Linear Beam Theory Research Articles

Influence of brush seal hysteresis effect on the nonlinear characteristics of rotor system

The hysteresis effect of brush seal is a key factor affecting the sealing performance of the rotor system, so it is particularly important to study the nonlinear behaviors of the rotor-bearing-brush seal system under the hysteresis effect. First, the entire circle of bristle pack is classified according to the position angle and contact condition of single bristle, and the rotor support force, the interaction force between the bristles, the air pressure and the friction of back plate are considered. It is proposed to use the loss rate to calculate the uniform load of the back plate, according to the linear beam theory, the bending force model of bristle under different rotor eccentricity is established. Second, the hysteresis angle is proposed to represent the degree of hysteresis effect of the whole circle of bristle pack, and the seal force model under hysteresis effect is established. Third, considering the interaction of seal force and oil-film force, the dynamic model of Jeffcott rotor-bearing-brush seal system is established. The influence of hysteresis effect on the dynamic characteristics of rotor system is studied by using bifurcation diagram, axis orbit, Poincaré map, spectrum diagram and time history. It is found that the hysteresis effect has different degrees of influence on the nonlinear dynamic characteristics of the rotor system in different rotational velocity intervals. In addition, the influence of seal interference on the motion state of rotor system cannot be ignored.

Large displacement analysis of stiffened plates with parallel ribs under lateral pressure using FE modeling with shell elements

This study provides the load–displacement behavior of stiffened plates with parallel ribs when modeled using shell elements with and without considering the large-displacement effects in comparison with the conventional equivalent beam analysis (EBA) in which such stiffened plates are simplified into an imaginary beam with the geometrical properties of an equivalent built-up cross-section. This study highlights the advantages of FE modeling with shell elements and large-displacement analysis (LDA) over the existing EBA method and then provides a path for using this method in the ultimate limit state (ULS) design of stiffened plates. The stress distributions in the panel plate part of the stiffened plates are presented to demonstrate the significance of considering the large-displacement effects in the analysis. The numerical investigation indicated that the linear beam theory does not correctly presume the true behavior of the stiffened plates and that the corresponding load–displacement estimation does not align with its behavior predicted by the LDA. The results also demonstrate that when shell element modeling is utilized, the internal forces and stresses in the panel plate are calculated correctly, only if the large-displacement effects are considered in the analysis. Finally, a method for estimating the ultimate load capacity of stiffened plates with parallel ribs is provided, based on the large-displacement FE analysis with shell elements, which can be used to establish empirical design equations.

Analysis of the applicability of composite steel beams using hot rolled I steel with Simple - Made - Continuous solution to bridge structures with small and medium spans in Vietnam

The article has proposed a typical structural form and cross-section for a composite beam span structure using hot rolled I steel with Simple-Made-Continuous solution for span structures with a length of less than 36m suitable to Vietnamese conditions. It also analyzed the span structure according to beam theory and nonlinear theory by finite element method, thereby determining the working capacity and the structural coefficients of beams according to linear beam theory. At the same time, it analyzed beam joints by finite element method using nonlinear element model for concrete and steel materials. Finally, the composite girder span structure using hot rolled I steel with Simple-Made-Continuous solution is also economically compared with another common span structure with the same span length, the simple steel beam.

Valorization of date palm wastes as sandwich panels using short rachis fibers in skin and petiole 'wood' as core

