Beam Theory Research Articles

Simplified modeling of elastic guided waves in a pre-stressed cable

Elastic waves can be employed to probe the mechanical state of pre-stressed cables and look for defects in critical areas. Contacts between wires have been shown to play a major role in wave propagation by giving rise to tension-dependent phenomena such as anomalous transmission at certain frequencies (the "notch frequency effect"). Here, we present a simplified way of modeling the free propagation of waves at low frequencies by analytically accounting for contacts. Wires are described within beam theory and connected with tension-dependent springs derived from Hertz law. In contrast to point-wise contacts, these line-wise contacts also induce second-neighbor coupling, i.e. they connect wires that are not in direct contact one another. The simplified model is compared with a reference model based on a finite elements representation of the cross section ("SAFE" method). The range of validity in frequency is discussed. The results provide a new interpretation for the first higher order modes in and their dependence with tension in terms of contact resonances. The drastic reduction of numerical costs suggests the method could be applicable to more complex civil engineering cables, composed of many wires.

Vibration analysis of functionally graded saturated porous beams using radiative energy transfer method

An energy flow model based on the radiative energy transfer method (RETM) is established for the high-frequency response prediction of finite functionally graded saturated porous beams. The Timoshenko beam theory and the Biot thoery are applied to derive the motion governing equations of the beam. The response of the beam loaded by a high-frequency point force is described by vibrational energy density and intensity. The energy response of the beam is contributed by the rays emitted from the actual source set at load point and then reflected through the fictitious sources at the boundaries. Fictitious sources are determined by the energy balance equations at the boundaries. Numerical examples show that the energy distributions of the saturated porous beams are well predicted by the proposed approach.

Performance of nodal regions of reinforced concrete frame corners subjected to opening bending moments

Design of frame corners in presence of opening moments is a task that has concentrated many research and engineering efforts in the past. It deals with the response of discontinuity regions, where beam theory does not hold valid and the use of equilibrium-based models (such as strut-and-tie models or stress fields) could be a suitable manner to understand the transfer of forces and to lead to consistent reinforcement detailing. However, the assessment of efficiency factors for equilibrium-based models is not straightforward for such details as their response has been observed to be governed in many cases by the tensile resistance of concrete and crack propagation. Furthermore, most efforts so far have been devoted to determining only the resistance of the region as a function of its reinforcement detailing, but limited works have been dedicated to their post-peak response. The post-peak response is nevertheless instrumental for redundant structures, as it governs the level of potential redistributions of internal forces in order to develop the full structural resistance of the system. Such aspect has become even more significant in present days, where the assessment of existing structures with potentially poor nodal detailing has become a need to avoid unnecessary retrofitting or replacement, in an effort to reduce carbon footprint of construction. In this paper, comprehensive research on this topic is presented. It is based on the results of a large database compiled by the authors comprising 126 tests. After filtering of the database, conclusions on the expected response of several details are obtained. Also, a general procedure grounded on an equilibrium-based model is proposed, providing tailored efficiency factors as a function of the mechanical reinforcement ratio and detailing type, but also providing a complete description of the post-peak response. This latter model is based on a kinematically-compatible displacement field for the post-peak phase, allowing to evaluate the opening of the cracks in the nodal region and the associated softening of resistance.

Nonlinear vibration and parametric excitation of magneto‐thermo elastic embedded nanobeam using homotopy perturbation technique

AbstractThe present study focuses on the modeling and analyzing the nonlinear vibration patterns and parametric excitation of embedded Euler–Bernoulli nanobeams subjected to thermo‐magneto‐mechanical loads. The Euler–Bernoulli nanobeam is developed with external parametric excitation. The governing equation of motion is derived by utilizing nonlocal continuum theory and nonlinear von Karman beam theory. Subsequently, the homotopy perturbation technique is employed to determine the vibration frequencies. Finally, the modulation equation of Euler–Bernoulli nanobeams is derived for simply supported boundary condition. In order to validate our findings, we conduct a comparative analysis against existing literature, thereby underscoring the effectiveness and robustness of our proposed methodology. The influence of stress, magnetic potential, temperature, damping coefficient, Winkler coefficient, and nonlocal parameters are tested numerically on nonlinear frequency‐amplitude and parametric excitation–amplitude responses. The numerical examples indicate the significant impact of physical variables on the nonlinear frequency and parametric excitation. The primary objective of this study is to investigate the effects of external physical variables on the dynamic behavior of nanobeams, particularly in nonlinear regimes. This study provides insights into the design and control of nanostructures under complex load conditions, contributing to developing advanced materials and nanosystems.

