Euler Beam Theory Research Articles

Symplectic method for the influence of surface effect on thermal buckling of graded porous nanobeams

The influences of surface effect on the thermal buckling behavior of graded porous nano-beams were analyzed. Based on the Grutin-Murdoch surface elasticity theory and Euler beam theory, the thermal buckling of graded porous nano-beams was studied by using Hamilton system. In the symplectic space, the buckling problem of gradient porous beams was reduced to the zero eigenvalue problem of the system, and the critical buckling temperature and buckling mode of nano-beams correspond to the symplectic eigenvalue and eigensolution of the Hamiltonian system. It is assumed that the performance of gradient porous materials varies continuously throughout the thickness, and two cosine forms of uneven distribution of porosity along its thickness are considered. By using bifurcation conditions and normalization methods, the critical buckling temperature rise for buckling modes and nano-beams was analytically determined. Finally, the influence of surface effects on the thermal buckling of gradient porous nano-beams was presented in the form of a chart. The results indicate that graded porous nano-beams have significant surface effects. Considering surface effects, the critical thermal load and critical buckling temperature of porous nano-beams will be increased. At the same time, appropriate porosity coefficients and pore distribution can effectively improve the mechanical properties of porous nano-beams.

Detection of Crack Location and Crack Depth in Leaf Spring

The objective of this work is to find the crack location and crack depth in leaf spring of an army jeep. The spring is subjected to heavy jerks and vibration during the military operation. Therefore it is necessary to find the crack in a leaf spring. Crack in a leaf spring introduces local flexibility that would affect the vibration response of the leaf spring. Main problem is to detect existence of a crack together with its location and depth in the leaf spring .We used a natural frequency a basic criteria for crack detection. Analytical calculations are done to find out natural frequency of leaf spring by using Euler's beam theory. Ansys 16.0 is used to find out natural frequency along with experimentation on FFT analyzer. Now, We propose a method based on some set of fuzzy rules obtained from the information supplemented by Numerical Analysis. Fuzzy controller use comprises of three input variables (first, second and third natural frequency) and two output variables (relative crack length and relative crack depth) are generated with Triangular MF.

Modeling Method of Meshing Stiffness of High Speed Spur Gears

As one of the most important dynamic excitation sources, the gear time-varying mesh stiffness is the key parameter of gear system dynamic model. So how to calculate the gear mesh stiffness accurately is of great importance. Based on Euler beam theory, this paper proposes a primitive calculation algorithm (PCA) to calculate the time-varying mesh stiffness of spur gears. A simplified gear model is developed based on nonlinear strain-displacement relationships, and a finite element program is used to simulate the gear's meshing stiffness. The rationality of this method was verified through finite element analysis. The results indicate that mesh stiffness plays a crucial role in controlling mesh deformation. In view of this, this study lays a theoretical foundation for further analysis of the vibration and noise of gears.

Research on the equivalent stretching mechanical properties of Nomex honeycomb core considering the effect of resin coating

Abstract In this article, the equivalent elastic modulus of the resin-impregnated Nomex honeycomb core is derived by considering the bending and stretching deformation of the honeycomb wall, based on the Euler beam theory. The elastic modulus obtained by the proposed theoretical method is proved to be in good accordance with both the experimental results and numerical results. Furthermore, the effect of the geometric parameters on the equivalent mechanical properties of the honeycomb core is discussed using a dimensionless method.

An experimental and numerical effort on the vibration behavior of additively manufactured recycled polyethylene terephthalate glycol components

