Higher-order Beam Model Research Articles

Mesh objective characteristic element length for higher-order finite beam elements

The use of fracture energy regularization techniques can effectively mitigate the mesh dependency of numerical solutions caused by the strain softening behavior of quasi-brittle materials. However, the successful regularization depends on the correct estimation of the crack bandwidth in Finite Element solutions. This paper aims to present an enhanced crack band formulation to overcome the strain localization instability especially for the higher-order elements developed in the framework of Carrera Unified Formulation (CUF). Besides, a modified Mazars damage method incorporating fracture energy regularization is employed to describe the nonlinear damage behavior of the concrete. To evaluate the efficiency of the proposed crack band formulation, three experimental concrete benchmarks are selected for the numerical damage analysis. By comparing numerical and experimental results, the proposed method can guarantee mesh objectivity despite varying finite element numbers and orders, indicating perseved fracture energy consumption within proposed higher-order beam models.

A data-driven procedure for one-dimensional dynamic analysis of thin-walled beams with arbitrary cross-sections

This paper develops a one-dimensional dynamic model for thin-walled beams with arbitrary complex cross-sections in a data-driven way. In order to consider complicated deformations in the framework of a beam theory, a universal node system is first created to fit the deformed shape of a thin-walled cross-section, whether prismatic or curved, with or without symmetry. This helps to reduce the three-dimensional displacement field of the thin-walled beam to one dimension, and results in a preliminary higher-order beam model with limited precision loss. For the purpose of largely condense DOFs of the new model, a data-driven approach is proposed to identify cross-section deformation modes through the principal component analysis of free vibration deformation data of an unconstrained thin-walled beam. As a result, a compact set of core deformation modes are obtained and selected to update the preliminary model, leading to the refined higher-order beam model of high accuracy and efficiency. Examples are presented to illustrate the concrete implementation and check the effects of data-related controlling parameters of the proposed procedure. We also verify through numerical examples that the proposed model agrees well with plate/shell and other beam theories, and has remarkable advantages in physical interpretation, modeling simplicity and computation efficiency.

Thermomechanical analysis of carbon fiber-reinforced polymer representative volume elements with voids

This work presents results of numerical simulations to investigate the effect of different void percentages on composite materials’ Coefficient of Thermal Expansion (CTE) and local stress fields. A random distribution of voids is considered within the Representative Volume Element (RVE) matrix, and different types of microstructures are considered, including square-packed and randomly distributed fibers. The use of a higher-order beam model within the framework of Carrera Unified Formulation (CUF) leads to a Component-Wise (CW) approach, resulting in an accurate, 3D description of the cross-section although using a 1D formulation. Numerical results for different fiber volume fractions and void concentration percentages demonstrate the agreement of the computed effective coefficients of thermal expansion of the present micromechanical thermoelastic model with references from the literature. The local stress fields are affected by voids, with higher effects over the matrix. Furthermore, higher void fractions lead to higher variability of stress peaks.

Variable-stiffness curved laminated-beams by curvilinear fibers with arbitrarily layup – Vibrational features by sine-based higher-order beam model with renewed-constitutive relations and improved-kinematics

The vibration characteristics of variable-stiffness based composite curved beams with general layups are examined using a higher-order beam theory that accounts for three-dimensional structural response features. The variable stiffness is spatially considered in the laminated beam by introducing the curvilinear fibres-based layers. The structural model is here represented by a sine-based shear deformable model considering zig-zag function for improving in-plane response and higher order polynomial for capturing through-thickness stretching effect. The appropriate constitutive relationships for laminated composite curved beam with arbitrary layup producing stiffness coupling effects are deduced from three-dimensional elasticity equations. The beam governing equations are then formed applying Hamilton’s principle and the free vibration features are predicted employing the eigenvalue analysis. The effectiveness of the curved laminated beam element developed here is tested against the available analytical solutions in the literature. A detailed numerical analysis by opting for different structural parameters like curved beam angle, thickness, ply-angle, layup, and beam end condition are made to investigate the vibration characteristics of general layup constant stiffness laminated curved beams and by including additional parameter, curvilinear fibre angle variation from centre-to-edge, for variable stiffness laminates.

