Beam Bending Theory Research Articles

Stiffness estimation of planar spiral spring based on Gaussian process regression

Planar spiral spring is important for the dimensional miniaturisation of motor-based elastic actuators. However, when the stiffness calculation of the spring arm is based on simple beam bending theory, the results possess substantial errors compared with the stiffness obtained from finite-element analysis (FEA). It deems that the errors arise from the spiral length term in the calculation formula. Two Gaussian process regression models are trained to amend this term in the stiffness calculation of spring arm and complete spring. For the former, 216 spring arms’ data sets, including different spiral radiuses, pitches, wrap angles and the stiffness from FEA, are employed for training. The latter engages 180 double-arm springs’ data sets, including widths instead of wrap angles. The simulation of five spring arms and five planar spiral springs with arbitrary dimensional parameters verifies that the absolute values of errors between the predicted stiffness and the stiffness from FEA are reduced to be less than 0.5% and 2.8%, respectively. A planar spiral spring for a powered ankle–foot prosthesis is designed and manufactured to verify further, of which the predicted value possesses a 3.25% error compared with the measured stiffness. Therefore, the amendment based on the prediction of trained models is available.

On the Modeling of Ship Stiffened Panels Subjected to Uniform Pressure Loads

Stiffened panels constitute structural assemblies of the entire ship hull, i.e., double bottom, side shell, deck areas, etc. Prescriptive dimensioning of the stiffeners (web thickness and height and flange thickness and breadth) is solely based on the application of beam bending theories. This work is divided into two parts. The first part involves the assessment of the structural response of one-way (single-bay) stiffened panels under uniform pressure. The objective is to evaluate the effectiveness of alternative approaches in obtaining accurate secondary stress fields. Both state-of-the-art analytical solutions (Paik, Schade, CSR, Miller) and numerical calculation tools (finite element analysis (FEA)) are employed and compared for this purpose. When it comes to cross-stiffened panels, numerical methods are usually used within the design process which is time demanding. The second part of this work focuses on the development of a fast, yet effective, prescriptive approach. This approach will allow the dimensioning of the longitudinal stiffeners by considering the secondary stress field. Combining finite element analysis and the Euler–Bernoulli bending theory, the effect of the transverse stiffeners to the longitudinal stiffeners is examined in order to estimate the type of support on the boundaries of the transverse stiffeners. Determining the type of support, will make it possible to apply the classical formula of bending stress instead of using finite element analysis, thus limiting the computational cost. Preliminary calculations show that most of the examined cases may be treated as fully clamped beams subjected to uniform pressure.

Buckling Electrothermal NEMS Actuators: Analytic Design for Very Slender Beams

Analytic approximations are presented for the response of buckling-mode electrothermal actuators with very slender beams with a width-to-length ratio of W/L≤0.001 of the type found in nanoelectromechanical systems (NEMS). The results are found as closed-form solutions to the Euler beam bending theory rather than by an iterative numerical solution or a time-consuming finite element analysis. Expressions for transverse deflections and stiffness are presented for actuators with the common raised cosine and chevron pre-buckled shapes. The approximations are valid when the effects of bending dominate over those of axial compression. A few higher-order approximations are also presented for less slender beams with 0.001≤W/L≤0.01.

Improvement of the coiling stress calculation model for a mandrel-sleeve-coil

The steel rolling process employs a coiling-uncoiling process in which a steel sheet is wound and unwound in a coil shape using a coiler to efficiently produce a long steel sheet with a constant thickness. As front and rear tension is required when the steel sheet enters and exits the rolling mill, the coiler introduces tension in the steel sheet through the control of the rotational speed. As the coil is produced, coiling tension accumulates, and pressure is applied to the inside of the coil. Finite element analysis and stress calculation analysis were derived from previous studies to prevent such pressure increases in the sleeves and coils. However, the radial and circumferential stresses at arbitrary positions inside the coil cannot be accurately determined by considering without the stresses’ difference in the thickness direction based on the assumption that the coil’s thickness is thin. In this study, an analytical model that can accurately calculate the sleeve and coil stress during elastic deformation was established by improving the internal circumferential stress generated when the steel sheet is bent into a coil and the radial stress equation associated with the beam bending theory. In addition, by comparing the finite element analysis model results reflecting the same coiling condition, this model’s validity was verified by confirming the consistency of the results.

