Order Beam Theory Research Articles

Stress resultants governing equations of any beam theory by direct manipulation of 3D equilibrium

In this article, the author shows that the equilibrium equations of any one-dimensional theory can be obtained by direct manipulation of the indefinite 3D elasticity equations: It is found that there is no need to refer to Newton-type or Lagrange-type equations to write the equilibrium in terms of stress resultants. Such ‘direct manipulation’ is illustrated by considering various problems, from the simplest to the most complex. First, the bar case is analyzed and the well-known equilibrium normal stress resultant equation is obtained: second, the Timoshenko beam theory is tackled; finally, the case of arbitrary higher order beam theories based on the Carrera Unified Formulation is treated. The equations of a new, unprecedented higher-order beam theory are derived directly from the indefinite equilibrium equations as an example of the capabilities of the proposed method. The manipulation introduced consists of the following points: (i) the 3D elasticity is first written for each degree of freedom (dof) of the beam theories; (ii) the respective stress resultants are used to replace stress components in the latter equations; (iii) the derivatives on the beam axis z do not affect the nature of the indefinite equilibrium equations and these derivatives are applied directly to the respective stress resultants; (iv) the derivatives on the section coordinate x – y should 1 – change the sign of the respective stress resultants, 2 – be applied to the functions of x, y used to define the stress resultants. These points have been proved by referring to the principle of virtual work and splitting the differential operators of the strain-displacement relations into two parts: over the section derivatives along the beam axis derivatives. Some possible further generalizations are discussed in the appendix; in particular, the extension to is briefly discussed.

Bending Analysis of Steel-Concrete Composite Beams with Porosity

This study investigated the bending behavior of a simply supported pervious composite beam using the finite element method based on cubic order beam theory. The cubic order axial displacement was assumed and solved using shear stress-free conditions on the extreme surfaces of the composite beam. For simplicity, this study considered one-dimensional axial displacement. Three nodded finite elements were assumed, and each node had eight unknowns. Shear locking was eliminated in the present model by numerical integration of the stiffness matrix. A uniform porosity distribution was implemented in the upper layer of the composite beam. The damping ratio was also considered for the bending analysis. The accuracy of the present model makes it robust for the analysis of composite beams.

Free Vibration Analysis of Steel-Concrete Pervious Beams

This study investigated the free vibration analysis of steel-concrete porous beams with partial or complete shear interface using a finite-element model based on the cubic order beam theory. The present model assumes uniform porosity distribution along the beam thickness. It is assumed that the axial displacement will vary cubically along the thickness of the layer. The cubic order beam theory is implemented using a continuous C0 finite element containing three nodes and each node has eight degrees of freedom. Shear locking is eliminated in the present model by the numerical integration of the stiffness matrix. Comparing the present model with the published literature, it is found that the present model is robust in predicting the free vibration of the steel-concrete porous beam.

Nonlocal state-space strain gradient wave propagation of magneto thermo piezoelectric functionally graded nanobeam

Abstract In this work, the state -space nonlocal strain gradient theory is used for the vibration analysis of magneto thermo piezoelectric functionally graded material (FGM) nanobeam. An analysis of FGM constituent properties is stated by using the power law relations. The refined higher order beam theory and Hamilton’s principle have been used to obtain the motion equations. Besides, the governing equations of the magneto thermo piezoelectric nanobeam are extracted by developed nonlocal state-space theory. And to solve the wave propagation problems, the analytical wave dispersion method is used. The effect of magnetic potential, temperature gradient, and electric voltage in variant parameters are presented in graph.