AbstractThe importance of recycling engineering components and thus obtaining low‐cost production solutions has become prominent in today's world. In this study, the mechanical and dynamic behaviors of three‐dimensional‐printed recycled polyethylene terephthalate glycol (RePET‐G) beams were investigated numerically and experimentally for the first time in the literature. Initially, the governing equations of the beams were determined according to the Bernoulli–Euler beam theory, and these equations were numerically solved using the differential quadrature method and ANSYS program. Subsequently, to validate the accuracy of the numerical models, the obtained natural frequencies were compared with experimental results. It was observed that the numerical results showed good agreement with the experimental results. Finally, the effects of beam length, infill rate, and building direction on the natural frequencies of RePET‐G beams were investigated. The outcomes showed that as the beam length changed, natural frequencies were significantly affected. Increasing the infill rate, especially for beams with vertical building direction, from 20% to 100% led to a slight decrease in the natural frequency values of the structure. Moreover, it was found that for beams with an infill rate of 100%, the natural frequency values obtained in the horizontal building direction were higher than those obtained in the vertical building direction.Highlights Printable recycled filaments have great potential for vibration applications. Sample length affects the first natural frequency value of RePETG parts. Differential quadrature and ANSYS methods can be utilized for the vibration. For the 3D‐printed samples, rising infill rate causes a natural frequency drop.

Nonlinear vibrations of variable speed rotating graphene platelets reinforced blades subjected to combined parametric and forced excitation

It is generally acknowledged that the disturbance of rotating speed will cause parametric resonance for blades, and the presence of rotor displacement also can make the blade vibrate transversely. However, the coupled resonance mechanism of blades under the combined action of rotating speed disturbance and rotor displacement is not yet clear. To answer this question, this article investigates the nonlinear coupled resonance behavior of rotating blades in two cases: typical tuned (the frequency ratio of parametric and forced excitations equals 2:1) and frequency ratio detuned. A nonlinear vibration equation for rotating blades is established based on Euler beam theory in conjunction with geometric nonlinearity. The mixing rule and modified Halpin-Tsai model are used to calculate the effective properties of GPLRMF materials. Subsequently, using the method of varying amplitudes (MVA), an approximate analytical solution of the coupled resonance response is derived, the Jacobian matrix is used to determine the stability of non-trivial solutions, and the accuracy of the analytical results is verified with the aid of numerical solutions. Finally, the steady-state response of typical tuned and detuned coupled resonance is analyzed, and the influence mechanism of factors such as excitation amplitude, excitation phase angle and other parameters on coupled resonance is studied in detail. The results indicate that the supercritical coupled resonance curve exhibits double resonance peaks and jumps. Meanwhile, the steady-state response of coupled resonance highly depends on the excitation phase angle.

Dynamics and Stability of Double-Walled Carbon Nanotube Cantilevers Conveying Fluid in an Elastic Medium

The paper concerns the dynamics and stability of double-walled carbon nanotubes conveying fluid. The equations of motion adopted in the current study to describe the dynamics of such nano-pipes stem from the classical Bernoulli–Euler beam theory. Several additional terms are included in the basic equations in order to take into account the influence of the conveyed fluid, the impact of the surrounding medium and the effect of the van der Waals interaction between the inner and outer single-walled carbon nanotubes constituting a double-walled one. In the present work, the flow-induced vibrations of the considered nano-pipes are studied for different values of the length of the pipe, its inner radius, the characteristics of the ambient medium and the velocity of the fluid flow, which is assumed to be constant. The critical fluid flow velocities are obtained at which such a cantilevered double-walled carbon nanotube embedded in an elastic medium loses stability.

In-Plane Elastic Properties of 3D-Printed Graded Hierarchical Hexagonal Honeycombs.

In this study, the graded hierarchical hexagonal honeycomb (GHHH) integrating gradient design and hierarchical design was fabricated using the 3D-printing technique, and its in-plane elastic properties were investigated theoretically, experimentally, and numerically. Theoretical solutions were developed based on the Euler beam theory to predict the effective elastic modulus and Poisson's ratio of GHHH, and theoretical values were in good agreement with the experimental and numerical results. The effect of gradient design and hierarchical design on the in-plane elastic properties of GHHH was also analyzed and compared. Results showed that the hierarchical design has a more significant effect on Poisson's ratio and adjusting the internal forces of GHHH compared with the gradient design. In addition, it was found that GHHH exhibited higher stiffness compared with regular hexagonal honeycomb (RHH), graded hexagonal honeycomb (GHH), and vertex-based hierarchical hexagonal honeycomb (VHHH) under the constraint of the same relative density, respectively. Specifically, the effective elastic modulus of GHHH can be enhanced by 119.82% compared to that of RHH. This research will help to reveal the effect of integrating hierarchical design and gradient design on the in-plane elastic properties of honeycombs.

