Shear Deformation Beam Research Articles

A Higher-Order Theory for Nonlinear Dynamic of an FG Porous Piezoelectric Microtube Exposed to a Periodic Load

This paper investigates the nonlinear dynamic deflection, natural frequency, and wave propagation in functionally graded (FG) porous piezoelectric microscale tubes under periodic load, hygrothermal conditions, and an external electric field. The piezoelectric material used to make the smart microtubes has pores that may be smoothly changed or uniformly distributed over the tube wall. Here, three types of porosity distribution are taken into consideration. The nonlinear motion equations are constructed using a novel shear deformation beam theory and the modified couple stress theory (MCST). The nonlinear motion equations are solved using the fourth-order Runge–Kutta technique and the Galerkin approach. The effects of various geometric parameters, porosity distribution type, porosity factor, periodic load amplitude and frequency, material length scale parameter, moisture, and temperature on the nonlinear dynamic deflection, natural frequency, and wave frequency of FG porous piezoelectric microtubes are explored through a number of parametric investigations.

Active control for higher order micro sandwich beam with various cores integrated by piezoelectric layers based on MSGT on the Pasternak foundation

In the present article, the vibration suppression of higher order micro sandwich beams based on modified strain gradient beam theory (MSGT) and Reddy/Timoshenko beam theories is developed. Three types of core, including honeycomb, foam, and porous for a micro sandwich beam, are considered. The core’s top and bottom are integrated by piezoelectric layers as actuator and sensor, respectively. Displacement fields are defined via third-order shear deformation beam theory (TSDBT) (or Reddy beam theory (RBT)) and first-order shear deformation beam theory (FSDBT) (or Timoshenko beam theory (TBT)). The governing equations of motion are derived using MSGT to consider the small-scale effect. To optimal approaches for the controller, state feedback is designed based on a linear quadratic regulator (LQR) in the state-space form. Moreover, the effect of various parameters, including small-scale parameters, three types of core, temperature changes, Pasternak foundation, density of core, dimensionless cell thickness ([Formula: see text]), and internal aspect ratio ([Formula: see text]), angles ([Formula: see text]) of honeycomb, and the porosity parameter is performed on the active control of the smart micro sandwich beam. The numerical results indicate that the parameters mentioned significantly affect the optimal vibration control of the smart micro sandwich beam. It is found that the lightweight cores have more attenuation of the transient response and faster settling time. By analyzing the effect of temperature changes on active vibration control of lighter core, it was found that the frequency decreases as the temperature increases. Also, by considering the Pasternak foundation, it is found that the time period without considering the elastic foundation is longer than when considering the elastic foundation, and the natural frequency is vice versa. Also, the settling time by considering the elastic foundation occurs earlier. In the active control field of smart micro beams, this theoretical investigation developed widely as a benchmark for experimental research and the manufacturing process.

First Principles Variational Formulation and Analytical Solutions for Third Order Shear Deformable Beams with Simple Supports using Fourier Series Method

The Fourier series method for solving bending problems of third-order shear deformable beams (TSBD) is presented in this paper. The theory accounts for transverse shear deformation and is suitable for moderately thick beams. Transverse shear stress-free conditions are valid at the beam surfaces for TSDB. The field equations are two coupled ordinary differential equations in terms of two unknown displacement functions – transverse deflection w(x) and warping function For simply supported ends considered, the loading and unknown functions w(x) and are represented in Fourier series that satisfies the boundary conditions. The problem is reduced to a system of algebraic equations in terms of the Fourier coefficients wn and which is solved to obtain wn and Axial and transverse displacements fields, axial bending stress and transverse shear stress are then determined for the cases of point load at midspan, uniformly and linearly distributed loads over the span. The results are identical with previous results obtained by other scholars who used Ritz variational methods and other mathematical tools. For moderately thick beams under uniform load with l/h = 4, the results obtained for is 0.516% greater than the exact result by Timoshenko and Goodier while for thick beams under uniform load with l/h = 2, the result for is 1.96% greater than the exact result by Timoshenko and Goodier. Similar acceptable variation was obtained for for beams under linearly distributed load with the variations being less than 2% for l/h = 2. However, unacceptable variations of 68.37% were found for in thick beams under point load, for l/h = 2.0. The variation for however reduced to 12.513% for moderately thick beams under point load for l/h =4.

