Equivalent Bending Research Articles

A size‐dependent model of strain gradient elastic laminated micro‐beams with a weak adhesive layer

AbstractA size‐dependent model for a laminated micro‐beam with a soft adhesive is developed in the framework of strain gradient elasticity theory. The layered beam is constituted by two Euler–Bernoulli strain gradient elastic isotropic beams, joined through a strain gradient spring‐type contact law at the adhesive level. The governing bending and extensional equations and boundary conditions are obtained by using the variational principle. The differential system shows a coupling between the flexural and axial behaviors of the upper and lower beams, due to the presence of interface terms related to shear and peeling stresses. Two benchmark problems have been presented through their closed‐form solutions, namely a simply‐supported laminated beam subjected to a uniform distributed load and a mode 2‐type loading configuration of a layered axially deformable beam. Size effects and non‐local phenomena, due to high strain concentrations, are highlighted. Though simple in their features, the examples prove the efficiency of the proposed approach in designing micro‐scale layered beams.

Spatial and frequency identification of the dynamic properties of thin plates with the Frequency-Adapted Virtual Fields Method

Vibroacoustic inverse methods use the measured response of a vibrating structure to identify a structural parameter or a dynamic load. Two inverse methods are considered, the Force Analysis Technique (FAT) and the Virtual Fields Method (VFM). The Corrected Force Analysis Technique (CFAT) is a variant of FAT that corrects its singularity. This correction allows the method to be applied in the high-frequency domain, when the number of measurement points per flexural wavelength becomes small. In this study, the proposed novelty is the development of a Frequency-Adapted VFM (FA VFM) to the case of a Love–Kirchhoff plate. Thanks to this method, the VFM can now be applied to identify the equivalent bending stiffness and structural damping of a thin plate when the number of measurement points per wavelength is small. The method has previously been developed for an Euler–Bernoulli beam. An experimental identification of the complex bending stiffness of an locally damped aluminium plate using Laser Doppler Velocimetry (LDV) data and the developed method is performed. The experimental study shows for the first time that the FA VFM can be used to map the equivalent bending stiffness and structural damping as a function of position on a plate and identify these parameters as a function of frequency over a large frequency band. The results of the Frequency-Adapted VFM are compared with those of CFAT and the classical VFM approach. FA VFM results are more accurate than those of classical VFM and similar to those of CFAT.

Advances in longitudinal equivalent bending stiffness analysis-novel nonlinear characteristics in the assembly process of prefabricated subway stations

The longitudinal bending stiffness of quasi-rectangular prefabricated subway stations plays a great important role in studying their mechanical behavior and conducting safety assessments. To address the issue of immature stress assumptions for existing prefabricated subway stations, three stress assumptions and five bending modes are proposed while considering the coupling effect between axial force and bending moment. The study investigates novel nonlinear characteristics in the longitudinal equivalent bending stiffness and the influencing factors during the assembly of an existing subway station. An effective method for identifying the longitudinal bending modes suitable for the prefabricated subway station is established, the proposed solution has a good agreement with measured longitudinal bending stiffness efficiency. The study demonstrates that the longitudinal bending mode of the prefabricated subway station shifts from "upward convex" to "downward concave" due to the stepwise assembly method. The longitudinal bending stiffness efficiency displays a significant hysteresis curve, eventually stabilizing at approximately 1 %. Upon completion of the monitoring ring assembly, the concrete bears both longitudinal tensile and compressive stress. When assembling 10 rings from the monitoring ring, the longitudinal reinforcing bars bear tensile stress. The longitudinal bending stiffness of the prefabricated subway station exhibits significant time-variable characteristics throughout the assembly process. The research findings can provide effective references for the analysis of longitudinal mechanics and bending performance during the assembly process.