Rotordynamic analyses with variable-kinematic beam and shell finite elements

This work investigates the dynamic response of various rotating cylindrical structures using low- and high-fidelity one-dimensional (1D) and two-dimensional (2D) finite element models. The adopted mathematical formalism is based on the Carrera Unified Formulation (CUF). CUF offers a procedure to obtain higher-order beam and shell models hierarchically and automatically. These theories are formulated by expanding the unknown variables over the beam cross-section or along the shell thickness. Various beam and shell models can be implemented depending on the choice of the polynomial employed in the expansion. Both Taylor-like and Lagrange polynomials are considered for developing different kinematic models. The linearized equations of motion include the Coriolis and initial stress contributions. Various thick and thin cylinders and disk structures with different boundary conditions are considered. The beam and shell results are compared with analytical and three-dimensional (3D) finite element (FE) solutions from the literature.

An Intuitive Derivation of Beam Models of Arbitrary Order

This article presents a new beam model that employs a recursive derivation procedure that enables the user to set the order of the governing differential equations as an input parameter, without the need for ad hoc assumptions or methodologies. This article employs a novel system of kinematic variables, section constants, and section functions that facilitate the development of higher-order beam models that retain a clear philosophical link to classical beam models such as Euler–Bernoulli beam theory and Timoshenko beam theory. The present beam model is a type of equivalent single layer beam model, wherein section constants are used to model the global stiffness characteristics of the beam, and section functions are used to recover sectional fields of displacements, strains, and stresses. The present beam model is solved for several example beams, and the results are compared to the results of finite element analyses. It is shown that the present beam model can accurately predict the deformed shapes and stress fields of each of the example beams. This article also reveals an interesting peculiarity of the elastic potential energy that pertains to any unidimensional beam model that is governed by differential equations that are of finite order.

A unified high-order model for size-dependent vibration of nanobeam based on nonlocal strain/stress gradient elasticity with surface effect

In this work, a unified high-order nanobeam model considering various high-order shear deformation beam theories is established to investigate the vibration response of nanobeam on the basis of two-phase local/nonlocal strain and stress gradient theory, as well as surface elasticity theory. The unified model also includes the Euler-Bernoulli beam model and Timoshenko beam model. Herein, the elastic dynamics governing equations and boundary conditions are derived using Hamilton's principle, and the analytical solutions, such as exact formulas for natural frequencies, are obtained by employing the Navier method for simply supported boundary conditions. The effects of local volume fraction, nonlocal parameter, material length scale parameter, shear deformation and surface energy in stress and strain-driven models are analyzed in detail, respectively. The parametric studies reveal that the two scale parameters (nonlocal parameters and material length characteristic parameters) have opposite effects on the stiffness of the nanobeams in the two driving models, while the surface parameters have the same effect on the stiffness of the two driving models. The influence of the slenderness ratio on the surface effect and scale effect is opposite, meaning that the increase of the slenderness ratio deepens the influence of the surface effect but weakens the influence of the scale effect. There are also differences in the effects of higher-order modes on the two effects. Higher modes lead to more significant scale effects, but the effect of higher modes on surface effects depends on the surface elastic properties of the material. We also find that the introduction of surface elasticity increases the gap between the TBT and other higher-order beams, which indicates the prediction results of the higher-order beam model are more accurate when both surface and nonlocal effects are considered. In addition, it is represented that the surface elasticity makes aluminum nanobeams exhibit a stiffness softening effect, while the effect of surface elasticity on the stiffness of silicon nanobeams is significantly dependent on the slenderness ratio and the number of modes.

A 1D Ritz–Jacobi formulation for the modal analysis of 3D anisotropic laminated composite and soft-core sandwich beam structures through 2D polynomials

The present article deals with the computation of natural frequencies of 3D fibre-reinforced anisotropic laminated composite and soft-core sandwich beams by using a 1D Ritz–Jacobi formulation. Generalised higher-order beam models, based on both 2D Taylor and 2D Chebyshev polynomials, are employed in the investigation. The weak-form of the governing equations (GEs) is derived via the classical Hamilton’s principle. Jacobi polynomials are selected to be the admissible functions used to solve the GEs. Various subsets of these polynomials are considered: Legendre polynomials, Lobatto polynomials, Gegenbauer polynomials (or Ultraspherical polynomials) and Chebyshev polynomials of 1st, 2nd, 3rd and 4th kind. Convergence and stability of both cross sectional functions and admissible functions are thoroughly analysed. Several parametric analyses are performed and the effect of some relevant parameters, e.g. lamination angle, boundary condition, stacking sequence, slenderness ratio and composite type, is studied.