On the bending, buckling and free vibration analysis of bio-inspired helicoidal laminated composite shear and normal deformable beams

The mechanical behaviours of bio-inspired helicoidal symmetric laminated composite (BIHLC) beams are investigated via the Ritz method. By exploiting the variational formulation, equations of motion along with element stiffness, geometrical stiffness, and mass matrices are derived. The study conducts a thorough examination, covering bending, buckling stability, and free vibration analyses of BIHLC beams with various lamination schemes. The developed model is verified against existing literature on conventional composite laminated and BIHLC beams. The study also examines the mechanical response of BIHLCs, considering boundary conditions, lamination schemes, orthotropy ratios, and aspect ratios. Notably, deflections, critical buckling loads, and fundamental frequencies demonstrate variations dependent on the specific lamination scheme, boundary condition, and aspect ratio. Novel findings, presented for the first time, offer valuable insights for future studies in this area.

Deriving Ritz formulation for the Static Flexural Solutions of Sinusoidal Shear Deformable Beams

This paper presents the Ritz variational method for the static bending analysis of sinusoidal shear deformable beams. The theory accounts for transverse shear deformation and satisfies transverse shear stress-free conditions at the top and bottom surfaces of the beam. The total potential energy functional for the thick beam bending problem is formulated and minimized using a Ritz procedure. The problem considered simply supported boundary conditions and two cases of loading – uniformly distributed load over the span and point load at the center. The function is a function of two unknown displacement functions constructed in terms of unknown generalized displacement parameters and shape functions that satisfy the boundary conditions. Ritz minimization of the functional is used to find the generalized displacement parameters and then the displacements w(x) and The displacements and stresses are found for the loading distributions considered. It is found that the results obtained agree remarkably well with the exact results obtained using the theory of elasticity. The differences between the present results and the exact solutions are less than 0.3% for maximum transverse displacement for both cases of loading considered. For uniform load over the beam, the result for l/t = 10 is 0.112% greater than the exact solution. For the point load at the center, the result for l/t = 10 is 4.468% greater than the exact solution. The increased difference in the point load case is due to the singular nature of the point load problem.

A New Model for Thermal Buckling of FG-MEE Microbeams Based on a Non-Classical Third-Order Shear Deformation Beam Theory

A New Model for Thermal Buckling of FG-MEE Microbeams Based on a Non-Classical Third-Order Shear Deformation Beam Theory

Framework of Physically Consistent Homogenization and Multiscale Modeling of Beam Structures

In this work, a framework of physically consistent homogenization and multiscale modeling (PCHMM) is developed to analyze complex slender structures with homogenized one-dimensional beam models. Firstly, it is highlighted that periodic boundary conditions are not valid for the representative volume element (RVE)-based homogenizations if a shear-deformable beam model is to be obtained. Addressing this issue, physically consistent boundary conditions (PCBCs) are discussed in detail, which accounts for the influence of shear forces on the behavior of the slender structure and the corresponding RVE. A procedure of enforcing such boundary conditions is also given, applying sectional stresses to the RVE with fictitious DoFs. Additionally, we introduce a newly developed correct formula to calculate effective sectional properties which guarantee the cross-scale energy conservation though a modified Hill’s condition. The PCHMM framework, with homogenization/dehomogenization procedures using PCBC and the modified Hill’s condition, is also conveniently integrated into a commercial finite element software. The correctness and effectiveness of the developed framework of homogenization and multiscale analysis are demonstrated through several numerical examples, including lattice beams, twist-morphing metamaterial spars, and a realistic wing. The extension of this theory to plate/shell structures will be presented in our future work.