Therapeutic radiation directly alters bone fatigue strength and microdamage accumulation

Radiotherapy (RTx) is an essential and efficacious oncologic treatment, however, post-RTx bone fragility fractures present a challenging clinical problem. Cancer survivors treated with RTx are at variable risk for these late-onset, complex fragility fractures. Little data exists regarding the effects of RTx on bone fatigue properties despite the likelihood of fatigue loading as a mechanism leading up to atraumatic fracture. In this study, femurs collected from adult male rats were irradiated ex vivo with a therapeutic dose of x-irradiation (20 Gy), and then fatigued using a three-point bend setup. Femurs positioned in an isotonic bath at room temperature were loaded to a range of prescribed initial strain levels (based on beam theory equations, prior to any fatigue damage) at 3 Hz in force control. The goals of this study were to determine the feasibility of assessing RTx-induced alterations in 1) femur fatigue strength, 2) structural microdamage (creep and stiffness), and 3) tissue damage (diffuse damage and/or linear microcracking). Mid-diaphyseal morphology and tissue mineral density were not different between the RTx and Sham groups (p ≥ 0.35). With increasing applied apparent strain, the number of cycles to failure was reduced for the RTx femurs when compared to the Sham femurs (treatment x εapp, p = 0.041). RTx femurs had a greater phase II (steady state) creep rate (p = 0.0462) compared to Sham femurs. For femurs that reached 500k cycles, the RTx group had greater diffuse damage area (p = 0.015) than the Sham. This study provides evidence that radiation at therapeutic doses can directly diminish bone fatigue properties. This loss of fatigue properties is associated with increased structural fatigue damage and diffuse microdamage, without alterations in morphology or tissue mineral density, indicating a reduction in bone quality.

The vibration response of five beam theories under eccentric moving harmonic loads

The shear rotary inertia caused by the shear deformation affects the natural frequencies and the vibration characteristics of bridges. To guarantee the safety of bridges, it is necessary to investigate the shear rotary inertia as a crucial content. Unlike previous studies, the motivation of this paper is to improve the Timoshenko beam model and further explore the influence of the shear rotary inertia on flexural–torsional natural frequencies and flexural–torsional vibration properties of bridges. Based on the Timoshenko beam theory, a modified dynamic equation of the Timoshenko beam is established. Its natural frequencies are obtained, and its analytical solutions under moving harmonic loads are derived using the integral transform method. In order to further understand the shear rotary inertia, the other four beams (namely, the Euler–Bernoulli beam, the shear beam, the Rayleigh beam, and the Timoshenko beam) are introduced to make comparisons with the modified Timoshenko beam. Subsequently, in the parameter analysis, the natural frequencies and vibration responses of five different beams with various lengths are analyzed numerically. It is discovered that the shear deformation and the shear rotary inertia have great effect on the high-mode vertical and bridge flexural–torsional frequencies, and it can improve the accuracy of calculating the natural frequencies of beams by about 6.00%. Furthermore, the shear deformation effect on vertical dynamic responses under harmonic moving loads intensifies with the beam spans increasing. The shear beam theory and the Timoshenko beam theory are appropriate for the vibration analyses of short beams, and the Rayleigh beam theory and the Timoshenko beam theory are suitable for the vibration analyses of long beams, while the modified Timoshenko beam theory, being more generally, is suitable for both beams.

Three-dimensional configuration tip rotor dynamics analysis

Abstract In order to analyze and study the dynamic characteristics of the 3D blade tip rotor, an aeroelastic dynamic model for a complex 3D blade tip rotor is established based on mechanics principles. The theory of medium deformation beam or large deformation beam is adopted as the blade structural dynamic model, and the quasi-steady and unsteady aerodynamic models (ONERA model) are adopted as the aerodynamic model. The rotor dynamics, aeroelastic stability and rotor loads of three-dimensional blade tip and rectangular blade tip are compared by examples. The analysis shows that both the forward and backward sweep angles have a certain effect on the rotor blade frequency, but the downward dihedral angle has little effect. The 3D tip configuration has the greatest influence on torsion mode frequency, followed by lagging mode frequency, which has little impact on flapping frequency. Reasonable design of forward and backward sweep angles and downward dihedral angles of 3D tip configuration can effectively reduce rotor loads, increase damping margin for aeroelastic stability, and improve stability.

