Nonlinear Bending Research Articles

Nonlinear bending analysis of anisotropic laminated tubular beams based on higher-order theory subjected to different kinds of distributed loads

Pipelines, and more generally long tubular structures, are major oil and gas industry tools used in exploration, drilling, production, and transmission. Technical challenges of the field have spawned significant research and development efforts in the mechanical behavior and the modes of failure of long tubes and pipes under various loads. Nonlinear bending analysis of anisotropic laminated composite tubular beams with different kinds of distributed loads resting on a two-parameter elastic foundation is investigated. Based on Hamilton's principle and the displacement field by Laurent series expansion form with a von Kármán-type of kinematic nonlinearity, the governing differential equations are obtained. Composite tubular beams with clamped-clamped, clamped-hinged, and hinged-hinged boundary conditions including two kinds of end conditions, namely movable and immovable are considered. A numerical solution of the transverse deflection of tubular beams by using Galerkin's method is employed to determine the relations between distributed loads and deflections of a composite beam with or without initial axial loads. The results of the present formulation concern the large-deflection bending behavior of laminated beams with different geometric and material parameters, distributed loads, end conditions and effect on elastic foundation, the obtained numerical results are also compared with the results of nonlinear finite element method. Some new results for anisotropic laminated tubular beams, such as effects of the end conditions, load characteristics and foundation stiffness are obtained for future references in the design of the civil, mechanical, aeronautical and aerospace industries.

Nonlinear bending vibration of a prestressed thick plate

In this paper, the flexural vibration frequency in the antisymmetric mode of a thick plate as a function of the amplitude of the vibration and the axial force applied is investigated. With this aim, the theory of geometrically nonlinear deformation of second order and an optimized three-dimensional Ritz method are used. The plate is homogeneous, elastically linear, free from any constraints, and subjected to axial forces uniformly distributed on two of its opposite sides. Several approaches are discussed. First, the problem based on finite stress and infinitesimal strains is solved. Second, the deformation energy is assumed as the energy in the initial state plus the vibration energy of small or large amplitude. Third, without assumptions about the size of the deformation and of the vibration amplitude, the theory of nonlinear deformation is employed. Finally, numerical calculations for free vibration are compared with experimental results, including their systematic uncertainties.

Nonlinear finite element analysis of a multiwalled carbon nanotube resting on a Pasternak foundation

ABSTRACTIn conjunction with the Reissner's mixed variational theorem and nonlocal Timoshenko beam theory, we develop a finite element method for the geometrically nonlinear bending analysis of a multiwalled carbon nanotube resting on an elastic foundation and with various boundary conditions. The effects of the van der Waals interaction and the interaction between the multiwalled carbon nanotube and its surrounding medium are considered. Numerical implementation shows that the finite element solutions converge rapidly, and their convergent solutions closely agree with those obtained using the differential quadrature method, based on a strong-form formulation, available in the literature.

Nonlinear bending and instability analysis of bioceramics composed with magnetite nanoparticles: Fabrication, characterization, and simulation

Magnetite is a bioresorbable material that has been widely employed in bioengineering domain, drug-delivery operations, and hyperthermia treatment. In the biomedical engineering, the major focus is on the mechanical properties of materials used in orthopedic and dental implants. In the present investigation, beam-type bone implant made of bredigite composed with magnetite nanoparticles (MNPs) is fabricated with a customized three dimensional (3D) printing, the nonlinear bending and postbuckling behaviors of it are studied. To accomplish this purpose, the Young's modulus of the prepared bio-nanocomposite material is obtained experimentally corresponding to different MNP weight fractions and sizes of the nanoparticles as well as their morphology shapes. In order to prepare the bio-nanocomposite material, the planetary high energy ball milling (HEBM) is put to use to mix the bredigite bioceramic powder as a bioactive matrix with various weight fractions of MNPs. After that, based upon a developed analytical solution, the nonlinear bending and instability response of the bio-nanocomposite beam-type bone implant are analyzed corresponding to various MNP weight fractions and morphology shapes of the material. It is found that for the bio-nanocomposite implant with smaller composite crystallite size, the maximum and minimum buckling loads are for irregular and mesoporous morphology shapes, respectively. On the hand, for the bio-nanocomposite bone implant with larger composite particle size, the maximum and minimum buckling loads are related to, respectively, the spongy and porous morphology shapes.