Examination of Beam Theories for Buckling and Free Vibration of Functionally Graded Porous Beams.

This paper examines the accuracy and effectiveness of various beam theories in predicting the critical buckling loads and fundamental frequencies of functionally graded porous (FGP) beams whose material properties change continuously across the thickness. The beam theories considered are classical beam theory (CBT), first-order shear deformation beam theory (FSDBT), third-order shear deformation beam theory (TSDBT), and the broken-line hypothesis-based shear deformation beam theory (BSDBT). Governing equations for those beam theories are formulated by using the Hamilton's principle and are then solved by means of the generalised differential quadrature method. Finite element simulation solutions are provided as reference results to assess the predictions of those beam theories. Comprehensive numerical results are presented to evaluate the influences of the porosity distribution and coefficient, slenderness ratio, and boundary condition on the difference between theoretical predictions and simulation results. It is found that the differences significantly increase as the porosity coefficient rises, and this effect becomes more noticeable for the rigid beam with a smaller slenderness ratio. Nonetheless, the results produced by the BSDBT are always the closest to simulation ones. The findings in this paper will contribute to the establishment of more refined theories for the mechanical analysis of FGP structures.

Vibration and buckling analysis of porous metal foam thin-walled beams with closed section

This paper investigates the vibration and buckling analysis of porous metal foam thin-walled box beams. These beams exhibit a unique structural configuration with symmetrical and asymmetrical porosity distributions along their wall thickness, thereby altering the effective mechanical properties. The first-order shear deformable beam theory is employed and the governing equations are derived using the Hamilton’s principle. Numerical results are presented for porous metal foam thin-walled box beams under simply-supported, clamped–clamped and clamped-free boundary conditions. The effects of various porosity parameters, length-to-side and side-to-wall-thickness ratios on the beams’ performance are also examined. A comprehensive comparison between the porous metal foam thin-walled box beams and their counterparts in the form of equivalent homogeneous are presented.

The Effect of Hyperelasticity and Nonlinearity on the Dynamic Behaviors of Hyperelastic Functionally Graded Beams on Nonlinear Elastic Foundation

Hyperelastic functionally graded materials have a wide range of application prospects in soft robotics and biomedical fields. This paper investigates the nonlinear free and forced vibrations of a hyperelastic functionally graded beam (HFGB) based on higher-order shear deformation beam theory. The geometrical nonlinearity is considered by using the von-Kármán’s nonlinear theory. The three-material-parameter free energy function named as Ishihara model is employed to characterize the hyperelastic material. The power-law gradient form along the thickness direction is adopted. The HFGB is resting on the elastic foundation. The Winkler, Pasternak and nonlinear stiffness coefficients are considered. The time-harmonic external force is applied to the HFGB. The nonlinear governing equations for the vibration of the HFGB are derived by using Hamilton’s principle, and are subsequently transformed into ordinary differential equations via Galerkin’s method. The nonlinear free vibration and primary resonance of the HFGB are investigated analytically by employing the extended Hamiltonian method and multiple scales method, respectively. The results indicate that the power-law index, slenderness ratio, material properties, and elastic foundation parameters have significant influences on the nonlinear frequency of free vibration as well as the frequency–response and force–response curves of forced vibration. The phase plane method is employed to analyze the system’s stability states under various excitation amplitudes. The relative error between the results of the current computational model and the published literature is less than 0.1 percent.