An accurate equation for the gravitational bending of light by a static massive object

An accurate equation for the gravitational bending of light by a static massive object

Mechanical behaviour of rigid-flexible combined structures: aluminium-inflated membrane beams for application in floating photovoltaic platform

This study aims to investigate the bending and failure behaviour of aluminium-inflated membrane beams for their applications in floating photovoltaic platforms. Four-point bending tests are conducted for a range of inflated pressures and two different deck heights to assess their effect on structural stiffness and ultimate bearing capacity. Meanwhile, the surface-based fluid cavity method is employed to develop the finite element model with the material properties determined by independent coupon-level tests. The bearing capacity of the aluminium-inflated membrane beam is positively correlated with the internal pressure and deck height. The midspan strain distribution is similar to those of the traditional four-point bending beam with the upper part undergoing compression and the lower part experiencing tension, however, the structural behaviour at the failure stage is different. Failure typically occurs due to localised depressions at the loading points on the aluminium deck, ultimately leading to structural failure. The numerical model closely matches the experimental data for the initial inflated and bending configurations, exhibiting a deviation of only 0.10% to 0.46% in diameter and 0.32% to 5.57% in equivalent bending stiffness. A parametric study shows that the loading properties of the beam are more sensitive to the internal pressure than the deck height and thickness.

ANALYTICAL CALCULATION OF A BEAM BASED ON AN ELASTIC WINKLER FOUNDATION WITH RANGE INHOMOGENITY

The aim of the study is the further development of analytical methods for calculating the bending of beams resting on a non-homogeneous continuous Winkler elastic foundation. This paper considers the case when the beam is under the influence of a uniformly distributed constant transverse load, and the inhomogeneity of the elastic foundation is given by a power function with an arbitrary non-negative power exponent . Fundamental functions and a partial solution of the corresponding differential equation of beam bending are found in an explicit closed form. These functions are dimensionless and are represented by absolutely and uniformly convergent power series. In turn, the formulas for the parameters of the stress-strain state of the beam – deflection, angle of rotation, bending moment and transverse force – are expressed through the indicated functions. The unknown constants of integration in these formulas are expressed in terms of the initial parameters, which are after the implementation of the specified boundary conditions. Thus, the calculation of the beam for bending is reduced to the procedure of numerical implementation of explicit analytical formulas for the parameters of the stress-strain state. An example demonstrates the practical application of the obtained solutions. A prismatic concrete beam based on a cubic variable elastic foundation is considered. This case corresponds to the power value . The results of the calculation by the author's method are presented in numerical and graphical formats for the case when the left end of the beam is hinged and the right end is clamped. The numerical values obtained by the author's method are accurate, since the applied calculation method is based on the exact solution of the corresponding differential equation. The availability of such solutions makes it possible to evaluate the accuracy of solutions obtained using various approximate methods by comparison. For the purpose of such a comparison, the paper presents the calculation results obtained by the finite element method (FEM). The absolute error of the FEM method when calculating this design was determined.

Analyses of bending performance of innovative paper-based sandwich beams with cup-box core

Paper-based materials are being alternative candidates to build load-bearing components for the high demanding building sector to be committed to the green transition, but more knowledge of structural mechanics of such components is needed. In this work, three categories of innovative load-bearing sandwich beams with cup-box core fully made of different paper materials were produced, tested and analyzed in four-point bending. The first failure mode was observed at the top facesheets due to the low compressive strength of paper materials; Beams with thin facesheets had premature buckling failure, whereas those with thicker facesheets exhibited ductility reaching higher deflection before the compressive failure. The developed finite element model, calibrated with the experiments for the equivalent bending and shear rigidities, provided figures of the modulus of the facesheet as well as the properties of the core. Furthermore, the compressive plasticity behavior of the facesheets was assessed by fitting the model with load-deflection curves from the tests. Using the model for the structural optimization of the thickness and height of the core, the work suggests optimal values 3 times higher than the original ones.

Comparison of Bending Stiffness between String of Pearls Plate-Bone Substitute Constructs with and without Bending Tees in a Fracture Gap Model