Application of two-steps perturbation technique to geometrically nonlinear analysis of long FGM cylindrical panels on elastic foundation under thermal load

ABSTRACTPresent research deals with the geometrically nonlinear bending of a long cylindrical panel made of a through-the-thickness functionally graded material subjected to thermal load. A panel under the action of uniform temperature rise loading is considered. Formulation of the shell is based on the third-order shear deformation shell theory, where the first-order shear deformation and classical shell theory may be extracted as special cases. Thermomechanical properties of the shell are assumed to be temperature dependent and are estimated according to a power law function across the shell thickness. Also, it is assumed that shell is in contact with an elastic foundation which acts in tension as well as in compression. The nonlinear governing equations of the shell are obtained using the von Kármán type of geometrical nonlinearity. The obtained governing equations are solved for two cases, i.e., simply supported shells and clamped shells. The developed equations are solved using a two-step perturbation technique. Accurate closed-form expressions are provided to obtain the mid-span deflection of the shell as a function of temperature elevation. Numerical results are provided to analyze the effects of power law exponent, boundary conditions, temperature dependency, side to radius ratio, and side to thickness ratio.

Differential quadrature method of nonlinear bending of functionally graded beam

Using the third-order shear deflection beam theory (TBT), nonlinear bending of functionally graded (FG) beams composed with various amounts of ceramic and metal is analyzed utilizing the differential quadrature method (DQM). The properties of beam material are supposed to accord with the power law index along to thickness. First, according to the principle of stationary potential energy, the partial differential control formulae of the FG beams subjected to a distributed lateral force are derived. To obtain numerical results of the nonlinear bending, non-dimensional boundary conditions and control formulae are dispersed by applying the DQM. To verify the present solution, several examples are analyzed for nonlinear bending of homogeneous beams with various edges. A minute parametric research is in progress about the effect of the law index, transverse shear deformation, distributed lateral force and boundary conditions.

Geometrically nonlinear bending analysis of functionally graded beam with variable thickness by a meshless method

Geometrically nonlinear bending deformation of Functionally Graded Beams (FGBs) with variable thickness is simulated by a meshless Smoothed Hydrodynamic Particle (SPH) method. The material properties of FGB is assumed to be varied smoothly in the thickness according to exponent-law distribution. To prevent the mesh-distortion in element-based numerical method, meshless SPH method is adopted, where corrective smoothed particle method and total-Lagrangian formulation are employed to improve its precision and stability. To validate the present SPH method, several numerical examples are performed and compared to analytical and finite element solutions.

A micromechanical mass sensing method based on amplitude tracking within an ultra-wide broadband resonance

We propose a mass sensing scheme in which amplitude shifts within a nonlinear ultra-wide broadband resonance serve as indicators for mass detection. To achieve the broad resonance bandwidth, we considered a nonlinear design of the resonator comprised of a doubly clamped beam with a concentrated mass at its center. A reduced-order model of the beam system was constructed in the form of a discrete spring-mass system that contains cubic stiffness due to axial stretching of the beam in addition to linear stiffness (Duffing equation). The cubic nonlinearity has a stiffening effect on the frequency response causing nonlinear bending of the frequency response toward higher frequencies. Interestingly, we found that the presence of the concentrated mass broadens the resonant bandwidth significantly, allowing for an ultra-wide operational range of frequencies and response amplitudes in the proposed mass sensing scheme. A secondary effect of the cubic nonlinearity is strong amplification of the third harmonic in the beam’s response. We computationally study the sensitivity of the first and third harmonic amplitudes to mass addition and find that both metrics are more sensitive than the linearized natural frequency and that in particular, the third harmonic amplitude is most sensitive. This type of open-loop mass sensing avoids complex feedback control and time-consuming frequency sweeping. Moreover, the mass resolution is within a functional range, and the design parameters of the resonator are reasonable from a manufacturing perspective.

