Beam Bending Theory Research Articles

The Use of Nonlocal Theory for Bending Vibrations of Single-Walled Carbon Nanotubes

In the paper are investigated the eigenfrequencies of single-walled carbon nanotubes (SWCNTs) by the analytical method based on nonlocal theory of beam bending. A continuum approach is applied for eigenfrequency computation of SWCNTs with four types of beam end conditions: clamped-free (C-F), simply-simply supported (S-S), clamped-simply supported (C-S) and clamped-clamped (C-C).

Investigation of the Effectiveness and Robustness of an MEMS-Based Structural Health Monitoring System for Composite Laminates

Failure of layered composites can be triggered by small inner defects; such defects can be present inside the composite from the beginning, or can be incepted by external actions during the life cycle of the structure. Defects generally evolve, under repeated loadings, into a delamination or debonding between adjacent laminae, and cannot be easily detected as they are masked by the composite skin: ad hoc health monitoring systems are therefore required. To prevent any distortion of the composite microstructure, which may trigger by itself the nucleation of the aforementioned defects, sensors should not be embedded inside the composite; in addition, to avoid a mechanical interaction between the monitoring system and the structure, sensors should be as lighter as possible. To comply with these two major requirements, we have recently investigated a microelectromechanical system-based, surface-mounted health monitoring strategy for laminates. In this paper, we provide the results of an experimental campaign to show the effectiveness of this strategy, in terms of sensitivity to the length of delamination in standard specimens (independently of the loading conditions at the tip of the delamination), and its robustness, in terms of repeatability of the outcomes relevant to the monitored state. The experimental data are compared with analytical predictions based on beam bending theory, showing a good accuracy.

Control method for centering rolls in radial-axial ring rolling process

The control method of centering rolls was developed to ensure the circularity of rings and remove the process instability during a radial-axial ring rolling (RARR) process. The position control method is determined by the growth velocity of rolling rings and mathematical correlations among the key process variables, such as ring geometries, roll shapes, and feed rates of rolls. The moving coordinate of centering rolls were calculated by the growing velocity of an outer radius of the ring, VR, and the angle between the links of the ring’s center with the mandrel’s center and with the centering roll’s center, a, in the RARR process. The growth behavior of a ring and the angle provided moving coordinates for the centering rolls in real time. The hydraulic adjustment mechanism for establishing a successful RARR process is also proposed using the parameter design of the linkage assembly and beam bending theory. In order to verify the reliability of the control method, the FE-simulation under a FORGE software environment and ring rolling experiments of 42CrMo4 rings with an outer diameter of 510mm. The results show that these methods provided a high ring circularity and process stability.

An operational calculus-based approach to a general bending theory of nonlocal elastic beams

Abstract This paper is concerned with obtaining exact solutions for the bending problem of an elastic nanobeam by using the Lurie's operational method. Within the framework of nonlocal elasticity theory, a general governing equation, capable of capturing the size effect, is first constructed in a systematic and straightforward manner. Then a solution methodology is described. Some explicit solutions involving trigonometric expansions are also presented and compared with other well known beam theories. The results indicate that this general beam theory can provide more accurate results, which can be served as benchmarks for other theoretical or numerical methods.

Large deflection of a non-linear, elastic, asymmetric Ludwick cantilever beam subjected to horizontal force, vertical force and bending torque at the free end

The investigated cantilever beam is characterized by a constant rectangular cross-section and is subjected to a concentrated constant vertical load, to a concentrated constant horizontal load and to a concentrated constant bending torque at the free end. The same beam is made by an elastic non-linear asymmetric Ludwick type material with different behavior in tension and compression. Namely the constitutive law of the proposed material is characterized by two different elastic moduli and two different strain exponential coefficients. The aim of this study is to describe the deformation of the beam neutral surface and particularly the horizontal and vertical displacements of the free end cross-section. The analysis of large deflection is based on the Euler–Bernoulli bending beam theory, for which cross-sections, after the deformation, remain plain and perpendicular to the neutral surface; furthermore their shape and area do not change. On the stress viewpoint, the shear stress effect and the axial force effect are considered negligible in comparison with the bending effect. The mechanical model deduced from the identified hypotheses includes two kind of non-linearity: the first due to the material and the latter due to large deformations. The mathematical problem associated with the mechanical model, i.e. to compute the bending deformations, consists in solving a non-linear algebraic system and a non-liner second order ordinary differential equation. Thus a numerical algorithm is developed and some examples of specific results are shown in this paper.