An analysis of vibration and buckling behaviors of nano-composite beams reinforced with agglomerated carbon nanotubes via differential quadrature finite element method

This article aims to investigate the buckling and free vibration behaviors of nanocomposite beams reinforced with functionally graded (FG) agglomerated carbon nanotubes (CNTs) in polymeric beams matrix. In order to estimate the material properties of the FG CNTs reinforced beam, the Eshelby–Mori–Tanaka rule on the basis of equivalent fiber is employed. Moreover, the Lagrange’s principle is used to achieve the governing equations under supposition of the refined higher order beam theory without need of any shear correction factor. The differential quadrature finite element method (DQFEM) as a robust and precise numerical procedure is utilized to achieve the natural frequencies and buckling loads with various boundary supports. Then, the assessment of the effects of different factors such as of CNTs agglomeration parameters, different patterns of CNTs distribution, length to thickness quotients, and mixed boundary conditions on the vibrational and buckling responses of agglomerated CNTs reinforced beam are explored. It is concluded that the current numerical solution based DQFEM is efficient and accurate for predicting the mechanical behavior of the nanocomposite structures. Moreover, the stability and vibration responses of the NC beams can be significantly affected by tailoring the agglomeration parameters.

FLEXURAL ANALYSIS OF THERMALLY LOADED SYMMETRIC SANDWICH BEAM

In the present paper, an attempt has been made to study the effect of temperature gradient on simply supported symmetric sandwich beam. A Navier’s solution technique is used. The temperature profile is assumed to be linear across the thickness of a sandwich beam. A higher order beam theory (HBT) is used to include the effect of shear deformation on thermal flexural response of the sandwich beam. The theory satisfies the shear stress free boundary condition at the top and bottom surfaces of the sandwich beam. No shear correction factor is required. The principle of virtual work is used to obtain the governing equations and boundary conditions. A program has been developed in FORTRAN-77 to obtain thermal stresses and displacements in the sandwich beam for various aspect ratios. The numerical results are presented for moderately thick and thin sandwich beams to assess the performance of the theory. The validity of the present theory is verified by comparing the results with the results available in the literature. The present results are in good agreement with the results of other theories.

Vibration analysis of cracked beam based on Reddy beam theory by finite element method

The lateral vibration of cracked isotropic thick beams is investigated. Generally, the analysis of thick beam based on line elements can be undertaken using either Timoshenko beam theory or a third order beam theory (TSDT) such as Reddy beam theory. TSDT is superior to Timoshenko beam theory, as it eliminates the need for a shear correction coefficient. However, there is no available solution for a cracked beam based on Reddy beam theory which is the main focus of this paper. The investigated beam is divided into several elements where their stiffness and mass matrices are derived using Reddy beam theory. Each element has two nodes with 3 degrees of freedom (1 lateral and 2 rotational) at each node. The crack was modeled using two rotating springs with stiffnesses that varied with crack depth. These elements are used to connect the adjacent beam elements. The impact of boundary conditions, slenderness ratio, crack location, and crack depth are investigated. In addition, experimental and finite element analyses of cracked steel beams is undertaken to validate the results of the present model. The results show that the maximum deviation between the analytical and the experimental results is less than 3% up to the third mode shape.

Novel bi-layer beam elements for elastic fracture analysis of delaminated composite beams

A general finite element method is proposed for linear elastic fracture analysis of delaminated bilayer composite beams. An interface deformable bilayer beam model is adopted in modeling the virgin portions of bilayer beams, and it is capable of guaranteeing the continuity condition of internal forces at the tip of delamination because the bilayers are modeled as two separate shear deformable sub-beams and both the interface normal and shear compliances are introduced to account for interface deformation. The stiffness matrices of bilayer beams using the linear and quadratic interpolation polynomials are both derived using the principle of minimum potential energy. Then, a finite element program is developed by assembling the element stiffness matrices of virgin and delaminated portions of bilayer beams and applying the corresponding equivalent nodal forces. Two typical fracture specimens (i.e., single leg bending and end-notched flexure bilayer beams) are analyzed using the proposed bilayer beam elements. The computed displacement and energy release rate are compared well with those of analytical solutions, and the mode components of energy release rate for single leg bending specimen are also obtained using the virtual crack closure technique. It demonstrates that the proposed novel bilayer beam elements are capable of effectively and efficiently predicting the displacement and energy release rate (including its mode components) of delaminated bilayer beams. The developed bilayer beam element is general in nature and accurate and efficient in particular, because it can achieve the same accuracy as the high order beam theory and be easily implemented in the existing commercial finite element software.

