Higher-order Beam Model Research Articles

A new higher-order elastoplastic beam model for reinforced concrete

The present paper introduces a new elastoplastic beam model for reinforced concrete based on a higher-order beam model previously developed (Int J Numer Methods Eng. https://doi.org/10.1002/nme.5926, 2018). Steel and concrete are both defined as elastoplastic materials. The beam model represents the concrete body whereas rebars are given a specific discretization. A Rankine criterion is used for concrete in both tension and compression, and a closed-form solution for the local projection of the trial stress on the yield surface is formulated. Steel rebars are modelled with 1D bar elements and added to the global stiffness of the concrete beam model. The kinematics of the higher-order beam model is enriched by a systematic method with displacement modes. This extension of the kinematics leads to local accuracy and yields results comparable to 3D computations. The present reinforced concrete model is validated through a set of case studies. Implemented within the software programs of the company Strains Engineering, the objective is to develop a fast computing and efficient model that can be directly used by engineers.

A novel porosity-dependent homogenization procedure for wave dispersion in nonlocal strain gradient inhomogeneous nanobeams

Present research aims to consider the reciprocal impacts of Young moduli and mass density while investigating the wave propagation behaviors of functionally graded (FG) nanobeams via a nonlocal strain gradient based shear deformable beam theory for the first time. The porosity influences are regarded within the framework of a newly developed method. The motion equations are obtained by extending the dynamic form of the principle of virtual work for a higher-order beam model. Thereafter, the constitutive relations are modified based on the nonlocal strain gradient theory in order to account for the effects of small scale. At the end, the dispersion curves are achieved by solving the problem with an analytical method. The results show that there is a remarkable difference between the results of this homogenization method and those of former simple methods that were utilized to capture porosity effects. Also, it is proven that the presented methodology is efficient enough to estimate the dynamic behaviors of FG nanobeams.

Vibration analysis of graphene oxide powder-/carbon fiber-reinforced multi-scale porous nanocomposite beams: A finite-element study

Application of the Rayleigh-Ritz method for the vibration problem of a porous multi-scale hybrid nanocomposite graphene oxide powder (GOP)/carbon fiber (CF)-reinforced beam is shown here for the first time on the basis of a new refined higher-order shear deformation beam theory. The structure consists of an initial matrix which is strengthened via both macro- and nano-scale reinforcements. Herein, GOPs and CFs are selected to be dispersed inside the resin. Moreover, the influences of porosity are included, too. The governing equations of the problem are achieved in the framework of a new refined higher-order beam model. Afterward, the Rayleigh-Ritz well-known finite-element method (FEM) is implemented to solve the problem for various boundary conditions (BCs). The validity of the presented formulation is checked by comparing the results of the employed FEM with those achieved from the Navier solution. it is shown that hybrid nanocomposites are able to support higher natural frequencies in comparison with either conventional fiber-reinforced composites or common two-phase GOP-reinforced nanocomposites.

Accurate evaluation of 3D stress fields in adhesive bonded joints via higher-order FE models

The accurate representation of the 3D stress fields at the bonded areas of adhesive joints is essential for their design and strength evaluation. In the present study, higher-order beam models developed in the framework of the Carrera Unified Formulation are employed to reduce the complexity and computational cost of numerical simulations on adhesive joints. The different components of the adhesive joint, i.e. adherends and adhesive, are modeled as beams with independent kinematics based on the Hierarchical Legendre Expansion (HLE). HLE models make use of a hierarchical polynomial expansion over the cross-section of the beam, thus allowing for the control of the accuracy of the stress solutions via the polynomial expansion. Recalling the Finite Element method, the beam axis is discretized by means of 1D elements. In this manner, generic geometries of the adhesive bonded joints can be studied. The proposed model is assessed through comparison against numerical and analytical references from the literature for single lap and double lap joints. Finally, a detailed 3D analysis is performed on the single lap joint problem, showing that the stress gradients along the adhesive are correctly and efficiently described if the proposed methodology is employed.

