Beam Bending Theory Research Articles

Experimental study on rotary inter-module connections with corrosion-resistant metal materials: Under axial tension

Modular construction is an innovative construction technology that has attracted considerable attention for its pronounced advantages in reducing reliance on on-site labor, speeding up construction, enhancing productivity, and ensuring superior quality. To extend the applicability of modular construction to structures in corrosive environments, such as ocean floating structures for exploring renewable energy, a rotary inter-module connection employing corrosion-resistant metal materials was proposed herein. Six full-scale inter-module connection specimens made of grade S30408 austenitic stainless steel, grade S220503 duplex stainless steel, and grade A96061-T6 aluminum alloy were tested under axial tension. The load-displacement relationships of each specimen were recorded to enable a thorough analysis of the strength, stiffness, and ductility of the tested specimens. As compared to the conventional carbon steel counterparts, the stainless steel specimens showed superior load-bearing capacity with improved ductility, while the aluminum alloy specimens experienced brittle fracture at the bolt shank-bottom plate junction with reduced load-bearing capacity. Three typical failure modes were identified in the experimental study, namely, plastification of the connector and lower corner fitting, plastification of the connector and both corner fittings, and fracture of the bolt. The load-strain relationships of each specimen were also discussed to analyze the load transfer mechanism and structural behavior of the connection. In addition, the applicability and accuracy of the simplified calculation methods based on beam bending theory and Navier solution for rectangular thin plates, respectively, were discussed based on the test results.

Susceptibility versus flexural stiffness in the stability of hybrid laminate columns with a rectangular cross section for a 1D model

Hybrid columns of FGM/FML type with inhomogeneous transverse structure perpendicular to their length are discussed. After that, laminated columns with different layups of laminate plies are discussed, with focus put on the effect exerted by many coupling of the stiffness submatrix on the lowest eigenvalue load. All columns were simply supported The study uses three methods for determining the lowest eigenvalues. The first two are analytical beam methods (i.e. 1D modelling approaches) based on the use of the Euler-Bernoulli theory. The proposed 1D methods are a generalisation of the well-known approach of determining the lowest eigenloads of isotropic columns. The first approach is to solve the eigenproblem directly from the differential equation of the deflection line. The second approach assumes that the curvature of the beam depends on the susceptibility of the beam. This is a fundamental assumption in the bending theory of beams and thin plates. The third method is the FEM (3D shell approach) which is used for the verification of the previous calculations. A comparison of the obtained results showed that the proposed 1D (one-dimensional) approach based on the full susceptibility matrix, which is the inverse stiffness matrix was as accurate as the 3D (three-dimensional) FEM shell model. The relative difference does not exceed 3%. Second 1D approach based on the stiffness matrix yields many times unacceptable differences relative to the reference results obtained by the FEM method. This comparison clearly demonstrates that an approach based on the assumption that curvature is directly proportional to susceptibility in flexural composite structures produces accurate results for all types of coupling.The existing literature extensively explores the impacts of stiffness matrix elements on both the linear and non-linear stability of laminate columns. However, there is a notable absence of studies focusing on the influence of susceptibility. This crucial aspect has been overlooked in prior studies. The present paper endeavors to fill this gap by shedding light on the influence of susceptibility, thereby emphasizing its significance in the analysis of laminate column stability.

Multi-objective optimization of composite sandwich structures using Artificial Neural Networks and Genetic Algorithm

