Euler-Bernoulli Beam Research Articles

An improved analytical model of a thick defective phononic crystal for bending wave excitation

This research aims to advance the existing analytical model, which was based on the Euler–Bernoulli beam theory, for predicting the velocity amplitude of bending waves excited while amplified by defective phononic crystals (PnCs) with piezoelectric elements. The previous analytical model falls short for thick defective PnCs and is limited to low frequencies. Hence, the Timoshenko beam theory is newly incorporated to address these issues, considering shear deformations and rotational inertia. Its performance is validated against a finite element model. It confirms that the proposed analytical model outperforms its predecessor, which was suitable for thin defective PnCs with a slenderness ratio above 10 and frequencies around 2 kHz. Remarkably, the improved analytical model demonstrates excellent predictive capabilities up to 40 kHz for thick defective PnCs with a slenderness ratio of around 2.5. With a relative error of less than 1% compared to the finite element model, this advanced approach also demonstrates time-efficient and stable computational performance. This work introduces three contributions. First, it pioneers the proposal of an electromechanically coupled analytical model for predicting the performance of defect-mode-enabled bending wave excitation. Second, this enhanced analytical model surpasses existing counterparts in terms of both accuracy and computational stability, particularly for thick defective PnCs, even at high frequencies. Last, the analytical model dramatically reduces the computation time.

Refined quasi-3D shear deformation theory for buckling analysis of functionally graded curved nanobeam rested on Winkler/Pasternak/Kerr foundation

In the present work, a novel refined three-variable quasi-3D shear deformation theory incorporates a correction factor is developed to analyze the buckling behavior of multi-directional functionally graded (FG) curved beams. The proposed displacement field is formulated in accordance with the Euler-Bernoulli beam theory. The research investigated two types of coated Functionally Graded (FG) nanobeams: Hardcore (HC) FG curved nanobeams and Softcore (SC) FG curved nanobeams. Three different material distributions are taken into consideration: a bidirectional material distribution referred to as “2D-FG,” a unidirectional transverse material distribution known as “T-FG,” and a unidirectional axial material distribution called “A-FG.” Eringen’s nonlocal elasticity theory is employed to capture small-scale effects. The total potential energy principle is utilized to derive the equilibrium equations of curved nanobeams. A novel solution, utilizing Galerkin’s method, has been developed to effectively address a range of boundary conditions. The curved FG beam is supported by an elastic foundation following the Winkler/Pasternak/Kerr model. A comprehensive analysis has been conducted to examine the impacts of various FG schemes, curved beam geometry, nonlocal parameter, elastic foundations, and various boundary conditions on the dimensionless critical buckling loads. This analysis aims to provide a comprehensive understanding of how each of these factors influences the critical buckling loads of the nanobeams.

Diffuse domain approach for flexible needle insertion and relaxation.

Needle insertion simulations play an important role in medical training and surgical planning. Most simulations require boundary conforming meshes, while the diffuse domain approach, currently limited to stiff needles, eliminates the need for meshing geometries. In this article the diffuse domain approach for needle insertion simulations is first extended to the use of flexible needles with bevel needle tips, which are represented by an Euler-Bernoulli beam. The model parameters are tuned and the model is evaluated on a real-world phantom experiment. Second, a new method for the relaxation of the needle-tissue system after the user releases the needle is introduced. The equilibrium state of the system is determined by minimizing the potential energy. The convergence rate of the coupled Laplace equations for solving the Euler-Bernoulli beam is 1.92 0.14 for decreasing cell size. The diffuse penalty method for the application of Dirichlet boundary conditions results in a convergence rate of 0.73 0.21 for decreasing phase field width. The simulated needle deviates on average by 0.29 mm compared to the phantom experiment. The error of the tissue deformation is below 1 mm for 97.5% of the attached markers. Two additional experiments demonstrate the feasibility of the relaxation process. The simulation method presented here is a valuable tool for patient-specific medical simulations using flexible needles without the need for boundary conforming meshing. To the best of the authors' knowledge this is the first work to introduce a relaxation model, which is a major step for simulating accurate needle-tissue positioning during realistic medical interventions.