Algerian agriculture offers free by-products that can be transformed in biobased-materials. We use one of them, the date palm ’wood’, to locally develop sandwich panels. We demonstrate by tensile tests that rachis fibers are more efficient than petiole fibers. Then, petiole ’wood’ is used as sandwich core and short rachis fibers (5%, 10%, 15%) are used as epoxy skin reinforcement. The skins are tested by tensile and bending tests, 15%Mf untreated short fibers have been selected as a best choice for stiffness of the skins. Three parameters are studied to optimize the design of the core: the orientation (longitudinal, transverse, radial), the length of the pieces of ’wood’ (30 mm, 100 mm), and the thickness (10 mm, 15 mm, 20 mm). The sandwich efficiency are studied by 3-points bending tests combined to linear elastic beam theory and failure mode analysis. The dominant failure modes are bottom skin failure and core shear, the latter being not activated in case of transverse orientation of core. Both other parameters (thickness, pieces of core length) are not really determinant in mechanical point of view, depending on the future use. We conclude that making sandwich board by using untreated short rachis fibers as reinforcement for skins and untreated transverse cut petiole ’wood’ as core is a very efficient choice to give technical and cheap valorization of palm leaf wastes for non-structural use.

Dynamic analysis in beam element of wave-piercing Catamarans undergoing slamming load based on mathematical modelling

Dynamic analysis of wave-piercing Catamarans undergoing slamming has tracked widespread attention in the design of marine structures. Hence, our investigation on the dynamic response relies heavily on the beam element of wave-piercing Catamarans subjected to slamming load. In the present paper, a numerical method and a mathematical modelling are developed to analyze the dynamic response of beam elements in wave-piercing Catamarans due to impacting moving fluid load. With assuming the initial phase of slamming force, an elastic linear hyperbolic shear deformation beam theory (HSDBT) with uniform thickness is utilized for modelling. For presenting a more realistic model, the porosity and structural damping of the Aluminum beam are considered. In addition, for improving the stiffness of the mentioned structure against the slamming load, graphene platelet (GPLs) are used for reinforcement of the beam. To capture the dynamic deflection of the porous GPL-reinforced beam, differential quadrature method (DQM) along with Newmark method is utilized. The aim of this work is to illustrate the influences of deadrise angle, vertical speed of impact, GPLs volume percent and distribution, geometrical parameters, porosity, structural damping and boundary conditions on the dynamic deflection of the structure. The results indicated that reinforcing the facesheets by CNTs reduces the dynamic deflection up to 59%.

Hydroelastic interaction of nonlinear waves with floating sheets

Hydroelastic responses of floating elastic surfaces to incident nonlinear waves of solitary and cnoidal type are studied. There are N number of the deformable surfaces, and these are represented by thin elastic plates of variable properties and different sizes and rigidity. The coupled motion of the elastic surfaces and the fluid are solved simultaneously within the framework of linear beam theory for the structures and the nonlinear Level I Green–Naghdi theory for the fluid. The water surface elevation, deformations of the elastic surfaces, velocity and pressure fields, wave reflection and transmission coefficients are calculated and presented. Results of the model are compared with existing laboratory measurements and other numerical solutions. In the absence of any restriction on the nonlinearity of the wave field, number of surfaces, their sizes and rigidities, a wide range of wave–structure conditions are considered. It is found that wave reflection from an elastic surface changes significantly with the rigidity, and the highest reflection is observed when the plate is rigid (not elastic). It is also found that due to the wave–structure interaction, local wave fields with different length and celerity are formed under the plates. In the case of multiple floating surfaces, it is observed that the spacing between plates has more significant effect on the wave field than their lengths. Also, the presence of relatively smaller floating plates upwave modifies remarkably the deformation and response of the downwave floating surface.

Theoretical study on the instability mechanism of flutter generated on a cantilevered flexible plate in three-dimensional uniform flow

The flutter of a thin flexible plate in uniform flow is an interesting theoretical problem in fluid–structure interaction phenomena, as well as an important engineering problem. Therefore, understanding of the instability mechanism is crucial to preventing fatal defects caused by the flutter. In this study, the instability mechanism of a flexible plate in three-dimensional uniform flow is investigated in terms of the energy transfer between the plate motion and fluid flow. A theoretical aerodynamic model of the plate is developed using the unsteady lifting surface theory and linear beam theory. The aerodynamic stability of the plate is investigated by eigenvalue analysis. The developed model is validated by comparison with previous experimental results and other theoretical models. Furthermore, to understand the instability mechanism, the work done by the fluid force on the plate surface is determined, and its influence on the stability of the system is examined.