Accurate stress fields of post-buckled laminated composite beams accounting for various kinematics

Highly flexible laminated composite structures, prone to suffering large-deflection and post-buckling, have been successfully employed in a number of scenarios. Therefore, accurate predictions of their stress distributions in the geometrically nonlinear analysis are of paramount importance for their design and failure evaluation. In this paper, for composite beams subjected to large-deflection and post-buckling, we investigate the effectiveness of different geometrically nonlinear strain approximations for the description of their nonlinear static response and for the determination of stress distributions. For this purpose, a unified formulation of geometrically nonlinear refined beam theory based on the Carrera Unified Formulation (CUF) and a total Lagrangian approach constitutes the basis of our analysis. Accordingly, various kinematics of one-dimensional structures are formulated via an appropriate index notation and an arbitrary cross-section expansion of the generalized variables, leading to lower- to higher-order beam models with only pure displacement variables for laminated composite beams. In view of the intrinsic scalable nature of CUF and by exploiting the principle of virtual work and a finite element approximation, nonlinear governing equations corresponding to various nonlinear strain assumptions can be straightforwardly and easily formulated in terms of fundamental nuclei, which are independent of the theory approximation order. Several numerical assessments are conducted, including large-deflection and post-buckling analyses of asymmetric and symmetric laminated beams under compression loadings. The numerical solutions are solved by using a Newton–Raphson linearization scheme along with a path-following method based on the arc-length constraint. Our numerical findings demonstrate the capabilities of the CUF model to calculate the large-deflection and post-buckling equilibrium curves as well as the stress distributions with high accuracy, which could be a basis to assess the validation ranges of various kinematics and different nonlinear strain approximations.

Buckling analysis of graphene oxide powder-reinforced nanocomposite beams subjected to non-uniform magnetic field

Present article deals with the static stability analysis of compositionally graded nanocomposite beams reinforced with graphene oxide powder (GOP) is undertaken once the beam is subjected to an induced force caused by nonuniform magnetic field. The homogenized material properties of the constituent material are approximated through Halpin-Tsai micromechanical scheme. Three distribution types of GOPs are considered, namely uniform, X and O. Also, a higher-order refined beam model is incorporated with the dynamic form of the virtual work\'s principle to derive the partial differential motion equations of the problem. The governing equations are solved via Galerkin\'s method. The introduced mathematical model is numerically validated presenting a comparison between the results of present work with responses obtained from previous articles. New results for the buckling load of GOP reinforced nanocomposites are presented regarding for different values of magnetic field intensity. Besides, other investigations are performed to show the impacts of other variants, such as slenderness ratio, boundary condition, distribution type and so on, on the critical stability limit of beams made from nanocomposites.

Nonlocal dynamic response of double-nanotube-systems for delivery of lagged-inertial-nanoparticles

This paper deals with transverse vibrations of doubly parallel nanotubes acted upon by doubly lagged-moving nanoparticles accounting for nonlocality. By exploiting the nonlocal Rayleigh and higher-order beam models, equations of motion of the nanosystem are obtained by considering the nonlocal inertial force as well as the lag of moving nanoparticles. For the case of ignorance of lateral inertia of moving nanoparticles, an analytical solution based on the Laplace transform method is suggested and the exact deformation fields of the nanosystem are presented for two patterns. In the case of consideration of the lateral inertia of moving nanoparticles, a numerical approach based on the Galerkin approach and admissible modes is suggested and the elasto-dynamic fields of the nanosystem are appropriately determined at each time. By comparing the predicted results by the proposed nonlocal numerical models with those of another work and the analytical approach, a reasonably good agreement is achieved. Subsequently, the roles of shear deformation, nonlocality, lag effect, geometry of the nanosystem, and kinematic properties of the moving nanoparticles on the dynamic deflections of the constitutive tubes are displayed. This scrutiny is aimed to be a solid base for examining vibrations of a membrane of nanotubes for delivery of multiple nanoparticles through their pores.

A Simplified Approach to Identify Sectional Deformation Modes of Thin-Walled Beams with Prismatic Cross-Sections

In this paper, a simplified approach to identify sectional deformation modes of prismatic cross-sections is presented and utilized in the establishment of a higher-order beam model for the dynamic analyses of thin-walled structures. The model considers the displacement field through a linear superposition of a set of basis functions whose amplitudes vary along the beam axis. These basis functions, which describe basis deformation modes, are approximated from nodal displacements on the discretized cross-section midline, with interpolation polynomials. Their amplitudes acting in the object vibration shapes are extracted through a modal analysis. A procedure similar to combining like terms is then implemented to superpose basis deformation modes, with equal or opposite amplitude, to produce primary deformation modes. The final set of the sectional deformation modes are assembled with primary deformation modes, excluding the ones constituting conventional modes. The derived sectional deformation modes, hierarchically organized and physically meaningful, are used to update the basis functions in the higher-order beam model. Numerical examples have also been presented and the comparison with ANSYS shell model showed its accuracy, efficiency, and applicability in reproducing three-dimensional behaviors of thin-walled structures.

