Beam Bending Theory Research Articles

Incremental Bending of Three-Dimensional Free Form Metal Plates Using Minimum Energy Principle and Model-Less Control

Bending 3D free form metal plates is a common process used in many heavy industries such as shipbuilding. The traditional method is the so-called line heating method, which is not only labor intensive but also inefficient and error-prone. This paper presents a new incremental bending method based on minimum energy principle and model-less control. First, the sheet metal is discretized into a number of strips connected through virtual springs. Next, by applying the minimum energy principle, the punching and supporting points are calculated for the strip. Then, the bended shape of the strip is computed based on the beam bending theory. This process is continued until the final shape is reached. To compensate the bending error, the computer vision-based model-less control is applied. The computer vision detects the bending error based on which additional bending steps are calculated. The new method is tested in a custom build incremental bending machine. Different metal plates are formed. For a metal plate of 1000 × 800 × 5 mm3, the average bending error is less than 3 mm. In comparison with the existing methods, the new method has a number of advantages, including simple, fast, and highly energy efficient.

Single crystal diamond cantilever for micro-electromechanical systems

A single crystal diamond (SCD) cantilever has been fabricated on HPHT diamond substrate by using ion implantation and patterning technique. Firstly, the carbon ions were vertically implanted into diamond substrate to create a non-diamond layer. Secondly, the diamond substrate was treated with photolithography and metal evaporation techniques to fabricate the desired cantilever pattern. Thirdly, a dry etching technique was used to selectively remove the defined regions. Then an electrochemical etching method was used to etch off non-diamond layer from the substrate. Furthermore, the mechanical properties of the suspended SCD cantilever have been investigated by atomic force microscopy. A Young's modulus of 907GPa was estimated using the beam bending theory. At the final stage, an electrostatic actuator of SCD cantilever as a vertical switch device was realized.

Bending of thin-walled beams of open section with influence of shear, part I: Theory

This two-part contribution presents a novel theory of bending of thin-walled beams with influence of shear (TBTS). The theory is based on the Vlasov’s general beam theory as well as on the Timoshenko’s beam bending theory. The theory is valid for general thin-walled open beam cross-sections. Part I is devoted to the theoretical developments and part II discusses analytical and obtained numerical results. The theory is based on a kinematics assuming that the cross-section maintains its shape and including three independent warping parameters due to shear. Poison’s effect is ignored, as well as warping constrains due to shear (as it known, those effect have small and for engineering praxis neglected influence on the stresses and displacements). Closed-form analytical results are obtained for three-dimensional expressions of the normal and shear stresses. Under general transverse loads, reduced to the cross-section principal pole, the beam will be subjected to bending with influence of shear and in addition to torsion due to shear with respect to the cross-section principal pole and to tension/compression due to shear, in the case of non-symmetrical cross-sections. The beam will be subjected to bending with influence of shear (i) in the plane of symmetry under the loads in that plane and in addition to tension/compression due to shear, (ii) in the plane through the principal pole orthogonal to the plane of symmetry under the loads in that plane and in addition to torsion with respect to the principal pole, in the case of the mono-symmetrical cross-sections. The beam will be subjected to bending with influence of shear in the principal planes, in the case of the bi-symmetrical cross-sections. The principal cross-section axes as well as the principal pole are defined by the classical Vlasov’s theory of thin-walled beams of open section. The analytical and numerical analyses presented in part II include comparisons with the classical beam theory, Euler-Bernoulli’ theory (EBBT) as well as comparisons with the finite element method (FEM).

A framework for estimating residual stress profile in seam welded pipe and vessel components Part II: Outside of weld region

Part I of this study has identified two key parameters (r/t ratio and characteristic heat input Qˆ) that dominate important through-thickness residual stress distribution characteristics, with which a residual stress profile estimation scheme has been developed for weld region, i.e., at weld centerline and weld toe. In Part II, we present an analytical model for achieving residual stress profile estimation for through-thickness sections away from the weld region in seam welded pipe and vessel components. A curved beam bending theory based model is analytically constructed through an assembly of two parts: One is weld fusion zone region and the other is the rest of the component section along circumferential direction. The final assembly of the two parts leads to a closed form solution to both axial (longitudinal) and hoop (transverse) residual stress components as a function of circumferential angular position away from weld toe. The effectiveness of the full-field residual stress estimation scheme is demonstrated by comparing with finite element modeling results over a broad range of weld geometries and welding conditions. The present development should provide a consistent and effective means for estimating through-thickness residual stress profile as a continuous function of pipe geometry and welding heat input.