Immersed boundary simulations of fluid shear-induced deformation of a cantilever beam

A biofilm is a colony of microorganisms that adheres and grows on a surface typically in contact with a stagnant or flowing fluid environment. The hydrodynamic interaction between the fluid and the film structure plays a key role in the various phases of the biofilm life-cycle — formation, growth, detachment, and reestablishment. The fluid-flow conditions determine the shear stresses exerted upon the film structure causing it to deform up to a critical point, beyond which the biofilm is uprooted from the surface into a planktonic state in the flowing fluid. To develop a fundamental understanding of the effects of the fluid-flow parameters on the mechanics of the biofilm, this work considered an ideal 2D model scenario of ‘a rectangular cantilever beam immersed in a channel filled with viscous, incompressible fluid, where one end of the beam is fixed to the channel wall, bending in response to a shear flow.’ The Immersed boundary (IB) method was employed to simulate numerically the fluid–structure interaction. The effect of physical parameters of the problem — stiffness and height of the cantilever and flow velocity, along with the numerical parameters on the equilibrium structure of the elastic beam was investigated and the simulation results were qualitatively compared with respect to linear beam theory. A careful analysis of the “corner effects” — irregularities in beam shape near the free and fixed ends, has been presented. It was shown that this issue can be effectively eliminated by smoothing out the corners with a “fillet” or rounded shape. Finally, the IB formulation was extended to include porosity in the beam to investigate the effect of the resultant porous flow on the beam deflection.

A micromechanical model of fiber bridging including effects of large deflections of the bridging fibers

A micromechanical model of cross-over fiber bridging is developed for the prediction of macroscopic mixed-mode bridging laws (traction-separation laws). The model is based on non-linear beam theory and takes into account debonding between fiber and matrix as well as buckling of fibers in compression. Further, it is shown how failure of the bridging fibers can be taken into account through a Weibull distributed failure strain. Predictions made by the proposed model are compared with predictions made by detailed 3D finite element models, and a very good agreement was observed. It is shown that models based on linear beam theory are only valid for small transverse deflections of the bridging ligament and greatly underestimate the force transferred by ligaments subjected to moderately large deflections. The novel model, on the other hand, is applicable in the entire range where the bridging problem transitions from a beam bending problem to a bar-like problem. Finally, an example of how the proposed model can be used for parameter/sensitivity studies is given. A conclusion from this study is that reducing the fracture toughness, Gc, of the interface between fibers and matrix may lead to increased energy dissipation through cross-over fiber bridging as more fibres remain intact longer.

Geometrically Non-Linear Vibration of a Cantilever Interacting with Rarefied Gas Flow

Abstract The work is devoted to study 2D pressure driven rarefied gas flow in a microchannel having an elastic obstacle. The elastic obstacle is clamped at the bottom channel wall and its length is half of the channel height. The gas flow is simulated by Direct Simulation Monte Carlo (DSMC) method applying the advanced Simplified Bernoulli Trial (SBT) collision scheme. The elastic obstacle is modelled as geometrically nonlinear Euler Bernoulli beam. A reduced 3 modes reduction model of the beam is created. The influence of the gas flow on the beam vibration is studied, considering the linear and nonlinear beam theories.

The six geometries revisited.

Forces and moments delivered by a straight wire connecting two orthodontic brackets are statically indeterminate and cannot be estimated using the classical equations of static equilibrium. To identify the mechanics of such two-bracket systems, Burstone and Koenig used the principles of linear beam theory to estimate the resulting force systems. In the original publication, however, it remains unclear how the force systems were calculated because no reference or computational details on the underlying principles have been provided. Using the moment carry-over principle and the relative angulation of the brackets, a formula was derived to calculate the relative moments of the two brackets. Because of the moment equilibrium, the vertical forces that exist as a force-couple on the two brackets can also be calculated. The accuracy of the proposed approach can be validated using previously published empirical data.