Abstract Objective The aim of this study was to compare the bending properties of String of Pearls plate-bone substitute constructs with and without bending tees in the nodes over a simulated fracture gap. It is hypothesized that the constructs with tees will have higher bending stiffness. Study Design Acetal polymer tubes and 12-hole, 3.5-mm String of Pearls plates were used to create plate-bone substitute constructs simulating stabilization in a bridging fashion over a 45-mm gap. Twenty-four constructs were made with 12 containing tees in the nodes over the fracture gap. Single-cycle load-to-failure 4-point bending was performed in mediolateral and craniocaudal planes. Bending stiffness was compared with a t-test (p < 0.05). Results All plate-bone substitute constructs had a permanent loss of structural integrity via plastic deformation of the plate. The bending stiffness (mean ± standard deviation) of the craniocaudal group was 59.11 ± 1.98 N/mm with tees and 59.25 ± 1.69 N/mm without tees (p = 0.88). In the mediolateral group, the bending stiffness was 43.17 ± 0.75 N/mm with tees and 41.09 ± 0.91 N/mm without tees (p = 0.0042). Conclusion In 4-point bending, the plate-bone substitute constructs with tees had equivalent bending stiffness in the craniocaudal plane and increased bending stiffness in the mediolateral plane. However, with a small absolute difference in values, the clinical significance is unclear. Future studies for cyclic bending, torsional, and axial compression tests should be performed to further investigate the value of tees in the nodes over a comminuted or gap fracture repaired in a bridging fashion.

A comprehensive analysis on coast side contact behaviour of thermoplastic gears against steel gear: a FEA approach

PurposeThe research paper comprehensively investigates the gear tooth deflection of standard thermoplastic gears with steel gear as the driver and driven companions. An accurate mapping of characteristic contact regions between the meshing gears was done, and the behaviour of the gear tooth in the premature and prolonged contact zones was studied.Design/methodology/approachThe study employs the finite element method to conduct a quasi-static 2D analysis of meshing gear teeth. The finite element model was created in AutoCAD and analysed using the ANSYS 19.1 simulation package.FindingsIn the polymer-polymer gear combinations, premature and prolonged contact primarily occurs along the addendum radii of meshing gears, whereas a novel contact phenomenon was observed in the coast side for polymer-metal and metal-polymer combinations, exhibiting a path perpendicular to the standard drive side contact. As well, the deflection of the tooth alters the load distribution across the interlocking gears, leading to a decrement in the root stresses.Originality/valueThe Lewis bending equation demonstrates that bending stresses depend solely on the applied load and the geometry of the tooth. It does not consider the effects of deflection. However, the computational results showed that the gear tooth deflection caused by different gear pair combinations also affects the bending stresses. The contact stresses observed in the polymer-polymer gear combination were observed to be within the material’s proportional limit. However, when a steel gear is paired with a polymer gear, the contact stresses exceed the proportional limit due to coast side contact.

Static and Dynamic Characteristics of 3D-Printed Orthogonal Hybrid Honeycomb Panels with Tunable Poisson’s Ratio

By adjusting the two wall angles of the orthogonal hybrid honeycomb (OHH), the tunable Poisson’s ratio change from negative to positive values and the variation in stiffness can be achieved. To effectively analyze its static and dynamic characteristics, a two-dimensional equivalent Kirchhoff–Love model (2D-EKM) is established based on the variational asymptotic method (VAM).This model aids in effectively addressing the complexity arising from anisotropy. The obtained equivalent orthotropic properties are validated through unit-cell uniaxial compression tests and three-point bending experiments on 3D-printed specimens. The numerical simulation results suggest that the VAM-based 2D-EKM can predict the in-plane and out-of-plane static behaviors of OHH panels, with a maximum error below 10%. Particularly in the dynamic analysis of a four-sided fixed OHH panel, the analysis time required by 2D-EKM is only 0.37% of that needed for the 3D FE model. The OHH-ZPR panel exhibits exceptional resistance to deformation, with a maximum deformation under in-plane tension reaching only 27% of that in the OHH-PPR panel. Moreover, each 1% increase in the height–length ratio results in a respective increase of 275.62% and 281.93% in equivalent bending stiffness along both directions. This highlights that enhancing this ratio effectively boosts the fundamental frequency compared to the elastic modulus ratio, effectively prevents low-frequency resonance occurrences, and offers vital insights for the design and optimization of OHH panels.