An analytical method for free vibrations of the fuel rod with non-uniform mass of small modular reactor

The intricate internal structure of fuel rods results in a non-uniform mass distribution, making it imperative to employ analytical methods for accurate assessment. The study utilizes Euler beam theory to derive the transverse vibration equation for beams with varying mass distribution. The approach involves transforming the non-uniform mass beam into a multi-segment beam with concentrated mass points. Modal function relationships between adjacent uniform segments are established based on continuous conditions at connection points. This transformation leads to the conversion of the variable coefficient differential equation into a nonlinear matrix equation. The Newton-Raphson method is then applied to calculate the characteristic equation and mode shapes, essential for determining natural frequencies. To validate precision, the results obtained are compared with those derived from the finite element method. Furthermore, the developed method is employed to assess the impact of gas plenum location and length on the natural frequency of fuel rods. The proposed methodology serves as a rapid design tool, particularly beneficial during the design phase of fuel rods with non-uniform mass distribution, aiding in configuring structural aspects effectively.

TZM Alloy Job Carrier for High Temperature Vacuum Furnace

RRCAT has a cold walled dry pumped controlled atmosphere furnace for brazing, diffusion bonding and heat treatment of components required for particle accelerators. This furnace has a work zone of 500 mm diameter and 1.5 meter length and it can be operated up to a temperature of 1100 deg C. A Titanium-Zirconium-Molybdenum (TZM) alloy job carrier has been designed for a load carrying capacity of 400 kg for better utilization of the vertical space. Due to alloy’s outstanding combination of properties e.g. high electrical and thermal conductivity, low coefficient of thermal expansion and high rupture as well as creep strength; it is a preferred choice in a wide variety of high-temperature applications in spite of its higher cost. The structural design of the job carrier needs optimization in order to reduce its weight as well as its cost without compromising on design constraints. In this paper, a detailed design of job-carrier has been presented using plate theory and Euler beam theory. Subsequently, finite element analysis (FEA) in ANSYSTM Workbench is performed to ensure that resultant stresses and deformation are within acceptable limits. Additionally, a limit load analysis has been performed to determine the maximum load capacity and factor of safety. FEA results are then used for fatigue life and creep assessment. The job-carrier has been successfully installed inside RRCAT’s controlled atmosphere furnace and has been tested at full load. The effort has also led to 70% reduction in the procurement cost with respect to commercially available products.

A two-stage based model for partially-buried pile group response in cross-anisotropic saturated media adjacent to moving harmonic loads

The pile foundations are frequently affected by adjacent traffic loads except for general active loads. However, the dynamic responses of the pile groups under moving loads have been rarely reported before. In addition, the influence of material anisotropy is often neglected. In this paper, a three-dimensional (3D) analytical model for the dynamic analysis of partially buried pile groups in stratified saturated cross-anisotropic media under adjacent moving harmonic loadings is developed. Specifically, a 3D Bernoulli–Euler beam theory is adopted to simulate the monopile and then superimposed into the finite element (FE) formulation for the partially embedded pile groups. Subsequently, the basic solutions of layered saturated cross-anisotropic soils are derived by the analytical layer-element approach (ALEA). By combination with ALEA, the two-stage based boundary element (BE) equations for the pile-soil interface are obtained. Finally, by coupling the FE and BE formulations, the dynamical response equation of pile-soil is established. Based on the verification of the accuracy of the proposed methodology by comparisons with existing solutions and FE results from ABAQUS, the impacts of material anisotropy, pile buried length, loading velocity, and pile stiffness on the time-domain dynamic behavior of pile groups are analyzed. The results show that increasing the pile-buried ratio or pile stiffness reduces the magnitude of the dynamic response. Moreover, with the increase of material anisotropic parameters, the peak pile response decreases. And the peak dynamic response first increases and then reduces as the loading velocity increases.