The Adhesion of Mica Nanolayers on a Silicon Substrate in Air

AbstractMica nanolayers (MNL) are a new dielectric material that can improve the electron transport properties of Si based microelectronic devices. However, the mechanical reliability of such devices is not well understood because measurement of the adhesion between an MNL and a Si substrate is insufficiently accurate. The recent work reports a bridging method that is developed based on the linear beam theory and can characterize the adhesion of MNL on a mica substrate in a reasonably accuracy. In this work, the accuracy of the bridging method is improved by using a nonlinear mechanical model. 15 MNL specimens are measured on a Si substrate in air and the new model gives an average adhesion energy of 119.69 ± 20.47 mJ m–2. The effect of environment on the adhesion is also investigated. The increase in temperature from 25 to 300 °C enhances adhesion, attributed to the increased bonding effect of interfacial contaminants. When humidity is below a threshold value, 80% in this case, the adhesion energy remains unchanged, but above the threshold, the nanobridges collapse during testing. Molecular dynamics simulation reveal that the collapse is because the increase of interfacial water molecules reduces the frictional force between the MNL and Si substrate.

Small scale fracture and strength of high-entropy carbide grains during microcantilever bending experiments

The fracture behaviour of (Hf-Ta-Zr-Nb)C high-entropy carbide grains was investigated by microcantilever bending experiments, and fracture related properties (e.g. strength, toughness) were determined using linear beam theory. Microcantilevers were FIB-milled from large grains with {001} and {101} orientations and were subjected to micro-bending experiments. SEM based fractographic analysis of the broken cantilevers revealed that approximately half of them fractured at the fixing position at FIB-induced surface cracks, while the rest of the beams failed at small cracks located at submicron size pores or inclusions. In all cases, fracture occurred on the {001} cleavage plane. The fracture strength of beams fractured at the fixing position was 11.8 ± 0.2 GPa, while the strength of the beams that failed at submicron defects was in the range of 3.8-8.9 GPa. The calculation of stress concentration in the vicinity of pores revealed that local stress field exceeded the value that induced cracking in ‘defect free’ beams.

Evaluation of Three-Point Bending Strength of Thin Silicon Die With a Consideration of Geometric Nonlinearity

With the trends of electronic packaging development toward small size, low-profile features, high-pin count, and high performance, the 3D IC (Three-dimensional Integrated Circuits) or stacked-die packages have been gaining popularity. For such package applications, IC silicon wafers have to be ground and processed to be relatively thin and then the thin silicon dies cut from these wafers have to gain sufficient strength in order to bear high stresses resulting from process handling, reliability testing, and operations. Hence, the strength of the thin dies has to be determined to ensure the good reliability of the packages. Three-point bending test is commonly used for measuring die strength; however, the feasibility of the test is still questionable for determining the strength of relatively thin dies. Therefore, the feasibility of the linear beam theory is evaluated by a nonlinear finite element method (NFEM) with taking into account geometric nonlinearity in this study. The results show that this nonlinearity would cause errors of the strength of thin dies if calculated by the linear beam theory. The fitting equations of the correction factors ( $\eta $ ) to linear solutions, extracted from the NFEM simulation, are proposed and proved to be workable with very good accuracy. It is also found that the correction factor highly depends on the deflection ( $\delta $ ), span length (L) and radius of roller support (r), but not elastic modulus (E) and thickness (t) of test specimens. After the nonlinearity has to be taken into account for the relatively thin silicon dies, the consistent statistical strength data have been obtained for various thicknesses of the test die specimens.