Numerical evaluation for the effective slab width of continuous composite beams

Basic beam bending theory and the effective width concept are commonly used in the design and analysis of steel–concrete composite beams to evaluate deflections, stresses, and strengths. The shear-lag effect is partially accountedfor by adopting a reduced “effective” slab width rather than the actual slab width. Besides the precise numerical values obtained via numerical analysis of a composite beam, Code provisions should provide a simplified practical techniquefor calculating effective slab width that is both simple and precise. Current design codes propose effective width values that are essentially a function of the beam span, ignoring the effects of other essential factors. Finite element analysis of continuous composite beams is a complex task, as large numbers of contacts and meshing elements are required to catch all the physical phenomena; this leads to potential convergence difficulties using a general static analysis. In this paper, several 3D numerical simulations of a composite continuous beam are conducted using the RIKS method available in ABAQUS. Comparisons to available experimental results are used to validate the proposed FE model. Then, the so-validated numerical model is utilized to perform a parametrical study to assess the effect of the longitudinal reinforcement at the hogging moment region and the influence of the concrete slab width on the effective width evaluation. Finally, the proposed finite element model and EC4 specification results are compared to draft a useable design guide.

Consistent Euler – Bernoulli beam theories in statics for classical and explicit gradient elasticities

The Euler – Bernoulli beam bending theory in elementary (engineering) mechanics uses two fundamental assumptions, first that the material behaviour is isotropic elastic and secondly that plane cross sections remain plane, rigid and perpendicular to the beam axis. It is well – known, that this theory suffers from the inconsistency that, e.g., the shear strain is always vanishing, whereas the shear stress is supposed to be not vanishing. In the present paper we remove this inconsistency in engineering mechanics by postulating the material response to be a limiting case of anisotropic elasticity. We show that all results within the “inconsistent” isotropic elasticity remain valid. Then, the anisotropic elasticity approach is extended to model bending of Euler – Bernoulli beam in explicit gradient elasticity. Especially, attention is focused on deriving distributions of stress components and discussing limiting responses in dependence of a material length parameter inherent in the elasticity law.

Coupled vortex-induced modeling for spatially large-curved beam with elastic support

The existing studies on mathematical modeling for the forced vibration response of naturally curved beams under elastic boundary conditions mainly focus on slightly curved structures. This work proposes the mathematical model for naturally curved beam with a large curvature under the elastic boundary. Under the assumption of small deformations, the six degrees of freedom are reduced to four degrees of freedom through spatial bending beam theory and differential geometry. The dynamics governing equations of the naturally curved beam are derived using Hamilton’s principle and Frenet–Serret formula. An improved Van der Pol model is introduced to investigate the vortex induced vibration (VIV) or vortex induced divergence (VID) response. Hammer test, wind tunnel experiment and finite element method validate the proposed model for spatially curved helical beam. It is found that under the fixed boundary condition, transposition occurs in the first two coupled modes in the normal direction of the helical beam. The errors of the first two nature frequencies are 0.016% and 0.088%, respectively. The modal analysis under the elastic boundary shows that the first-order decoupled modes in each direction will “disappear” with the variation of stiffness. Under the boundary condition of small stiffness, the first two modes of the normal direction change abruptly at the subtended angle of 76∘∼90∘. This abrupt change produces negative damping and leads to structural unstability. When the first two frequencies are close to each other, the water velocity region of VIV or VID response will intersect and the orthogonal modes in normal direction should be calculated coupled. In this case, the amplitude of the first order mode in the normal direction will always be greater than that of the second order mode. The maximum amplitude in u direction can reach 0.014 m. The obtained results provide useful information for further research on the fluid-induced vibration response of natural large-curved beam .

Nonlinear mechanics of a thin-walled honeycomb with zero Poisson’s ratio

The nonlinear, in-plane mechanics of a thin-walled honeycomb with zero Poisson’s ratio under large deformation is investigated in this paper. A theoretical method for calculating in-plane tensile modulus, modified factors of linear constitutive relations of the honeycomb structures with zero Poisson’s ratio is proposed based on the theory of Euler-Bernoulli beam and the bending theory of beam in large deflection, and a finite element simulation is given to validate. In addition, parametric analysis for revealing the impacts of geometrical configurations and material parameters on in-plane mechanical properties of the honeycombs have been studied systematically. These findings suggest that geometric and/or material parameters provide different contributions to the effective mechanical properties and lead to a separate design for the in-plane mechanical properties. After that, the effects of geometric and/or material nonlinearities on mechanical properties of the honeycomb structures with zero Poisson’s ratio are revealed by considering the dimensionless tangent stiffness of the honeycombs.