An enhanced quasi-3D HSDT for free vibration analysis of porous FG-CNT beams on a new concept of orthotropic VE-foundations

The original primary objective of this study is to conduct a comprehensive investigation into the free vibration behavior of beams with porosity, reinforced by carbon nanotubes (CNTs). These beams are supported by an arbitrary orthotropic variable elastic foundation (AOVEF). The focus lies on understanding how the presence of porosity and CNT reinforcement, coupled with the complex support provided by the AOVEF, influences the vibration characteristics of the beams. In addition, we intend to examine the impact of a number of micromechanical models on the vibration properties of these beams. Four carbon nanotube variables are introduced to symbolize several CNT variations. CNTs’ mechanical properties are examined through a variety of micromechanical models. Shear deformation effects are considered into account in the framework of higher-order shear deformation (HSDT) beam theory. The equations of motion are constructed using Hamilton’s concept and the equation system for the FG-CNT beam with supported ends is solved via the Navier methodology. A new approach in elastic foundation modeling is the Arbitrarily Orthotropic Variable Elastic (AOP-VE) foundation. It builds on the Winkler layer concept but introduces the idea of a variable response along the length of the beam foundation. Unlike the Winkler-Pasternak model, AOP-VE allows for control over the directional properties of the Pasternak foundation. The study looks into multiple aspects, involving various distribution kinds of CNTs, the impact of Winkler and Pasternak factors, mode values, side-to-length ratio, porosity, and angle variation. The results indicate that these parameters have a major effect on the inherent vibration features of FG-CNT beams. The study results in various novel findings that improve the understanding of the subject topic and set a standard for future studies.

Higher‐order effect of axial force on free vibration and buckling of functionally graded sandwich beams

AbstractThis paper presents a novel higher‐order shear deformation beam theory for analyzing the stability and free vibration of functionally graded (FG) sandwich beams with emphasis on the effect of higher‐order moment (HOM) and cross‐sectional warping. The governing equation of axially loaded FG sandwich beams is derived from three‐dimensional equations of the theory of elastic waves in bodies with homogeneous initial stresses. The characteristic equations for typical end conditions are obtained exactly. The numerical results of the natural frequencies and critical loads are calculated and verified for special cases by comparing them with the existing solutions. The effects of gradient index, core thickness, HOM, and warping shapes on the natural frequencies and critical buckling loads are elucidated for different slenderness ratios and end constraints.

Accurate free and forced vibration behavior prediction of functionally graded sandwich beams with variable cross-section: A finite element assessment

The current investigation proposes an enhanced finite element beam model to assess the dynamic behavior of functionally graded sandwich beams with variable cross-sections. By integrating the shear effect, this beam model integrates both C0 and C1 continuities to generate elementary matrices using an efficient three-unknown shear deformation beam theory. The model focuses on advanced sandwich beams with functionally graded material face-sheets and metallic cores, exploring various geometric arrangements. The originality of this study addresses variable cross-sections in sandwich beams, evaluating the influence of material composition and geometric configurations on free and forced vibration responses, including different boundary conditions. The methodology uses the eigenvalue method for examining free vibration and applies Newmark’s algorithm for solving forced vibration problems. The current model expands the scope of analysis compared to previous models that primarily focus on uniform cross-sections and conventional material distributions. The developed model incorporates non-uniform geometrics, material gradient parameters, and advanced finite element analysis to enhance precision and efficacy through detailed parametric investigations and comparisons with existing literature. This contribution significantly improves the mechanical modeling of composite sandwich structures, particularly those incorporating functionally graded materials and complex geometrical aspects.

Dynamic response of sandwich beam with porous core excited by two harmonic loads travelling in opposite directions

This paper presents the time history response of a sandwich beam with a porous core subjected to two moving harmonic loads with opposite directions. In the modelling, the beam is combined of two isotropic face sheets and a porous core with symmetric porosity distribution. The quasi-3D shear deformation beam theory in conjunction with Hamilton's variational principle is utilized to set up the governing equations of motion. The Navier solution is used to obtain the displacement field. The accuracy of the study is validated by comparing with existing results in the literature for specific cases. Effects of the velocity and excitation frequency of the moving loads on the deflection-time history are investigated and discussed. Numerical results reveal that in the case of the double-moving harmonic loads in the antiphase, the resonance phenomenon cannot occur when the excitation frequency approaches the natural frequency of the beam.