A framework for estimating residual stress profile in seam-welded pipe and vessel components part I: Weld region

A recent comprehensive investigation into residual stress distributions in pipe and vessel longitudinal seam welds is presented in this paper, covering component wall thickness from 1/4″ (6.35 mm) to 10″ (254 mm), component radius to wall thickness ratio from 2 to 20, and linear welding heat input from low (50 J/mm) to high (6000 J/mm). Through the use of a residual stress decomposition technique, two key parameters that govern through-thickness residual stress distributions in terms of their membrane and bending content have been identified. One is component radius to wall thickness ratio (r/t) and the other is a characteristic heat input density (Q^) having a unit of J/mm3. With these two parameters, a unified functional form for estimating through-thickness residual stress profile in seam welded components is proposed in this paper (Part I) for supporting fitness for service assessment for crack-like flaws in weld region. A curved beam bending theory based model is introduced in Part II as a means of analytically describing through-thickness residual stress profile as a function of circumferential position away from the weld region until residual stresses become zero. The effectiveness of this proposed framework for achieving residual stress profile estimation within weld region (Part I) for longitudinal seam welds in pressure vessel and piping components has been confirmed by finite element residual stress analysis results on a large number of component configurations and different welding conditions.

Theory of micropolar gyroelastic continua

This paper is devoted to the dynamic modeling of micropolar gyroelastic continua and explores some of the modeling and analysis issues related to them. It can be considered as an extension of the previous studies on equivalent continuum modeling of truss structures with or without angular momentum devices. Assuming unrestricted or large attitude changes for the axes of the gyros and utilizing the micropolar theory of elasticity, the energy expressions and equations of motion for undamped micropolar gyroelastic continua are derived. Whereas the micropolar gyroelastic continuum model with extra coefficients and degrees of freedom is primarily developed to account for the asymmetric stress–strain analysis in the gyroelastic continua, it also proves to be beneficial for a more comprehensive representation of the actual gyroelastic structure. The dynamic equations of the general gyroelastic continua are reduced to the case of one-dimensional gyroelastic beams. Simplified micropolar beam torsion and bending theories are used to derive the governing dynamic equations of micropolar gyroelastic beams from Hamilton’s principle. A finite element model corresponding to the micropolar gyrobeams is built in MATLAB\({^{\circledR}}\) and is used in numerical examples to study the spectral and modal behavior of simply supported micropolar gyroelastic beams.

An Application of the Inventive Design Methodology to Thermal Systems: Improvement of the Insertion of Thermocouples in a Combustor

Although technical innovation is increasingly essential to meet the new challenges in the field of thermal engineering, it is rare the application of means supporting systematic innovation commonly used in other areas. This paper aims stimulating the use of the Inventive Design methodology in the field of thermal engineering. It discusses the improvement of a thermocouple’s mounting arrangement in a research combustor. The original design was based solely on the designer’s creativity and understanding and the solution found to conciliate wall temperature measurement with the possibility and easiness of combustor disassembly for overhaul made sacrifices in both these design criteria. The paper explores a quantitative relation obtained from beam bending theory to improve the solution as to the later criterion. However, only through the application of the theory of inventive problem solving (TRIZ), specifically the contradiction matrix tool, was it possible to evolve beyond the current thermocouple arrangement paradigm and identify two new designs with potential for noticeable improvements in: reduction of measurement error, prevention of probe damage, and readiness in combustor servicing. This case study shows that Inventive Design is worth of being included in the toolbox of thermal equipment designers.