Effects of two-dimensional reed oscillation on airflow and sound generation in a single-reed instrument

While two-dimensional (2D) reed oscillation modes of single-reed woodwind instruments have been reported in previous studies, little is known about their effects on airflow and sound generation. In this study, we conducted aeroacoustic simulations of a clarinet mouthpiece and resonator coupled with one-dimensional (1D) and 2D reed deformation models and investigated the changes in flow and sound generation due to the 2D reed vibration. The 1D and 2D reeds were modeled using 1D beam and thin plate theories, respectively, whereas the three-dimensional airflow was simulated by solving the compressible Navier–Stokes equations. The self-sustained oscillations of the 2D reed model mainly exhibited a flexural mode at the fundamental frequency, which is consistent with previous observations. Complex torsional modes were observed only at higher harmonic frequencies. A comparison between the 1D and 2D reed models demonstrated that the 2D reed opened later at the side face of the mouthpiece than the 1D reed owing to the torsional mode, which changed the time variation of the flow rate into the mouthpiece and the far-field sound in the high-frequency range. These results suggest the importance of the torsional deformation characteristics of reeds on the timbre of single-reed instruments.

Stability analysis of a vertical cantilever pipe with lumped masses conveying two-phase flow

This study investigates the vibration behavior of a vertical cantilever pipe conveying gas-liquid two-phase flow, focusing on the influence of lumped masses attached to the vertical cantilevered pipe. The governing motion equation based on small deflection Euler–Bernoulli beam theory is solved by using the generalized integral transforms technique. The proposed solution approach was first validated against available numerical and experimental results in the literature. The effects of the mass ratios, number and position of lumped masses on the stability of the pipe are investigated. Numerical results show that the parameters of the lumped masses affect significantly the stability of the pipe conveying two-phase flow, by altering the fluid–structure interaction dynamics and impacting natural frequencies and vibration modes of the pipe. Specifically, as the position of a single lumped mass moves downward from the fixed end to the free end, the critical flow velocity initially increases and subsequently decreases, thereby reducing the stability of pipe. Moreover, increasing the number of lumped masses significantly impacts the critical flow velocity due to the mass ratios and locations. Notably, modal “jumping” phenomena are observed, which demonstrate continuous shifts between equilibrium and non-equilibrium states in the cantilever pipes. These findings are crucial for ensuring the safe operation of pipes with discrete masses across various engineering applications.

Exact closed-form solution for buckling and free vibration of pipes conveying fluid with intermediate elastic supports

In this paper, the exact closed-form solution is given to investigate the influence of the intermediate elastic support on the buckling and free vibration of an elastically supported pipe. According to the Euler–Bernoulli beam theory, the mechanical model of the pipe is established. The exact equilibrium configuration is derived using the generalised function method without enforcing continuity conditions. A simple solution to the eigenvalue problem is formulated using the methods of complex mode superposition and Laplace transformation. The comparative study shows the differences in the supercritical vibration characteristics and highlights the limitations of previous studies. Parametric studies are carried out to investigate the influence of elastic support and intermediate support conditions on the equilibrium configuration, critical flow velocity, and natural frequency. The results demonstrate that the proposed closed-form solution can determine the support conditions that lead to the maximum critical flow velocity and natural frequency of a pipe with multiple intermediate supports. The maximum values are required to adjust the support conditions leading to the nodes of higher-order equilibrium configurations and complex modes. Furthermore, the natural frequencies of the pipe conveying supercritical fluid no longer satisfy the monotonicity for the support stiffness, the symmetry for the support position, and the ‘zero-point’ property for the support number.

A 3-DOF spreading precision positional stage for ductile end-face fly-cutting of quartz glass

Ductile removal is widely employed to eliminate subsurface damage in brittle materials. Achieving this requires the cutting depth to be set extremely low, presenting significant challenges for error compensation and precise feeding of the machine tool. In this paper, a novel three-degree-of-freedom spreading precision positioning stage is developed to mitigate the effects of workpiece deflection errors and feed resolution on the depth of cut during the end-face fly-cutting process. First, a bridge and half-bridge composite structure is designed to facilitate the planar spreading of the spatial motion mechanism. A mathematical model of the composite structure is developed based on elastic beam theory. Second, the effects of various structural parameters on the amplification ratio of the structure are investigated. The accuracy of the theoretical model is verified by finite element analysis. Finally, ductile fly-cutting experiments on quartz glass are conducted using a precision 5-axis machine tool.

