Beam Theory Research Articles

Curvature programming of freestanding 3D mesostructures and flexible electronics based on bilayer ribbon networks

Three-dimensional (3D) buckling assembly of flexible electronics from strategically designed two-dimensional (2D) precursor structures has enabled important applications in a variety of areas, owing to its versatile applicability to a broad range of length scales and high-performance materials, as well as to a rich diversity of 3D topologies. Rational design methods that allow direct mapping of 3D mesostructures onto unknown 2D precursor structures and loading parameters are foundational to these assembly technologies, but face scientific challenges, such as the high nonlinearity of spatial deformations and tricky bifurcation behaviors. While a few inverse design methods based on the beam theory, topology optimization and machine learning algorithms have been reported, the shape programming of freestanding 3D mesostructures/electronics with highly complex curvature distributions remains elusive. In this work, we propose a curvature programming method based on bilayer ribbon networks, along with a mold-assisted assembly strategy, as a new route to customizable freestanding 3D mesostructures and electronics. Combined mechanics modeling, finite element analyses and experimental measurements allow a clear understanding of nonlinear bending-stretching coupled deformations of bilayer ribbon networks during the 2D-to-3D transformation. A parameter domain with one-to-one mapping of the dimensionless curvature and the bending stiffness ratio is identified, offering a theoretical basis of the curvature programming. By introducing a discretization strategy, a variety of regular (e.g., circles, ellipses, spirals and toroids) and biomimetic 3D curved ribbons and mesosurfaces (e.g., mimicking wavy vines, diatoms and arbitrarily curled leaves) were inversely designed and experimentally realized. A device demonstration capable of strain/temperature sensing and micro-LEDs indication suggests application opportunities in bioelectronics and microelectromechanical systems.

Numerical Simulation Method for the Aeroelasticity of Flexible Wind Turbine Blades under Standstill Conditions

With the trend towards larger and lighter designs of wind turbines, blades are progressively being developed to have longer and more flexible configurations. Under standstill conditions, the separated flow induced by a wide range of incident flow angles can cause complex aerodynamic elastic phenomena on blades. However, classical momentum blade element theory methods show limited applicability at high angles of attack, leading to significant inaccuracies in wind turbine performance prediction. In this paper, the geometrically accurate beam theory and high-fidelity CFD method are combined to establish a bidirectional fluid–structure coupling model, which can be used for the prediction of the aeroelastic response of wind turbine blades and the analysis of fluid–structure coupling. Aeroelastic calculations are carried out for a single blade under different working conditions to analyze the influence of turbulence, gravity and other parameters on the aeroelastic response of the blade. The results show that the dominant frequency of the vibration deformation response in the edgewise direction is always the same as the first-order edgewise frequency of the blade when the incoming flow condition is changed. The loading of gravity will make the aeroelastic destabilization of the blade more significant, which indicates that the influence of gravity should be taken into account in the design of the aeroelasticity of the wind turbine. Increasing the turbulence intensity will change the dominant frequency of the vibration response in the edgewise direction, and at the same time, it will be beneficial to the stabilization of the aeroelasticity response.

An effective investigation of chatter prediction system on Al6061 alloy in an end milling process

Visual examination of the surface topography, in conjunction with the other sensors, may confirm the existence of chatter. Online chatter detection during real machining operations is possible with the use of sensors, and the presence of noise in their output and restricted bandwidth are the major drawbacks of these sensors. Productivity drops and manufacturing costs go up when there is a lot of chatter in the machining process. In the present paper, an integrated spindle tool system is modeled using finite element method with Timoshenko beam theory including rotational and shear deformation effects. To maximize the average stable depth of cut in an end milling process while simultaneously minimizing the chatter vibration levels, real time and offline strategies have been investigated. Machining experiments performed on Al6061-alloy specimens provide an empirical confirmation of the stability boundaries. The surface topography methods such as scanning electron microscope (SEM) and optical microscope images along with vibration levels are considered, to identify chatter marks under various machining conditions, which helps to assure cutting process stability. Stability lobe diagrams are plotted with these derived conditions and observed at an incremental level in the axial depths of the cut. The methodology shown in this paper improves the machining stability of the end milling with the reduction in the tool tip vibrations.