A higher-order morphoelastic beam model for tubes and filaments subjected to biological growth

Slender structures, in which width and thickness are much smaller than the length, are common in the natural world, such as wheat straws, bean sprouts, and biofilaments. Due to biological growth, these structures exhibit diverse morphological profiles with curling hairs, curling leaves, and twining plants as some examples. Accurate prediction of growth induced instabilities and corresponding shear deformation require a higher-order morphoelastic beam theory for growing tubes and filaments, for which we present a theory here. In the theory, the deformation gradient is decomposed into elastic deformation and growth using multiplicative decomposition. Appropriate assumptions for slender structures with circular cross-section are introduced and the displacement field and constitutive relations are derived from three-dimensional morphoelasticity. Corresponding variational principle is established and equilibrium equations are obtained. Our higher-order beam with growth theory can model growth-induced instability and shear deformation of growing tubes and filaments with circular cross-sections. Importantly, it can predict shear stress distribution and locations of maximum transverse shear stress in the circular cross-section. The shear stress, critical growth and post-buckling predictions of our proposed model and analytical solutions were validated by comparing its predictions against results from three-dimensional finite element simulations. Growth-induced instability and corresponding shear stress distributions were analyzed and the results are discussed. Our model can provide improved prediction for critical growth induced instabilities which are overestimated if the shear deformation is not considered.

Vibration analysis of functionally graded beams using a higher-order shear deformable beam model with rational shear stress distribution

This paper extends the higher-order shear deformable mixed beam element model with rational shear stress distribution to vibration analysis of functionally graded (FG) beams. In the element model proposed, the stress equilibrium equation is employed to derive the rational distribution of transverse shear stress. The beam element formulation oriented with vibration analysis is derived on the basis of the variational principle with mixed energy. To visualise the effectiveness and applicability of the proposed mixed beam element model in vibration analysis, numerical examples have been studied. According to the numerical results, in comparison with the classical beam element model and the beam element models based on the conventional first-order or higher-order shear deformation theory, the vibration frequencies obtained using the mixed beam element model are significantly more satisfactory. Moreover, the comparative study demonstrates that, for vibration analysis of FG sandwich beams, the rational distribution of transverse shear stress is of decisive significance to achieve high-precision solutions. With the mixed beam element used, the load–frequency relationship of FG sandwich beams is discussed. As indicated by the results, in addition to the axial force, the bending moment exerts a significant difference upon the vibration frequencies of the FG beams.

A higher-order size-dependent beam model for nonlinear mechanics of fluid-conveying FG nanotubes incorporating surface energy

Fluid-conveying tubes in nano scale present different mechanical behaviors from the macro tubes. A higher-order size-dependent beam model is developed coupling with the nonlocal stress, strain gradient effects, surface energy effects for flow-inducing post-buckling and free vibration analysis of the functionally graded nanotubes. Also, the slip flow effect is considered to modify the kinetic energy of the fluid. The governing equations of post-buckling and vibration are derived by the Hamilton variational principle in which the obtained post-buckling tube configuration is also the initial configuration for vibration analysis. The two-step perturbation method is extended to the nonlinear analysis of the fluid-conveying tubes, and a perturbation scheme is proposed. Subsequently, the influences of nano effects on post buckling, natural frequency and nonlinear vibration of the fluid-conveying nanotubes are studied. The imperfection sensitivity of the static and dynamic behaviors is also discussed. Numerical results reveal the dual influences of nano effects on the nonlinear behaviors and indicate the significance to consider the effects of surface energy, small scale and slip flow in nonlinear analyses of the fluid-conveying nanotubes in a comprehensive way.