Interfacial Competitive Debonding of a Bilayer Elastic Film on a Rigid Substrate

Abstract Different from the system of a single-layer elastic film on a rigid substrate, the debonding interface is difficult to determine in a bilayer or multilayer film-substrate system. A peeling model of a bilayer elastic film on a rigid substrate is established in the present paper, in order to predict which interface debonding occurs first. The interfacial competitive debonding mechanism is theoretically analyzed with the help of the beam bending theory. A criterion of which interface debonding occurs first is proposed. It is found that the interfacial debonding path is mainly controlled by five dimensionless parameters, i.e., the strength ratio and the critical separation distance ratio of the upper and lower interfaces, the Young's modulus ratio and the thickness ratio of the upper and lower films, and the possible initial cantilever length for ease of loading. The corresponding competitive debonding map is well obtained. From the map, which interface debonding occurs first can be easily predicted. It is interesting to find that the interfacial debonding path can be well tuned by any one of the five parameters. The results of the finite element calculation further confirm the theoretical predictions. The present work can not only provide a theoretical method to determine the interfacial debonding path but also be helpful for the optimal design of multilayer film-substrate systems in practical applications.

Model tests on horizontal load-bearing and deformation resistance capacities of pile-bucket foundation in soft seabed soil

Large-diameter pile-bucket foundations are a new type of foundation used for offshore wind turbines. In this study, model tests were performed on pile-bucket foundations to investigate their load-bearing and deformation resistance capacities. The strains of the pile and bucket and the horizontal displacements at the loading point were measured using strain gauges and displacement transducers, respectively. The variations in moments of the pile and bucket with depth were calculated and fitted into polynomial functions of sixth to eighth orders. Based on the moment polynomials, the distributions of the horizontal displacements and soil reaction of the pile and bucket were determined according to the beam bending theory. Improvements in the load-bearing and deformation resistance capacities were quantitatively determined by comparing the test results for the pile-bucket foundation with those of the monopile foundation. The initial stiffness and ultimate horizontal bearing capacity of the pile-bucket foundation were 50% and 20%, respectively, higher than those of the monopile foundation. For sections within the bearing depth of the bucket, the ultimate soil reaction force of the pile-bucket foundation was 20 times that of the monopile foundation. However, a 10%–20% improvement in the ultimate soil reaction was observed for deeper sections.

Statistical characterization of non-linear microscopic mechanical deformation through randomly oriented fibrous porous transport layers for advanced electrochemical energy systems

A robust compression model based on the unit-cell beam bending theory is presented to predict the mechanical behavior of stochastically generated fibrous porous transport layers (PTLs) at various clamping pressures. For this purpose, three-dimensional PTLs were constructed by piling random paper-type carbon-fiber structures. A representative elementary volume was determined based on the relative porosity gradient errors with a 95% confidence level for statistical analyses. Subsequently, a mechanical compression model based on the beam bending theory was developed to determine the microscale deformation characteristics of the PTLs for electrochemical energy systems. Based on the beam bending theory, carbon fibers are modeled as beams, and the bending of fibers is considered to be the main contributor to deformation of the PTLs. The numerical model shows good agreement with published experimental data in literature, i.e., a nonlinear stress–strain relationship. Next, the model was applied to feature bulk and local mechanical variations of the PTLs as functions of the number of carbon-fiber layers, porosity, polytetrafluoroethylene (PTFE) loading, and external clamping pressure. It was found that the addition of binder/PTFE into fibrous substrates results in the decreased porosity and increased mechanical strength of the PTLs. The detailed three-dimensional microscale deformation simulations revealed that the statistical mean strain of the PTLs was exponentially proportional to the porosity in the range 0.7–0.9 and decreased on addition of PTFE in the fibrous carbon substrate at a stack clamping pressure of 1 MPa. Moreover, the statistically estimated local strain distribution along the in- and through-planes of the PTLs indicated that the microscopic local deformation was approximately uniform through the PTLs. This modeling study can be utilized to understand the mechanical behavior of heterogeneous PTLs during external compression for advanced electrochemical systems.

Flexural analysis of thin-walled laminated composite beams with arbitrary open sections and shear influence

Analytic model for the flexural analysis of thin-walled laminated composite beams with arbitrary open sections and shear influence is presented. Model is based on general Vlasov’s theory of thin-walled beams of open sections, as well as on Timoshenko’s beam bending theory. The shear strain in the beam middle surface is taken into analysis, i.e. the out-of-plane warping of the cross-section due to shear. Analytic expressions for the stresses and displacements with the shear influence are derived in closed analytic form. Beams subjected to general transverse load reduced to principal pole are considered. The influence of shear deformations and fiber angle on stress and displacement distributions is investigated. Beams with different height-to-span ratios and different boundary conditions are analyzed. The accuracy of the presented model is verified by comparing the results with the finite element solutions utilizing shell elements, as well as with the results from the available literature. Very good agreement between analytical and numerical results is obtained.