Experimental, analytical, and numerical studies of the energy absorption capacity of bi-material lattice structures based on quadrilateral bipyramid unit cell

The present study investigates the mechanical performance and energy absorption capacity of bi-material 3D lattice structures via experimental, analytical, and numerical approaches. The analytical model has been developed to obtain the effective parameters in energy absorption, such as equivalent elastic modulus and yield stress. Analytical relations based on hyperbolic shear deformation beam theory were extended to calculate the mechanical properties of an extracted unit cell from the lattice structure subjected to applied loads and appropriate boundary conditions. Then, experimental and nonlinear numerical studies have been conducted to analyze the energy absorption properties of the bi-material lattice structure. Quasi-static compression tests were conducted to analyze this lattice structure's mechanical properties and energy absorption capacity. As the numerical study, the elastic-plastic damage behavior was implemented in finite element analyses to examine the nonlinear response of considered structures. The obtained results reveal that the numerical models exhibit an acceptable prediction. According to the results, not only does the use of hybrid structures provide more energy absorption and improve mechanical properties, but also, in comparison to the usual lattice structures fabricated with a single material, the rational combination of two materials makes the bi-material three-dimensional lattice structure to inherit the optimum energy absorption and stiffness.

Investigation on internal resonance of fluid conveying pipes with initial geometric imperfection

This paper shows numerical research on the 1:2 internal resonance and chaotic bifurcation phenomena of the graphene platelets reinforced metal foams (GPLRMF) pipes, with emphasis on the effect of initial geometric imperfection. To this end, combining the high order shear deformation beam theory and Kelvin-Voigt viscoelasticity, a mathematical model that considers geometric imperfection is established. The fluid in pipeline is supposed to be incompressible and inviscid. Subsequently, the Galerkin discretization and multiscale method are utilized for solving the nonlinear governing equations in the case of 1:2 internal resonance. And the nonlinear dynamic behavior of the fluid conveying pipes can be analyzed through the frequency amplitude diagrams, bifurcation curves, time course curves, phase trajectories, and Poincaré maps. The double jump phenomena are excited to demonstrate the dynamic characteristics of the pipes with 1:2 internal resonance. In the case of small disturbance coefficients, various internal resonance and chaos phenomena can be captured in the pipeline system. Research shows that the coupled effects between initial geometric imperfection and nonlinear geometric relationships is non-monotonic. That is to say, for bimodal internal resonance, when the initial geometric imperfection increase, the unstable region will be compressed, and the response amplitude will increase.

Radial point interpolation‐Chebychev method with asymptotic numerical method for nonlinear buckling analysis of graphene oxide powder‐reinforced composite beams

SummaryThis study aims to analyze the buckling and post‐buckling behaviors of multilayer nanocomposite beams reinforced with graphene oxide powders (GOPs) at low concentrations. The GOPs are randomly oriented and evenly distributed throughout the composite layers, with their weight fraction varying in the thickness direction. The Halpin‐Tsai model is employed to estimate each layer's effective material properties. The nonlinear governing equations for the FG‐GOPRC beam are derived based on the first‐order shear deformation beam theory, and a nonlinear algebraic system is formulated using the principle of potential energy. In this research, a comprehensive parametric analysis is conducted to examine the influence of various factors on the buckling and post‐buckling behaviors of the composite beams. These factors include the distribution pattern and weight fraction of graphene platelets (GPLs) nanofillers, as well as the geometry, size, slenderness ratio, and boundary conditions of the beams. Through the analysis, the study highlights the significant strengthening effect of these factors on the buckling and post‐buckling performance of the composite beams. The findings from this investigation contribute to a deeper understanding of the behavior of multilayer nanocomposite beams reinforced with GOPs. This knowledge can be valuable in designing and optimizing of composite structures for enhanced buckling resistance and post‐buckling performance.