Heterogeneous Beam Element for Multiscale Modeling of Non-prismatic Composite Beam-like Structures

Anisotropy, heterogeneity, and geometry make full system-level three-dimensional (3D) finite element analysis (FEA) simulation of real composite beam-like structures, such as wind blades, impossible due to limitations on computing resources. A popular approach is to cut the structure into multiple cross-sections, from which the beam properties based on the Euler-Bernoulli or Timoshenko beam models are extracted using two-dimensional (2D) cross-sectional analyses. These properties are then utilized in beam elements for system-level analyses. However, the mechanics behind 2D cross-sectional analyses are based on prismatic beams, limiting the accuracy of this approach on non-prismatic structures, especially in predicting stresses. In this work, we present a new multiscale method based on Mechanics of Structure Genome to analyze beam-like non-prismatic composite structures. A beam-like structure is homogenized into a series of 3-node Heterogeneous Beam Elements (HBE) with effective beam element stiffness matrices, which are then used as input for one-dimensional (1D) beam analysis using Abaqus User Element subroutine (UEL). Using the macroscopic beam analysis results as input, we can also perform dehomogenization to predict the stresses and strains in the original structure. We use three examples, a prismatic composite beam, an isotropic homogeneous tapered beam, and a composite tapered beam, to demonstrate the capability of HBE and show its advantages over the traditional cross-sectional analysis approach. HBE can capture macroscopic behavior and detailed stresses due to non-prismatic geometry. HBE provides a new concept to model non-prismatic composite beams by modeling heterogeneous beam-like structures as equivalent beam elements rather than beam properties of material points of the reference line.

Natural frequencies of prismatic and tapered composite girders with corrugated steel webs

The present study proposed an analytical formula to predict the natural frequencies of prismatic and tapered composite girders with corrugated steel webs (CSWs), and rigorously provides the derivation process. It is experimentally and numerically verified that the proposed formula is more applicable for predicting the natural frequencies of girders with CSWs, in comparison to the traditional formula based on Euler-Bernoulli beam theory. The study demonstrates that the traditional formula used for determining the natural frequencies of prismatic girders with CSWs leads to significant overestimation, mainly because it ignores the effects of shear deformation. The finite element (FE) simulation suggests that the free vibration mode shapes of tapered girders with CSWs exhibit sinusoidal patterns. Consequently, an equivalent stiffness method has been developed for accurately and conveniently estimating the natural frequencies of tapered girders with CSWs. This paper establishes a theoretical foundation for modifying the natural frequency calculation formula of prismatic and tapered girders with CSWs in current specifications, the research findings hold significant implications for the structural dynamic analysis and design of such girders.

Novel vibration self-suppression of periodic pipes conveying fluid based on acoustic black hole effect

The pipes conveying fluid are concerned as extremely significant fluid-solid coupling structures due to the extensive application in the many engineering fields, but poor operating performances induced by unexpected vibration may destroy the related devices in the piping structures. To effectively attenuate the fluid-solid coupling vibration, a novel control technique is proposed to realize the vibration self-suppression of pipes conveying fluid with integration of the periodic acoustic black hole (ABH) wedges inspired by the extraordinary band gaps (BGs) behaviors of the metamaterials. On the basis of the Euler-Bernoulli (E-B) beam model, the mechanical model and dynamic equations are established by using Hamilton's principle. By applying the spectral element method (SEM) in conjunction with the transfer matrix method (TMM), as well as considering the Bloch theorem, the mechanism of elastic-wave distributions of such periodic ABH pipes conveying fluid are disclosed through investigating the BGs characteristic, frequency response and wave shapes. The results show the proposed technique can effectively form BGs in low frequency bands, and most of the energies are absorbed and scattered by the wedge edge of the pipes in the wave propagation. More importantly, by adjusting the system parameters, the proposed structure can enlarge the BGs regions and transfer it to the low frequency bands. It provides a simple design scheme for the manufacture of pipe conveying fluid with ultra-wide BGs.