On the Nonlinear Vibrational Responses of a Large Vessel with a Broad Bow Flare under Wave Excitation: Theory and Experiment

A fully coupled nonlinear three‐dimensional (3D) hydroelastic method is developed to investigate vibrational responses of a large ship with a pronounced bow flare subjected to high seas. This numerical model consists of a 3D boundary element method, 1D Euler‐Bernoulli beam model, and a 2D generalized Wagner model. Green water loads were considered. Experimental study was carried out in a towing tank on a self‐propelled segmented model with nonuniform steel backbones. The ship model was tested in regular incident waves of large amplitude. Impact pressure and nonlinear vertical bending moments were measured and compared with numerical predictions. The proposed nonlinear model produced similar results to the experimental model. Furthermore, the effects of elastic modes and nonlinearities on the numerical results were analyzed.

Asymptotic method for investigating resonant regimes of nonlinear bending vibrations of elastic shaft

Asymptotic method for investigating resonant regimes of nonlinear bending vibrations of elastic shaft

Nonlinear bending analysis of FG-GRC laminated cylindrical panels on elastic foundations in thermal environments

Abstract This paper presents an investigation on the nonlinear bending behaviors of nanocomposite laminated cylindrical panels made of graphene-reinforced composites (GRCs) supported by an elastic foundation under thermal environmental conditions. The extended Halpin-Tsai micromechanical model together with results from MD simulations is employed to obtain the material properties of GRCs which are temperature dependent. A piece-wise functionally graded distribution of graphene fillers along the panel thickness direction is adopted in order to maximize the effect of graphene reinforcement to the panels. The Reddy's third order shear deformable shell theory is applied to derive the governing equations for the bending problem of the panels and the von Karman geometric nonlinearity, the foundation support and the temperature effect are also included in the derivation. The governing equations are solved by applying a two-step perturbation approach to obtain the panel load-deflection and load-bending moment curves of FG-GRC laminated cylindrical panels subjected to a transverse uniform or sinusoidal load. The influences of loading conditions and temperature variation on the nonlinear bending behaviors of FG-GRC laminated cylindrical panels are discussed in details.

Nonlinear analyses of FGM plates in bending by using a modified radial point interpolation mesh-free method

A modified mesh-free radial point interpolation method (RPIM) which employs a new radial basis function capable of building the shape functions without any supporting fixing parameters is developed for the nonlinear bending analyses of functionally graded material (FGM) plates. The shape functions constructed by the RPIM possess Kronecker delta function property and thus special schemes for imposing boundary conditions are not needed. The nonlinear bending formulations of FGM plates are based on a higher-order shear deformation plate theory considering the von Kármán large deflection assumption. The nonlinear equations are solved by the modified Newton–Raphson iterative technique. The computed results by the modified RPIM are compared with the other numerical solutions reported in the literature to verify the effectiveness and accuracy of the proposed mesh-free method. Detailed parametric studies are then carried out to scrutinize the effects that the volume fraction, plate length-to-thickness ratio, boundary condition and initial deflection have on the nonlinear flexural responses of FGM plates. Results illustrate that the modified mesh-free RPIM can effectively predict the nonlinear bending behavior of FGM plates.

Non-linear bending analysis of nanocomposites reinforced by graphene-nanotubes with finite shell element and membrane enhancement

The purpose of this paper is to investigate the non-linear bending behavior of functionally graded shell structures reinforced by carbon nanotubes (FG-CNTRC) using non-linear double directors shell model and membrane enhancement. This model induces a high order variation of the displacement field along the thickness direction and imposes a zero transverse shear stresses condition in the top and bottom surfaces. The effective material properties of FG-CNTRC are estimated by the modified rule of mixtures using some efficiency parameters. Uniformly (UD) and three graded distributions, known as FG-V, FG-O and FG-X, are considered to show the effects of carbon nanotubes (CNTs) profiles and their volume fractions on non-linear deflections of various shapes of shell structures. To illustrate these effects, a set of numerical examples including beams, plates, cylindrical panels and hemispherical shells are presented. The effects of some geometrical parameters are also examined. A convergence studies are carried out to validate the proposed non-linear model, then non-linear results for FG-CNTRC shell structures are provided leading hence to outline the applicability and the efficiency of the present model.