Symplectic analysis of thin-walled curved box girders with torsion, distortion and shear lag warping effects

Shear lag effect and torsional and distortional warping can significantly affect the performance of thin-walled curved box girders used in modern bridge engineering. The structural behaviour of these bridges exhibits complexity due to coupled bending and torsion together with warping effects of non-uniform torsion, distortion and shear lag. A practical method of analysis, based on the symplectic approach, with the same perspective as the Higher Order Beam Theories, is presented for overcoming the difficulties of numerical approaches via the Finite Element Method. In this paper, the Hamiltonian Structural Analysis method implements the analysis of the shear lag effect together with non-uniform torsion and distortion, for curved box girder bridges. The examples given can allow engineers to evaluate the effects of the negative shear lag phenomenon and the distribution of axial stresses in box slabs and webs, due to different warping effects, which strongly influence thin-walled box sections subjected to point loads and restrained areas.

Design and Characterization of Asymmetric Cell Structure of Auxetic Material for Predictable Directional Mechanical Response.

A three-dimensional auxetic structure based on a known planar configuration including a design parameter producing asymmetry is proposed in this study. The auxetic cell is designed by topology analysis using classical Timoshenko beam theory in order to obtain the required orthotropic elastic properties. Samples of the structure are fabricated using the ABSplus fused filament technique and subsequently tested under quasi-static compression to statistically determine the Poisson’s ratio and Young’s modulus. The experimental results show good agreement with the topological analysis and reveal that the proposed structure can adequately provide different elastic properties in its three orthogonal directions. In addition, three point bending tests were carried out to determine the mechanical behavior of this cellular structure. The results show that this auxetic cell influences the macrostructure to exhibit different stiffness behavior in three working directions.

Vibration analysis of thin-walled functionally graded sandwich beams with non-uniform polygonal cross-sections

In the paper, an analysis of thin-walled functionally graded straight and curved beams for general non-uniform polygonal cross-sections has been introduced for vibration problems. Higher order beam theory that has fully taken into account major deformations, e.g. in-plane distortion, out-of-plane warping, is adopted by means of beam frame modal approach. In addition with the effects of geometric parameters, the study also deals with some consequences resulting from material anisotropy which in turn tackles better in evaluation behaviors of thin-walled beams that enables analytical modeling to have the potential for further extension. Subsequently, the analysis can be applied for general closed cross-section beams with arbitrary curvatures whereas materials properties are assumed to be graded across the wall thickness following a predefined shape function. A numerical verification for the overall analysis and examples have been carried out based on the available results created from benchmark problems and finite element software simulation found in literature. Then, natural frequencies and vibrational mode shapes have been obtained through a robust framework of finite element method. Finally, a parametric study and discussion towards vibration phenomena are fully investigated referring to gradual law, skin-core-skin effects, etc.

Influence of magnetic field on the wave propagation response of functionally graded (FG) beam lying on elastic foundation in thermal environment

In the current work, wave propagation in simply supported (S-S) functionally graded (FG) beam is investigated while the beam is exposed to magneto-thermal environments and embedded on two-parameter elastic foundation. The mechanical properties of the structure are supposed to vary through the thickness via the exponential and power law function. The governing equations of the motion are derived by implementing Hamilton’s principle on the basis of refined higher order beam theory which can capture the shear deformation effect. Two novel metal and ceramic with interesting properties named Osmium and Silicon Carbide (SiC) are used as materials of this structure. The thermal loadings are applied to the structure with various temperature changes namely uniform, linear and sinusoidal, through the thickness. Influences of the wide range of parameters such as magnetic field intensity, slenderness ratio, various thermal loadings, exponential- and power-law exponents and elastic foundation coefficient on the wave frequency and phase velocity of the FG beam have been illustrated comparatively and the results are thoroughly discussed to obtain the highlights of each figure.