Sound radiation characteristics of a beam under supersonic airflow and non-uniform temperature field

This paper investigates vibro-acoustic behaviour of an isotropic beam subjected to in-plane varying temperature field and supersonic flow. For a given elevated temperature and non-uniform temperature (NUFT) field, the aerodynamic pressure (ADP) is varied in fraction of its critical value, and the consequent effect on sound radiation characteristics is investigated. Shear and normal deformable beam theory (SNDBT) based kinematics is used, and equations of motion are obtained using Hamilton's principle. Following that, the Ritz method is used to solve the equations. To start with, aeroelastic flutter and vibro-acoustic response characteristics (as a function of critical aerodynamic pressure (λcr)) are analyzed without considering any thermal stress effect. Following that, for a given temperature field, the critical buckling temperature (ΔTcr) of the beam is obtained. Then, the impact of thermal stress caused by the NUFT field on the flutter and sound radiation characteristics is studied. The study observed that the combined effects of the NUFT field and variable ADP significantly impacted the beam's flutter and vibro-acoustic responses.

Exact closed-form dynamic stiffness matrix of damaged frames comprising Timoshenko–Ehrenfest beams

The present work proposes the extension of an exact distributional model for cracked beams to include the local influence on axial deformation on shear deformable beams. Axial, shear and flexural concentrated flexibilities are taken into account to model the damaged sections. Shear deformability is accounted for by adopting the Timoshenko–Ehrenfest model and exact closed-form solution of the mode shapes for multiple cracked beams are derived. On this ground, the final achievement of the work consists in the original formulation of the Dynamic Stiffness Matrix (DSM) with multiple cracked sections in exact closed form. The closed-form expression of the DSM allows the assemblage of the global DSM of a structure to conduct the free vibration analysis of damaged framed structures as well as for the study of frames with semi-rigid joints. This approach has the advantage to avoid any numerical procedure for the construction of the DSM as well as the introduction of further degrees of freedom (dofs) in correspondence of the damaged sections, keeping the size of the problem the same as in the intact configuration. Natural frequencies of the frames are determined by means of the Wittrick–Williams algorithm (W–W), then the mode shapes and modal forces are subsequently obtained in closed form. The proposed approach is validated with results available in literature for two portal frames characterized by the presence of cracks and semi-rigid joints, respectively. Finally, the potentiality of the approach, also in view of practical applications, is demonstrated analysing a multi-storey frame with semi-rigid connections.

Nonlinear isogeometric analysis of axially functionally graded graphene platelet-reinforced composite curved beams

The linear and nonlinear isogeometric finite element models of an axially functionally graded graphene platelet-reinforced composite (AFG-GPLRC) curved beam are established within the framework of the third-order shear deformation beam theory (TSDT) and von-Kármán’s nonlinear geometric relation. The AFG-GPLRC curved beams can be seen as composite structures in which the graphene platelets (GPLs) are continuously distributed in the matrix along the length direction of the curved beam according to different patterns. The modified Halpin-Tsai parallel model and the rule of mixture are implemented to predict the effective Young’s modulus and mass density as well as Poisson’s ratio, respectively. Hamilton's principle, TSDT, and von-Kármán’s strain-displacement relation are combined to derive the governing partial differential equation of motion and corresponding boundary conditions. Furthermore, the Non-Uniform Rational B-splines (NURBS)-based isogeometric analysis (IGA) approach together with a direct iterative technique are utilized to solve the nonlinear governing equation. The accuracy and efficiency of the proposed IGA framework are confirmed by comparing corresponding numerical solutions with other available results. The parametric investigations, such as the curved beam’s geometric parameters, boundary conditions, and GPL’s distribution patterns, on the nonlinear bending and vibration responses of the AFG-GPLRC curved beams are carried out by some illustrative examples.