Analytical modelling for residual stress prediction in multi-step side milling of GH4169 thin-wall parts based on deformation

GH4169 thin-walled parts are widely used in aerospace due to their high-temperature, corrosion resistance, and fatigue strength. However, they often deform from machining-induced residual stress, which is a significant unresolved manufacturing challenge. Additionally, initial residual stress from previous steps critically impacts subsequent processes. This study found that the residual stress of side milling of thin-walled parts under smaller cutting parameters is mainly caused by mechanical effects, and the influence of milling heat can be ignored. In this study, an analytical prediction model for residual stress of multi-step side milling thin-walled parts based on the deformation of thin-walled parts is proposed for the first time, which is suitable for smaller cutting parameters. Then, the proposed model is examined the mechanisms through which residual stresses develop during side milling and defined the applicability of model based on specific assumptions. To validate these assumptions, an experimental setup was devloped to simulate the movement of the heat source during milling. Subsequent two-step side milling experiments on GH4169 confirmed the accuracy of the analytical prediction model in predicting the residual stresses of multi-step side milling in thin-walled parts. It was observed that higher equivalent bending stiffness in thin-walled parts correlates with reduced machining deformation. The analytical model also provides a strategic approach to control machining deformation by managing the equivalent bending stiffness of the parts.

The stick-slip bending behavior of the multilevel helical structures: A 3D thin rod model with frictional contact

The multilevel helical structures in various engineering and natural fields offer excellent deformation flexibility and load bearing capabilities. Understanding the interplay between the local frictional contact and the geometric characteristics of the helical structure under complex external loads has attracted considerable interest. In this work, the effect of local frictional contact behaviors on the bending in multilevel helical structures is investigated by using a combination of theoretical modeling, finite element (FE) simulations, and experiments. In the case of pure bending, the kinematic parameters of the bent multi-stage helix are derived concisely by the idea of the kinematic analogy. The bending stiffness of the multi-stage helix is further obtained. In the case of the combined tension/torsion and bending, the 3D thin rod model incorporating Coulomb’s friction is established to describe the mechanical responses. It is found that the relationship between equivalent bending stiffness and the laying angle exhibits nonlinearity. A comparison with the classical Papailiou model reveals that, for helical structures at large laying angles, the influence of friction is primarily determined by the internal force in the tangential direction, which is the core assumption of the Papailiou model. However, in the case of small laying angles, the helical twisting characteristics and the contribution of the internal forces and moments in the other two directions (normal and binormal directions) to the friction cannot be ignored. Subsequently, a multilevel frictional contact transmission formulation is proposed according to the force action–reaction principle. Based on the above formulation, the non-simplified thin rod equations with Coulomb’s friction are extended to describe the multilevel stick-slip bending behaviors of the second stage cable (3*3). The dissipation capacity of helical structures is evaluated quantitatively under the hysteretic bending. Finally, the theoretical model is verified by FE simulations and experimental results. This work provides insights for unveiling the intrinsic relationship between the nonlinear bending and local frictional contact behaviors in the multilevel helical structures.

Broadening bandgaps in a multi-resonant piezoelectric metamaterial plate via bandgap merging phenomena

Locally resonant metamaterials usually have narrow bandgaps, which significantly limits their applications in realistic engineering environments. In this paper, an optimization method based on the genetic algorithm is proposed to broaden bandgaps in multi-resonant piezoelectric metamaterial through the merging of multiple separated bandgaps. Using the effective medium theory, the equivalent bending stiffness and dispersion relationship of a metamaterial plate are first obtained. Then, the criteria for determining the bandgap ranges for the two cases with and without damping are provided and analyzed. Furthermore, based on the bandgap merging phenomena, an optimization method for widening the bandgap is proposed based on the genetic algorithm. By investigating the bandgap widening effects in cases without and with damping, it is found that, when there is no damping, the bandgap can only be slightly widened; while after introducing damping into the transfer functions, the bandgap can be significantly widened by more than 200%. The bandgap widening effects are verified by comparing with finite element simulation results.

Theoretical and experimental research on the rocking truss system with post-tensioned rocking connection