On the Finite Element Free Vibration Analysis of Delaminated Layered Beams: A New Assembly Technique

The dynamic analysis of flexible delaminated layered beams is revisited. Exploiting Boolean vectors, a novel assembly scheme is developed which can be used to enforce the continuity requirements at the edges of delamination region, leading to a delamination stiffness term. The proposed assembly technique can be used to form various beam configurations with through-width delaminations, irrespective of the formulation used to model each beam segment. The proposed assembly system and the Galerkin Finite Element Method (FEM) formulation are subsequently used to investigate the natural frequencies and modes of 2- and 3-layer beam configurations. Using the Euler-Bernoulli bending beam theory and free mode delamination, the governing differential equations are exploited and two beam finite elements are developed. The free bending vibration of three illustrative example problems, characterized by delamination zones of variable length, is investigated. The intact and defective beam natural frequencies and modes obtained from the proposed assembly/FEM beam formulations are presented along with the analytical results and those available in the literature.

Torsional warping eigenmodes including the effect of the secondary torsion moment on the deformations

In this paper the influence of torsional warping of cross-sections of twisted beams on their eigenvibrations is investigated, considering the secondary deformations due to the angle of twist. The investigation is based on the analogy of second-order bending beam theory, including the effect of the deformations due to the shear forces on the deformations, and the non-uniform torsion theory of thin-walled open and closed cross-sections, including the secondary torsion moment deformation effect. The solutions of the underlying differential equations are used for setting up the transfer matrix relations. The numerical investigation is focused on modal analysis of straight beams with and without consideration of warping of doubly symmetric open and closed cross-sections. The influence of the secondary torsion moment on the non-uniform torsional eigenfrequency is also studied and evaluated. The obtained results are compared with the ones calculated by standard solid and warping beam finite elements.

Error analysis of a twin-cantilever sensor for measuring external diameter

The error characteristics of the twin-cantilever sensor for measuring external diameter of tensile test piece were discussed. By using of beam bending theory and strain electric measurement theory, the origins of the errors of the twin-cantilever measuring system were analyzed, and a synthetic error formula of the system was established, which contains four components, the contact offset error, the tilt error, the Abbe error and the implicated error. A experimental system was designed, and the calibration and measurement experiments were carried out. The synthetic error formula was used to analyze the errors of the experimental system, and the error control method, including a set of error correction formula, was discussed. The analytical result shows that the resolution of the experimental system is smaller than 0.0002mm; through error correction, the absolute error limit of the experimental system can be determined as ±0.001mm.

Effect of thickness stretching and porosity on mechanical response of a functionally graded beams resting on elastic foundations

The novelty of this paper is the use of an efficient beam theory for bending, free vibration and buckling analysis of functionally graded material (FGM) beams on two-parameter elastic foundation. The present theory accounts for both shear deformation and thickness stretching effects by a parabolic variation of all displacements across the thickness, and satisfies the stress-free boundary conditions on the upper and lower surfaces of the beam without requiring any shear correction factor. Due to porosities, possibly occurring inside FGMs during fabrication, it is therefore necessary to consider the vibration, bending and buckling behaviors of beams having porosities in this work. The equation of motion for FGM beams is obtained through Hamilton’s principle. The closed form solutions are obtained by using Navier technique, and then fundamental frequencies are found by solving the results of eigenvalue problems. The validity of the present theory is investigated by comparing some of the present in literature. It can be concluded that the proposed theory is accurate and simple in solving the bending, free vibration and buckling behaviors of FGM sandwich beams.

The internal loads, moments, and stresses in rod-like particles in a low-speed, vertical axis mixer

A discrete element method (DEM) model is used to predict the internal load and moment distribution within rod-like particles in a low-speed, vertical axis mixer. The internal loads and moments are combined with small deformation beam bending theory to determine the internal stress distributions. Parametric studies using the model examined the influence of particle aspect ratio, blade rotational speed, and material properties. The spatial distributions of loads and moments, averaged over all particles and time steps, are symmetric about the particle center-plane with a maximum at the particle center-plane. In addition, the largest average maximum absolute principal stress is observed to occur along the particle circumference at the center-plane of the particle. These results indicate that particle failure is not only most likely to occur at the center-plane of the particle, but the failure will begin at the particle’s circumference. The largest average loads, moments, and maximum absolute principal stress increase with particle aspect ratio. The frequency distributions of maximum absolute principal stress at the high stress range are fit well with a Weibull distribution. Increasing blade speed, bed height and particle–particle friction coefficient generally lead to an increase in internal loads, moments, and stresses. The largest maximum absolute principal stresses occur at the base of the mixer and in front of the blades near the mixer circumference where the bed depth is the greatest.

