Classical Beam Theory Research Articles

Multiple-fibre interferometry setup for probe sample interaction measurements in atomic force microscopy

Atomic force microscopy (AFM) often relies on the assumption that cantilever bending can be described by simple beam theory and that the displacement of the tip can be evaluated from the cantilever angle. Some more advanced metrological instruments use free-space or fibre interferometers for measuring the position of the cantilever apex directly, thereby simplifying the metrology traceability chain. The next logical development, covering measurements of both the cantilever apex position and its deformation due to lateral forces acting during different AFM measurement regimes, is presented in this paper. It is based on using a set of closely packed fibre interferometers that can be used to determine localised bending of the cantilever at different positions along the cantilever. This can be used for detection of cantilever deformation beyond classical beam theory, and can yield both better understanding of sources of uncertainty in individual AFM force–distance measurements and more accurate scanning in constant height mode in high-speed AFM applications.

A new approach to the mechanics of DNA: Atoms-to-beam homogenization

It is useful to describe the deformation characteristics of long biological macromolecules, such as deoxyribonucleic acid (DNA), by means of terms such as “bending”, “stretching”, or “twisting”. These terms are borrowed from classical beam theory, a traditional and widely known subfield of continuum mechanics, whereas the standard numerical modeling procedure for macromolecules, which is molecular dynamics, does not allow for explicit introduction of the aforementioned deformation modes. This somehow puts some limit to the mechanical understanding of biological macromolecules. As a remedy, we here propose an upscaling (or homogenization) approach, spanning a new conceptual bridge from molecular dynamics to beam theory. Firstly, we apply the principle of virtual power (PVP) to classical continuum beams subjected to stretching and twisting, as well as to atomic compounds represented as discrete systems of mass points in the framework of molecular dynamics. Equating virtual power densities associated with continuum and discrete representations provides homogenization rules from the atomic compounds to the continuum beam line elements. Secondly, the forces acting on the aforementioned mass points are derived from energy potentials associated with bond stretching, valence and torsion angle variations, as well as electrostatic and van der Waals interactions. Application of this strategy to a specific DNA sequence consisting of 20 base pairs reveals deformation-dependent conformational changes, as well as paradox phenomena such as “stretching due to overwinding”, in line with known experimental observations.

Nonlinear vibration of smart nonlocal magneto-electro-elastic beams resting on nonlinear elastic substrate with geometrical imperfection and various piezoelectric effects

This paper studies nonlinear free vibration characteristics of nonlocal magneto-electro-elastic (MEE) nanobeams resting on nonlinear elastic substrate having geometrical imperfection by considering piezoelectric reinforcement scheme. The piezoelectric reinforcement can cause an enhanced vibration behavior of smart nanobeams under magnetic field. All of previously reported studies on MEE nanobeams ignore the influences of geometric imperfections which are very substantial due to the reason that a nanobeam cannot be always perfect. Nonlinear governing equations of a smart nanobeam are derived based on classical beam theory and an analytical trend is provided to obtained nonlinear vibration frequency. This research shows that changing the volume fraction of piezoelectric constituent in the material has a great influence on vibration behavior of smart nanobeam under electric and magnetic fields. Also, it can be seen that nonlinear vibration behaviors of smart nanobeam are dependent on the magnitude of exerted electric voltage, magnetic potential, hardening elastic foundation and geometrical imperfection.

Coupled bending-torsion flutter investigation of MRE tapered sandwich blades in a turbomachinery cascade

This paper studies the effects of bending-torsion coupling on the flutter stability boundaries of a turbomachinery cascade with Magnetorheological Elastomer (MRE) based sandwich blades. The blade structure is considered as a non-uniform sandwich beam with an embedded MRE core. The governing equations of bending and torsional motions are obtained based on the classical sandwich beam theory and the unsteady Whitehead aerodynamic theory is applied for modeling of the aerodynamic flow. The equations of motion governing on the coupled aeroelastic system have been derived in a discrete form by Lagrange's equations and using the assumed modes method. The stability analysis is performed and the effects of various parameters such as coupling, magnetic field, taper ratios and different mistuning patterns on the flutter characteristics have been investigated. The results indicate that the flutter stability is clearly affected by the interaction between the bending and torsional motions which is in agreement with the results previously obtained for a typical cascade. The results also indicate that the change of blade cross sectional area has a significant effect on the flutter boundaries.