Comparative Study of Beam Theories on the Effect of Span-Depth Ratio for Symmetric and Un-symmetric Loadings

In the present work, a comparative study is done between Shear-indeformable (Euler-Bernoulli) model and Shear-deformable (Timoshenko) model for two different types of beams (simply supported and fixed beam) under uniformly distributed loading condition and analysed for different Span-Depth ratio (L/D). For un-symmetric loading, a simply supported beam subjected to external point moment is also analysed for different L/D ratios to check the behaviour of beam. Transverse Displacement is taken as a yardstick. Shape functions for Euler-Bernoulli and Timoshenko beam models are developed in MATLAB software by using the Finite Element Method for calculating deflection at different points on the beam. It was observed that for large L/D ratio, the deflections are same for both the beam theory models but as L/D ratio decreases, the deflections for both the beam theories vary from each other. For small L/D ratio, Timoshenko beam model gives more accurate results, since the Timoshenko beam theory is a higher order beam theory than the Euler-Bernoulli beam theory, it is known to be superior in predicting the response of the deep beam.

On the static deformation of FG sandwich beams curved in elevation using a new higher order beam theory

This article presents the static analysis of FG sandwich beams curved in elevation. Navier-type semi-analytical solutions are obtained based on polynomial type fifth order shear and normal deformation theory. The beam has FG skins and isotropic core. Material properties of FG skins are graded in z-direction according to the power-law distribution. The present theory accounts for a fifth-order distribution of axial displacement and fourth-order distribution of transverse displacement. The present theory considers the effect of thickness stretching and gives a realistic variation of transverse shear stress through the thickness of the beam. The governing equations are obtained within the framework of the principle of virtual work. Semi-analytical static solutions for the simply supported FG sandwich beams curved in elevation are obtained using Navier’s technique. The beam is subjected to uniformly distributed load. The non-dimensional numerical values for displacements and stresses are obtained for various power-law index and thickness of the core. The present results are found in good agreement with previously published results.

Considering Bending and Vibration of Homogeneous Nanobeam Coated by a FG Layer

In this research static deflection and free vibration of homogeneous nanobeams coated by a functionally graded (FG) layer is investigated according to the nonlocal elasticity theory. A higher order beam theory is used that does not need the shear correction factor. The equations of motion (equilibrium equations) are extracted by using Hamilton’s principle. The material properties are considered to vary in the thickness direction of FG coated layer. This nonlocal nanobeam model incorporates the length scale parameter (nonlocal parameter) that can capture the small scale effects. In the numerical results section, the effects of different parameters, especially the ratio of thickness of FG layer to the total thickness of the beam are considered and discussed. The results reveal that the frequency is maximum for a special value of material power index. Also, increasing the ratio of thickness of FG layer to the total thickness of the beam increases the static deflection and decreases the natural frequencies. These results help with the understanding such coated structures and designing them carefully. The results also show that the new nonlocal FG nanobeam model produces larger vibration and smaller deflection than homogeneous nonlocal nanobeam.

Assessment of first and third order shear deformation beam theories for the buckling and vibration analysis of nanobeams incorporating surface stress effects

A novel approach for the vibration and buckling analysis of nanobeams with surface stress effects is presented in this paper. The material model is derived in which the bulk core of a beam is assumed to be a single crystalline material and have certain crystallographic directions. The beams are assumed to be covered by atomic layers with the same crystallographic directions on the top and bottom surfaces. A generic third order shear deformable beam theory is employed to derive the total energy functional of the beams. Minimizing the total potential energy functional by applying the p-Ritz method, the eigenvalue equations for the buckling and vibration of nanobeams with surface stress effects are obtained. The buckling and vibration behaviours of the beams considering the surface stress effects are discussed in detail and are compared with the results obtained from the first order beam theory and the Reddy's third order beam theory. It is found that in certain cases, both the first order and the Reddy's third order beam theories may not be able to capture the surface effect accurately on the vibration and buckling behaviours of the nanobeams. The effects of surface stresses on the buckling and vibration behaviours for nanobeams made of Ni single crystals obtained by the generic third order beam theory showed opposite trends to the ones obtained by the first order or the Reddy's third beam theory.