Analytical solution for seismic response of vertical shaft with primary and secondary linings under SH wave excitation

This paper presents a novel mechanical model for seismic response analysis of vertical shafts. The model is based on elastic foundation beam theory and simplifies the shaft linings as parallel Euler Bernoulli beams. Their interaction is simulated using uniformly distributed springs. The governing differential equation for shaft vibration is derived for SH wave excitation. The seismic input motion is applied to the primary lining by converting the free field displacement into harmonic form. The closed-form solutions for seismic response of both primary and secondary linings are obtained via the transfer function method. The accuracy of analytical solution is validated by comparing with finite element analysis results. Additionally, the impact of foundation stiffness, secondary lining thickness, shaft outer diameter, and relative stiffness between linings on seismic response are investigated in the parametric analysis. The analytical solutions of the seismic response analysis of vertical shafts indicates that the relative stiffness between linings significantly affects the secondary lining's peak bending moment. Furthermore, higher stiffness of the elastic connection layer between the two linings can effectively reduce the peak displacement response of the secondary lining.

Experimental measurements of stress in an acoustic black hole using a laser Doppler vibrometer

Acoustic Black Holes (ABHs) make use of modifications to a structure to effectively decrease the structural wavespeed, thus increasing the effect of damping material applied in the ABH taper region, resulting in greater vibration attenuation. The most common way in which this is implemented is by gradually reducing the thickness of the structure over a finite interval, to a very thin tip. The focusing effect of the ABH results in high amplitude vibrations occurring in the thin part of the structure, resulting in high stresses and raising significant concerns about fatigue life. This paper presents an experimental assessment of stress in the taper section of an ABH used to terminate a uniform beam, using laser Doppler vibrometer measurements to avoid the mass loading associated with accelerometers or strain gauges. A calculation of stress using Euler–Bernoulli beam theory is then presented, and the validity of this approach is assessed for a thick damping layer applied to a thin structure. A comparison is then made to predictions from a numerical model, in order to validate the results from the experimental measurements.

A new model of Crossflow fluid-structure instabilities for helicopter blades

Crossflows that induce fluid-structure instabilities arise in many engineering applications, such as flow over a road bridge or an aircraft wing. At a certain flow speed, an aeroelastic flutter instability is induced arising from the coupled translational and rotational movement of the structure. This instability is modified in a subgroup of these applications that include the additional effect of rotation, e.g., propellers. The structure now experiences a linearly varying effective mean flow (faster near the tip than at the root) and an induced stiffness (via tension) due to the centrifugal force that is also spatially varying. The flutter instability can create vibration fatigue issues and can also create a significant acoustic signature. This paper reports a newlinearised numerical model of the problem that fully couples a finite-difference Euler–Bernoulli beam model to a Theodorsen unsteady lift model that includes these effects of rotation. Comparisons with theoretical predictions in non-rotating flow demonstrate the effect on the trends for the onset of linear instability. The model is then applied to the problem of helicopter blade flutter, and the effects of various strategies for delaying the onset of linear instability are modelled.

A unique and comprehensive approach to investigate the transverse free vibration of non-uniform and functionally graded Euler–Bernoulli beams

A unique and comprehensive approach to investigate the transverse free vibration of non-uniform and functionally graded Euler–Bernoulli beams

Analytical Solution for the Two-Layered Composite Beam-Column with Interlayer Slip and Constant Axial Load

The authors present an analytical solution for the two-layered composite beams with imperfect shear connections. The considered beam is simply supported at both ends. The beam is subjected to transverse and axial loads. The kinematic assumptions of the Euler-Bernoulli beam theory are used. The connection of the beam components is perfect in normal direction, but the axial displacement field may have jump. The shear axial force derived from the imperfect connection is proportional to the relative slip occurring between the layers. The determination of the analytical solution is based on the Fourier method. Two examples illustrate the application of the presented analytical method.