Nonlinear bending analysis of nonlocal nanoplates with general shapes and boundary conditions by the boundary-only method

In this paper, the geometrically nonlinear bending analysis of nanoplates with general shapes and boundary conditions is highlighted. The governing equations are derived based on the classical plate theory using nonlocal differential constitutive relation of Eringen and von Kármán's nonlinear strains. The boundary-only method is developed by using the principle of the analog equation (PAE). According to the PAE, the original governing differential equations are replaced by three uncoupled equations with fictitious sources under the same boundary conditions, namely two Poisson equations and one biharmonic equation. Subsequently, the fictitious sources are established using a technique based on the boundary element method and approximated by using the radial basis functions. The solution of the actual problem is attained from the known integral representations of the potential and plate problems. Therefore, the kernels of the boundary integral equations are conveniently established and readily calculated that the complex nanoplates can be easily analyzed. The accuracy of the proposed methodology is evaluated by comparing the obtained results with available solutions. Moreover, the influences of nonlocal parameter on the various characteristics of effective distributed loads are elucidated. Finally, the effects of nonlocal parameter, von Kármán's nonlinearity and aspect ratio on nonlinear bending responses are studied.

Study on out-of-plane flexural behavior of aluminum alloy gusset joints at elevated temperatures

Aluminum alloy gusset (AAG) joint is the most widely applied joint system in aluminum alloy reticulated shells, and the elevated-temperature mechanical behavior of joints have serious influence on the safety of structures. This paper investigated the out-of-plane flexural behavior of AAG joint by means of experimental and numerical analysis, and theoretical formulae of its bearing capacity and bending stiffness were derived. Firstly, elevated-temperature experiments on 9 AAG joints were conducted in a special-shaped oven. The test results revealed that: (1) the failure modes of thin-plate AAG joints at temperatures under 300°C are block tearing and local buckling of the gusset plate, which remain the same as those at room temperature; (2) when the gusset plate is thick enough (usually as thick or thicker than the flange of the member), the joint zone will not collapse under 300°C; (3) the initial stiffness approximately remains unvaried, while the ultimate flexural bearing capacity drops with the elevation of temperature. Subsequently, finite element (FE) models using ABAQUS were established and verified by the test results. It is confirmed that the FE models can accurately describe the mechanical performance of the AAG joints at elevated temperatures. Based on the simplified FE model, a parametric study was carried out, considering various aluminum alloy brands, thickness of the gusset plate, radius of the center region of the gusset plate and different temperatures. Finally, the block tearing resistance, the local buckling resistance, and parameters of four-linear bending stiffness model of the FE models were presented, and the theoretical formulae were derived using statistical regression technology. The theoretical formulae were verified to be accurate enough against the experimental and FE results, to estimate the bearing capacity and the nonlinear bending stiffness of AAG joints at elevated temperatures under 300°C, which can serve for practical engineering.

Nonlinear bending of functionally graded porous micro/nano-beams reinforced with graphene platelets based upon nonlocal strain gradient theory

Fast improvements in technology causes to provide the ability to manipulate porous graded materials for using in directed design of micro/nano-structures. The prime objective of the current study is to anticipate the size-dependent nonlinear bending of functionally graded porous micro/nano-beams reinforced with graphene platelets, and subjected to the uniform distributed load together with an axial compressive load. Via the nonlocal strain gradient theory of elasticity, two entirely different features of size effects are incorporated in the third-order shear deformable beam model. In addition to the uniform distribution of porosity, three different functionally graded porosity distributions along the thickness of micro/nano-beams are supposed in such a way that the relationship between coefficients related to the relative density and porosity is considered for an open-cell metal foam. On the basis of Hamilton's principle, the non-classical governing differential equations of motion are established. After that, the Galerkin method in conjunction with an improved perturbation technique is utilized to attain explicit analytical expressions for nonlocal strain gradient load-deflection paths of the functionally graded porous micro/nano-beams reinforced with the graphene nanofillers. It is observed that the type of porosity dispersion pattern has no considerable influence on the significance of size effects. However, by approaching the axial compressive load to the critical buckling value, the significance of the both nonlocality and strain gradient size dependencies on the nonlinear bending behavior of functionally graded porous micro/nano-beams increases, and this increment in the nonlocal small scale effect is more considerable than in the strain gradient one.