Fibre direction strain measurement in a composite ply under pure bending using Digital Volume Correlation and Micro-focus Computed Tomography

This paper presents an experimental demonstration and validation of high-resolution three-dimensional experimental strain measurement using Digital Volume Correlation (DVC) on Carbon Fibre-Reinforced Polymers, via through-thickness strain analysis under a state of pure bending. To permit the application of DVC to displacements and/or strain measurements parallel to the fibre direction in well-aligned unidirectional materials at high volume fractions, a methodology was developed for the insertion of sparse populations of 400 nm BaTiO3 particles within the matrix to act as displacement trackers (i.e. fiducial markers). For this novel material system, measurement sensitivity and noise are considered, along with the spatial filtering intrinsic to established DVC data processing. In conjunction with Micro-focus Computed Tomography, the technique was applied to a simple standard specimen subjected to a four-point flexural test, which resulted in a linear strain distribution through the beam thickness. The high-resolution, fibre-level strain distributions (imaged at a voxel resolution of ∼0.64 µm) were compared against the classical beam theory (Euler–Bernoulli) in incrementally decreasing averaging schemes and different sub-set sizes. Different sampling and averaging strategies are reported, showing that DVC outputs can be obtained that are in very good agreement with the analytical solution. A practical lower limit for the spatial resolution of strain is discerned for the present materials and methods. This study demonstrates the effectiveness of DVC in measuring local strains parallel to the fibre direction, with corresponding potential for calibration and validation of micromechanical models predicting various fibre-dominated damage mechanisms.

Forced axial vibration of micro and nanobeam under axial harmonic moving and constant distributed forces via nonlocal strain gradient theory

In this paper the forced axial vibration of a nanobeam by deploying the nonlocal strain gradient theory exposed to a distributed moving force, is investigated. The assumed nanobeam in this paper is under the axial constant distributed load. The governing equations of the vibration of the nanobeam are obtained via the classic beam theory and the Hamilton principle. The determined partial differential equations are converted to differential equations by applying the assumed modes method. The dynamic axial displacement of the nanobeam along its length, is obtained by solving the differential equations via the convolution integral. The effect of length scale parameter, nonlocal parameter, velocity parameter, the constant distributed load and the excitation frequency on the maximum nondimensional dynamic axial displacement of the nanobeam are analyzed. Accordingly, the effect of these two parameters on the maximum nondimensional axial displacement of the nanobeam under the harmonic moving load is of significance importance.

Vibration analysis of a rotating cantilever double-tapered AFGM nanobeam

Nano-structures such as carbon nanotube, nanobeams, nanorods, nanoplates, nanowires, and nanorings are tremendously used in various small-scale devices and investigating their dynamical behavior has been a hot research topic and can be beneficial in manufacturing and designing new devices. Therefore, free vibration analysis of a rotating cantilever double-tapered axially functionally graded material nanobeam is presented in this work. Due to the nano-scale dimension, classical beam theories are incompetent at describing the behavior of nanobeams. Thus, the nonlocal Eringen elasticity theory is adopted which considers the nonlocal scale effect for the small dimension effect of the structure of a nano-scale mechanism. The proposed nanobeam structure is assumed to taper linearly in two different axes, and its material is changing nonlinearly along its length. The equation of motion of the proposed system is found utilizing the nonlocal Eringen theory, and it is solved using a semi-analytical technique, differential transform method. Mode shapes and natural frequencies are extracted as the solution of the equation of motion of the system. Furthermore, the effects of several parameters such as nonlocal scale effect, rotational speed, hub radius, and taper ratios on the natural frequencies are investigated. Finally, a comparison between the presented work and other reported results show an excellent agreement.

The Boundary Element Method Applied to the Analysis of Euler–Bernoulli and Timoshenko Continuous Beams

This article is concerned with the analysis of continuous beams under static loading by the Boundary Element Method. The problem can be governed by the Euler–Bernoulli theory, also called classical beam theory, or by the Timoshenko theory; consequently, two different formulations arise. In order to obtain the analytical solutions for the Timoshenko continuous beams, an equivalent version of the three moments equation is presented. Two examples are included: the first consists of a two-span beam; the second, of a three-span beam. The results for the displacement, rotation, bending moment and shear force are always compared with the corresponding analytical solutions.