A novel approach of bending behavior of carbon nanotubes by combining the effects of higher-order boundary conditions and coupling through doublet mechanics

This article presents the bending behavior of carbon nanotubes (CNTs) based on doublet mechanics (DM) and Euler–Bernoulli beam theory to analyze how deflection changes with higher-order boundary conditions and the coupling, which have not been previously investigated. The governing equations and associated boundary conditions are obtained using the principle of minimum potential. Numerical solutions show an important effect of higher-order boundary conditions on the deflection of cantilever CNTs, unlike simply supported CNTs. A paradox in which DM shows a stiffening behavior for cantilever CNTs is resolved using higher-order boundary conditions. The coupling significantly effects the bending behavior of armchair CNTs.

A Unified Numerical Approach to the Dynamics of Beams with Longitudinally Varying Cross-Sections, Materials, Foundations, and Loads Using Chebyshev Spectral Approximation

Structures with inhomogeneous materials, non-uniform cross-sections, non-uniform supports, and subject to non-uniform loads are increasingly common in aerospace applications. This paper presents a simple and unified numerical dynamics model for all beams with arbitrarily axially varying cross-sections, materials, foundations, loads, and general boundary conditions. These spatially varying properties are all approximated by high-order Chebyshev expansions, and discretized by Gauss–Lobatto sampling. The discrete governing equation of non-uniform axially functionally graded beams resting on variable Winkler–Pasternak foundations subjected to non-uniformly distributed loads is derived based on the Euler–Bernoulli beam theory. A projection matrix method is employed to simultaneously assemble spectral elements and impose general boundary conditions. Numerical experiments are performed to validate the proposed method, considering different inhomogeneous materials, boundary conditions, foundations, cross-sections, and loads. The results are compared with those reported in the literature and obtained by the finite element method, and excellent agreement is observed. The convergence, accuracy, and efficiency of the proposed method are demonstrated.

An Inertial Noncontact Piezoelectric Rotary Energy Harvester with Linear Reciprocating Motion

Herein, an inertial noncontact piezoelectric rotary energy harvester with linear reciprocating motion (L‐PREH) is presented. The existing piezoelectric rotary harvester employing gravity excitation has limited performance when the rotation speed is high due to the negative influence of centrifugal force. L‐PREH translates rotational motion into linear motion via the transmission chain and employs inertial force excitation to overcome high‐speed performance limitations. Using the Euler–Bernoulli beam theory, the motion governing equations of piezoelectric transducers have been derived, and an electromechanical coupling model has been constructed. Moreover, the piezoelectric transducer is simulated and analyzed. The control variable approach is used to explore the key parameters impacting output performance. When the mass is positioned in symmetrical method, the guide rod is fixed in noncentral place, the limiter is fixed in longest distance, the distance between the mass's center and the main frame is the maximum, and the rotating speed is 450 RPM, the maximum peak‐to‐peak output voltage of an L‐PREH single transducer is 24 V. The highest power of two piezoelectric transducers linked in parallel with a load resistance of 400 kΩ is 0.27 mW, which can light up more than 70 light‐emitting diodes. The L‐PREH can drive low‐power devices.