A mechanical response of functionally graded nanoscale beam: an assessment of a refined nonlocal shear deformation theory beam theory

This paper presents a nonlocal shear deformation beam theory for bending, buckling, and vibration of functionally graded (FG) nanobeams using the nonlocal differential constitutive relations of Eringen. The developed theory account for higher-order variation of transverse shear strain through the depth of the nanobeam, and satisfy the stress-free boundary conditions on the top and bottom surfaces of the nanobeam. A shear correction factor, therefore, is not required. In addition, this nonlocal nanobeam model incorporates the length scale parameter which can capture the small scale effect and it has strong similarities with Euler–Bernoulli beam model in some aspects such as equations of motion, boundary conditions, and stress resultant expressions. The material properties of the FG nanobeam are assumed to vary in the thickness direction. The equations of motion are derived from Hamilton\'s principle. Analytical solutions are presented for a simply supported FG nanobeam, and the obtained results compare well with those predicted by the nonlocal Timoshenko beam theory.

Vibration of Single-Walled Carbon Nanotubes by Using Nonlocal Theory

The free bending vibration of single-walled carbon nanotubes (SWCNTs) is investigated in the present paper. A continuum approach based on nonlocal theory of beam bending is used for natural frequency computation. Analytical solutions of frequency equations are given for four types boundary conditions: clamped-free (C-F), simply-simply supported (S-S), clamped-simply supported (C-S) and clamped-clamped (C-C). The graphical representations of numerical results are shown for the first case of boundary conditions clamped end - free end.

Buckling analysis of thin-walled functionally graded sandwich box beams

Buckling analysis of thin-walled functionally graded (FG) sandwich box beams is investigated. Material properties of the beam are assumed to be graded through the wall thickness. The Euler-Bernoully beam theory for bending and the Vlasov theory for torsion are applied. The non-linear stability analysis is performed in framework of updated Lagrangian formulation. In order to insure the geometric potential of semitangental type for internal bending and torsion moments, the non-linear displacement field of thin-walled cross-section is adopted. Numerical results are obtained for FG sandwich box beams with simply–supported, clamped–free and clamped–clamped boundary conditions to investigate effects of the power-law index and skin-core-skin thickness ratios on the critical buckling loads and post-buckling responses. Numerical results show that the above-mentioned effects play very important role on the buckling analysis of sandwich box beams.

Remote monitoring of bi-axial loads on a lifting surface moving unsteadily in water

A system of measuring the bi-axial load on a lifting surface (blade) which is freely moving and operates submerged in water at the laboratory scale is described. A blade with a span of 500 mm, a chord of 60 mm and a thickness of 9 mm (15% of the chord) was employed and the lift/drag forces were measured using a bespoke strain-gauge based load cell located at the mid-span of the blade, measuring bending moments in two independent directions. The requirement to move freely dictated that the load cell was encapsulated within the blade, along with signal conditioning circuitry, power supply and a data logger with wireless transmission. Submerged operation in water resulted in very short transmission distances, meaning that data were recorded and subsequently transferred using an aerial placed close to the blade while it was stationary. Assumptions based on Euler–Bernoulli beam bending theory were used to infer the total load from measurements of the bending moment at the mid-span and example data from a freely moving aerofoil on a Darrieus-type tidal energy extraction device are presented. The novelty of this system lies in its combination of free movement, submerged operation and small scale.