Exact solutions for the macro-, meso- and micro-scale analysis of composite laminates and sandwich structures

The present work proposes a closed-form solution based on refined beam theories for the static analysis of fiber-reinforced composite and sandwich beams under simply supported boundary conditions. The higher-order beam models are developed by employing Carrera Unified Formulation, which uses Lagrange-polynomials expansions to approximate the kinematic field over the cross section. The proposed methodology allows to carry out analysis of composite structure analysis through a single formulation in global-local sense, i.e. homogenized laminates at a global scale and fiber-matrix constituents at a local scale, leading to component-wise analysis. Therefore, three-dimensional stress/displacement fields at different scales can be successfully detected by increasing the order of Lagrange polynomials opportunely. The governing equations are derived in a strong-form and solved in a Navier-type sense. Three benchmark numerical assessments are carried out on a single-layer transversely isotropic beam, a cross-ply laminate [Formula: see text] beam and a sandwich beam. The results show that accurate displacement and stress values can be obtained in different parts of the structure with lower computational cost in comparison with traditional, enhanced as well as three-dimensional finite element methods. Besides, this study may serve as benchmarks for future assessments in this field.

Geometrically nonlinear inelastic analysis of steel–concrete composite beams with partial interaction using a higher-order beam theory

A comprehensive finite element model based on a higher-order beam theory (HBT) is developed for an accurate prediction of the response of steel–concrete composite beams with partial shear interaction. The formulation of the proposed one dimensional finite element model incorporated nonlinearities due to large deformations of the beam as well as inelastic material behaviour of its constituent components. The higher-order beam model is achieved by taking a third order variation of the longitudinal displacement over the beam depth for the steel and concrete layers separately. The deformable shear studs used for connecting the concrete slab with the steel girder are modelled as distributed shear springs along the interface between these two material layers. The Green–Lagrange strain vector is used to capture the effect of geometric nonlinearity due to large deflections. The von Mises plasticity theory with an isotropic hardening rule and a damage mechanics model are incorporated within the proposed finite element model for simulating the inelastic response of the beam materials. The nonlinear governing equations are solved by an incremental-iterative technique following the Newton–Raphson method. A dissipation based arc-length method is employed to capture the post peak response of these beams successfully. The capability of the proposed model is assessed through its validation and verification using existing experimental results and numerical results produced by detailed finite element modelling of these beams.

Stability analysis of porous multi-phase nanocrystalline nonlocal beams based on a general higher-order couple-stress beam model

This article investigates buckling behavior of a multi-phase nanocrystalline nanobeam resting on Winkler-Pasternak foundation in the framework of nonlocal couple stress elasticity and a higher order refined beam model. In this model, the essential measures to describe the real material structure of nanocrystalline nanobeams and the size effects were incorporated. This non-classical nanobeam model contains couple stress effect to capture grains micro-rotations. Moreover, the nonlocal elasticity theory is employed to study the nonlocal and long-range interactions between the particles. The present model can degenerate into the classical model if the nonlocal parameter, and couple stress effects are omitted. Hamilton\'s principle is employed to derive the governing equations and the related boundary conditions which are solved applying an analytical approach. The buckling loads are compared with those of nonlocal couple stress-based beams. It is showed that buckling loads of a nanocrystalline nanobeam depend on the grain size, grain rotations, porosities, interface, elastic foundation, shear deformation, surface effect, nonlocality and boundary conditions.

Analysis of thin-walled beam-shell structures for concept modeling based on higher-order beam theory

Many engineering structures consist of thin-walled beams and shells. Especially for fast design in an early design stage, a simplified analysis using beams for load-carrying members and shells for panels is very useful, but there appears no accurate beam-shell combined finite element model. The main reason is that the Timoshenko or Euler beam elements are incapable of representing significant sectional deformations near beam joints or near beam-panel interfaces. Although some progress has been made in developing higher-order beam elements that can accurately capture the sectional deformations, there is no investigation to develop higher-order beam and shell combined models useful to analyze various engineering structures. The main contribution of this work is to present the first attempt to model structures made of thin-walled closed beams and shells in terms of higher-order beam elements and shell elements and to establish the matching conditions between the dissimilar field variables of higher-order beam and shell elements along their interfaces. For the finite element analysis of a whole structure, the interface matching conditions are imposed through Lagrange multipliers. High accuracy of the proposed higher-order beam-shell method is demonstrated through static and modal analyses of various structures including a simplified model of a vehicle body-in-white (BIW).