Sandwich structures offer significant opportunities to improve the performance of many industrial applications such as aerospace, automotive, and marine. The design of a composite sandwich structure is often challenging because it is driven by the balance between the weight and cost of these structures. In this paper, a multi-objective optimization model using a Genetic Algorithm (GA) and an Artificial Neural Network (ANN) is presented to evaluate a new optimization approach in terms of weight and cost minimization for the composite sandwich structures. Classical lamination and beam bending theories were used, along with Monte Carlo simulation, to generate the design data for the proposed composite sandwich structure, which is applied as a floor panel for a high-speed train. Multilayer feedforward neural networks were used for predicting safety factors, cost, and weight of the designed structure based on the following inputs: core density, core thickness, face sheet materials’ combinations, and applied load. The trained Neural Network model was able to predict the considered results with a good performance metric, namely the coefficient of determination (R2 = 0.99) and the Mean Square Error (MSE = 1.3·10−5).Multi-objective optimization for cost and weight minimization was performed with a Genetic Algorithm using the derived ANN model. The obtained Pareto front provided several non-dominated optimal points leading to insights on the optimization process. The Finite Element Method (FEM) was used to model the key points of the optimal designs (i.e. the design with the lowest cost, weight, and Pareto optimal points). The FEM and optimization results had a maximum deviation of about 8.9%, indicating a good agreement between the two techniques.The newly elaborated methodology demonstrates a new approach for obtaining the optimum design of the investigated composite sandwich structure constructed from honeycomb core and laminated face sheets in terms the cost and weight. The study concluded that the use of Carbon Fiber-Reinforced Plastic (CFRP) or Fiber-Metal Laminate (FML) face sheets results in significant weight savings of about 59.5% and 48.6%, respectively, compared to an all-aluminum sandwich structure.

Honeycomb structure filling morphing wing trailing edge: Design strategy, deformation feedback, and active control

Abstract Pursuing highly efficient aerodynamic efficiency in aircraft has driven the development of morphing wing technology. However, there are still limitations to morphing wing technology, including adaptation of load and deformation, and deformation monitoring and control. This work introduces an intelligent trailing edge structure that balances deformation and load-bearing and achieves deformation monitoring and active control. Firstly, we employ a honeycomb structure for non-uniform filling of the trailing edge. The filling method is obtained through inverse design using a genetic algorithm based on neural networks, allowing the device to undergo continuous deformation while meeting load-bearing requirements. The bending deformation of the wing is achieved using shape memory alloy (SMA) wire. Additionally, we design and fabricate a metal-based multichannel flexible sensor, and based on beam bending theory, we establish the strain–displacement relationship. These sensors are affixed to the trailing edge surface, enabling real-time monitoring and active control of trailing edge deformation. Building an experimental platform to test this system, the results show that the sensors can accurately give feedback on the degree of wing deformation, and the error of active deformation control technology is less than 4%. This provides a new method for the deformation feedback control closed-loop system of intelligent variant wings.

DCB Bonding Energy Measurement Using Confocal IR Microscopy

Direct bonding is a widely used technique in the semiconductor industry. Many materials can be assembled and stacked together with this technique, enabling the fabrication of many useful structures. The direct bonding technique is, for instance, a major step of the Smart Cut ™ process flow enabling the industrial-scale production of Silicon-On-Insulator (SOI) substrates [1][2]. Material diversification in the microelectronics industry (i.e. “More than Moore” approach) imposes new technological constraints. The substrate bonding process control has to be improved calling for more and more accurate bonding energy measurement protocols.In this work, a new technique is assessed in order to measure, at the wafer scale, direct bonding energies. It is derived from the standard Double Cantilever Beam (DCB) method [3] and uses interferometry in confocal IR laser source microscopy to measure crack openings (Fig. 1A) [4][5][6]. Such a bonding energy measurement protocol shows a better accuracy compared to other techniques (Fig. 1B & 1D). This is due to a better confocal microscopy resolution and the high intensity of the laser source. The elastic energy stored in bent wafers is obtained by measuring the beam curvature [7][8]. DCB deformation models are discussed from short-range crack opening to long distance beam-bending theories. Comparison is done between results from analytic models, Finite Element Model (FEM) and experiments (Fig. 1C).Near-bonding edge zone modeled with FEM simulation accurately matches experimental data. A new mechanical and universal bending model, valid at both small and large scales, is proposed. With this model, bonding energy extraction is then reduced to a simple scaling parameter extraction directly linked to the stored elastic strain energy (Fig 1C & 1D) [9]. This work opens prospects for deformation and bonding energy measurements in various configurations including bonded heterostructures, polymer, and metal bonding. [1] Bruel, Electron. Lett. 31, 1201 (1995).[2] K. Celler, A.-J. Auberton-Hervé, B. Aspar, C. Lagahe-Blanchard, and C. Maleville, in Wafer Bonding Applications and Technology, Springer Series in Materials Science, edited by M. Alexe and U. Gosele, Springer, Berlin (2004).[3] Maszara, G. Goetz, A. Caviglia, and J. B. Mckitterick, J. Appl. Phys. 64, 4943 (1988).[4] Wu, S. Gowrishankar, R. Huang et al., Int. J. Fract. 202, 1–19 (2016).[5] Gowrishankar, H. Mei, K. M. Liechti et al., Int. J. Fract. 177, 109–128 (2012).[6] Pallares, L. Ponson, A. Grimaldi, M. George, G. Prevot, and M. Ciccotti, Int. J. Fract. 156, 11–20 (2009).[7] Bertholet, “Measurement, optimization and multiscale modeling of silicon wafer bonding interface fracture resistance,” Ph.D. thesis (Catholique de Louvain, 2006).[8] Olbrechts, B. Lejeune, Y. Bertholet, T. Pardoen, and J.-P. Raskin, ECS Trans. 3(6), 279–289 (2006).[9] Colonel, A. Calvez, F. Fournel et al., J. Appl. Phys. 132, 215106 (2022). Fig. 1: A – (Top) Four pasted IR confocal microscope images acquired during the DCB testing (the vertical red dashed line shows the closing point’s fitted position). (Bottom) Air gap height measured as a function of pixel position.B – Simulated reflected intensity (left y-axis) and experimental reflected intensity (right y-axis), both as functions of the estimated distance from the closure point x. The vertical dashed grey line shows the real experimental closing point.C – Air gap height as a function of the position/thickness ratio (in log/log scales) for different analytical and numerical models and experimental data.D – Bonding energies including errors obtained with the standard length measurement method, the curvature interpolation method, the scaling method, or J-integral calculation from FEM as functions of the annealing temperature. Figure 1

