Showing papers on "Timoshenko beam theory published in 2022"

Impact response of inclined self-weighted functionally graded porous beams reinforced by graphene platelets

Abstract: This study presents a theoretical analysis on the low-velocity impact response of inclined porous nanocomposite beams under various impulsive loads. The laminated beam model consists of multiple layers modelled as closed-cell cellular solids with identical thickness, where each layer contains uniformly distributed internal pores and is reinforced by dispersing graphene platelets into the matrix. The layer-wise continuous variations in both internal pore size/density and graphene fraction result in functionally graded lightweight beams with controllable density distributions and varying elastic moduli across the thickness direction. The material properties of each layer are determined according to Halpin–Tsai micromechanics model and the extended rule of mixture. The governing equations of the inclined beam are derived based on Timoshenko beam theory then solved by employing Ritz method for space domain and Newmark method for time domain. The static bending due to the self-weight of the beam is examined first, and then imported into the dynamic analysis as the initial stress state for the beam under impulsive impacts with six different pulse shapes. A comprehensive parametric study is conducted with a special focus on the combined effects of graded material distributions and inclined angle on the beam behaviour. Results show that a larger inclined angle reduces the beam deflection, and the rectangular impulsive load can lead to the largest mid-span deflection of fully clamped graded beams that may reach over 70% more than those under some of other impact load types. This study should provide insights into the design of lighter and stiffer inclined structural components subjected to various impulsive loading conditions.

Nonlinear continuum mechanics of thick hyperelastic sandwich beams using various shear deformable beam theories

Abstract: Abstract In this study, the time-dependent mechanics of multilayered thick hyperelastic beams are investigated for the first time using five different types of shear deformation models for modelling the beam (i.e. the Euler–Bernoulli, Timoshenko, third-order, trigonometric and exponential shear deformable models), together with the von Kármán geometrical nonlinearity and Mooney–Rivlin hyperelastic strain energy density. The laminated hyperelastic beam is assumed to be resting on a nonlinear foundation and undergoing a time-dependent external force. The coupled highly nonlinear hyperelastic equations of motion are obtained by considering the longitudinal, transverse and rotation motions and are solved using a dynamic equilibrium technique. Both the linear and nonlinear time-dependent mechanics of the structure are analysed for clamped–clamped and pinned–pinned boundaries, and the impact of considering the shear effect using different shear deformation theories is discussed in detail. The influence of layering, each layer’s thickness, hyperelastic material positioning and many other parameters on the nonlinear frequency response is analysed, and it is shown that the resonance position, maximum amplitude, coupled motion and natural frequencies vary significantly for various hyperelastic and layer properties. The results of this study should be useful when studying layered soft structures, such as multilayer plastic packaging and laminated tubes, as well as modelling layered soft tissues.

Dynamic response analysis of Euler–Bernoulli beam on spatially random transversely isotropic viscoelastic soil

Abstract: A novel dynamic soil-structure interaction model is developed for analysis for Euler–Bernoulli beam rests on a spatially random transversely isotropic viscoelastic foundation subjected to moving and oscillating loads. The dynamic equilibrium equation of beam-soil system is established using the extended Hamilton's principle, and the corresponding partial differential equations describing the displacement of beam and soil and boundary conditions are further obtained by the variational principles. These partial differential equations are discretized in spatial and time domains and solved by the finite difference (FD) method. After the differential equations of beam and soil are discretized in the spatial domain, the implicit iterative scheme is used to solve the equations in the time domain. The solving result shows the FD method is effective and convenient for solving the differential equations of beam-soil system. The spring foundation model adopted the modified Vlasov model, which is a two-parameter model considering the compression and shear of soil. The advantage of the present foundation model is avoided estimating input parameters of the modified Vlasov model using prior knowledge. The present solution is verified by publishing solution and equivalent three-dimensional FE analysis. The present model produced an accurate, faster, and effective displacement response. A few examples are carried out to analyze the parameter variation influence for beam on spatially random transversely isotropic viscoelastic soil under moving loads.