An effective investigation of chatter prediction system on Al6061 alloy in an end milling process

Visual examination of the surface topography, in conjunction with the other sensors, may confirm the existence of chatter. Online chatter detection during real machining operations is possible with the use of sensors, and the presence of noise in their output and restricted bandwidth are the major drawbacks of these sensors. Productivity drops and manufacturing costs go up when there is a lot of chatter in the machining process. In the present paper, an integrated spindle tool system is modeled using finite element method with Timoshenko beam theory including rotational and shear deformation effects. To maximize the average stable depth of cut in an end milling process while simultaneously minimizing the chatter vibration levels, real time and offline strategies have been investigated. Machining experiments performed on Al6061-alloy specimens provide an empirical confirmation of the stability boundaries. The surface topography methods such as scanning electron microscope (SEM) and optical microscope images along with vibration levels are considered, to identify chatter marks under various machining conditions, which helps to assure cutting process stability. Stability lobe diagrams are plotted with these derived conditions and observed at an incremental level in the axial depths of the cut. The methodology shown in this paper improves the machining stability of the end milling with the reduction in the tool tip vibrations.

Machine learning and traditional approaches in shear reliability of steel fiber reinforced concrete beams

In the field of structural engineering, the exploration of steel fibre reinforced concrete (SFRC) beams has recently intensified, particularly due to their improved tension and shear performance of structure. This study pioneers a novel reliability analysis of shear capacity predictions for SFRC beams, distinctively classifying the datasets into high-strength (HSFRC) and normal-strength (NSFRC) categories. A comprehensive database of 142 HSFRC and 265 NSFRC beams serves as the foundation for this analysis, which critically examines the standard Gaussian distribution in shear design models and proposes the Lognormal and Weibull distributions as more precise alternatives. Employing advanced First-order (FORM) and Second-order Reliability Methods (SORM), the study covers a broad spectrum of load conditions, including dead, live, snow, wind, and seismic loads, to evaluate various empirical, semi-empirical and machine learning proposed shear capacity prediction formulas. One of the key innovations of this research is the development of differentiated resistance coefficients for various risk levels in the reliability analysis, allowing future structural designers to tailor their designs according to specific risk profiles. This approach significantly enhances the balance between economic efficiency and structural safety based on the evolution of different target reliability indexes. The study reveals that existing design equations for SFRC beams generally lean towards conservatism. This study found that formulas derived from machine learning exhibited superior prediction ability compared to traditional theoretically derived regression formulas. When compared the proposed machine learning formulas, the Tarawneh's formula demonstrated better prediction ability than the Kara's formula when applied to larger datasets. However, high predictive power does not necessarily equate to high reliability. Machine learning formulas prioritise predictive accuracy, often at the expense of insufficient redundancy for ensuring safety. Overall, it introduces Kara's formula for NSFRC beams, which stands out with its superior predictive performance, offering an optimal balance of safety and cost-efficiency. Ashour's formula is also identified as a more effective and safer option for HSFRC beams. Complementing these findings, the extensive sensitivity analysis of the collected data not only confirms the robustness of the conclusions but also prepares for the integration of broader datasets in the future.

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.

Damage identification in scale sensitive beam-like nano-joints in the framework of space-fractional Euler–Bernoulli theory

In the paper, a fundamental question about damage identification in scale-sensitive bodies is posed. For this purpose, nano-scale beam-like joints are analyzed using dedicated computational schemes. The theoretical description is led in the framework of non-local space-fractional Euler–Bernoulli (s-FEBB) theory — a special case of space-Fractional Continuum Mechanical model generalizing classical continuum mechanics utilizing fractional calculus. The problem of damage identification is stated as an optimization task where the original numerical strategy is implemented. What is crucial, the s-FEBB model assumes constitutive parameters identified and validated for the silver nanobeam in previous studies. Based on tens of thousands of analyses, we arrive at the general conclusion that localization of fracture/damage/deterioration in material bodies whose dimensions are of the order of intrinsic microstructure is possible but requires advanced modeling — otherwise, damage can be identified in a wrong place or more places, or local damage can become distributed one and/or with wrong intensity.

Symmetry-Optimized Dynamical Analysis of Optical Soliton Patterns in the Flexibly Supported Euler–Bernoulli Beam Equation: A Semi-Analytical Solution Approach

This study employs spatial optimization principles to investigate the nonlinear vibration of a flexibly supported Euler–Bernoulli beam, a (1 + 1)-dimensional system subjected to axial loads. The modified Khater method, a crucial tool in mechanical engineering, is utilized to analyze analytical solutions, which include a symmetric spatial representation of the waveform as an integral part of each solution. Notably, periodic soliton solutions for the nonlinear model closely align with numerical and approximate analytical solutions, demonstrating the accuracy of our modeling approach. Density diagrams, contour diagrams, and Poincaré maps depicting the obtained analytical solutions are presented to elucidate their accuracy and provide visual confirmation of the optimized engineering model’s physical significance. The planar dynamical system is derived through the Galilean transformation by employing mathematical models and appropriate parameter values, thereby further refining problem understanding. Sensitivity analysis is conducted, and phase portraits with equilibrium points are illustrated by analyzing a special case of the investigated dynamical system, emphasizing its symmetrical properties. Lastly, we perform a global analysis to identify periodic, quasi-periodic, and chaotic behaviors, with an extra weak forcing term confirmed by Poincaré maps and a two-dimensional symmetric basin of the largest Lyapunov exponent.