Numerical Analysis of Cracked Double-Beam Systems

Based on elasticity theory, this paper discusses the static analysis of a cracked double-beam system in the presence of a Winkler-type medium. It is further assumed that the double-beam system is constrained at both ends by elastically flexible springs with transverse and rotational stiffness. Using a variational formulation, the governing static equations are derived and solved using analytical and numerical approaches. In the first approach, closed-form solutions for the displacement functions are obtained based on the Euler–Bernoulli beam theory. In the second approach, the Cell Discretisation Method (CDM) is performed, whereby the two beams are reduced to a set of rigid bars connected by elastic constraints, in which the flexural stiffness of the bars is concentrated. The resulting stiffness matrix is easily deduced, and the governing equations of the static problem can be immediately solved. A comparative analysis is performed to verify the accuracy and validity of the proposed method. The study focuses on the effect of various parameters, including crack depth and position, boundary conditions, elastic medium and slenderness. The validity of the proposed analysis is confirmed by comparing the current results with those obtained from other approaches. In particular, the results obtained by closed-form solution and CDM are compared with the Finite Element Method (FEM). The accuracy of the results was assessed by making comparisons with results found in the literature and reported in the bibliography. It was shown that the proposed algorithm provides a simple and powerful tool for dealing with the static analysis of a double-beam system. Finally, some concluding remarks are made.

Porosity-dependent stability analysis of bio-inspired cellular nanocomposite shells

This paper presents an innovative numerical model to investigate buckling behaviour of bio-inspired continuously graded porous (CGP) nanocomposite cylindrical shells. It is postulated that the shell subjected to combined lateral pressure and axial compressive load is constructed from metal foams with closed-cell structures that possess graded internal pores, which exhibit three types of continuously graded porosity profiles based on a power-law distribution. A scaling relation for the effective Young’s modulus of the cellular structure determined by a variational finite element method (FEM) is used. The effective constitutive law of an elastic isotropic metal matrix containing distributed elastic carbon-nanotubes (CNTs) is estimated in consideration of the impact of CNTs agglomeration using a continuum model based on the Eshelby–Mori–Tanaka (EMT) approach. In contrast to conventional approaches, the study employs Euler–Bernoulli beams to model stiffeners within the CGP shells. This choice allows for a more realistic representation of stiffener effects, as opposed to the prevalent approach of uniform smearing across the shell’s surface. The equilibrium equations of the CGP shell, based on the Reddy higher-order shell theory (RHST), is obtained through the application of the Euler equation. Subsequently, the equations for stability are obtained through the utilization of the variational method. This study emphasizes the effects of geometrical parameters, porosity variability, and distribution of CNTs on the buckling performance of the CGP shells. The intricate interplay between CNTs and porosity distributions critically influences the stability behaviour of CGP shells. CNTs arrangement remarkably impacts buckling behaviour at higher length-to-mean radius ratios, while symmetric porosity near the mid-surface significantly enhances stiffness. These findings provide valuable insights for designing closed-cell cellular stiffened shells with optimal porosity to enhance stability.

A pile–soil interaction model for ground-borne vibration problems based on the singular boundary method

An efficient three-dimensional approach for solving pile–soil interaction problems is proposed. In the approach, the soil is modelled as an elastic half-space, and its response in the presence of the pile’s corresponding cavity is computed by employing the singular boundary method. The pile is modelled analytically using the classic rod and Euler–Bernoulli beam theories. For the coupling with the soil, the pile is divided in a set of rigid segments that interact along their circumference with the soil. The methodology allows the rotational motions and reaction torques at these segments to be accounted for and their contribution in the accuracy of the scheme is assessed. A criterion to define the minimum number of collocation points that offers an acceptable trade-off between accuracy and numerical performance is also proposed. The method is validated against well-established methodologies and using the reciprocity principle that relates the wave radiation from the pile to the ground field with the incident wave problem due to a load on the ground surface. Results are shown for different soil stiffnesses and different pile length to diameter ratios. The employment of the singular boundary method is shown to provide strong computational advantages to detailed modelling approaches such as the three-dimensional finite element-boundary element method, as well as overcoming the fundamental limitations of plane-strain and axisymmetric methods.