Mechanical behavior of solid expandable tubular in curved borehole based on transverse beam bending theory

With the development of shale gas, casing losses in horizontal and curved borehole sections become more and more serious. It is a new attempt to repair horizontal casing with solid expandable tubular (SET), but the passability and expansibility of SET in curved hole will determine the success of this technique. In this paper, based on the bending theory of beam and finite element technique, the theoretical expression of the limit length of SET through curved borehole is derived, and a three-dimensional dynamics model considering friction and contact nonlinear wellbore- SET-expansion cone coupling is established by CAE. Finally, the dynamic load coefficient is introduced to modify the theoretical calculation results, and the extensibility of the SET is studied. The results show that: (1) The theoretical calculation formula of SET flexible passage has high reliability. Finite element technology is used to introduce dynamic load coefficient to modify it, and the dynamic load coefficient is 0.9. (2) SET flexible pass is more in line with practical construction than rigid pass, and the limit length of flexible pass is far greater than that of rigid pass. (3) It is recommended that the expansion rate should not exceed 12% when the SET length meets the limit for SET in curved holes. This value is about 50% lower than the ultimate expansion rate in vertical wells. The results of this paper provide theoretical support for the SET's passability and expansibility in curved borehole sections.

Controller Design of a Wire-Driven Flexible Robot

Robot tasks and functions are advanced by IoT, remote working, and new human lifestyle requirements in tedious or accurate occupations such as surgery, product inspections, or agriculture harvesting. Numerous robots are technologically advanced by scientists for diverse tasks. Flexible robots are developed based on stated applications since they can adjust their geometry to the working situations. The present study introduces a wire-driven flexible robot stimulated by animal trunks. It can make a motion in planar and space based on the construction of that. Initially, a kinematic model was developed to predict end-effector trajectory, and then a dynamic model was established to compute the required tension of tendon wires based on bending beam theory. Models are simulated by MATLAB/SIMULINK Software and implemented by a generic controller. Various input functions are applied to the model, and outputs show good stability and accuracy. Based on these results, in future work, a WDFR will be developed and controlled by the current presented PID controller.