Assessment of classical, advanced, and layer-wise theories for the vibration of rotating composite anisotropic blades

The present paper aims at studying composite cambered structures, tracking the seminar work “Rotating Blade Vibration Analysis Using Shells” presented by Leissa in Journal of Engineering for Power, in 1982, devoted to homogeneous metallic blades. A refined unidimensional (1D) formulation is here implemented to overcome the limitations of classical beam theories. With an appropriate choice of cross-sectional expansions, it is possible to make the 1D model suitable for analyzing shallow blades. This approach enables one to generate both layerwise (LW) and equivalent single layer (ESL) descriptions of the problem unknowns. Furthermore, it is possible to implement classic theories as special cases. Natural frequencies are determined for isotropic and composite blades, showing the effects of changing the fiber lamination angle of symmetric and unsymmetric configurations. Besides, this study investigates the effects of thickness and rotational speed over the structure. Significant differences between classic and high-order theories are found, concerning the accuracy and the computational costs. The causes of these differences are discussed, and the results can be used as a benchmark for future studies.

A comparison of the Bolotin and incremental harmonic balance methods in the dynamic stability analysis of an Euler–Bernoulli nanobeam based on the nonlocal strain gradient theory and surface effects

This study addresses the dynamic stability of an Euler–Bernoulli nanobeam under time-dependent axial loading based on the nonlocal strain gradient theory (NSGT) and considering the surface stress effects. The studied nanobeam cross-section was rectangular, and simply-supported boundary conditions were assumed. Moreover, a uniform thermal gradient was applied to the nanobeam. The elastic medium was modeled based on the Pasternak theory. The strain–displacement relations were derived using the Von Kármán equations. The governing equations were obtained by the energy method and applying the Hamilton's principle. Furthermore, the Bolotin and Incremental Harmonic Balance (IHB) methods were used to solve the differential equations. This study investigates the impact of such parameters as the small-scale parameter, the material length scale, surface effects, elastic medium parameters, temperature variations, geometry, and the static loading factor on the Dynamic Instability Region (DIR). The results are suggestive of the shift of the DIR to lower frequency zone by increasing the small-scale Eringen's nonlocal theory parameter, whereas an increase in the material length scale from the strain gradient theory moves the region to higher frequencies. In case the said parameters are equal, the result conforms to the classical beam theory. In addition, assuming a Pasternak medium and taking into account the effects of surface stress (Young's modulus and the residual stress of the surface) shifts the DIR to higher frequencies, whereas applying a compressive static load moves the region to lower frequencies. Moreover, depending on the thermal expansion coefficient of the medium, temperature variations can also displace the DIR.

Static Examination of Simply Supported Laminated Composite Beam with Varying Load Using Trigonometric Shear Deformation Theory

Objectives: To study the trigonometric shear deformation theory for the evolution of displacements and stresses of crossply simply supported laminated beam subjected to varying load. Methodology: A trigonometric shear deformation is used. The inplain displacement field uses a sinusoidal function in terms of the thickness coordinate to include the shear deformation effect. The theory satisfies the shear stress free boundary conditions on the top and bottom surfaces of the plate. The present theory obviates the need of a shear correction factor. Governing equations and boundary conditions of the theory are obtained using the principle of virtual work. Finding: Stresses and displacements for orthotropic, single-layer, three-layer symmetric square cross-ply laminated beam subjected to varying load. Novelty: The numerical results of the present theory for displacement and stresses are compared with those of classical (ETB), first-order (FSDT), and higher-order shear deformation beam theories. Keywords: Layerwise Composite Beam, Trigonometric Shear Deformation Theory, Varying Load, Simply Supported Beam.

Modeling and Analysis of a Planar Soft Panel Continuum Mechanism

Abstract Continuum mechanisms have drawn wide attention to scholars due to their salient advantages including compliance and dexterity. In this paper, a planar continuum mechanism made of soft panels is proposed. This mechanism has a reduced degree-of-freedom (DOF) compared with some existing continuum mechanisms capable of 3D motion. However, it can meet some application requirements in the field of robot and aerospace due to its characteristics of small stiffness in the motion plane and large stiffness perpendicular to the motion plane. Besides, a combined kinematics and statics modeling approach is presented for this mechanism by using the classical beam theory and a constrained optimization method. In order to ensure the model accuracy, a hybrid approach is proposed to consider gravity depending on the deformation under study. By comparing our results with those from the commonly used constant-curvature method, it is shown that our model is more accurate in predicting the deformation shapes.