Micromechanics of periodically heterogeneous materials using higher-order beam theories and the mechanics of structure genome

A novel approach for the micromechanical analysis of periodically heterogeneous composite materials is proposed in this paper. It is based on the use of refined beam theories for the modeling of the microstructure and the mechanics of structure genome (MSG) for the derivation of the governing equations of the unit cell problem. On the one hand, MSG is recalled to decouple the multiscale problem into global and local analysis, providing the constitutive information and the local fields with no need of ad hoc assumptions, nor multiple loading steps. On the other hand, the Carrera Unified Formulation (CUF) is employed to generate higher-order beam models that show the same accuracy as conventional solid elements with reduced computational efforts. Accordingly, the main direction of the constituents (e.g. the fibre direction) is discretized by means of one-dimensional finite elements whereas the cross-section is hierarchically enriched with a set of Legendre-based polynomials with non-local capabilities. In addition, the implementation of a non-isoparametric mapping technique permits the representation of the exact geometry of the constituents with no additional costs. The validity and efficiency of the proposed model is assessed through comparison with several benchmark solutions of fibre reinforced and particle reinforced composites.

A unified nonlocal strain gradient model for nanobeams and the importance of higher order terms

In present paper, a unified size-dependent high-order beam model which contains various higher-order shear deformation beam models as well as Euler–Bernoulli and Timoshenko beam models is developed to study the simultaneous effects of nonlocal stress and strain gradient on the bending and buckling behaviors of nanobeams by using the nonlocal strain gradient theory. For this objective, a nonlocal parameter is introduced to capture the nonlocal effect and a material length scale parameter is involved to evaluate the strain gradient effect. The governing equations and the associated boundary conditions are formulated by using Hamilton's principle. Navier's method is utilized to obtain the analytical solutions for bending and buckling of a simply supported nanobeam. The present model is validated by comparing the obtained results with those available in literature. The influences of nonlocal parameter, material length scale parameter, slenderness ratio and shear deformation on the bending and buckling behaviors of the nanobeam are examined in detail. Results reveal that within the framework of nonlocal strain gradient theory, results predicted by Timoshenko beam model and various higher-order beam models are almost same with some negligible differences. Moreover, it is found that the nanobeam could exhibit either stiffness-softening effect or stiffness-hardening effect, which depends on the relative magnitude of the nonlocal parameter and the material length scale parameter.

Nonlocal excitation and potential instability of embedded slender and stocky single-walled carbon nanotubes under harmonically vibrated matrix

Until now various aspects of vibrations of single-walled carbon nanotubes (SWCNTs) have been explored; however, their dynamics and possible instabilities because of the excitation of matrix have not been addressed methodically. By considering a harmonic transverse excitation, the explicit expressions of elastic fields are obtained based on the nonlocal Rayleigh, Timoshenko, and higher-order beam models. The roles of the nonlocality, slenderness ratio, amplitude and frequency of matrix excitation and interactional behavior of the embedded nanotube on the dynamic transverse displacements of SWCNTs are comprehensively displayed. The capabilities of the Rayleigh model as well as the Timoshenko model in capturing the deflection of the nanostructure based on the higher-order beam theory are also explained in some detail. The nonlocal elastodynamic fields of the nanostructure in the resonance state as well as the critical values of lateral and rotational stiffness of the matrix are also introduced and the influences of crucial factors on such parameters are explained and discussed carefully.

Nonlinear Free Vibration Analysis of Fluid Conveying FG Tube Via Energy Balance Method

Hereby, a fluid conveying tube made by FG (functionally graded) materials was considered. For which a higher-order beam model for tubes which can satisfy the shear stress boundary conditions on the inner and outer surfaces was applied. Through the Hamilton’s principle, the governing equations were gained. By keeping the first order mode shape and using energy balance method, the linear and nonlinear frequencies were obtained. Numerical simulation justified the correctness of these results. Discussions on the relationship between the frequencies and the FG tube’s parameter were also performed.