Nonuniform bending theory of hyperelastic beams in finite elasticity

This paper deals with the equilibrium problem of slender beams inflexed under variable curvature in the framework of fully nonlinear elasticity. For the specific case of uniform flexion, the authors have recently proposed a mathematical model. In that analysis, the complete three-dimensional kinematics of the beam is taken into account and both deformations and displacements are considered large. In the present paper, the kinematics of the aforementioned model has been reformulated taking into account beams under variable curvature. Subsequently, focusing on the local determination of the curvature, new equilibrium conditions on cross sections are introduced in the mathematical formulation. The governing equations take the form of a coupled system of three equations in integral form, which is solved numerically through an iterative procedure. Therefore, for the generic class of hyperelastic and isotropic materials, explicit formulae for the displacement field, the stretches and stresses in every point of the beam, following both Lagrangian and Eulerian descriptions, are derived. The analysis allows studying a very wide class of equilibrium problems for nonlinear beams under different restraint conditions and subject to generic external load systems. By way of example, the Euler beam has been considered and the formulae obtained have been specialized for a specific neoprene rubber material, the constitutive constants of which have been determined experimentally. The shapes assumed by the beam as the load multiplier increases are shown through some graphs. The distributions of stretches and Cauchy stresses are plotted for the most stressed cross section. Some comparisons are made using a FE code. In addition, the accuracy of the obtained solution is estimated by evaluating a posteriori that the equilibrium equations are locally satisfied.

Higher-order beam bending theory for static, free vibration, and buckling analysis of thin-walled rectangular hollow section beams

In higher-order beam theories, cross-sectional deformations causing complex responses of thin-walled beams are considered as additional degrees of freedom. To fully capture their bending responses, enriched sectional modes departing from Vlasov’s assumptions have been utilized in recent studies. However, due to these bending-related modes, no available higher-order beam bending theory has established explicit stress-generalized force relations that are fully consistent with those by the classical beam theories and earlier studies based on Vlasov’s assumptions. If they are available, physical significance of the bending-related generalized forces can be readily understood. In addition, equilibrium conditions at a joint of multiple thin-walled beams can be explicitly derived. Here, we newly propose a higher-order beam bending theory that not only includes as many bending-related sectional modes as desired, but also provides the desired explicit stress-generalized force relations. To this end, we establish a recursive analysis method that derives hierarchical bending-related sectional modes. We show that this method can yield certain relations among the sectional mode shapes, which are critical in establishing the desired explicit relations. The validity of the present theory is confirmed by calculating the static, free vibration, and buckling responses of several thin-walled rectangular hollow section beams.

Measurement of the Elastic Modulus and Residual Stress of Thermal Barrier Coatings Using a Digital Image Correlation Technique

This paper presents an experimental study on simultaneously measuring the elastic modulus and residual stress of a thermal barrier coating (TBC) after different isothermal heat treatments. The elastic modulus and residual stress of TBCs were theoretically analyzed based on composite beam bending theory. Thereafter, an experimental setup was established combining the 3D digital image correlation method with the bending test to obtain the curvature changes in the TBC sample. Finally, the elastic modulus and residual stress of the ceramic layer with different isothermal heat treatments were obtained. The results show that the elastic modulus of the ceramic layer measured under compression is greater than that under tension, and the elastic modulus of the ceramic layer increases first and then tends to be stable as the heat treatment time increases. In addition, the residual stress of the TBCs ceramic layer quickly changes from compressive stress to tensile stress with heat treatment, and the tensile stress increases with the increase in thermal exposure time. Furthermore, the reasons for the change tendency were analyzed according to the variation in porosity and microstructures by processing the scanning electron microscope (SEM) figures. The results demonstrate that simultaneously determining the elastic modulus and residual stress of TBC based on combining the 3D digital image correlation method with the bending test is effective and reliable.