Vibration analysis of shear actuated bi-directional FGP beam using generalized differential quadrature method

The primary objective of this study is to exhibit the shear mode vibrations of a bidirectional functionally graded piezoelectric beam. All material properties are assumed to vary bidirectional in accordance with an exponential law. Using Timoshenko beam theory, the governing equations and boundary conditions are derived using the Hamilton principle. The equations of motion are then solved using the generalized differential quadrature method. Since no previous solutions existed for the current problem, the obtained results are compared with the findings from a simplified case in previous research. The effects of exponent parameters, slenderness ratios, and boundary conditions on natural frequencies of the bidirectional FGP beam are discussed in detail.

Buckling analysis of nanoscale beams based on nonlocal Timoshenko beam theory

Abstract Eigen-buckling problems of nanoscale beams continue to be of great research interest, and the nonlocal theory is widely used. However, the existing research generally adopted some simplified assumptions of nonlocal effects. This article studies the buckling behaviors of the nonlocal Timoshenko beam, where the nonlocal effects are considered both on the governing equations and boundary conditions. The variational principle is adopted to obtain the nonlocal governing equations and boundary conditions. The buckling solutions for nanoscale beams are obtained analytically. Numerical comparisons validate the correctness of the present results. Parameter study shows that the buckling characteristic of nonlocal beams is different from that of classical beams, and the nonlocal effect is dependent on boundary conditions and geometry size of nanoscale beams.

Unified model of blazars and radio galaxies: synthesizing observational data with relativistic beaming theory

Unified model of blazars and radio galaxies: synthesizing observational data with relativistic beaming theory

Subsidence Prediction Method Based on Elastic Foundation Beam and Equivalent Mining Height Theory and Its Application

Grouting technology in overburden separation is recognized as an effective method to prevent surface subsidence and reuse solid waste. This study used mechanical analysis to explore deflection characteristics of key strata and accurately predict and control surface subsidence. Conceptualizing the coal–rock mass beneath the key strata as an elastic foundation, we developed a method to calculate the elastic foundation coefficients for various regions and established an equation for key strata deflection, validated through discrete element numerical simulations. This simulation also examined subsidence behavior under different grout injection–extraction ratios. Additionally, combining the equivalent mining height theory with the probability integral method, we formulated a predictive model for surface subsidence during grouting. Applied to the 8006 working face of the Wuyang Coal Mine, this model was supported by numerical simulations and field data, which showed a maximum surface subsidence of 546 mm at a 33% injection–extraction ratio, closely matching the theoretical value of 557 mm and demonstrating a nominal error of 2%. Post-grouting, the surface tilt was reduced to below 3 mm/m, meeting regulatory standards and eliminating the need for ongoing surface structure maintenance. These results confirm the model’s effectiveness in forecasting and controlling surface subsidence with grouting. The study can provide a basis for determining the grouting injection–extraction ratios and evaluating the effectiveness of surface subsidence control in grouting into overburden separation projects.

Energy dissipation and power flow analysis based on acoustic black hole laminated beams

To enhance the understanding of the dynamic performance of composite beams, assessing their energy accumulation and dissipation capabilities is crucial. The purpose of this paper is to establish a dynamic analysis model of acoustic black hole (ABH) laminated beams based on the Timoshenko beam theory and isogeometric method and to analyze structural intensity and power flow. The vibration governing equation for the ABH laminated beam is derived by calculating its potential and kinetic energy. The convergence of the results is confirmed through multiple mesh refinements, while the model's accuracy and applicability are substantiated through comparisons with the literature, traditional finite element simulations, and experimental data. This investigation into the power flow and structural intensity under varying parameter conditions elucidates the energy transfer mechanisms within ABH laminated beams, offering valuable insights for the deployment of ABH technologies in practical engineering applications.