Symmetry-Optimized Dynamical Analysis of Optical Soliton Patterns in the Flexibly Supported Euler–Bernoulli Beam Equation: A Semi-Analytical Solution Approach

This study employs spatial optimization principles to investigate the nonlinear vibration of a flexibly supported Euler–Bernoulli beam, a (1 + 1)-dimensional system subjected to axial loads. The modified Khater method, a crucial tool in mechanical engineering, is utilized to analyze analytical solutions, which include a symmetric spatial representation of the waveform as an integral part of each solution. Notably, periodic soliton solutions for the nonlinear model closely align with numerical and approximate analytical solutions, demonstrating the accuracy of our modeling approach. Density diagrams, contour diagrams, and Poincaré maps depicting the obtained analytical solutions are presented to elucidate their accuracy and provide visual confirmation of the optimized engineering model’s physical significance. The planar dynamical system is derived through the Galilean transformation by employing mathematical models and appropriate parameter values, thereby further refining problem understanding. Sensitivity analysis is conducted, and phase portraits with equilibrium points are illustrated by analyzing a special case of the investigated dynamical system, emphasizing its symmetrical properties. Lastly, we perform a global analysis to identify periodic, quasi-periodic, and chaotic behaviors, with an extra weak forcing term confirmed by Poincaré maps and a two-dimensional symmetric basin of the largest Lyapunov exponent.

Characterization of pressure-dependent nonlinear bending behavior of yarns and its application in modeling the compression of 2D woven fabrics

This paper investigates the pressure-dependent nonlinear bending behavior of yarns, which is essential for the application of the virtual fiber modeling (VFM) method in the mechanical analyses of fabrics. An experimental method, along with a theoretical model based on classical beam theory, is presented to characterize the varying bending stiffness of yarns under different pressures. A tailored beam user element is then developed, incorporating the nonlinear bending behavior and combined with a truss element to create a physics-based virtual fiber formulation. Utilizing this formulation, the original kinematic VFM method is extended for modeling the mechanical response of 2D woven fabrics under compression. The predicted results of the proposed model closely match the reported experiment, demonstrating the significance of introducing the nonlinear bending behavior of yarns. This method can be a valuable tool for the fabric compression process and generating realistic mesoscale geometries for textile composites.

Mechanical metastructure with embedded phononic crystal for flexural wave attenuation

Destructive interference-based metamaterials have shown excellent characteristics in elastic wave manipulation and vibration attenuation. Nevertheless, challenges persist in the application due to limited space and lightweight design, as current metastructures require additional beam structure. To simplify the design of metamaterials for flexural wave manipulation, this paper presents a new class of embedded phononic crystal for manipulating flexural wave propagation in both one and two-dimensional space by taking advantage of destructive interference, which can effectively suppress the mechanical vibration of a beam structure with a broad band gap. The flexural wave dispersion characteristic in a non-uniform beam structure is derived based on the Euler–Bernoulli beam theory, and an embedded phononic structure with the mechanism of destructive interference is presented to demonstrate its effectiveness in mitigating mechanical vibration. Subsequently, four typical units of embedded phononic structures are designed for attenuating flexural wave propagation in a beam structure. Finally, both numerical simulations, including one and two-dimensional phononic crystals, and physical experiments are implemented to evaluate the performance of the presented metastructure for flexural wave manipulation, which indicates that the proposed embedded phononic structures can effectively mitigate structural vibration in the low-frequency domain. To the best of our knowledge, it is the first attempt to design the metabeam with embedded phononic structures by taking advantage of destructive interference.

Vibration Analysis of Porous Cu-Si Microcantilever Beams in Fluids Based on Modified Couple Stress Theory.