Bending behavior of diamane and twisted bilayer graphene: Insights from four-point bending deformation

The intriguing physical properties of two-dimensional (2D) nanomaterials make them promising building blocks for flexible electronics. Using a four-point bending approach, this work establishes a comprehensive understanding of the bending behavior of diamane – a 2D diamond nanostructure, from elastic deformation to structural failure through atomistic simulations. The four-point bending method accurately reproduces the pure bending of the sample, and the obtained force-displacement curve fit well with the classical Euler beam theory. Structural failure is observed from diamane under bending when its thickness or the number of layers increases. Atomic insights reveal that the crack initiates from the tension side of the sample, resulting in a tension-induced bending failure. Specifically, the bending limit is found to be slightly larger than the fracture strain under tensile deformation. Additionally, the bending behaviour of the diamane analogous – twisted bilayer graphene with interlayer-bonding (TBGIB), has been investigated. Different from diamane, TBGIB bends elastically at the initial stage and then experiences structural failures with increasing bending strain. Higher interlayer bonding density is observed to result in a higher bending stiffness. Meanwhile, significant interlayer shear strain is detected during bending, which leads to interlayer bond breakage, rippling, and buckling of the graphene layer. This work provides a full description of the pure bending behavior of diamane and its analogous structure, which could be beneficial for their applications in flexible electronics.

Comparative study on free vibration analysis of rotating bi-directional functionally graded beams using multiple beam theories with uncertainty considerations

The present study investigates the free vibration behavior of rotating beams made of functionally graded materials (FGMs) with a tapered geometry. The material properties of the beams are characterized by an exponential distribution model. The stiffness and mass matrices of the beams are derived using the principle of virtual energy. These matrices are then evaluated using three different beam theories: Bernoulli–Euler (BE) or Classical Beam Theory (CBT), Timoshenko (T) or First-order Shear Deformation Theory (FSDT), and Reddy (R) or Third-order Shear Deformation Theory (TSDT). Additionally, the study incorporates uncertainties in the model parameters, including rotational velocity, beam material properties, and material distribution. The mean-centered second-order perturbation method is employed to account for the randomness of these properties. To ensure the robustness and accuracy of the probabilistic framework, numerical examples are presented, and the results are compared with those obtained through the Monte Carlo simulation technique. The investigation explores the impact of critical parameters, including material distribution, taper ratios, aspect ratio, hub radius, and rotational speed, on the natural frequencies of the beams is explored within the scope of this investigation. The outcomes are compared not only with previously published research findings but also with the results of 3-Dimensional Finite Element (3D-FE) simulations conducted using ANSYS to validate the model’s effectiveness. The comparisons demonstrate a strong agreement across all evaluations. Specifically, it is observed that for thick beams, the results obtained from FSDT and TSDT exhibit a greater agreement with the 3D-FE simulations compared to CBT. It is shown that the coefficient of variation (C.O.V.) of first mode eigenvalue of TSDT, FSDT and CBT are approximately identical for random rotational velocity and discernible deviations are noted in CBT compared to FSDT and TSDT in the case of random material properties. The findings suggest that TSDT outperforms FSDT by eliminating the need for a shear correction coefficient, thereby establishing its superiority in accurately predicting the natural frequencies of rotating, tapered beams composed of FGMs.

Vibration damping behavior of 1D periodic aluminum/epoxy resin composite laminated structures under bending vibration excitation

A novel model considering slip effect of 1D periodic aluminum/epoxy resin composite laminated structures (PAERCLS) is proposed to investigate the vibration damping behavior under bending vibration excitation. Dynamic equation and characteristic equation of the 1D PAERCLS are derived respectively based on Euler beam theory, Bloch's theorem and transfer matrix method. Rationality of the model is verified by amplitude-frequency response analysis. Furthermore, the influence of connection stiffness ks, height ratio h1/h and length ratio la /l on the bandgap distribution characteristics of the 1D PAERCLS is studied in detail. Taking the calculated width of each bandgap in the condition of ks=0 N/m as a reference value, results show that when ks increases from 0 N/m to 1.0 × 109 N/m, the widths of the 1st, 2nd, 3rd and 5th bandgap decrease to 18.4 %, 74.2 %, 68.9 % and 100. 0 % of the reference value, respectively, while the width of the 4th bandgap increases to 363.4 % of the reference value. The maximum widths of the first four bandgaps affected by ratio la /l decrease respectively to 44.4 %, 79.0 %, 86.2 % and 90.5 % of the reference value, while the ratio of h1/h has almost no effect on it, when ks increases from 0 N/m to 1.0 × 109 N/m.