Numerical Algorithm for Beam Residual Stress Identification

The goal of the paper is the numerical study of the original hysteresis computational material models and their application in beam's bending theory. The paper presents the chosen differential material models and approaches based on step by step solution respecting the elastic and plastic conditions. In the centre of authors´ interest is the numerical method study of the plastic zones identification or study of residual stresses distribution in the beam's cross-section. The results obtained from new MATLAB's programs are compared with FEM models in ADINA.

Buckling analysis of cold-formed steel channel-section beams at elevated temperatures

This paper presents a numerical investigation on the buckling behaviour of plasterboard protected CFS channel-section beams subjected to uniformly distributed loads when exposed to fire on its one side. The work involves three phases, namely heat transfer analysis, pre-buckling analysis and buckling analysis. The heat transfer analysis is accomplished using two-dimensional finite element analysis methods, from which the temperature fields of the channel-section beams are obtained. The pre-buckling analysis is completed using the Bernoulli bending theory of beams with considering the effects of temperature on strain and mechanical properties. The buckling analysis is performed using combined finite strip analysis and classical Fourier series solutions, in which the mechanical properties are considered to be temperature dependent. The results show that there are significant temperature variations in web, fire exposed flange and lip. Also, it is found that the buckling behaviour of the beam with temperature variation in its section is quite different from that of the beam with a constant uniform temperature in its section.

Numerical analysis on springing and whipping using fully-coupled FSI models

This paper focuses on ship springing and whipping analysis using a three-dimensional (3-D) Rankine panel method combined with a beam-element-based 1-D structural model and a shell-element-based 3-D structural model. In addition, slamming loads are considered by 2-D generalized Wagner model (GWM). The beam model is a classical idealization of a ship structure, which is based on Timoshenko beam theory for bending and Vlasov beam theory for non-uniform torsion. The 3-D model consists of beam and shell elements, and its motions are approximated by a few of lower modes. Different coupling schemes are applied to couple the 3-D panel method and the different structure models. Whereas the 1-D beam model is coupled in a Cartesian coordinate system, the 3-D finite element (FE) model is coupled in a generalized coordinate system. The difference in coordinate systems leads to different numerical implementations of coupling. Agreement and discrepancy between the coupled models are discussed regarding results for a 60m barge, a 6500 TEU containership, and an experimental model of a virtual 10,000 TEU containership. The bulkheads in the barge and the 6500 TEU containership are properly considered in beam modeling according to relative stiffness between the bulkheads and hulls. In linear responses to waves, good agreement is obtained between all the models. However, differences between the models are found in nonlinear springing and whipping responses.

Stress intensity factor solutions for adhesive-bonded lap-shear specimens of magnesium and steel sheets with and without kinked cracks for fatigue life estimations

In this paper, stress intensity factor solutions for adhesive-bonded lap-shear specimens of magnesium alloy AZ31 and hot-dip-galvanized (HDG) mild steel sheets with and without kinked cracks are investigated for fatigue life estimations. First, the kinked fatigue crack failure mode of the adhesive-bonded lap-shear specimens is briefly reviewed. Then, the analytical global J integral and effective stress intensity factor solutions for main cracks in lap-shear specimens of three dissimilar sheets under plane strain conditions are developed based on the beam bending theory. The global effective stress intensity factor solutions for the main cracks in the lap-shear specimens from the corresponding finite element analyses are then presented and validated by the analytical solutions. Next, the local stress intensity factor solutions for kinked cracks with the experimentally observed kink angle as functions of the kink length from the corresponding finite element analyses are presented and the computational solutions are also compared with the analytical solutions at small kink lengths. The results indicate that the computational local stress intensity factor solutions for kinked cracks approach to the analytical solutions as the kink length decreases to a small value and the kinked crack is under dominant mode I loading conditions. The computational results also indicate that the local stress intensity factor solutions at a small kink length of microstructural significance may be used as the stress intensity factor solutions for zero or near zero kink length for fatigue life estimations when the computational results are not available. The computational local stress intensity factor solutions are then adopted to estimate the fatigue lives of the lap-shear specimens based on a kinked crack growth model and available material constants for the Paris law. The fatigue life estimations are lower than the experimental results. However, the general trend of fatigue life estimations agrees with that of the experimental results.