Experimental and Numerical Investigation of Mode I and Mode II Interlaminar Behavior of Ultra-Thin Chopped Carbon Fiber Tape-Reinforced Thermoplastics

This manuscript describes an experimental and numerical study conducted to investigate two representative cases of delamination in ultra-thin chopped carbon fiber tape-reinforced thermoplastics (UT-CTT): delamination in a double cantilever beam (DCB, mode I) and delamination in an end-notched flexure (ENF, mode II). In examples of mode I delamination, unstable crack growth resulted in sawtooth-like load–displacement curves after the initial stage of increasing load, while some specimens displaying mode II delamination exhibited stable crack propagation. The crack surfaces of DCB and ENF UT-CTT specimens were observed by a three-dimensional measurement macroscope and a scanning electron microscope. The values of mode I and mode II interlaminar fracture toughness were obtained based on linear elastic fracture mechanics and beam theory in reference to JIS K 7086 standard, respectively. Furthermore, numerical analysis utilizing the surface-based cohesive zone model based on the triangular traction–separation law was used to predict the delamination behavior in UT-CTT. The minimum and maximum values of critical strain energy release rate were used to define the range of load–displacement curves corresponding to unstable crack propagation under mode I, while the parameters including interlaminar shear strength and friction were taken into account for the predictions of delamination propagation under mode II. The numerically predicted load–displacement curves showed good correlation with the theoretical and experimental results. The research provides interlaminar mechanical properties for damage modeling and numerical simulation method to express fracture behaviors of UT-CTT structures.

Design and Modeling of a New Biomimetic Soft Robotic Jellyfish Using IPMC-Based Electroactive Polymers.

Smart materials and soft robotics have been seen to be particularly well-suited for developing biomimetic devices and are active fields of research. In this study, the design and modeling of a new biomimetic soft robot is described. Initial work was made in the modeling of a biomimetic robot based on the locomotion and kinematics of jellyfish. Modifications were made to the governing equations for jellyfish locomotion that accounted for geometric differences between biology and the robotic design. In particular, the capability of the model to account for the mass and geometry of the robot design has been added for better flexibility in the model setup. A simple geometrically defined model is developed and used to show the feasibility of a proposed biomimetic robot under a prescribed geometric deformation to the robot structure. A more robust mechanics model is then developed which uses linear beam theory is coupled to an equivalent circuit model to simulate actuation of the robot with ionic polymer-metal composite (IPMC) actuators. The mechanics model of the soft robot is compared to that of the geometric model as well as biological jellyfish swimming to highlight its improved efficiency. The design models are characterized against a biological jellyfish model in terms of propulsive efficiency. Using the mechanics model, the locomotive energetics as modeled in literature on biological jellyfish are explored. Locomotive efficiency and cost as a function of swimming cycles are examined for various swimming modes developed, followed by an analysis of the initial transient and steady-state swimming velocities. Applications for fluid pumping or thrust vectoring utilizing the same basic robot design are also proposed. The new design shows a clear advantage over its purely biological counterpart for a soft-robot, with the newly proposed biomimetic swimming mode offering enhanced swimming efficiency and steady-state velocities for a given size and volume exchange.

Analysis of Efficiency of Uniform-Strength Composite Leaf Springs under Various Loading Conditions

To obtain analytical estimations of possible mass reduction, the simplest case of a cantilever rectangular beam loaded by a concentrated force or a distributed load is considered. Five geometrical parameters of designing are distinguished, which can be optimally defined from the five requirements: by strength, by possible accumulated elastic energy, by uniform strength, by connection of cross-sectional sizes, and by shearing force resistance owing to known strength at the interlaminar shear. The validity limits are determined, outside of which the linear beam theory leads to incorrect results for deflection of uniform-strength beams when loaded with distributed forces. The analytical dependences of reducing the desired mass of uniform-strength beams on the behavior of applied forces are the following: the more nonuniformly the applied load changes, the greater the advantage by mass ensures uniform-strength profiling under the given conditions by the strength and accumulated elastic energy.