The weak story failure often occurs in steel frame structures and the collapse risk of the frames will increase because of concentrated plastic deformation in the soft story. The work presented in this paper aims to avoid the soft story mechanisms and the improve self-centering capacity of steel frames by employing the innovative rocking truss. Compared with the previous rocking system, it is more flexible in application and suitable for retrofit of existing buildings. The theoretical approach to quantifying the stiffness demand of the rocking truss was proposed and presented two critical parameters (equivalent bending stiffness ratio η and the rotation stiffness ratio λ) for the design of the rocking truss. The 1/3 scaled steel frame with rocking truss (SFRT) was tested and theoretical analysis results were validated by test results. The finite element model of SFRT was built following the boundary conditions of the test specimen. The numerical results are consistent with the experimental observations. The nonlinear static and dynamic analysis results of the SFRT were compared with those of the same steel frame (SF) with a bracing system. It was found that the rocking truss can reduce the maximum and residual drifts of the main frame as expected, and the plastic development of the RBS at the beam end in SFRT is less than that in SF with a bracing system. The parametric analysis of the rocking connections with different values of prestressing force was carried out. The results confirm that the rotational stiffness and load-bearing capacity of the rocking connection can be accurately evaluated by the theoretical force-displacement curve. The effect of the equivalent bending stiffness ratio η and the rotation stiffness ratio λ on the lateral deformation pattern of structures was discussed by theoretical analysis, and the design procedure of the rocking truss is recommended to satisfy the demand of uniform inter-story drift and self-centering capacity.

Experimental analysis of frost resistance and failure models in engineered cementitious composites with the integration of Yellow River sand

Abstract This study investigates the potential use of Yellow River sand (YRS) sourced from the lower reaches of the Yellow River in China as a sustainable and cost-effective substitute for quartz sand in engineered cementitious composites (ECCs). This region accumulates around 400 million tons of sand annually. The study evaluates the impact of different YRS replacement percentages (0, 25, 50, 75, and 100%) on mechanical and microstructure properties under freeze-thaw conditions, focusing on assessing the ECC durability during cooling cycles. The results show that YRS exhibits a smaller normal distribution of particle sizes compared to that of quartz sand and a 5.77 times greater specific surface area, affecting the ECC particle size distribution. After 300 cooling cycles, the R25 group maintains 97.5% of the initial mass and 79.4% of flexural strength, indicating superior durability. The R25 group also demonstrates a minimal decrease of 11.5% in equivalent bending strength, reaching a level of 104.4% compared to R0. The R25 group’s porosity is 30.80%, with an average pore size of 20.47 mm, showing 1.3% and 6.7% decreases compared to the R0 group. Additionally, this study establishes a failure progression equation using the Weibull probability distribution model, with calculated values closely aligning with measured values. Overall, this study recommends using YRS as a sustainable ECC material.

Improved calculation method for corresponding characteristics of foundation pit excavation on the diaphragm wall

The excavation of the pit causes displacement of the surrounding soil, and the excessive deformation causes damage to the existing support structure, which in turn affects the safety of the pit. Reasonable calculation of structural deformation and internal force is crucial for design and construction. Most of the existing theoretical methods simplify the diaphragm wall(DW) as an Euler-Bernoulli beam acting on the Winkler foundation, consider beam-soil interaction, and simplify the soil as isotropic and continuous. However, the shear effects due to the differential deformation of the structure, the unloading stresses acting on the structure due to foundation excavation, and the discontinuous nature of the multilayered soil are neglected. In this paper, an improved analysis method is proposed based on the elastic foundation beam theory. The DW is simplified as a Timoshenko beam and the foundation is simplified as a Vlasov two-parameter model, and a proposed model considering the shear effect of the DW and the interaction of adjacent springs is established, and the proposed method for the deformation and internal force of the DW is obtained by the finite-difference method. The correctness and applicability of the proposed method are verified by numerical simulation and field monitoring data. The effects of equivalent bending stiffness, equivalent shear stiffness, soil elastic modulus, and excavation depth on the deformation and internal force of the DW were further analyzed. The results show that the proposed method can accurately solve the deformation and internal force of the DW, and the maximum errors between the proposed method and the numerical simulation results are only 4.5 % and 1.3 %, respectively. The equivalent bending stiffness of the DW and the elastic modulus of the soil have more significant effects on the horizontal deformation and internal force. The excavation depth is more sensitive to the deformation of the DW, and there is an exponential decay trend between the two. When the equivalent shear and bending stiffnesses reach 6.8×107 kN·m2 and 2.9×107 kN/m, the effect on the horizontal deformation is no longer obvious. The proposed method in this paper can accurately calculate the internal force and deformation of the DW.