Hydrodynamic analysis of a floating platform coupled with an array of oscillating bodies

In this paper, based on multi-body dynamics and beam bending theory, a frequency-domain model is developed to investigate the hydrodynamic performance of a hybrid floating platform (HFP). The HFP investigated here is characterized by hydroelasticity and complex interconnection conditions. The constraint matrix is derived to handle the couple response of an elastic platform and an array of rigid bodies, representing wave energy converters. The hydroelasticity and coupled dynamics of the floating platform in waves and the wave power extraction performance is analyzed. The numerical results reveal that the platform deformations have a significant impact on the relative pitch response between the buoy array and the platform. Neglecting hydroelasticity leads to an overestimation of the hydrodynamic efficiency of the buoy array when 0.4 < λ/L < 1.0, as demonstrated in our calculations. Influence of key parameters such as spacing and incident wave angle is examined numerically. As a result of these investigations, suggestions are identified for an improved design of such kind of multi-use floating platform.

Consistent Euler–Bernoulli beam theory in statics for gradient elasticity based on Laplacians of stress and strain

The Euler–Bernoulli beam bending theory in engineering mechanics assumes that the material behavior is isotropic elastic and that plane cross sections remain plane and rigid. It is well-known that this theory suffers from inconsistencies that, e.g., the shear strain is always vanishing, whereas the shear stress does not vanish. In recent work, consistent Euler–Bernoulli beam theories in classical and explicit gradient elasticities were accomplished by assuming the constitutive response to be anisotropic elastic, subject to internal constraints. This approach is extended in the present paper to get consistent Euler–Bernoulli beam theory for gradient elasticity based on Laplacians of stress and strain. The developed beam theory is employed to discuss bending of cantilever beams.

Study on Sinusoidal Post-Buckling Deformation of Coiled Tubing in Horizontal Wells Based on the Separation Constant Method

In this paper, a set of partial differential equations considering the dynamic effects of the coiled tubing (CT) is established based on the bending theory of slender beams considering the axial loads. The analytical solution of sinusoidal deformation with the time term is obtained. The critical load of coiled tubing during sinusoidal buckling and the change of half-wave number during sinusoidal post-buckling are studied through the introduction of the necessary conditions, solution and the separation constant. The contact force of coiled tubing with sinusoidal post-buckling on the horizontal well wall is analyzed. The results show that the critical load for sinusoidal buckling of fixed-size CT is related to the section angular acceleration coefficient. The half-wave number produced by the sinusoidal post-buckling bending of the CT gradually decreases during compression. The contact force of the deformed CT to the borehole wall is related to the compression speed of the CT. By introducing the dynamic term and the separation constant, this research model can provide a theoretical basis for studying the transformation of the CT from sinusoidal buckling to helical buckling.

Research on welding deformation of hollow thin-walled complex structural parts based on plane bending theory of constant section beam

In order to effectively forecast the welding deformation and deformation pattern of hollow thin-walled aluminum alloy complicated structural parts, based on the plane deformation theory of constant section beam and workpiece constraint conditions, the complex feature structure was split and the deflection curve equation of each position was constructed. Moreover, the inherent strain method was used to solve the welding equivalent load, and the welding deformation was calculated by combining the equivalent load with the deflection curve equation. On the basis of the theoretical load value, the welding deformation at different positions during and after welding was numerically simulated by finite element simulation. What's more, the theoretical deformation of each plate is verified by the simulated deformation results. Besides, the simulation deformation results of post welding deformation are verified by welding experiment. The results show that the shrinkage deformation of the welding position is the largest, the transverse and longitudinal deformation are 0.007 mm and 0.018 mm respectively, the error rates of theory and simulation are 5.2 % and 5.5 % respectively. What's more, the welding deformation at the non-welding position is mainly reflected in the vertical direction of the upper skin plate. After the external constraints are released, the actual maximal deformation values of the two sides of the upper skin plate are 0.08 mm and 0.064 mm respectively. The deformation presents an M-shaped distribution, which is comparable to the deformation distributions of theory and simulation. The deformation value caused by the welding process accounts for 10 % of the total deformation after the removal of external constraints. In addition, the welding process is the main stage of forming the deformation distributions at each position, and the release of external constraints is the main factor leading to the final deformation.