Bending Vibrations of Carbon Nanotubes by Using Nonlocal Theory

The topic of this article is the investigation of the free bending vibration of single-walled carbon nanotubes (SWCNTs). The underlying theory of frequency computation is based on nonlocal theory of beam bending. The solutions of frequency equation are given for four types of boundary conditions: clamped-free (C-F), simply-simply supported (S-S), clamped-simply supported (C-S) and clamped-clamped (C-C). The graphical outputs of computations are given only for limited number of cases that are able to characterize influences of parameters in question.

Experimental Tests of Wind Turbine Main Shaft Motion on a Laboratory Test Rig


 
 
 This paper introduces research and discusses findings dealing with failure modes of gearboxes in wind turbines. These gearboxes fail in general after five years which is far below the expected design life of twenty years of a wind turbine. The research is taking a more holistic approach towards finding typical behaviour of the main shaft taking the transient nature of the wind into consideration. In this research, a small scale wind turbine test rig has been designed and manufactured with displacement sensors installed to observe the displacement of the main shaft at specific points, namely the main bearing locations of the forward framework of a wind turbine nacelle, where the main shaft is installed. The experimental data measured from the test rig is being analysed with common beam bending, statistical and fatigue theories to draw conclusions for long term loading in service. Aspects of the turbulent nature of the wind driving the wind turbine have been taken into consideration as being part of the aerodynamic loading onto the rotor and eventually the gearbox, transmitted through the main shaft. The purpose of the test rig at this stage is to obtain a quantitative insight into the motion of the main shaft. The deliberately chosen softer aluminum material and the more slender geometry for the components should provide exaggerated displacements which help to make motion and deformations more obvious. At this point, no resemblance to real size wind turbines has been established.
 
 

Prediction of Updated Cutting Parameters for a Spindle Subjected to Bearing Wear: A Free Vibration-Based Approach

The effects of spindles vibrational behavior on the stability lobes and the chatter behavior of machine tools are discussed. Multi-segment spinning spindle models, developed based on the Euler-Bernoulli beam bending theory, have revealed that the system exhibits coupled Bending-Bending (B-B) vibration and its natural frequencies are found to decrease with increasing spinning speed. It has also been observed from the experimental data that an average spindle goes through three stages of operation, namely settling, normal operation and failure. As spindle fundamental frequency changes, the stability lobes change, i.e., the originally selected cutting parameters could lead to chatter. It is shown that using the experimental results, it is possible to establish an expression for the variation of spindle's fundamental frequency in terms of machine hours, which can in turn be used to predict chatter-free cutting parameters through calibrated models and the stability lobes.

The effect of film thickness on polypyrrole actuation assessed using novel non-contact strain measurements

Micro-actuators have been developed that exploit the electrochemically induced volume change of the electro-active polymer polypyrrole. The strain regime is inherently complex at a physical level and whilst volume change can be estimated indirectly using, for instance, bending beam theory, such methods become unreliable for large deflections owing to limitations in the mathematical model. A new non-contact measuring technique based on laser micrometry is presented to characterize the time-dependent expansion of electro-active films such as polypyrrole. Measurements have been made which demonstrate that the observed strain is dependent on film thickness. The new measurement technique is straightforward to perform and it is anticipated that it can be used for future materials development and performance assessment, including long-term stability evaluations and operational failure studies of the films.

Finite element modelling versus classic beam theory: comparing methods for stress estimation in a morphologically diverse sample of vertebrate long bones

Classic beam theory is frequently used in biomechanics to model the stress behaviour of vertebrate long bones, particularly when creating intraspecific scaling models. Although methodologically straightforward, classic beam theory requires complex irregular bones to be approximated as slender beams, and the errors associated with simplifying complex organic structures to such an extent are unknown. Alternative approaches, such as finite element analysis (FEA), while much more time-consuming to perform, require no such assumptions. This study compares the results obtained using classic beam theory with those from FEA to quantify the beam theory errors and to provide recommendations about when a full FEA is essential for reasonable biomechanical predictions. High-resolution computed tomographic scans of eight vertebrate long bones were used to calculate diaphyseal stress owing to various loading regimes. Under compression, FEA values of minimum principal stress (σ(min)) were on average 142 per cent (±28% s.e.) larger than those predicted by beam theory, with deviation between the two models correlated to shaft curvature (two-tailed p = 0.03, r(2) = 0.56). Under bending, FEA values of maximum principal stress (σ(max)) and beam theory values differed on average by 12 per cent (±4% s.e.), with deviation between the models significantly correlated to cross-sectional asymmetry at midshaft (two-tailed p = 0.02, r(2) = 0.62). In torsion, assuming maximum stress values occurred at the location of minimum cortical thickness brought beam theory and FEA values closest in line, and in this case FEA values of τ(torsion) were on average 14 per cent (±5% s.e.) higher than beam theory. Therefore, FEA is the preferred modelling solution when estimates of absolute diaphyseal stress are required, although values calculated by beam theory for bending may be acceptable in some situations.

