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In this paper, the bending, buckling, and free vibration of functionally graded porous (FGP) beams are studied based on two beam theories (with or without considering thickness stretching, respectively). The effect of thickness stretching is obtained by comparing the results of the two theories. Two symmetrical distributions and one asymmetrical distribution of pores are considered. Both Young’s modulus and mass density of the FGP beams are in gradient variation in the thickness direction. The governing equations are constructed using Hamilton’s principle. The analytical solutions are obtained by Navier’s method. The effects of slenderness ratios, pore distribution, porosity and thickness stretching on FGP beams have been investigated. The results show that the inhomogeneity of FGP beams in the thickness direction is positively correlated with the effect of thickness stretching.

This study presents a quasi-two-dimensional higher-order shear deformation theory for stress and displacement analysis of isotropic, laminated, and sandwich beams subjected to concentrated load. The assumed displacement field includes the effect of transverse shear and normal deformations. The condition of zero transverse shear stresses on the upper and lower surface of beams is satisfied, hence the present formulation does not require the shear correction factor generally associated with first-order models. By applying the principle of virtual work, the governing equations and boundary conditions for laminated and sandwich beams are derived. Transverse shear and normal stresses are determined from the stress-equilibrium equations of elasticity theory. The transverse stresses obtained using this approach satisfy the continuity condition at layer interfaces and the stress boundary conditions at the external surfaces. The outcomes of the present theory are compared with those of third-order, and first-order shear deformation models, and the classical beam model. The results highlight the significant deviation in displacements and stresses of laminated and sandwich beams when normal strain is included in the model, as compared to the prediction made by lower-order models.

According to a phenomenological phase transition mechanism introduced quite recently, we propose a systematic examination of the polymorphic contraction–expansion effects observed for a broad class of morphologies with nonvanishing surface (line) tension. In our mean-field approach, we are looking for the line order parameter behavior as a function of a thermodynamic control parameter at a fixed number of entities consituting the morphology as a whole. We also wish to somehow invert the problem, and will be looking for some evolution going via a set of quasi-equilibrium states while fixing the control parameter. Both the situations considered are of interest for studying the so-called morphology diagrams as well as, to some extent, for describing selected model evolutions in a two-dimensional microworld, e.g., in nanoscale, where biomembranes and interfaces are presently of special attention.

In this paper we present the results of a dynamical nanotube bending study. Carbon nanotube structures under constant rate bending have been studied using classical molecular dynamics simulations method, with the inter-atomic forces described by Brenner's empirical model potential. We show that the tube diameter and the prevalent temperature play an important role in the outcome. Some of the results have been assessed by letting the most deformed part of the nanotube further relax by using a dynamical tight binding method. Significant difference was found between the empirical and the tight binding models in predicting the structure.

This paper deals with the temperature and rate-dependent elasto-viscoplasticity behaviour of a highly ductile acrylic adhesive and its effect on plastic bending of adhesively bonded sheet metals. Tensile lap shear tests of aluminium single-lap joints were performed at various temperature of 10-40°C at several tensile speeds. Based on the experimental results, a new constitutive model of temperature and rate-dependent elasto-viscoplasticity of the adhesive is presented. From V-bending experiments and the corresponding numerical simulation, it was found that the gull-wing bend is suppressed by high-speed forming at a lower temperature.

Background: This paper is concerned with a nonlinear semi-continuum model for an ultrathin structure. The basic equations of the theoretical model for silicon micro/nanosheets are derived, and the geometric nonlinearity is introduced in the model. Methods: From two different approaches including the new strain energy and the new external potential energy, we establish the nonlinear semi-continuum theoretical model of silicon micro/nanosheets, respectively. A new dimensionless nonlinear semi-continuum parameter is defined. Based on the theoretical model, the characteristics of bending deformation and free vibration are revealed. Results: The relationships between bending deflection and atomic layers in thickness direction as well as the relaxation coefficient between atomic layers are analyzed. The resonance frequencies of free vibration and their relationship with atomic layers are calculated. By introducing the specific property parameters of silicon micro/nanomaterials, several numerical calculations have been carried out. Conclusion: The theoretical results are compared with other studies in the literature, such as nonlinear finite element method (FEM), experimental and classical results, to validate the semi-continuum model established in the present research. This work can provide new ideas for the mechanical analyses of micro/nanomaterials and structures, and the results could be foundations for the design and application of silicon micro/nanosheets.