Analysis of Casing String Running Characteristics in Negative-Displacement Horizontal Wells

Summary By creating a reverse deflection, negative-displacement horizontal well technology can successfully address the issue of construction challenges brought on by too little displacement in front of the horizontal well target. However, conventional horizontal well casing string running models are unsuitable for negative-displacement horizontal wells. This is because conventional horizontal well models assume that the casing string is in contact with the lower side of the borehole, which is inconsistent with the actual situation in negative-displacement horizontal wells. In this study, we examine the effects of fluid viscous resistance and internal and external fluid interaction forces on the running of the casing string using the Gaussian method and the complementary surface equivalence method. Based on the bending beam theory, we establish a deflection model to examine the interaction between the casing string and the borehole wall in the bending section. The friction and hookload calculation models of each well section and the strength check models are also established. The calculation results show that, according to the aforementioned model, there is an average error of 7.5% between the measured data and the calculated frictional force of the H1 and H2 negative-displacement horizontal wells. This error is within the reasonable range of field application and attests to the validity of the theoretical model. Finally, we study the influence of running factors on running ability and strength of casing string using the control variable method. The results indicate that the weight of the casing string is positively correlated with the variation law of the running ability of the casing string, the maximum offset distance and the length of the horizontal section are negatively correlated with the running ability of the casing string, and the borehole curvature is positively correlated with the variation law of the connection stress of the internal and external threads of the casing string. Therefore, in the process of running the casing string in negative-displacement horizontal wells, using heavier casing string and reducing the maximum offset distance and horizontal section length can improve the running ability of the casing string.

Functional Hazard Assessment of a Modular Re-Configurable Morphing Wing Using Taguchi and Finite Element Methodologies

Growing concerns over the CO2 footprint due the exponential demand of the aviation industry, along with the requirements for high aerodynamic performance, cost saving, and manoeuvrability during different phases of a flight, pave the path towards adaptable wing design. Morphing wing design encompasses most, if not all, of the flight condition variations, and can respond interactively. However, functional failure of the morphing wing might bring devastating impacts on the passengers, crew, and/or aircraft. In the present work, the dynamic characteristics of a re-configurable modular morphing wing developed in-house by a research group at the Toronto Metropolitan University, are investigated from the perspective of a functional hazard assessment (FHA). This modular morphing wing, developed based on the idea of a parallel robot, consists of a number of structural elements connected to each other and to the wing ribs through eye-bolt joints. Timoshenko’s bending beam theory, in conjunction with the finite element method (FEM), is exploited to model the structural members. Possible hazards, assumed here to be the structural failure of the beam components, have been identified and their failure conditions are assessed. Numerical simulations have been presented to show the impact of various combinations of the identified hazards on the vibration signature of the morphing wing in unmorphed and morphed configurations. Identification of changes in the wing’s vibration signature is a vital component in the fail-safe structural and aeroelastic design of an aircraft. The present study is geared towards the structural response of the system in the absence of any aerodynamic loads.

A Bilayer Thin-Film Strain Gauge With Temperature Self-Compensation

A Karma–CuNi bilayer strain gauge with a low-temperature coefficient of resistance (TCR) is designed for temperature self-compensation. The impact of heat treatment and structural parameters on its performance is investigated. The TCR of the bilayer thin-film strain gauge can be adjusted to near 0 ppm/°C by controlling the thickness ratio and the annealing temperature. Besides, the Karma–CuNi thin films had a gauge factor (GF) of 2.2. Finally, the Karma–CuNi bilayer thin-film strain gauge was fabricated on the metal substrates and bone-like metal elastomer. An experimental setup based on bending beam theory was developed to determine the GF of the strain gauges fabricated on metal substrate. It shows promising applications for in situ strain or torque measurement of some metal component when high sensitivity and high spatial resolution is required.

Distributed-parameter Bouc-Wen modeling and output feedback control of a piezoactuator for a manipulation task

The Bouc-Wen hysteresis model and an Euler-Bernouilli beam bending theory are extended to distributed parameter model for a piezoelectric actuator (piezo actuator). Then we use the separation principle for its control and two types of observers: a neural network-based observer and a high-gain observer. Simulations with various conditions show that better performances are obtained when using the neural network-based observer.