Multimode guided wave extraction capabilities using embedded thin film sensors in a composite laminated beam

A novel method to extract individual wave modes from the ultrasonic multimode guided wave response sensed by embedded piezoelectric sensors is presented. Multimode response involving two fundamental modes, i.e., flexural and axial, and two higher-order wave-modes, i.e., shear and thickness contraction have been considered in the present work. The model is based on higher order beam theory which uses wave kinematics to extract individual wave modes from the multimode response. Further, a numerical finite element model is employed to validate the deformation profile of this analytical model. The embedment of thin film sensors at the same location but within different layers produce signals that exhibit a relative difference in the phase and amplitude due to kinematics. The sensor signals are then post-processed to obtain individual Lamb wave modes. Experiments are carried out to validate the procedure for extracting individual guided wave mode response. The method is independent of sampling rate and works in the presence of complicated mode converted responses. Electronic analog circuits can also be used to perform operations on sensor signals before analog-to-digital conversion, thereby eliminating digital signal processing and the requirement of onboard computers for structural health monitoring especially in aircraft. Such capability minimizes power consumption and weight since we may be able to avoid the use of a computer.

Popular benchmarks of nonlinear shell analysis solved by 1D and 2D CUF-based finite elements

This research work deals with the analysis of elastic shell structures in the large displacement and rotation field adopting one-dimensional (1D) and two-dimensional (2D) unified models. Namely, higher order beam and shell theories accounting for geometrical nonlinearities are formulated by employing a unified framework based on the Carrera unified formulation (CUF) and a total Lagrangian approach. Thus, a finite element (FE) approximation is used along with a Newton-Raphson method and an arc-length path-following approach to perform nonlinear analyses. Low- to higher order beam and shell theories are used to evaluate the nonlinear equilibrium path, and results are compared between the two models, with reference solutions coming from literature or 2D and three-dimensional (3D) models from NASTRAN. Convergence analyses show how CUF 1D models are able to describe the geometrical nonlinear behavior of analyzed structure with a lower number of degrees of freedom (DoFs) than 2D and 3D models.

Static stability analysis of agglomerated multi-scale hybrid nanocomposites via a refined theory

Present paper is proposed to capture the influences of carbon nanotubes’ agglomeration on the stability behaviors of multi-scale hybrid nanocomposite beams within the frameworks of refined higher order beam theories for the first time. In this research, a mixture of macroscale and nanoscale fillers will be utilized to be dispersed in an initial matrix to possess a multi-scale hybrid nanocomposite. The equivalent material properties are seemed to be calculated coupling the Eshelby–Mori–Tanaka model with the rule of the mixture to consider the effects of carbon nanotubes inside the probably generated clusters while finding the mechanical properties of such novel hybrid nanocomposites. Furthermore, an energy-based approach is implemented to obtain the governing equations of the problem utilizing a refined higher order beam theorem. Next, the derived equations will be solved in the framework of Galerkin’s well-known analytical method to reach the critical buckling load. It is worth mentioning that influence of various boundary conditions is included, too. Once the validity of presented results is proven, a set of numerical examples are presented to explain how each variant can affect the structure’s stability endurance.

Existence of second spectrums of Timoshenko beam and Mindlin–Herrmann rod theories on the basis of atomistic studies

This paper considers a single-layer 2D triangular lattice system as an ideal higher-order rod and/or beam. The objective is to compare the solutions of these continuum models with the standard atomistic solutions. The accuracy of these higher-order rod and beam theories, in modeling several nano-systems, depends on the frequency of vibration. The existence of a second spectrum associated with these higher-order continuum theories and its contribution to the continuum solutions are studied on the basis of molecular dynamics (MD) results. It is shown that the second spectrum of a Timoshenko beam is less accurate with respect to the MD results but cannot be disregarded totally. Also, the existence of a second propagating mode associated with a Mindlin–Herrmann rod is substantiated by MD simulation results. The accuracy of these higher order rod and beam theories deteriorates beyond their cut-on frequencies. However, the use of higher-order rod and beam theories in several high-frequency dynamic and multiscale analyses is still justifiable.