The vibrations in functionally graded porous Cu-Si microcantilever beams are investigated based on physical neutral plane theory, modified coupled stress theory, and scale distribution theory (MCST&SDT). Porous microcantilever beams define four pore distributions. Considering the physical neutral plane theory, the material properties of the beams are computed through four different power-law distributions. The material properties of microcantilever beams are corrected by scale effects based on modified coupled stress theory. Considering the fluid driving force, the amplitude-frequency response spectra and resonant frequencies of the porous microcantilever beam in three different fluids are obtained based on the Euler-Bernoulli beam theory. The quality factors of porous microcantilever beams in three different fluids are derived by estimating the equation. The computational analysis shows that the presence of pores in microcantilever beams leads to a decrease in Young's modulus. Different pore distributions affect the material properties to different degrees. The gain effect of the scale effect is weakened, but the one-dimensional temperature field and amplitude-frequency response spectra show an increasing trend. The quality factor is decreased by porosity, and the degree of influence of porosity increases as the beam thickness increases. The gradient factor n has a greater effect on the resonant frequency. The effect of porosity on the resonant frequency is negatively correlated when the gradient factor is small (n<1) but positively correlated when the gradient factor is large (n>1).

Prestress Force and General Excitation Identification for Plate-like Concrete Bridges

Prestress force dominates the carrying capacity of concrete girders, and is vital for bridge health monitoring. Many vibrational-based methods have been proposed to determine prestress using bridge responses induced by known excitations, which means that they can barely use the normal traffic of in-service bridges as excitation to achieve long-term monitoring. Moreover, most studies are based on beam theory, which may not be precise for plate-like bridges. Hence, this paper establishes a motion equation for a prestressed slab via Kirchhoff’s plate theory and proposes a two-step procedure to assess the prestress and general excitation simultaneously through only bridge responses. The excitation is determined in the first step via the Load Shape function method and used as input for the prestress identification via state-space formulation in the second step. A numerical study on a prestressed plate subjected to a moving load is conducted. Considering different levels of measurement noise and load speed, the proposed method can determine the prestress and moving load with 7.33% and 10.18% error, respectively. A laboratory test on a prestressed box girder subjected to a fixed cyclic load is performed, the prestress and cyclic load are both determined to have good stability, and the errors are under 11.32%.

Dynamic simulation of a simply supported beam subjected by multiple constant moving forces

Abstract Dynamic simulation models, which are conducted with Simulink, are introduced to study behaviors of a simply supported beam under constant moving forces based on the vibration mechanics model established by Bernoulli-Euler beam theory. By utilizing the orthogonality of vibration, a set of decoupled forced vibration equations can be obtained, which can be constructed as a subsystem of the Simulink model. The generalized loads in each decoupled vibration equation are also separately formed into submodules to facilitate the extension of the action of a single moving load to multiple moving loads. Every partial and general simulation model in Matlab/Simulink are made and the simulation results indicate that dynamic responses are related to the distance between moving loads and beam span.

An experimental and analytical study of mode I fracture and crack kinking in thick adhesive joints

This study investigates the fracture behavior of thick adhesive joints manufactured with composite adherends and bonded with an epoxy-based structural adhesive common to the wind turbine industry. For that purpose, double cantilever beam specimens with an adhesive thickness of approximately 10 mm and different pre-crack lengths are manufactured and tested under mode I loading. Analytical approaches are compared to assess the energy release rate, including the simple beam theory, modified beam theory, compliance calibration method, and beam on an elastic and elastic-plastic foundation. In order to evaluate the applicability of the analytical approaches, an in-situ measurement method based on Digital Image Correlation is also employed to determine the energy release rate of the thick adhesive joints. The crack propagation angle is determined theoretically using the second-order crack kinking theory. A good correlation is observed between the theoretical predictions and experimental results. Furthermore, it is demonstrated that due to the T-stress, the crack tends to deviate from the middle of the joint and propagate towards the interface. By comparing different data reduction methods to evaluate the energy release rate of thick adhesive joints, recommendations for their fracture analysis are made, pinpointing the beam on an elastic and elastic-plastic foundation as the most suitable model.

Machine Learning and Traditional Approaches in Shear Reliability of Steel Fiber Reinforced Concrete Beams