Double cantilever beam bonding energy measurement using confocal IR microscopy

A new technique is assessed in order to measure, at the wafer scale, direct bonding energies. It is derived from the standard Double Cantilever Beam (DCB) method and uses interferometry in confocal IR laser source microscopy to measure crack openings. Such a bonding energy measurement protocol has better accuracy compared to other techniques. This is due to a better confocal microscopy resolution and the high intensity of the laser source. The elastic energy stored in bent wafers is obtained by measuring the beam curvature. DCB deformation models are discussed from the short-range crack opening theory to long distance beam-bending theories. Comparison is made between models, experimental results, and FEM simulations. Finally, the bonding energy error during standard measurements is estimated.

Mass matrices for elastic continua with micro-inertia

In this paper, the finite element discretization of non-classical continuum models with micro-inertia is analysed. The focus is on micro-inertia extensions of the one-dimensional rod model, the beam bending theories of Euler–Bernoulli and Rayleigh, and the two-dimensional membrane model. The performance of a variety of mass matrices is assessed by comparing the natural frequencies and their modes with those of the associated discrete systems, and it is demonstrated that the use of higher-order mass matrices reduces errors and improves convergence rates. Furthermore, finite element sizes larger than the corresponding physical length scale are shown to be sufficient to capture the natural frequencies, thus facilitating numerical models that are not only reliable but also computationally efficient.

(Digital Presentation) A Constant Deformation Modulus for the Simulation of Gas Diffusion Layer

The deformation of gas diffusion layer(GDL) directly affects gas diffusion and electron transfer in the fuel cell. The modulus of stress-strain relationship is the basic parameter in the process of deformation. In this paper, based on beam bending theory and geometric probability analysis, a nonlinear analytic model of stress-strain with physical meaning is established, and a constant deformation modulus of GDL is given. Based on the deformation modulus, we further simulate the inhomogeneous deformation of GDL within a half channel-rib model. The results indicate that the uneven deformation with deformation modulus is more significant than that with linear elastic modulus in the same condition. Finally, a shape-dependent bending radius design for the rib edge is proposed, in which the inhomogeneous deformation of GDL becomes smoother as the radius increases, thereby improving the service condition of GDL and prolonging its service life.

Modeling and Characterizing Two Dielectric Elastomer Folding Actuators for Origami-Inspired Robot

Origami-inspired robots are automated machines that achieve deformation and function through folding. This letter models and characterizes two dielectric elastomer folding actuators, and proposes a new method for driving origami-inspired robots. Multilayer bending dielectric elastomer actuators (MBDEAs) and minimum energy structural dielectric elastomer actuators (MESDEAs) are placed at the creases to drive a rigid origami-inspired robot joint. We establish an analytical model of MBDEAs based on beam bending theory. This model establishes the relationships between the deformation angle and output moment of the MBDEA and the input electric field intensity according to the design parameters. In addition, a design model of MESDEAs with the rectangular hollow area is presented for the equilibrium deformation angle and maximum output moment. The analytical model of MBDEAs for the deformation angle and the design model of MESDEAs are both preliminarily verified by experiments. Finally, we realize the drive of the joint based on MBDEAs and MESDEAs. The design and model are evaluated through experiments. This work paves the way for subsequent motion control of origami-inspired robots and further expands the application fields of dielectric elastomer actuators (DEAs).

(Digital Presentation) A Constant Deformation Modulus for the Simulation of Gas Diffusion Layer

The deformation of gas diffusion layer(GDL) directly affects gas diffusion and electron transfer in the fuel cell. The modulus of stress-strain relationship is the basic parameter in the process of deformation. In this paper, based on beam bending theory and geometric probability analysis, a nonlinear analytic model of stress-strain with physical meaning is established, and a constant deformation modulus of GDL is given. Based on the deformation modulus, we further simulate the inhomogeneous deformation of GDL within a half channel-rib model. The results indicate that the uneven deformation with deformation modulus is more significant than that with linear elastic modulus in the same condition. Finally, a shape-dependent bending radius design for the rib edge is proposed, in which the inhomogeneous deformation of GDL becomes smoother as the radius increases, thereby improving the service condition of GDL and prolonging its service life.