Static performances of scissor structures

Scissor structures have very promising applications and have attracted much attention due to their novel characteristics of being deployable and foldable. However, most of the current studies have focused on its geometric and kinematic characteristics, while neglecting the analysis of their mechanical properties as load-bearing structures. In this study, general formulas for calculating the internal forces and equivalent bending stiffness of scissor structures are derived and verified using finite element method. The results show that the analytical and numerical simulation solutions are in good agreement and the expansion angle is the main factor affecting the static performances of the scissor structures. In particular, the extreme increase in axial force at expansion angles above 60°will lead to instability problems. Therefore, special attention should be paid when determining the expansion angle at or above 60°. Herein, the bending moment dominates the internal forces and the stresses generated by the bending moment exceed the axial forces by a factor of about 10–30. This study is expected to contribute greatly to scissor structure design by quantitatively evaluating preliminary design, exploring the impact of various parameters, optimizing design, and understanding limitations and potential issues.

Aeroelastic analysis of a very large wind turbine in various atmospheric stability conditions

With the growing trend towards larger wind turbine rotor diameters, the impact of wind shear on rotor performance and loads becomes increasingly significant. Atmospheric stability strongly influences wind shear, leading to higher wind shear under stable atmospheric conditions. In this study, the aeroelastic performance of the IEA 22 MW rotor is assessed under inflow conditions generated by different methods. Inflow conditions were generated using turbulence models specified in the IEC Standards and also by Large Eddy simulations. Standalone OpenFAST simulations were conducted with the respective inflow conditions.It was found that at rated and above-rated wind speeds, the time-averaged wind turbine design loads were higher in stable atmospheric conditions, in comparison to the IEC NTM inflow conditions, while the opposite held for below-rated wind speeds. Specifically, the time-averaged root flapwise bending moment and rotor thrust were found to be higher by up to 7% in stable atmospheres. However, maximum design and fatigue loads were considerably higher in the IEC NTM case due to elevated turbulence levels. Compared to the IEC NTM case, the damage equivalent root flapwise bending moment was found to be 30% to 70% lower in the different scenarios.

Ribbed elastic metasurface with lateral scalability for flexural wave manipulation

With the goal of engineering applications, the scalability and structural portability of elastic metasurfaces have become significant and attractive. However, most existing elastic metasurfaces cannot be arbitrarily scalable, which will inevitably limit their applications. Here, a novel conceptual design of ribbed elastic metasurfaces (REMs) is proposed for anomalous wavefront manipulation of flexural waves in thin plates. Ribbed subunits, as functional subunits of REMs, are characterized by simple constructions. The phase shift mechanism for the flexural waves across the ribbed subunits is revealed by analyzing the lowest dispersion bands. The analytical model for ribbed subunits is also established based on the equivalent orthotropic plate theory and transfer matrix method (TMM) to accurately predict the phase shift and amplitude of the transmitted waves. In addition, lateral scalability can be observed when the ratio of rib width to subunit width is constant, owing to the constant equivalent bending stiffness. By considering the lateral scalability of subunits, the continuous adjustment of the refraction angle can be perfectly achieved. The deflecting and focusing functionalities of the designed REMs are numerically and experimentally demonstrated. Both simulated and experimental results are consistent with theoretical results. Since ribbed plates are typical load-bearing structures in engineering, our design has broad application prospects, including but not limited to vibration control, energy harvesting, and structural health monitoring.

Comparative Analysis and Unified Derivation of Reissner’s Equations For 2D Bending of Thick Plates and Timoshenko’s Equations for Bending of Beams

Currently, calculations of flexural deformations of lithospheric plates are carried out on the basis of Kirchhoff’s theory of bending of thin plates formulated about 170 years ago. The paper examines the possibility of refining these calculations based on the theory of bending of thick plates by Timoshenko and Reissner. A new unified derivation is presented of the Timoshenko equations for 2D bending of beams and the Reissner equations for bending of slabs by direct transformation of the general elasticity equations with a simple approximate replacement of power cubic functions with effective linear ones. This derivation offers a simpler and more detailed understanding of the difference between the equations and the meaning of the simplifications made in these theories. By comparing the analytical solutions of the Timoshenko and Reissner equations with the existing test analytical solutions of the exact elasticity equations, quantitative estimates of the accuracy of these theories are presented.