Anisotropic turbulence-induced statistical properties evolution of a radially polarized partially coherent pulsed beam on propagation

In this study, we introduce a new kind of partially coherent vector beam with Gaussian distribution in the spatial, temporal and correlation components named a radially polarized partially coherent Gaussian Schell-model pulsed (GSMP) beam as a natural extension of scalar Gaussian Schell-model pulsed beam. Analytical formulas for the two-frequency cross-spectral density matrix elements of this beam are derived based on the extended Huygens–Fresnel theory. Within the frame of extend anisotropic non-Kolmogorov spectrum, the effects of asymmetry anisotropic turbulence eddies on the statistical properties of a radially polarized partially coherent GSMP beam are investigated numerically. Our work improves the classical theory of stochastic electromagnetic beams on propagation and may be useful for applications involving free-space optical communications.

Analytical Study of the Snap-Through and Bistability of Beams With Arbitrarily Initial Shape

Abstract We derive the snap-through solution and the governing snapping force equations for an arbitrarily preshaped beam deflected under a mid-length lateral point force. The exact solution is obtained based on the classical theory of elastic beams as a superposition of the initial shape and the modes of buckling. Two kinds of solutions are identified depending on the axial force level. The two solutions, bifurcation conditions, bistability conditions, and the snapping force equations are derived and discussed. The snap-through and snapping force solutions are then calculated for two common beam initial shapes, the curved (first buckling shape) and the inclined one (V-shape). In both cases, explicit expressions are obtained describing the snap-through behavior. The analytical modeling results show excellent agreement with finite element simulations. The comparison between the two cases shows a similar snap-through behavior qualitatively, while several differences and similarities are noticed quantitatively.

Multiaxial deformations of ionic polymer metal composites

Ionic polymer metal composites (IPMCs) are a promising class of soft active materials. Their high compliance, low actuation voltage, and ability to operate in wet environments have motivated two decades of intensive research on IPMC actuators. While we have witnessed several breakthroughs in the technology of IPMCs, from additive manufacturing to IPMC-based robots, our understanding of the physical underpinnings of their actuation remains elusive. There is a paucity of continuum physically-based models to investigate IPMC actuation, where the literature relies on structural models that postulate admissible mechanical deformations from classical beam or plate theories. In this sense, we know little about multiaxial deformations elicited by counterions’ diffusion and electromigration through the ionomer. Here, we demonstrate that macroscopic actuation of IPMCs is accompanied by localized mechanical deformations in the vicinity of the electrodes, caused by osmotic pressure and Maxwell stress. Toward this aim, we put forward a comprehensive nonlinear finite element analysis, conducted in Abaqus™ through a newly developed user element that allows for testing hypotheses on the inner workings of IPMC actuation and exploring complex configurations. Alongside with computational advances, we establish an exact solution for the two-dimensional Saint-Venant problem of plane-strain actuation of an IPMC, based on linear elasticity and nonlinear electrochemistry. Verified against finite element results, the exact solution offers a mathematically-tractable treatment of localized phenomena in the vicinity of the electrodes. Our results unveil a rich dependence of through-the-thickness deformation on the electric double layers that are formed in the vicinity of the electrodes. Due to the asymmetry of the boundary layers in the vicinity of the cathode and the anode, the ionomer deforms asymmetrically with respect to its mid-axis that also experiences an axial stretch. The Poisson ratio of the ionomer is found to have a critical role in shaping the response of the IPMC, from the onset of actuation to its back-relaxation. This study constitutes a first modeling step toward illuminating complex, multiaxial deformations of IPMCs, whose understanding is critical toward the design and manufacturing of high performance IPMC actuators.