An explanation of damage and failure in cellulose aerogels: modeling approach

AbstractCellulose aerogels can be considered as foam‐like materials and have recently been modeled using micro‐mechanical approaches. The proposed microcell based models describe the constitutive response of these aerogels based primarily on the bending of the cell wall fibrils, and show good agreement with experimental data. Under compression, the cell walls are considered to undergo nonlinear bending. On the other hand, under tension the deformation in the cell walls is attributed to both, bending and stretching. Accordingly, the total aerogel network energy is divided into a bending one and a tension one. The cell walls undergoing such combined loading fail after reaching a critical normal stress value, and are then considered to no longer contribute to the total network energy. Material constants in these models are obtained from experimental observations, like for example, the pore‐size data analysis, thus resulting into physically motivated models. Finally, the models are also shown to be effective in predicting the constitutive response of other polysaccharidic aerogels. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A numerical study on the nonlinear behavior of corner supported flat and curved panels

The nonlinear behavior of corner supported plates and curved shell panels is investigated here using the first-order shear deformation theory based on Marguerre’s membrane strains for shallow shells and von Karman’s nonlinearity. The nonlinear differential equations are transformed into a set of nonlinear algebraic equations by using the element-free Galerkin method. The moving kriging shape function with two different types of correlation formulae (Gaussian and quartic spline) is employed here. After studying the effectiveness of the method, a detailed parametric study is conducted to examine the effect of support size on the displacements and bending moments of corner supported rectangular plates. Thereafter, the numerical study is extended to the nonlinear bending and stability behaviors of corner supported shallow cylindrical and spherical shell panels.

Bending and vibration analyses of coupled axially functionally graded tapered beams

The nonlinear bending and vibrations of tapered beams made of axially functionally graded (AFG) material are analysed numerically. For a clamped–clamped boundary conditions, Hamilton’s principle is employed so as to balance the potential and kinetic energies, the virtual work done by the damping, and that done by external distributed load. The nonlinear strain–displacement relations are employed to address the geometric nonlinearities originating from large deflections and induced nonlinear tension. Exponential distributions along the length are assumed for the mass density, moduli of elasticity, Poisson’s ratio, and cross-sectional area of the AFG tapered beam; the non-uniform mechanical properties and geometry of the beam along the length make the system asymmetric with respect to the axial coordinate. This non-uniform continuous system is discretised via the Galerkin modal decomposition approach, taking into account a large number of symmetric and asymmetric modes. The linear results are compared and validated with the published results in the literature. The nonlinear results are computed for both static and dynamic cases. The effect of different tapered ratios as well as the gradient index is investigated; the numerical results highlight the importance of employing a high-dimensional discretised model in the analysis of AFG tapered beams.

Nonlinear bending and snap-through instability analyses of conical shell panels using element free Galerkin method

The nonlinear bending and snap-though instability phenomenon of isotropic and composite conical shell panels are investigated here using the element free Galerkin (EFG) method with moving kriging (MK) shape function. Sanders’ shell theory along with von Kármán strain-displacement assumptions are employed to derive the nonlinear equations of equilibrium, which are solved by modified Riks technique in conjunction with Newton-Raphson method. The convergence and accuracy of the EFG method are examined for the linear and nonlinear bending behavior of conical shell panels. Thereafter, the effect of geometrical parameters on the nonlinear stability characteristics of conical panels is investigated under different loading conditions. New results for linear as well as nonlinear bending behavior of isotropic and laminated conical shell panels, hitherto not found in the literature, are presented for future reference.