Virtual Work Principle for Piezoelectric Semiconductors and Its Application on Extension and Bending of ZnO Nanowires

This paper presents the principle of virtual work (PVW) for piezoelectric semiconductors (PSs), which extends the piezoelectric dielectrics to involve the semiconducting effect. As an application of the PVW, a one-dimensional (1D) approximation theory for the extension and bending of PS nanowires is established by directly applying the PVW and Bernoulli–Euler beam theory with the aid of the second-order approximation of electrostatic potential. To illustrate the new model, the mechanical displacement, electrostatic potential, and concentration of electrons for extension and bending deformation of n-type ZnO nanowires are analytically determined. Additionally, numerical results show that, for n-type Zinc Oxide nanowires, the distribution of electrostatic potential is anti-symmetric along the thickness direction for extension deformation. In contrast, the bending deformation causes a symmetric distribution of electrostatic potential characterized by the zeroth-order and the second-order electrostatic potential. Furthermore, these two different deformations result in the redistribution of electrons. The electrostatic potential can be tuned by adjusting the amplitude of the applied mechanical load. Moreover, we find that the increase in doping level will reduce the magnitude of electrostatic potential due to the screening effect. The presented PVW provides a general approach to establishing structural theories and an effective way of implementing numerical methods.

Closed-Form Analytical Solutions for the Deflection of Elastic Beams in a Peridynamic Framework

Peridynamics is a continuum theory that operates with non-local deformation measures as well as long-range internal force/moment interactions. The resulting equations are of the integral type, in contrast to the classical theory, which deals with differential equations. The aim of this paper is to analyze peridynamic governing equations for elastic beams. To this end, the strain energy density is formulated as a function of the non-local curvature. By applying the Lagrange principle, the peridynamic equations of motion are derived. Examples of non-local boundary conditions, including simple support, clamped edge, roller clamped edge, and free edge, are presented by introducing the interaction domain. Novel closed-form analytical solutions to integral equations are presented for beams with various boundary conditions, including clamped—simply supported, clamped–clamped, simply supported–roller-clamped, and clamped–roller-clamped beams. Furthermore, different types of loadings, including uniformly distributed load, concentrated force, and concentrated moment, are considered. The results are validated by comparing the derived solutions against solutions to the classical Bernoulli–Euler beam theory. A very good agreement between the non-local and the classical theories is observed for the case of the small horizon sizes, which shows the capability of the derived equations of motion and proposed boundary conditions.

Tunability of “spring-mass” resonator on band gaps with surface effects of a piezoelectric phononic crystal double-layer nanobeam

A model of piezoelectric phononic crystal (PC) double-layer nanobeam periodically attached by “spring-mass” resonators is proposed. As a comparison, two other simplified models are also proposed. In order to calculate the band structure, the plane wave expansion (PWE) method is improved by introducing the Euler beam and surface piezoelectricity theories. All the results and analysis demonstrate that the tunable advantage is displayed in double-layer nanobeam on account of the antisymmetric and symmetric vibrational modes controlled by adding additional springs between two different nanobeams. Moreover, the influencing rules of parameters related to additional spring and “spring-mas” resonator are also revealed. The bandgap properties can be applied to nano electro-mechanical system (NEMS) to realize active control of vibration.