Analytical and numerical studies on the slope inertia-based Timoshenko beam

In this work, a comparative study is performed between the slope inertia-based Timoshenko (SIBT) beam and other classical beam theories. We prove that the SIBT beam comprehensively incorporates physical effects of shear deformation, rotary inertia, and shear inertia. The transcendental equations of frequency are derived from the initial parameter solutions, and then used to demonstrate that the frequency spectrum of the SIBT beam theory has only one branch. A finite element implementation of the SIBT beam is developed and verified using closed-form solutions. We find that the SIBT and Love theories predict the same mode shape but different frequencies. Comparisons of frequency spectra obtained from different beam theories also suggest that the shear deformation influences the frequency more than the rotary inertia and that the SIBT beam theory predicts lower frequency values due to its complete consideration of shear deformation, rotary inertia and shear inertia. Parametric studies demonstrate that boundary conditions influence lower order frequencies predicted by the SIBT beam theory significantly, but only marginally influence the high order frequencies; they also show that the frequency spectrum of the SIBT beam theory has an upper bound.

Adptive Multi-scale Finite Element Considering Local Buckling of Members for Latticed Frames

To consider the local buckling of the members in the reticulated frames, based on the multi-scale simulation, the part of member that having local buckling is divided by the shell elements as a micro-model, and the other part of the member was simulated by beam elements as a macro-model. The incremental displacement constraint equations for the nodes on the section between the two models are established based on the plane section premise of classical beam theory. A method to introduce the constraint equations is derived based on the Updated Largrangian method. The location of the micro-model is predicted by the deflection function of the beam, and the length of the collapsed part is estimated. Its accuracy proves to be satisfied by the pure shell element model.

Multi-scale Finite Element Method for Members for Pipe Frames

To consider the local buckling of the members in reticulated frames, based on the multi-scale simulation, the part of member may be collapsed were divided by the shell elements as a micro-model, and the other part of the member was simulated by beam elements as a macro-model. The incremental displacement constraint equations for the nodes on the section between the two models are established based on the plane section premise of classical beam theory. By constraint variational principle, the tangent stiffness matrixes and the nodal load vectors of the micro-model and the outside structure are combined, and then the equations are solved. The location of the collapsed part is predicted by the deflection function of the beam, and the length of the collapsed part is estimated. Two case studies about a single beam and a single-layer reticulated dome are presented to show the feasibility and the validity of this method.

Dynamics of size-dependant Timoshenko micro beams subjected to moving loads

In this study, dynamic behaviour of size dependant Timoshenko microbeams under the influence of moving loads is modelled and investigated using FEM method. Firstly, the stiffness and mass property matrices of small-scale size-dependant two-node finite element are obtained by using new shape functions derived from the static state of motion equations created using classical Timoshenko beam theory (TBT) and modified couple stress theory (MCST). Subsequently, the load-beam interaction is modelled and integrated with beam motion equations without neglecting the mass inertia effects of the moving load. The effects of different load velocities, mass of the load and material size parameter on the beam dynamics have been investigated in a wide context.

Nonlinear secondary resonance of FG porous silicon nanobeams under periodic hard excitations based on surface elasticity theory

To impart desirable material properties, functionally graded (FG) porous silicon has been produced in which the porosity changes gradually across the material volume. The prime objective of this work is to predict the influence of the surface free energy on the nonlinear secondary resonance of FG porous silicon nanobeams under external hard excitations. On the basis of the closed-cell Gaussian-random field scheme, the mechanical properties of the FG porous material are achieved corresponding to the uniform and three different FG patterns of porosity dispersion. The Gurtin–Murdoch theory of elasticity is implemented into the classical beam theory to construct a surface elastic beam model. Thereafter, with the aid of the method of multiple time-scales together with the Galerkin technique, the size-dependent nonlinear differential equations of motion are solved corresponding to various immovable boundary conditions and porosity dispersion patterns. The frequency response and amplitude response associated with the both subharmonic and superharmonic hard excitations are obtained including multiple vibration modes and interactions between them. It is found that for the subharmonic excitation, the nanobeam is excited within a specific range of the excitation amplitude, and this range shifts to higher excitation amplitude by incorporating the surface free energy effects. For the superharmonic excitation, by taking surface stress effect into account, the excitation amplitude associated with the peak of the vibration amplitude enhances. Moreover, in the subharmonic case, it is demonstrated that by increasing the porosity coefficient, the value of the excitation frequency at the joint point of the two branches of the frequency-response curve reduces. In the superharmonic case, it is revealed that an increment in the value of porosity coefficient leads to decrease the peak of the oscillation amplitude and the associated excitation frequency.