In the field of structural engineering, the exploration of steel fibre reinforced concrete (SFRC) beams has recently intensified, particularly due to their improved tension and shear performance of structure. This study pioneers a novel reliability analysis of shear capacity predictions for SFRC beams, distinctively classifying the datasets into high-strength (HSFRC) and normal-strength (NSFRC) categories. A comprehensive database of 142 HSFRC and 265 NSFRC beams serves as the foundation for this analysis, which critically examines the standard Gaussian distribution in shear design models and proposes the Lognormal and Weibull distributions as more precise alternatives. Employing advanced First-order (FORM) and Second-order Reliability Methods (SORM), the study covers a broad spectrum of load conditions, including dead, live, snow, wind, and seismic loads, to evaluate various empirical, semi-empirical and machine learning proposed shear capacity prediction formulas. One of the key innovations of this research is the development of differentiated resistance coefficients for various risk levels in the reliability analysis, allowing future structural designers to tailor their designs according to specific risk profiles. This approach significantly enhances the balance between economic efficiency and structural safety based on the evolution of different target reliability indexes. The study reveals that existing design equations for SFRC beams generally lean towards conservatism. This study found that formulas derived from machine learning exhibited superior prediction ability compared to traditional theoretically derived regression formulas. When compared the proposed machine learning formulas, the Tarawneh's formula demonstrated better prediction ability than the Kara's formula when applied to larger datasets. However, high predictive power does not necessarily equate to high reliability. Machine learning formulas prioritize predictive accuracy, often at the expense of insufficient redundancy for ensuring safety. Overall, it introduces Kara's formula for NSFRC beams, which stands out with its superior predictive performance, offering an optimal balance of safety and cost-efficiency. Ashour's formula is also identified as a more effective and safer option for HSFRC beams. Complementing these findings, the extensive sensitivity analysis of the collected data not only confirms the robustness of the conclusions but also prepares for the integration of broader datasets in the future.

$$ H^1$$-Galerkin mixed finite element method for the vibration equation of beam with structural damping

$$ H^1$$-Galerkin mixed finite element method for the vibration equation of beam with structural damping

Dynamic Response of Girder Bridges with Corrugated Steel Webs Subjected to Moving Loads

Girders with corrugated steel webs (CSWs), replacing traditional concrete webs in prestressed concrete box girders, exhibit a unique dynamic behavior due to the accordion effect and the thinner dimension of CSWs. The study aims to investigate the dynamic response of girder bridges with CSWs under moving loads. Utilizing the Timoshenko beam theory and modal analysis method, the analytical formula for the dynamic response, considering shear deformation and rotational inertia, was developed. The proposed analytical formula was validated through finite element simulation. Employing dimensionless parameters, a parametric analysis was conducted to examine the effects of multi-modes, loading position and shear deformation on the dynamic response of girders with CSWs. The results highlighted the significant influence of the shear parameter αS and the speed parameter SE1 on the magnitude of the impact factor. Precise calculations for deflection and bending moment only require consideration of low-order modes, whereas accurate estimation of shear force requires attention to higher-order modes. In particular, a high-frequency resonance phenomenon is identified when the speed parameter SE1 and the shear parameter αS conformed to the formula “πSE1αS = 1”. The suggested impact factor formulas for midpoint deflection, midpoint bending moment and end shear force have been develop and categorized into three groups based on the magnitude of the shear parameter αS.

First-order GBT for truncated conical shells with circular cross-section

This paper presents a new and general first-order Generalized Beam Theory (GBT) formulation for thin truncated conical shells with circular cross-section. With respect to the previous formulations (Nedelcu, 2011; Mureşan et al., 2019), which were developed for the linear stability analysis case and therefore only required the consideration of rather simple load cases, the present one keeps all membrane strain terms and therefore is able to enforce the standard GBT membrane strain assumptions exactly. Even if the resulting expressions and the boundary conditions are more involved, it is shown that it is possible to define the so-called GBT “Vlasov warping deformation modes” also in the conical case, which are essential for the overall performance of the formulation. For instance, these deformation modes allow finding the shell inextensible deformation shapes. Several numerical tests are presented to show the excellent performance of the resulting finite element, through comparison with refined shell finite element models, and the advantages offered by the GBT modal decomposition features.

Rainbow trapping of out-of-plane mechanical waves in spatially variant beam lattices

We numerically and experimentally investigate the propagation of mechanical waves in two-dimensional periodic and spatially graded elastic beam lattices. Experiments on metallic lattices admit the characterization of the linear elastic wave dispersion over a wide range of frequencies, resulting in complete, experimentally-constructed dispersion surfaces in excellent agreement with predictions obtained from finite element-based Bloch wave analysis. While Timoshenko beam theory is shown to be sufficiently accurate for predicting the lowest modes, experiments prove that solid finite elements are required to capture the dispersion relations at higher frequencies as well as when mode coupling occurs. Based on an improved numerical procedure, group velocity maps further highlight the directionality of wave dispersion and allow for the simple identification of bandgaps. In addition to classically studied periodic trusses, we extend the framework to spatially graded structures and demonstrate acoustic rainbow trapping in beam lattices undergoing out-of-plane vibrations. Our experiments confirm broadband vibration attenuation of the typical meta-wedge type previously observed only in optics and few mechanical studies. Results further show convincing agreement between Bloch theory-based predictions, finite element simulations, and experimental measurements. Such spatially-variant architected lattices show great promise for steering the motion of elastic waves in applications from wave guiding and wave shielding to energy harvesting.