A simple higher-order beam model that is represented by two kinematic variables and three section constants

This article presents a new higher-order beam model. The present beam model is governed by differential equations that are similar to those present in some existing higher-order beam models; however, the present beam model makes use of a novel method of calculating the transverse shear stiffness, which facilitates the calculation of a shear-warping stiffness without the need for an assumed warping displacement field, and without introducing any additional kinematic variables. The present beam model also facilitates the recovery of the distributions of longitudinal normal stresses and transverse shear stresses. The authors postulate that the bending and shear terms in first-order shear deformation theory represent the first two terms in an infinite series that would constitute an ideal one-dimensional beam model, and it is suggested that the present beam model constitutes the first four terms in this hypothetical infinite series. The present beam model is solved for several example beams, and the results are compared with those of existing classical and higher-order beam models, as well as computational results from finite element analyses. It is shown that the present beam model is able to accurately represent deformed shapes and stress distributions pertaining to beams that exhibit non-trivial shear compliance as well as non-trivial shear-warping stiffness. In the case of laminated composite beams comprising a large number of laminae, the present beam model offers a level of analytical fidelity that is comparable to that of existing zigzag beam models; however, unlike zigzag beam models, the present beam model is equally well suited for the analyses of beams comprising any number of laminae.

A higher-order beam model for the snap-buckling analysis of FG pipes conveying fluid

The snap buckling of the FG curved pipes conveying fluid has not been reported due to the existing research on the snap-buckling problem. Therefore, the purpose of this paper is to explore this issue. First, we adopt a new high-order shear theory model and consider the thermal and geometric nonlinearity effects, and assume that the density and modulus of elasticity of the liquid are independent of temperature. Based on the generalized variational principle, the governing equation of the FG curved pipes conveying fluid is derived. Then, we assume that the FG curved pipes conveying fluid has simply supported boundary or fixed supported boundary conditions, and use the two step perturbation method to obtain the expression of the relationship between load and deflection. Then, we investigate the influence of boundary conditions, shear deformation, temperature variation, functional gradient index parameters, liquid flow velocity and geometry size on the snap buckling problems of the FG curved pipes conveying fluid. The results show that these factors have significant influence on the fluidstructure interaction problems.

A Comparison of Nonlinear Bending and Vibration of Hybrid Metal/CNTRC Laminated Beams with Positive and Negative Poisson’s Ratios

Carbon nanotube reinforced composite (CNTRC) is one of the novel classes of advanced composite materials. This paper investigates the nonlinear bending and nonlinear vibration responses of hybrid laminated beams made of CNTRC layers bonded with metal layers on the top and bottom surfaces. We proposed a hybrid metal/CNTRC laminated beam for which out-of-plane Poisson’s ratio is negative. The effective material properties of CNTRC layers are graded in a piece-wise pattern across the thickness of the beam. The material properties of both CNTRC layer and metal layer are temperature dependent. The beams are rested on an elastic foundation and are located in thermal environments. Reddy’s higher-order beam model is used to establish the motion equations of the hybrid metal/CNTRC laminated beam. The motion equations include the von Kármán geometric nonlinearity, the thermal effects and the beam-foundation interaction. By employing a two-step perturbation approach, the bending and nonlinear vibration solutions are obtained. A comparison of nonlinear responses of hybrid metal/CNTRCl laminated beams with positive and negative Poisson’s ratios (NPRs) under different thermal environmental conditions is carried out and discussed in detail.

Convergence studies of finite element model for analysis of steel-concrete composite beam using a higher-order beam theory

The finite element model based on a higher-order beam theory (HBT) is the most favourable modelling technique for an accurate prediction response of steel-concrete composite beams. 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 with the interface between these two material layers. For the implementation of finite element formulation, the field consistent technique and full numerical integration of the stiffness matrix is carried out in order to avoid any shear locking and stress oscillation problem. There are four different types of beam elements incorporated to develop models to observe an optimal number of beam element required for good convergence. The proposed model is assessed through validation and verification using existing published results and numerical results produced by detailed 2D finite element modelling of steel-concrete composite beams. The validation and verification show a very good performance of the proposed finite element models.

