materials with important upcoming applications in the fields of flexible electronics, optoelectronics, and strain engineering, among others. Knowledge of their mechanical behavior is key to the correct design and enhanced functioning of advanced 2D devices and systems based on aluminum nitride and gallium nitride nanosheets. With this background, the surface Young's and shear moduli of AlN and GaN nanosheets over a wide range of aspect ratios were assessed using the nanoscale continuum model (NCM), also known as the molecular structural mechanics (MSM) approach. The NCM/MSM approach uses elastic beam elements to represent interatomic bonds and allows the elastic moduli of nanosheets to be evaluated in a simple way. The surface Young's and shear moduli calculated in the current study contribute to building a reference for the evaluation of the elastic moduli of AlN and GaN nanosheets using the theoretical method. The results show that an analytical methodology can be used to assess the Young's and shear moduli of aluminum nitride and gallium nitride nanosheets without the need for numerical simulation. An exploratory study was performed to adjust the input parameters of the numerical simulation, which led to good agreement with the results of elastic moduli available in the literature. The limitations of this method are also discussed.

State-of-the-art on the anchorage performance of rock bolts subjected to shear load

AbstractRock bolts are extensively utilized in underground engineering as a means of offering support and stability to rock masses in tunnels, mines, and other underground structures. In environments of high ground stress, faults or weak zones can frequently arise in rock formations, presenting a significant challenge for engineering and potentially leading to underground engineering collapse. Rock bolts serve as a crucial structural element for the transmission of tensile stress and are capable of withstanding shear loads to prevent sliding of weak zones within rock mass. Therefore, a complete understanding of the behavior of rock bolts subjected to shear loads is essential. This paper presents a state-of-the-art review of the research progress of rock bolts subjected to shear load in three categories: experiment, numerical simulation, and analytical model. The review focuses on the research studies and developments in this area since the 1970s, providing a comprehensive overview of numerous factors that influence the anchorage performance of rock bolts. These factors include the diameter and angle of the rock bolt installation, rock strength, grouting material, bolt material, borehole diameter, rock bolt preload, normal stress, joint surface roughness and joint expansion angle. The paper reviews the improvement of mechanical parameter setting in numerical simulation of rock bolt shear. Furthermore, it delves into the optimization of the analytical model concerning rock bolt shear theory, approached from the perspectives of both Elastic foundation beam theory coupled with Elastoplasticity theory and Structural mechanic methods. The significance of this review lies in its ability to provide insights into the mechanical behavior of rock bolts. The paper also highlights the limitations of current research and guidelines for further research of rock bolts.

Numerical study of interactions between Bingham fluids and flexible structures using the immersed boundary-lattice Boltzmann method

This paper studies the interactions between Bingham fluids and flexible structures under an immersed boundary lattice-Boltzmann framework. The objective of this paper is to understand the rheology effects of Bingham fluids on the structure responses and the flow characteristics within Bingham fluids–structure interactions. To this end, a numerical algorithm that integrates the lattice Boltzmann method and the smoothed point interpolation method is proposed. Alongside this, a hybrid multiple relaxation time scheme in the framework of the immersed boundary-lattice Boltzmann method is used to improve the numerical accuracy and stability. Then, the numerical algorithm is applied to three typical fluid–structure interactions cases in both Newtonian and Bingham fluids regarding an elastic beam, a self-induced elastic beam attached to a circular cylinder and a self-propelled fishlike body. Numerical results highlight that the yield stress of Bingham fluids influences predominantly on the structure responses, such as the solid deformation, the oscillation amplitude, the oscillation period, and the swimming velocity. In addition, the yielded/unyielded zones' evolution process of the Bingham fluids–structure interaction is also identified in this paper.

General Exact Analytical Expressions for Rotation and Displacement of a Timoshenko Beam Under Variable Loads with Validation

The researchers involved in the field of applied mathematics mostly project the efficient and viable solutions of those problems in which have practical applications in science and engineering. The Timoshenko beam model (TBM) is described by a system of ordinary differential equations where the expressions of rotation and displacement of the beam are ultimately required. Most of the work on analytical solutions of beam problems in literature focuses the elastic and Euler-Bernoulli beams, whereas for the TBM the numerical solutions are usually preferred. The analytical expressions for the TBM exist for only very basic load cases and are also not general for any load function. In this paper, we attempt to suggest a novel protocol based on expressing a load function by a polynomial or its power series development, and then use it to develop general analytical expressions for the rotation and displacement of a fixed TBM which is not load specific. The proposed general equations can provide ease of access and handling for the practitioners working in applied mathematics, structural engineering and mechanical vibrations as the developed equations can be used with only constant inputs to tailor particular expressions of rotation and displacement profiles of a fixed TBM under any variable load. For performance evaluations of the proposed general equations, we have also obtained particular expressions for some important variable loads, like: linearly varying loads ((LVLs): triangular and trapezoidal and quadratically varying loads (QVLs): parabolic/square and circular loads. Finally, the proposed protocol for generalization has been validated for fixed elastic beams under uniformly distributed loads (UDLs), and the results match exactly with those expressions available in literature. The contributions of this study, on one hand provide ready, direct and exact general expressions for the rotation and displacement profiles of a fixed TBM, while on the other hand the solution of such problem is achieved with quite negligible computational overhead, execution time and software implementations.