Analysis of Stresses in Double-Lap

Adhesively bonded joints are frequently used in aircraft and aerospace structures. A bonding technique called overlamination, where the parts to be joined are connected by adherends that are laminated directly upon the assembly, is more common in the marine industry, and there is a need to evaluate the stress distribution in such joints. In the present paper, the interfacial stress distribution in double-lap joints is investigated. A new analytical model of the shear and normal stresses are derived by using the beam bending theory. This method does not require an assumption of stress distribution in the lap. Both the shear stress and the normal stress along the adhesion layer can be evaluated by the present theoretical method. It is shown that both the shear stress and the normal stress depend on the thickness of adhesion layer. Its validity is verified by using numerical results of FEM. It is shown that our solutions provide a better prediction than the original theoretical solutions.

Vibration Modeling of Machine Tool Spindles: A Calibrated Dynamic Stiffness Matrix Method

The effects of spindles vibrational behavior on the stability lobes and the Chatter behavior of machine tools have been established. The service life has been observed to reducethe system natural frequencies. An analytical model of a multi-segment spinning spindle, based on the Dynamic Stiffness Matrix (DSM) formulation, exact within the limits of the Euler-Bernoulli beam bending theory, is developed. The system exhibits coupled Bending-Bending (B-B) vibration and its natural frequencies are found to decrease with increasing spinning speed. The bearings were included in the model usingboth rigid, simply supported, frictionless pins and flexible linear spring elements. The linear spring element stiffness is then calibrated so that the fundamental frequency of the system matches the nominal value.

Analytical modeling of nonlinear behavior of composite stub-girders

This paper presents an inelastic analytical model for modeling the nonlinear behavior of partially composite stub-girder under midspan concentrated load with partially end joint stiffness. The method treats the stub-girder as a beam with three distinct layers, the top and bottom layers behave according to the normal beam bending theory, while the middle layer is a shear layer without any bending stiffness. At the interface between the steel and concrete elements, the headed stud shear connectors are modeled as linear elastic built-in cantilevers. The present method smears the effects of all the contributing shear elements, stubs and studs, along the span. The differential equation for the stub-girder is solved to obtain the general expression of elastic deflection for the case when one concentrated load is applied at the mid-span. Based on the present approach a computer program called “PZA” is developed. The program accounts for nonlinear behavior of concrete and steel materials. The accuracy and reliability of the program are demonstrated by the analysis of three stub girders and one composite beam over the entire loading range up to ultimate load and the results are compared with published experiments. A relatively good agreements with the experimental findings are found. Then a parametric study is carried out to discuss the effect of the initial stiffness of shear connector and the stub height on the behavior of stub-girder.

Design considerations of Gudgeon pin in reciprocating air compressors by semi analytic approach

The deformation and stress experienced by the Gudgeon pin of a reciprocating compressor used in air brake system is scientifically predicted when the pin is fully floating with lubricating oil surrounding it and when starved of oil. Both semi analytical approach in finite element method and simple bending theory of beams are used. Inadequacy of the beam approach is highlighted. The results obtained by both the approaches are compared. Role of clearance in piston bore and small end of connecting rod and effectiveness of lubrication are examined. Factor of safety associated with the design of Gudgeon pin is also looked into. Key words: Gudgeon pin, semi and fully floating pin, semi analytic method, role of lubricant, factor of safety

Effect of Surface Viscosity on the Vibration of Microcantilevers

Using the theory of beam bending and the theory of surface rheology, the equation describing the effect of a surface viscous film on the vibration of an elastic microcantilever is derived. A generalized solution for the vibration of the microcantilever is obtained, which depends on the boundary conditions and the surface viscosity. There is a phase lag between the externally harmonic excitation and the microcantilever response due to the energy loss from surface flow. For small energy dissipation with negligible surface energy, the maximum energy loss occurs approximately at the natural frequencies of the microcantilever.