Flexoelectricity describes the coupling between polarization and strain gradients and presents a strong size dependence at nanoscale. In the current work, based on the extended linear piezoelectricity theory with flexoelectricity, we study bending of piezoelectric beams with consideration of flexoelectric effect. When a concentrated force at any position and electric voltage is exerted, the expression for bending deflection of simply-supported and clamped beams is derived. The obtained results show that flexoelectric effect can cause a softer elastic behavior of simply-supported and clamped beams. Sensitivity analysis of the transverse deflection and bending moment is made for two typical boundary conditions. Flexoelectric effect has a more significant effect on the bending response of a piezoelectric beam with smaller thickness.

In this work, multilayer graphene was chosen and modeled in ANSYS software. In the software, it was subjected to bending forces through various bend angles. In order to test for bending, the model is subjected to rigid support on the base side. Then a force is applied on the side on which the angle is provided to simulate Bending. Stress–strain plots were analyzed for various bending angles. The graphene sheet was also subjected to twist forces within the software. In order to test for torsion, the model is subjected to rigid support on both sides. Then a couple is applied on the opposite sides to simulate torsion. Torsional stress and strain measurements were analyzed. The trends were studied and correlations with the material properties of graphene were established. The quest is to identify a magic angle for bend and twist.

Well-posedness of a free boundary problem for electrostatic microelectromechanical systems (MEMS) is investigated when nonlinear bending effects are taken into account. The model describes the evolution of the deflection of an electrically conductive elastic plate suspended above a fixed ground plate together with the electrostatic potential in the free domain between the two plates. The electrostatic potential is harmonic in that domain and its values are held fixed along each plate. The equation for the elastic plate deflection is a parabolic quasilinear fourth-order equation, which is coupled to the gradient trace of the electrostatic potential on the elastic plate.

This work estimates the optimum process parameters for the double-layer deposition of aluminum 6063 over EN8 medium carbon steel using friction surfacing. The control process parameters like axial force and feed rate of the consumable rod on two different deposition surface conditions, i.e. a good interface and a course interface during the process were investigated. The second layer deposition was performed over the roughness developed after the first layer deposition and contributed largely to the mechanical anchoring mechanism at the second coating interface. Assessment of the mechanical strength at the first coating interface (substrate/ coating 1) and second coating interface (coating 1/ coating 2) through bending and microhardness tests at both the interfaces was performed. The bending test results disclosed the existence of small delamination with no cracking at the second interface with a feed rate of 150mm/min at axial force of 5kN. An increase of 16.2% in the hardness value of consumable aluminum was seen at the first coating interface region. In contrast, the mean hardness at the second coating interface region was nearly the same as that of the consumable rod. Extensive studies of the surface coating and the microstructural features at both interfaces were performed through FESEM and EDAX analyses.

Most of the biomechanical experiments, which validate vertebral finite-element models, deal with vertebral bodies in axial compression. In standing position, gravity loads can induce bending on the last thoracic and first lumbar vertebrae. Hence, the purpose of the study was to evaluate the strength of vertebrae submitted to anterior bending. The boundary conditions were carefully assessed for further validation of finite-element models. Fifteen vertebrae (T11–L2) were tested until failure. The load was applied on the anterior part of the vertebral body and distributed to the whole vertebral body through a polymethylmetacrylate layer. The 3D position of the origin of the force relative to the vertebra was assessed using a motion-capture system and 3D reconstructions of the testing device. The mean failure load was equal to 2098 N. Ninety-five percent of the vertebrae failed through a vertical displacement smaller than 1.5 mm. Rotations were weak during the test (< 1°), therefore the protocol can be simplified by removing the motion capture system if the initial 3D origin of the load is known. To our knowledge, it is the first protocol that quantifies the strength of whole vertebrae under anterior bending. It collects with accuracy all the data necessary for one-to-one vertebral finite-element model validations.

Bending Forwards: Blending Yoga into Our Modern Lifestyle.

In the Great Hanshin–Awaji earthquake of 1995, the phenomena of repetitive inelastic buckling were observed in many steel girders including horizontal girders of portal steel piers on elevated highways. The authors have been interested in the ability of steel girders to dissipate the hysteretic plastic strain energy due to such repetitive buckling of steel girders for earthquake-resistance design. This paper is focused on the repetitive buckling behavior of eight steel plate girders under inelastic shear or the combined shear and bending due to a concentrated point load adopting two independent cyclic loading patterns. The model girders were selected considering the combined variations of flange thickness, flange width and depth-to-thickness ratio of the web. Good correlations were found between the results of tests and finite element analyses using shell elements considering the material and geometrical nonlinearities in the repetitive inelastic buckling behavior of plate girders.