Investigation of failure initiation in curved composite laminates using a higher-order beam model

Curved laminates are prone to delamination failure from applied bending moments that straighten out the laminate and induce tensile stresses in the unreinforced radial direction. These non-classical through-thickness stresses are important even for thinner configurations and need to be accounted for in the design of lightweight composite structures, preferably in a computationally efficient manner. Here, we investigate failure-inducing critical stresses for a number of curved laminates using a higher-order beam model derived from the Hellinger–Reissner mixed variational principle, which guarantees that the hoop, interlaminar shear and radial stresses are equilibrated. By solving the governing equations of the theory in the strong form using the pseudo-spectral differential quadrature method, the model is capable of predicting accurate 3D stress fields in curved laminates, even in the vicinity of localised features such as supported edges. The model is used alongside commonly used failure criteria to reproduce experimental failure initiation results found in the literature, and the comparison suggests that failure mode, location and load are all predicted accurately. Finally, failure maps that highlight the critical stress component for failure initiation are constructed. As the thickness of the curved laminate increases, the critical stress component transitions from intralaminar hoop stress to interlaminar shear or radial stress depending on the specific laminate configuration. These findings and failure maps collectively provide insights into the mechanics of failure initiation that should also prove useful for design purposes.

Unified formulation of geometrically nonlinear refined beam theories

ABSTRACTBy using the Carrera Unified Formulation (CUF) and a total Lagrangian approach, the unified theory of beams including geometrical nonlinearities is introduced in this article. According to CUF, kinematics of one-dimensional structures are formulated by employing an index notation and a generalized expansion of the primary variables by arbitrary cross-section functions. Namely, in this work, low- to higher-order beam models with only pure displacement variables are implemented by utilizing Lagrange polynomial expansions of the unknowns on the cross section. The principle of virtual work and a finite element approximation are used to formulate the governing equations, whereas a Newton-Raphson linearization scheme along with a path-following method based on the arc-length constraint is employed to solve the geometrically nonlinear problem. By using CUF and three-dimensional Green-Lagrange strain components, the explicit forms of the secant and tangent stiffness matrices of the unified beam element are provided in terms of fundamental nuclei, which are invariants of the theory approximation order. A symmetric form of the secant matrix is provided as well by exploiting the linearization of the geometric stiffness terms. Various numerical assessments are proposed, including large deflection analysis, buckling, and postbuckling of slender solid cross-section beams. Thin-walled structures are also analyzed in order to show the enhanced capabilities of the present formulation. Whenever possible, the results are compared to those from the literature and finite element commercial software tools.

Higher-order beam model for stress predictions in curved beams made from anisotropic materials

A higher-order beam model for analyzing the flexural response of curved multilayered beams with constant curvature and arbitrary constant thickness is developed. The new model is derived from the Hellinger–Reissner mixed variational statement and predicts inherently equilibrated 3D stresses from an equivalent single-layer model. As a starting assumption, the hoop stress is formulated as a series of higher-order stress resultants multiplied by Legendre polynomials. The governing equations are derived in a generalized manner such that the modeling order can be adjusted and is not defined a priori. Hence, the highest order Legendre polynomial determines the modeling order. The through-thickness shear and normal stresses are derived by integrating the generalized hoop stress in Cauchy’s polar equilibrium equations. As a result, all stress fields are based on the same set of variables, thereby considerably reducing the computational effort. The three stress fields, and two displacements in the radial and hoop directions are used in the Hellinger–Reissner functional to derive a new set of stress-displacement relations. The enforcement of the classical membrane and bending equilibrium equations of curved beams in the Hellinger–Reissner functional guarantees that all interlaminar and surface traction equilibrium conditions are satisfied exactly. A validation study of a composite laminate using a high-fidelity 3D finite element model shows that the stresses are captured very accurately by the present model, but with much less computational effort than the finite element model. As a result, the developed model can provide rapid and accurate insights into the expected damage onset behavior of curved laminates.

Magnetic field effects on buckling behavior of smart size-dependent graded nanoscale beams

In this article, buckling behavior of nonlocal magneto-electro-elastic functionally graded (MEE-FG) beams is investigated based on a higher-order beam model. Material properties of smart nanobeam are supposed to change continuously throughout the thickness based on the power-law model. Eringen's nonlocal elasticity theory is adopted to capture the small size effects. Nonlocal governing equations of MEE-FG nanobeam are obtained employing Hamilton's principle and they are solved using the Navier solution. Numerical results are presented to indicate the effects of magnetic potential, electric voltage, nonlocal parameter and material composition on buckling behavior of MEE-FG nanobeams. Therefore, the present study makes the first attempt in analyzing the buckling responses of higher-order shear deformable (HOSD) MEE-FG nanobeams.