Bending analysis of partially restrained channel-section purlins subjected to up-lift loadings

Cold-formed steel section beams are widely used as the secondary structural members in buildings to support roof and side cladding or sheeting. These members are thus commonly treated as the restrained beams either fully or partially in its lateral and rotational directions. In this paper an analytical model is presented to describe the bending and twisting behaviour of partially restrained channel-section purlins when subjected to uplift loading. Formulae used to calculate the bending stresses of the roof purlins are derived by using the classical bending theory of thin-walled beams. Detailed comparisons are made between the present model and the simplified model proposed in Eurocodes (EN1993-1-3). To validate the accuracy of the present model, both available experimental data and finite element analysis results are used, from which the bending stress distributions along the lip, flange and web lines are compared with those obtained from the present and EN1993-1-3 models.

A tip deflection calculation method for a wind turbine blade using temperature compensated FBG sensors

The tip deflections of wind turbine blades should be monitored continuously to prevent catastrophic failures of wind turbine power plants caused by blades hitting the tower. In this paper, a calculation method for wind turbine blade tip deflection is proposed using a finite difference method based on arbitrary beam bending and moment theory using measured strains. The blade strains were measured using fiber optic Bragg grating sensors. In order to confirm this method, a 100 kW composite wind turbine blade was manufactured with epoxy molded fiber optic Bragg grating (FBG) sensors installed in the shear web of the blade. A number of these sensors, normal FBG probes, were fabricated to only measure strains and the other sensors, temperature compensated FBG probes, were prepared to also measure strain and temperature. Because the output signals of FBG sensors are dependent on strains as well as temperatures, the sensor output signals should be compensated by the temperatures to obtain accurate strains. These FBG sensors were attached on the lower and upper parts of the web at one meter intervals throughout the entire length of the blade. To evaluate the measurement accuracy of the FBG sensors, conventional electrical strain gauges were also bonded onto the surface of the web beside each FBG sensor. By performing a static load test of the blade, the calculated tip deflection of the blade was well determined within an average error of 2.25%.

Modified 4-Point Bending Test for Adhesion Measurement at the Interface of Iron Coated with Aluminum Casting Alloy

In order to reduce the weight in automotive or aeronautic components, aluminum and its alloys are being used more and more. However, their mechanical properties are not as good as those of steel and can sometimes be below the necessary requirement. To overcome this issue, reinforcement of some parts is required. In this case, the mechanical reliability of the interface between the aluminum alloy matrix and its reinforcement has to be insured. This can be done by controlling the process parameters involved in developing a strong metallurgical bond between the two metals. In this work an Fe/Al–7 wt% Si alloy interface was investigated, aimed at linking the chemistry of the reaction layers with the toughness of the bond. First, the chemistry of the interface was characterized after immersion of Fe plates in an aluminum alloy melt. Then, the fracture energy of these assemblies was measured using a modified 4-point bending test for the adhesion measurement. In the case of pure elastic deformation during the loading, an analytical approach based on beam bending theory can be used to quantify the fracture energy. Nevertheless, in bonding between metallic partners, plastic dissipation can be expected. This is why a numerical calculation, based on a cohesive zone mode was proposed. The use of cohesive elements to describe the interface behavior allows for modeling both the threshold of de-cohesion and the progression of damage at each point of the interface.

Recoil and Vibration in an Archery Bow Equipped with a Multi-Rod Stabilizer

The aim of this research is to create a mechanical and mathematical model of a multi-rod stabilizer for the sport archery bow and to analyze its capability to damp disagreeable recoil and vibration of the bow during internal ballistic motion. The research methods are based on the Euler-Bernoulli theory of beam bending, Lagrange equations of the second kind, the Cauchy problem, and the Runge-Kutta method. A mathematical software package was used to analyze the problem. The approach to the problem of sport-bow stabilization in the vertical plane that is proposed in this paper addresses the practical needs both of applied engineering mechanics and of the sport of archery. Numerical results from computer simulation are presented in both tabular and graphical form. The common motion of the string and arrow (internal ballistic motion) is accompanied by intense vibration which is caused by disruption of the static force balance at the moment of string release.

Recoil and Vibration in an Archery Bow Equipped with a Multi-Rod Stabilizer

The aim of this research is to create a mechanical and mathematical model of a multi-rod stabilizer for the sport archery bow and to analyze its capability to damp disagreeable recoil and vibration of the bow during internal ballistic motion. The research methods are based on the Euler-Bernoulli theory of beam bending, Lagrange equations of the second kind, the Cauchy problem, and the Runge-Kutta method. A mathematical software package was used to analyze the problem. The approach to the problem of sport-bow stabilization in the vertical plane that is proposed in this paper addresses the practical needs both of applied engineering mechanics and of the sport of archery. Numerical results from computer simulation are presented in both tabular and graphical form. The common motion of the string and arrow (internal ballistic motion) is accompanied by intense vibration which is caused by disruption of the static force balance at the moment of string release.