New optical quantum hyperbolic recursional ferromagnetic microscale

In this paper, we construct properties of quasi recursional normal electromagnetic flexible elastic quasi microscale beams in terms of quasi normalized operator. We give new characterizations for ferromagnetic electric normalized quasi optimistic density with quasi frame. Finally, we design optical applications for recursional electromagnetic flexible elastic quasi microscale beam with optical quasi resonator.

Control of the maglev system with a low stiffness elastic track beam by pole assignment and a reduced order state observer

A maglev train system is an open loop unstable and strongly nonlinear system. Self-excited coupling vibration between the magnet and the track beam can occur even when the train is stationary. To avoid this, the track beam is generally designed with high stiffness, which directly increases the construction cost of the maglev line. In this research, a new control method for the maglev system with a low stiffness elastic track is proposed based on full state feedback theory and pole assignment theory. In the newly designed controller, track beam DOF is introduced into the control loop with the use of the reduced order state observer (ROSO) to make the control system have the ability to suppress track vibration actively, thus reducing the over dependence of the system stability on track properties. Effectiveness of the proposed control strategy is verified through experiments. Results show that it exhibits a better response compared to the traditional control method, with smaller fluctuations in levitation air gap and magnet acceleration. Furthermore, it can maintain system stability with relatively lower track stiffness, reducing the minimum requirements for track stiffness by 53.4% in the test.

Mechanical response of tunnels crossing active fault zones under normal or reverse faulting: A refined nonlinear analytical approximate solution

Tunnels crossing active fault zones will be severely damaged under normal or reverse faulting. The tunnel-stratum interaction is nonlinear, and the fault zone width significantly affects the tunnel response. Nevertheless, the existing analytical models neglect the effect of nonlinear tunnel-stratum interaction or fault zone width and adopt unrealistic assumptions. This work proposes a refined nonlinear solution, considering the nonlinear tunnel-stratum interaction and fault zone width, and some widely employed assumptions are removed. Specifically, a tunnel under normal or reverse faulting is simplified to an elastic beam acting on a nonlinear foundation, the nonlinear tunnel-stratum interaction is considered by setting a series of nonlinear axial and vertical springs, and the fault zone width is considered by reducing the spring stiffness. Additionally, the nonlinear axial force and geometrical nonlinearity terms are considered in the governing equations. The comparison results indicate that the results obtained from the proposed model agree well with those from model tests and numerical models and have a better accuracy than the existing analytical models. Finally, the effects of fault, stratum, and lining parameters on the response of tunnels crossing normal or reverse faults are discussed regarding stress distribution and failure scope.

An Investigation of the Energy Harvesting Capabilities of a Novel Three-Dimensional Super-Cell Phononic Crystal with a Local Resonance Structure.

Using the piezoelectric (PZT) effect, energy-harvesting has become possible for phononic crystal (PnC). Low-frequency vibration energy harvesting is more of a challenge, which can be solved by local resonance phononic crystals (LRPnCs). A novel three-dimensional (3D) energy harvesting LRPnC is proposed and further analyzed using the finite element method (FEM) software COMSOL. The 3D LRPnC with spiral unit-cell structures is constructed with a low initial frequency and wide band gaps (BGs). According to the large vibration deformation of the elastic beam near the scatterer, a PZT sheet is mounted in the surface of that beam, to harvest the energy of elastic waves using the PZT effect. To further improve the energy-harvesting performance, a 5 × 5 super-cell is numerically constructed. Numerical simulations show that the present 3D super-cell PnC structure can make full use of the advantages of the large vibration deformation and the PZT effect, i.e., the BGs with a frequency range from 28.47 Hz to 194.21 Hz with a bandwidth of 142.7 Hz, and the maximum voltage output is about 29.3 V under effective sound pressure with a peak power of 11.5 µW. The present super-cell phononic crystal structure provides better support for low-frequency vibration energy harvesting, when designing PnCs, than that of the traditional Prague type.

Analysis of a Wind-Driven Power Generation System with Root Slapping Mechanism

This study introduces a groundbreaking slap-type Vibration Energy Harvesting (VEH) system, leveraging a rotating shaft with magnets to induce vibrations in an adjacent elastic steel sheet through magnetic repulsion. This unique design causes the elastic sheet to vibrate, initiating the oscillation of a seesaw-type rigid plate lever. The lever then slaps a piezoelectric patch (PZT) at the elastic steel sheet’s root, converting vibrations into electrical energy. Notably, the design enables the PZT to withstand deformation and flapping forces simultaneously, enhancing power conversion efficiency. The driving force for the rotating shaft is harnessed from the downstream flow field generated by moving objects like rotorcraft, fixed-wing aircraft, motorcycles, and bicycles. Beyond conventional vibration energy harvesting, this design taps into additional electric energy generated by the PZT’s slapping force. This study includes mathematical modeling of nonlinear elastic beams, utilizing the Method of Multiple Scales (MOMS) for in-depth vibration mode analysis. Experimental validation ensures the convergence of theory and practice, confirming the feasibility and superior voltage generation efficiency of this slap-type VEH concept compared to traditional VEH systems.