A Galerkin approach for analysing coupling effects in the piezoelectric semiconducting beams

The piezoelectric semiconductor combines the characteristics of both piezoelectric and semiconducting materials, making it highly promising for a variety of engineering applications. The present work adopts a Galerkin approach to derive a general weak form of the coupled governing equations for the analysis of piezoelectric semiconductors. The coupling effect between the strain, the electric field, and the gradients of the concentrations of holes and electrons has been considered. Piezoelectric semiconductors are often deployed in the shape of rods and beams and therefore commonly experience axial and transverse load. Thus, the Euler–Bernoulli beam theory is employed to formulate a numerical approach to analyse the coupling behaviour of such piezoelectric semiconductor beams. In order to satisfy the C1 continuity requirement of Euler–Bernoulli beam theory without introducing additional degrees of freedom, a discretisation with B-Splines is used. Non-mechanical variables, including electrical potential, holes and electrons concentrations, are decomposed into constant, linear and quadratic terms through series expansion along the thickness, which ensures compatibility with beam formulation. To validate the proposed method, an N-type piezoelectric semiconducting beam under a sinusoidal distributed load, for which an analytical solution is available, is initially examined. Subsequently, complex examples including a cantilever beam with generalised loading conditions and a beam with PN-junction are analysed to demonstrate the capabilities of the proposed method in handling more challenging scenarios.

Vibration response of viscoelastic nanobeams including cutouts under moving load

This manuscript presents a mathematical model and numerical solution to predict vibration responses of nonlocal strain gradient (NLS) perforated viscoelastic nanobeam under dynamic moving loads. The microstructure and size length scales are considered in the frame of NLS theory. The viscoelastic damping of the structure is considered by linear Kelvin Voigt viscoelastic model. Timoshenko and Euler Bernoulli beam theories are developed to consider both thin and thick structure response. The moving load is portrayed by point load and harmonic type. The dynamic equations of motion incorporating viscoelasticity and size dependency effects is derived by using Hamilton principle. A differential quadrature method (DQM) with Chebyshev–Gauss–Lobatto formula is used to discretize the space domain and solve the model numerically. The obtained results are validated and compared with available literature. Numerical studies are conducted to show the influence of the material viscosity parameter, perforation parameters, shear deformation effect, nonlocal strain gradient, moving load velocity, and beams slenderness ratio on the dynamic behavior of viscoelastic perforated nanobeams under moving load. The material viscosity parameter can effectively control the magnitude and stability of the magnification factor profiles. Higher filling ratios result in larger values of the dynamic magnification factor, whereas larger numbers of hole rows lead to smaller values. The proposed procedure is supportive in the analysis and design of perforated viscoelastic NEMS structures under moving load.

Analysis of Dynamic Characteristics of Tristable Exponential Section of Piezoelectric Energy Harvester

Variable-cross-section beams have better mass and strength distribution compared with constant cross-section beams, which can optimize the harvesting power of piezoelectric vibration energy harvesters, which are widely used in self-supplied and low-power electronic devices, providing more convenience and innovation for the development of micromechanical intelligence and portable mobile devices. This paper proposes a piezoelectric energy harvester with a tristable-exponential-decay cross section, which optimizes the strain distribution of the cantilever beam through exponential decay changes to improve the harvesting efficiency of the harvester in low-frequency environments. First, the nonlinear magnetic force is obtained based on the magnetic dipole, and the dynamic model is established by using the Euler–Bernoulli beam theory and Lagrangian equation. The influence of the structural parameters of the harvester on the system dynamics and output characteristics is analyzed in the two dimensions of time and frequency, and the influence of the exponential decay coefficient on the system dynamic response and output power is deeply studied. The research shows that the exponential decay section can reduce the first natural frequency of the cantilever beam; by changing the amplitude, frequency, d and dg of the excitation acceleration, the switching of the monostable, tristable and bistable states of the system can be realized. With a decrease in the exponential decay coefficient, under a low-frequency excitation of 0–7 Hz, the output power of the cantilever beam per unit volume is significantly improved, and under a 4 Hz low-frequency excitation, the acquisition output power per unit volume of the cantilever beam is increased by 7 times, thus realizing low-frequency, high-efficiency energy harvesting.