Accurate and efficient analytical simulation of free vibration for embedded nonlocal CNTRC beams with general boundary conditions

This study aims to evaluate the free vibrational response of embedded restrained nanobeams enriched by nanocomposites based on an exact Fourier series approach. In order to capture the small-scale effects on the dynamical response, Eringen's differential form of nonlocal elasticity is used which employs a scale (nonlocal) parameter. Within the framework of Rayleigh and Bernoulli-Euler beam theories, including the effect of nonlocality and employing the Fourier sine series together with Stokes' transformation, systems of linear equations are obtained and solved using the coefficient matrices. The combined effects of elastic boundary conditions, elastic foundation, dispersion patterns and volume fractions of carbon nanotubes, and nonlocal parameter are examined by solving eigenvalue problems constructed with Fourier infinite series. Free vibration frequencies are calculated for carbon nanotube-reinforced nanobeams under different rigid or restrained boundary conditions, including Winkler-Pasternak type elastic foundation. A comprehensive parametric study is performed, focusing on various effects for the free vibrational response of the composite nanobeam reinforced with carbon nanotubes. It is concluded that adding a small amount of carbon nanotube material can reinforce the stiffness of the composite nanobeam, and its free vibration performance is significantly affected by the distribution patterns, elastic medium, and boundary conditions.

Analysis of dynamic modeling and power flow of ABH laminated composite thin beam with elastic boundaries

To investigate the energy distribution characteristics of a laminated structure with an acoustic black hole (ABH), this study introduces the dynamic modeling and power flow analysis of an ABH laminated thin beam (ABH laminated beam) with elastic boundary conditions for the first time. ABH profile is defined by a power function in the dynamic modeling process. Utilizing the Euler-Bernoulli beam theory, the constitutive equation of ABH laminated beams is formulated by using the isogeometric method. Three sets of artificial springs are employed to replicate the elastic boundary conditions, incorporating the potential energy of the springs. Subsequently, the dynamic model of the ABH laminated beam is developed by solving the Lagrange equation concerning the total energy within the uniform region, ABH region, and boundary. Through numerical examples, the convergence and accuracy of the proposed modeling approach are validated by comparing the results with those obtained from COMSOL Multiphysics and experimental data. The analysis of power flow and structural intensity elucidates the energy propagation behavior in ABH laminated beams and the underlying mechanism of the ABH effect. The results demonstrate that the ABH laminated beam displays remarkable absorption and dissipation characteristics, introducing a new perspective for the design and application of ABH structures.

Effect of the transition piece on the natural frequencies of monopile-supported offshore wind turbines

The lateral natural frequency of an offshore wind turbine (OWT) is a crucial factor to consider in the design of OWT foundations. Eighty percent of currently installed OWTs are mounted on monopile foundations, which are connected to the superstructure via a transition piece (TP). To explore the differences in natural frequencies between grouted connection (GC) and TP-less designs, this study incorporates a transition piece into the calculation model of the OWT system. Based on the Timoshenko beam theory and transfer matrix method, the characteristic equations of the lateral natural frequency of the OWT are derived. The study found that the effect of the transition piece on the natural frequency is strongly influenced by the OWT type and the soil stiffness, and the influence of transition piece on different order natural frequencies is different for the same OWT. For the NREL offshore 5-MW baseline wind turbine, the TP-less model reduces the first natural frequency of the OWT by 1.31 % and elevates the second and third natural frequencies by 11.88 % and 2.72 %, respectively, compared to the GC model. The increase in soil shear wave velocity amplifies the influence of the transition piece on the first and third natural frequencies, while weakening its influence on the second natural frequency.

A numerical procedure for calculating ice-induced fatigue damage for ship hulls in level ice

Ship hulls advancing in level ice are exposed to considerable cyclic ice loadings, which might cause fatigue damage, threatening the structural integrity and endangering the crews’ lives. Ice-induced fatigue problem is of significance to ice-going ships. In this paper, a fatigue analysis for ship structures in level ice is performed based on a dedicated 6-degree-of-freedom (6-DOF) numerical model of ice-ship interaction, founded on a circumferential crack assumption. The numerical procedure considers the time-varying instantaneous waterline and the couplings between ship motions and external forces. The local ice loads used for fatigue assessment can be obtained from numerical simulations, and the random load peaks are represented by Weibull distribution. A case study of fatigue calculation for designated transverse frames is carried out, of which the applied stress is converted from ice loads by means of structural beam theory. Eventually, according to a proper S-N curve, the fatigue damages are evaluated on the basis of Palmgren–Miner rule, respectively under the operational scenarios of laterally symmetrical and asymmetrical initial ice edges.