Buckling and post-buckling of FRC laminated beams in thermal environment using a generalized higher-order shear deformation zig-zag beam model

The thermal buckling and post-buckling of fibre-reinforced composite (FRC) beams are analyzed based on a generalized higher-order shear deformation zig-zag beam theory (RHDZT). The current zig-zag beam model satisfies the stress boundary conditions on the upper and lower surfaces and the reality that the shear strain is discontinuous at the layer interfaces. The material properties of FRCs are assumed to be temperature independent and the in-plane boundary conditions supposed immovable. By using the Hamilton variation principle, the governing equations are established based on the mid-plane of the beam. A two-step perturbation method is employed to obtain the approximate analytical solutions of the critical thermal buckling loads and post-buckling behaviors with clamped-clamped boundary conditions. The critical buckling results based on the zig-zag model and its degenerate equivalent single-layer models are compared and verified with those in the literatures. The post-buckling results of the FRC beams with different slenderness ratios, elastic modulus ratios and ply angles are investigated. Numerical results show the remarkable accuracy of the current model and also demonstrate the significant roles the above-mentioned effects play in the buckling and post-buckling problem of FRC laminated beams.

Imperfection sensitivity of free vibration of symmetrically/anti-symmetrically laminated FRC beams in thermally pre-and post-buckling equilibrium states

Linear/nonlinear free vibration of imperfect symmetrically/anti-symmetrically laminated fibre-reinforced composite (FRC) beams in the pre-and post-buckling equilibrium states are studied. It is assumed that the initial defect has the same shape as the first-order buckling mode and the in-plane boundaries are immovable. A modified higher-order shear deformation zig-zag beam model is applied for refined modeling of the laminated beams. The Hamilton variational principle is utilized to obtain the governing equations of motion. The two-step perturbation technique is combined with the harmonic balance method to obtain the asymptotic analytic solutions. The results based on the zig-zag model and its degenerate equivalent single-layer models are compared and verified. Further, the sensitivity of vibration characteristics to the size of imperfections at different temperature rise is investigated. Numerical results suggest that there exists a load range near the critical buckling load of the perfect beams where the linear/nonlinear frequency is sensitive to the initial imperfections.

A material-based higher-order shear beam model for accurate analyses of FG beams with arbitrary material distribution

A material-based higher-order shear beam model is proposed for accurate analyses of functionally graded (FG) beams with arbitrary material distribution through thickness. Distinct from the existing higher-order shear beam models, the shear function of the present model is dependent on the material distribution. The specific form of this material-based shear function is described by a piecewise linear interpolation field and determined by ensuring the consistency of transverse shear stress distributions between Euler-Bernoulli beam theory and the higher-order shear beam theory. Based on the proposed beam model, a 3-node beam finite element is developed for static and buckling analyses. Numerical examples are studied to visualize the accuracy and effectiveness of the proposed model and finite element. Solutions of static displacements and critical loads obtained by various beam elements including the proposed element, the existing displacement-based beam elements and the mixed higher-order shear beam element are compared. The accuracy loss of the existing higher-order shear beam models is discussed. As demonstrated by the results, the proposed beam model and finite element have excellent solution accuracy and desirable applicability.

Large Deflection of Elastic Beams under Impact by Rigid Particles

The structural response of a slender beam subjected to impact by several rigid particles is investigated. A higher-order shear deformation beam model and the von-Karman geometric measure are employed. The equations of motion, continuity conditions at the points of impact by particles, and geometric and natural boundary conditions for the beam are derived by means of Hamilton’s principle. The generalized differential quadrature and Newmark time integration methods are utilized to carry out the numerical solution of the system of nonlinear partial differential equations comprised of the equations of motion and stress resultants. The durations of contact, displacements, and stress resultants are obtained for beams with various types of boundary conditions.

On wave propagation in two-dimensional functionally graded porous rotating nano-beams using a general nonlocal higher-order beam model

This paper studies the wave propagation of two-dimensional functionally graded (2D-FG) porous rotating nano-beams for the first time. The rotating nano-beams are made of two different materials, and the material properties of the nano-beams alter both in the thickness and length directions. The general nonlocal theory (GNT) in conjunction with Reddy's beam model are employed to formulate the size-dependent model. The GNT efficiently models the dispersions of acoustic waves when two independent nonlocal fields are modelled for the longitudinal and transverse acoustic waves. The governing equations of motion for the 2D-FG porous rotating nano-beams are established using Hamilton's principle as a function of the axial force due to centrifugal stiffening and displacement. The analytic solution is applied to obtain the results and solve the governing equations. The effect of the features of different parameters such as functionally graded power indexes, porosity, angular velocity, and material variation on the wave propagation characteristics of the rotating nano-beams are discussed in detail.