A semi-analytical method for bending, global buckling, and free vibration analyses of sandwich panels with square-honeycomb cores is presented. The discrete geometric nature of the square-honeycomb core is taken into account by treating the core sheets as thin beams and the sandwich panel as composite structure of plates and beams with proper displacement compatibility. Based on the classical model of sandwich panels, the governing equations of motion of the discrete structure are derived using Hamilton's principle. Closed-form solutions are developed for bending, global buckling, and free vibration of simply supported square-honeycomb sandwich panels by employing Fourier series and the Galerkin approach. Results from the proposed method agree well with available results in the literature and those from detailed finite element analysis. The effects of various geometric parameters of the sandwich panel on its behavior are investigated. The present method provides an efficient way of analysis and optimization of sandwich panels with square-honeycomb cores.

The present generation of international structural steel design codes treats material nonlinearity through simplified elastic-plastic or rigid-plastic material models. However, the actual stress–strain response of structural steel is more complex than this and features, in particular, strain hardening. Strain hardening refers to the increase in strength beyond yield because of plastic deformation. The influence of strain hardening on the behavior and design of steel structures is examined in this study through both the experimentation and the analysis of existing data, and a method to exploit the additional capacity that arises is outlined. Both determinate and indeterminate structures are considered. The proposed design method, referred to as the continuous strength method (CSM), is a deformation-based design approach employing a continuous relationship between cross-sectional slenderness and cross-sectional deformation capacity, together with a material model that allows for strain hardening. Comparisons are made between test results generated as part of the present study and collected from existing studies, and the predictions from the CSM and Eurocode 3 (EC3). For all cases considered, the CSM, through a rational exploitation of strain hardening, offers a more accurate prediction of observed physical behavior.

This paper presents an analytical study on the dynamic characteristics of castellated beams. The study focuses on the effect of web shear on the free vibration frequencies of castellated beams. By using the Hamilton's principle, a simple closed-form solution for determining the free vibration frequencies of simply supported castellated beams is developed. The results show that the shear effect on the free vibration frequencies increases with the cross-sectional area and distance between the centroids of the two tee sections of castellated beams, but decreases with respect to increasing web thickness or increasing beam length. The shear effect is also found to be greater in higher vibration modes.

This paper presents the Hencky bar-chain model (HBM) for buckling and vibration analyses of Euler–Bernoulli beams with elastic end restraints. The Hencky bar-chain comprises rigid beam segments (of length a = L/n where L is the total length of beam and n the number of beam segments) connected by frictionless hinges with elastic rotational springs of stiffness EI/a where EI is the flexural rigidity of the beam. The elasticity and the mass of the beam are concentrated at the hinges with rotational springs. The key contribution of this paper lies in the modeling of the elastic end restraints of the Hencky bar-chain that will simulate the same buckling and vibration results as that furnished by the first-order central finite difference beam model (FDM) which was earlier shown to be analogous to the HBM. The establishment of such a physical discrete beam model allows one to obtain solutions for beam-like structure with repetitive cells (or elements) as well as to calibrate the Eringen's coefficient e₀ in the nonlocal beam theory that captures the small length scale effect.

In this paper, the Ritz-based solutions are developed for the bending, buckling and vibration behaviors of laminated composite beams with arbitrary lay-ups. A quasi-3D theory, which accounts for a higher-order variation of both the axial and transverse displacements, is used to capture the effects of both shear and normal deformations on the behaviors of composite beams. Numerical results for various boundary conditions are presented and compared with existing ones available in the literature. Besides, the effects of fiber angle, span-to-height ratio, material anisotropy and Poisson’s ratio on the displacements, stresses, natural frequencies and buckling loads of the composite beams are investigated.

This paper presents a novel approach for determining the critical lateral-torsional buckling loads of beams subjected to arbitrarily transverse loads. This new method is developed based on the classical energy method. However, the difference of the present method from the traditional energy methods is the formulation of potential energy of external loads, which is expressed in terms of the internal bending moment and internal shear force in the pre-buckling stage regardless of the type of loading. Compared to the traditional formulations of the potential energy of external loads, not only is the present formula simple in the form, easy and convenient in the calculation, but it also provides a unified form for calculating accurate critical load of lateral-torsional buckling of the beams.

In this paper, the surface effect on the static bending behavior of functionally graded porous (FGP) nanobeams subjected to a concentrated transverse load is studied by using Reddy’s higher-order beam theory. Three types of porosity distributions are considered for the nanobeam, i.e. uniform porosity distribution, symmetric and asymmetric non-uniform porosity distributions. With the consideration of the surface effect, the nanobeams can be abstracted as a composite beam composed of a surface layer and a bulk volume. According to the generalized Young–Laplace equation, the normal stress discontinuity across a surface due to the effect of surface stress is taken into consideration. The analytical solutions of the static bending problem of FGP nanobeams are obtained for the beams with hinged-hinged, clamped-clamped and clamped-free boundary conditions. The effects of the residual surface stress, porosity distribution type, porosity coefficient and length-to-thickness ratio on the transverse displacement of the FGP beams are discussed.