Bending moment analysis of super large ore carrier based on graphical interface development

In order to improve the safety of shipping process, this paper designs a graphical user interface for calculating the static water bending moment of super large ore earner. Due to the complexity of the actual loading conditions, the long hull girder and the mobility and sinking of cargo, the irregular distribution of gravity and buoyancy along the direction of the captain of the VLCC is prominent. In the actual loading process of ships, compartment loading is adopted in many cases, and the position where the maximum shear force and bending moment appear is generally not in the corresponding position. Traditional ship strength analysis methods can be divided into two kinds. One is to establish the ship’s finite element model to calculate the ship’s strength. The other is to calculate the ship’s strength according to the simple beam bending theory. In this paper, a 360m VALE MAX ore carrier is taken as an example to calculate the local bending moment of the ship at a specific position by using QT software. Compared with the ship loading manual, the error of the program is small, and the program has guiding significance in the actual shipping process.

The influence of stiffener geometry on flexural properties of 3D printed polylactic acid (PLA) beams

We used finite element analyses (FEA) on Abaqus to study flexural properties of additive manufactured beams using polylactic acid (PLA) polymer. Experimental stress–strain data from flexural testing are used to define elastic–plastic properties of the material in the computation software. The flexural experiments are used to validate the FEA approach suggested. The method provides good results of deflection and stress with errors well below 10% in most of the cases. Therefore, by using the proposed approach, costs related to repeated experimental works can be avoided. In addition, the flexural rigidities of the additive manufactured beams are studied. Five different beam stiffener designs (diamond, honeycomb, square, triangular and wiggle) are studied based on beam bending theory. The force–deflection data from the flexural tests are used to determine the area moments of inertia of the beams. The honeycomb stiffener showed the highest force–deflection behaviour that led to the highest calculated area moment of inertia. However, with the lowest force–deflection behaviour, the square stiffener had the lowest calculated area moment of inertia.

Mechanical Analysis of Shape Memory Alloy Beams under Asymmetric Bending

Based on the bending theory of beams, combining with the constitutive relationship of shape memory alloy materials, the asymmetric bending of shape memory alloy beams under concentrated load was studied. Considering tension-compression asymmetry on both sides of the tension and compression in the process of bending, tension-compression asymmetry coefficient was introduced, by using the step-by-step method. The stress distribution, neutral axis displacement, curvature and phase boundary of the beam sections of the I-shaped cross-section shape memory alloy beam were analyzed. The results show that under the same load, the maximum offset of the neutral axis displacement increases with the increase of tension-compression asymmetry coefficient. Under the same tension-compression asymmetry coefficient, the displacement and curvature of the neutral axis increased with the increase of load; as tension-compression asymmetry coefficient increased, the proportion of mixed phase increased gradually, the proportion of martensite phase decreased, and the asymmetry of phase boundary became more obvious. Under the same conditions, the rectangular section is more suitable. The cross section of I shaped cross section was prone to occur phase transformation.

Numerical-experimental approach to assess the deflection behaviour of carbon fibre fishing rod

Deflection of a commercial hollow type fishing rod is investigated in this research. The fishing rod of interest is of Abu Garcia brand, made of carbon fibre. Analytical, experimental and numerical approach were performed to assess the bending of the rod under maximum 1.25 kg load located at the tip which represents the rod in pulling. The deflection is defined by longitudinal displacement of each point of guide rings located along the fishing rod. A mathematical model based on basic bending theory of cantilevered beam is formulated to obtain the deflection and then experiment was carried out by immobilizing one end horizontally on the wall and applying at tip force. The final position of the rings was marked, and the displacements were recorded. Finite element model was developed as an isotropic and elastic material via ANSYS software. Results from all approach were compared and it was found that analytical and numerical able to predict a similar trend of bending behaviour. However, large discrepancies were observed in the experimental curves due to several reasons.

A modified beam theory for bending of eccentrically supported beams

According to the conventional beam theories, the bending analysis of a beam under transverse loading depends on support conditions in addition to the other parameters. The conventional beam theories assume that supports are solely placed at the mid-plane of the beam. However, in practice, the beams often are supported at the point different from their centers. In this study, a modified beam theory is presented for the static bending analysis of an eccentrically simply-supported beam under transverse uniform loading by the application of the MacLaurin series. This study presents a theoretical approach to the analysis of eccentrically supported beams. To overcome the limitation of the Euler beam theory mainly applicable to concentrically supported beams both the transverse deflection and the axial deflection are considered in order to deal with eccentrically supported conditions. The effects of eccentric supports on the flexural rigidity of the beam have been investigated. Analytic equations derived were used to investigate the effect of varying support positions through the thickness on bending analysis of beams under transverse loading. The findings revealed that the flexural rigidity of beams is significantly influenced by eccentric pin–pin support. The accuracy of the equations was verified by comparing the results obtained with the finite element solutions. The results show that the derived equations are not only accurate but also simple in predicting the bending of beams. Communicated by Francesco Tornabene