Thermodynamic consistency of beam theories in the context of classical and non-classical continuum mechanics and a thermodynamically consistent new formulation

In order to enhance currently used beam theories in $$\mathbb {R}^2$$ and $$\mathbb {R}^3$$ to include mechanisms of dissipation and memory, it is necessary to establish if the mathematical models for these theories can be derived using the conservation and the balance laws of continuum mechanics in conjunction with the corresponding kinematic assumptions. This is referred to as thermodynamic consistency of the beam mathematical models. Thermodynamic consistency of the currently used beam models will permit use of entropy inequality to establish constitutive theories in the presence of dissipation and memory mechanism in the currently used beam theories. This is the main motivation for the work presented in this paper. The currently used beam theories are derived based on kinematic assumptions related to the axial and transverse displacement fields. These are then used to derive strain measures followed by constitutive relations. For linear beam theories, strain measures are linear functions of displacement gradients and stresses are linear functions of strain measures. Using these stress and strain measures, energy functional is constructed over the volume of the beam consisting of kinetic energy, strain energy and potential energy of loads. The Euler’s equation(s) extracted from the first variation of this energy functional set to zero yields the differential equations describing the evolution of the deforming beam. Alternatively, principle of virtual work can also be used to derive mathematical models for beams. For linear elastic behavior with small deformation and small strain, the two approaches yield same mathematical models. In this paper we examine whether the currently used beam mathematical models with the corresponding kinematic assumption (i) can be derived using the conservation and balance laws of classical continuum mechanics or (ii) are the conservation and balance laws of non-classical continuum mechanics necessary in their derivation. In order to ensure that the mathematical models for various beam theories result in deformation that is in thermodynamic equilibrium we must establish the consistency of the beam theories with regard to the conservation and the balance laws of continuum mechanics, classical or non-classical in conjunction with their corresponding kinematic assumptions. Currently used Euler–Bernoulli and Timoshenko beam mathematical models that are representative of most beam mathematical models are investigated. This is followed by details of general and higher-order thermodynamically consistent beam theory that is free of kinematic assumptions and other approximations and remains valid for slender as well as deep beams. Model problem studies are presented for slender as well as deep beams. The new formulation presented in this paper ensures thermodynamic equilibrium as it is derived using the conservation and the balance laws of continuum mechanics and remains valid for slender as well as non-slender beams.

Analyzing the Correctness of Equal Strength Composite Profiled Beam Bending Problems

This paper describes the simplest cases of a rectangular cantilever beam loaded by an concentrated force, distributed load, or dead weight. There are five geometric parameters whose optimal values can be calculated on the basis of five requirements: rigidity (or accumulated elastic energy), strength, equal strength, ratio of sectional sizes, and resistance to the shear force (strength condition in interlaminar shear). The parameter range is determined, outside of which the linear beam theory cannot be used to correctly calculate the deflection of equal strength beams loaded by distributed forces or dead weight.

Effect of flexoelectricity on the electromechanical response of graphene nanocomposite beam

Owing to its unique multifunctional and scale-dependent physical properties, graphene is emerged as promising reinforcement to enhance the overall response of nanotailored composite materials. Most recently, the piezoelectricity phenomena in graphene sheets was found through interplay between different non-centrosymmetric pores, curvature and flexoelectricity phenomena. This has added new multifunctionality to existing graphene and it seems the use of piezoelectric graphene in composites has yet to be fully explored. In this article, the mechanics of materials and finite element models were developed to predict the effective piezoelectric and elastic (piezoelastic) properties of the graphene reinforced nanocomposite material (GRNC). An analytical model based on the linear piezoelectricity and Euler beam theories was also developed to investigate the electromechanical response of GRNC cantilever beam under both electrical and mechanical loads accounting the flexoelectric effect. Furthermore, molecular dynamics simulations were carried out to determine the elastic properties of graphene which were used to develop the analytical and numerical models herein. The current results reveal that the flexoelectric effect on the elastic behavior of bending of nanocomposite beams is significant. The electromechanical behavior of GRNC cantilever beam can be tailored to achieve the desired response via a number of ways such as by varying the volume fraction of graphene layer and the application of electrical load. Our fundamental study highlights the possibility of developing lightweight and high performance piezoelectric graphene based nanoelectromechanical systems such as sensors, actuators, switches and smart electronics as compared with the existing heavy, brittle and toxic piezoelectric materials.