New analytical model for thermomechanical responses of multi-layered structures with imperfect interfaces

A new analytical model is developed for thermomechanical responses of multi-layered structures with an arbitrary number of layers and subjected to general thermal and mechanical loading. The formulation is based on an extended Bernoulli–Euler beam theory and a slip-interface model. The former includes Poisson’s effect and covers both the plane stress and plane strain deformations, and the latter allows slipping between two adjacent layers but no jump in the normal displacement or traction. An analytical solution for a multi-layered structure under general thermomechanical loading is derived by using a new approach that first determines one interfacial shear stress and the curvature of the deformed structure. To illustrate the newly developed model, three example problems for two-, three- and five-layer structures respectively are analytically solved by directly applying the new model. In all three cases, the solutions are obtained in closed-form expressions by considering both temperature changes and mechanical loads including body forces, distributed normal and shear stresses on the top and bottom surfaces, and normal forces, transverse shear forces and bending moments at the two ends, unlike existing ones. It is shown that the current solution for two-layer structures recovers an existing solution without considering Poisson’s effect and mechanical loading and the classical solution of Timoshenko for perfectly bonded bi-metal thermostats as two special cases. The closed-form solution for five-layer structures with imperfect interfaces is derived here for the first time. In addition, numerical results are provided for five- and seven-layer transistor stacks to quantitatively demonstrate the new model. It is found that the current results for the five-layer transistor stack agree well with those obtained by others, thereby further validating the new model.

Novel metamaterial structures with negative thermal expansion and tunable mechanical properties

The thermal expansion and stiffness characteristics are significant in determining the responses of lightweight mechanical metamaterials in a thermal environment. This paper proposed a novel trapezoid unit with negative thermal expansion (NTE) by three conventional materials with positive thermal expansion, and constructed a series of lightweight metamaterials by sharing edges and sharing vertexes. The effective linear coefficients of thermal expansion (CTE) of these metamaterial structures are theoretically explored by considering axial and bending deformations. Based on Euler beam theory, the effective linear elastic stiffnesses of these metamaterials are given accurately by the stiffness matrix method. The accuracy of the theoretical effective CTE and stiffness is well verified by numerical modeling. The effective linear CTE, elastic stiffness, and Poisson's ratio of these metamaterials can be programmable by controlling geometrical parameters and material properties. The NTE effect can be achieved in ET, EQ, EH, and VT metamaterials due to the presence of re-entrant structures, in which the ET metamaterial can fulfill the maximum magnitude NTE. The NTE and negative Poisson's ratio can simultaneously be realized in EQ metamaterial. Significantly, a broader range of CTE from negative to positive values can be fulfilled in these metamaterials compared with the conventional triangle unit. Furthermore, the high specific stiffness with extremely low thermal stress can be obtained in EQ, VT, and VH metamaterials, which benefits the stability controlling and enhancing natural frequencies of structures in a thermal environment.

Geometric nonlinear analysis of dielectric layer based on concave paper-cut structure with zero Poisson’s ratio

When the sensor works in a limited environment, its accuracy is easily affected by unnecessary strain loss. The key to improve accuracy is to reduce the transverse strain of the dielectric layer structure. It is an innovative technology to construct zero Poisson’s ratio dielectric layer to limit the lateral strain of dielectric layer under normal pressure. The porous metamaterial dielectric layer with zero Poisson’s ratio is constructed based on the paper-cutting theory. The equivalent nonlinear mechanical model is established by use of Bernoulli Euler beam theory and energy method. The analytical expressions of equivalent Poisson’s ratio and equivalent Young’s modulus are given, and the necessity of considering geometric nonlinear large deformation is revealed. An improved variable step iterative method is proposed in order to solve the problem of equivalent internal force analysis caused by geometric deformation nonlinearity. The key of this method is to determine the displacement at the free end under the premise of considering the nonlinear superposition of the rigid body motion of the curved bar of the metamaterial. Based on the equivalent nonlinear mechanical model, the structural deformation characteristics of the dielectric layer structure in the linear small deformation stage and the nonlinear large deformation stage are analyzed. The results of theoretical, finite element simulation and experimental research reveal the necessity of considering geometric nonlinear factors in the practical application of the structure, which means that the foundation is theoretically and experimentally laid for the design of porous elastic dielectric layer of flexible capacitive pressure sensor.