Application of Haar wavelet discretization and differential quadrature methods for free vibration of functionally graded micro-beam with porosity using modified couple stress theory

Abstract: The present investigation is aimed at the implementation of Haar Wavelet Discretization Method (HWDM) and Differential Quadrature Method (DQM) on the free vibration of a Functionally Graded (FG) micro-beam with uniformly distributed porosity along the thickness. As per the power-law exponent model, the material properties such as Young's modulus, and mass density are varied along the thickness of the FG micro-beam and the beam is made up of Aluminum (Al) as metal constituent and Alumina (Al2O3) as ceramic constituent. Modified couple stress theory is employed to capture the small scale effect and pointwise convergence studies for HWDM as well as DQM have also been carried out to exhibit the effectiveness of the methods with respect to the undertaken problem. The results obtained by both methods are compared to demonstrate the accuracy of the present model, revealing excellent accuracy. The effect of power-law exponent, porosity volume fraction index, and thickness to material length scale parameter on the natural frequencies is thoroughly investigated with proper physical explanations for Hinged-Hinged (H-H), Clamped-Hinged (C-H), Clamped-Clamped (C-C), and Clamped-Free (C-F) boundary conditions. Further, mode shapes are also plotted for qualitatively assessing the dynamics of the structural component.

Dynamic analysis of soil-structure interaction shear model for beams on transversely isotropic viscoelastic soil

Abstract: Dynamic response problem of the Timoshenko shear beam resting on the transversely isotropic viscoelastic foundation and subjected to a moving load is investigated. The extended Hamilton's principle is used to obtain the dynamic response equations of the beam-soil system. The corresponding partial differential equations are derived using variational principle. Using the analytical solutions, finite-element method, and central difference method, these differential equations are solved and mutually verified. The Newmark-β iterative algorithm is employed to decouple the dynamic equations of the beam-soil system. A modified two-parameter spring foundation model is used to simulate the dynamic characteristics of soil medium. Finite-element analysis demonstrates that the developed shear model of beam-soil is effective and accurate. Other numerical examples are carried out to analyze the effect of the shear beam, the load speed, and transverse isotropy of the dynamic medium model.

Free Vibration Analysis and Numerical Simulation of Slightly Curved Pipe Conveying Fluid Based on Timoshenko Beam Theory

Abstract: The modeling of a slightly curved pipe conveying fluid usually adopts the Euler–Bernoulli beam theory. In this paper, a dynamic model of the slightly curved pipe conveying fluid based on the Timoshenko beam theory is established for the first time. The complex mode method is used to obtain the frequencies, the modes, and the first critical velocity of the slightly curved pipe. Two kinds of initial configurations of the pipe with fixed–fixed boundary conditions are studied. Based on the Galerkin truncation method, the natural frequencies of the slightly curved pipe are also obtained with the generalized eigenvalue method. Moreover, the Coriolis force caused by the fluid is equivalent to the damping matrix. Therefore, a novel finite element model of the curved pipe considering fluid influence is developed. The numerical simulation method is extended to calculate the mode and frequency of the slightly curved pipe. Numerical results show that all the three methods have high accuracy when calculating the natural frequencies of the transverse vibration of the slightly curved pipe conveying fluid. However, the developed finite element method does not show the effect of flow velocity when determining the modes. Moreover, the initial bending cannot be ignored when analyzing the vibration characteristics of the slightly curved pipe conveying fluid.

Wave propagation in Timoshenko-Ehrenfest nanobeam: A mixture unified gradient theory

Abstract: A size-dependent elasticity theory, founded on variationally consistent formulations, is developed to analyze the wave propagation in nano-sized beams. The mixture unified gradient theory of elasticity, integrating the stress gradient theory, the strain gradient model, and the traditional elasticity theory, is invoked to realize the size-effects at the ultra-small scale. Compatible with the kinematics of the Timoshenko–Ehrenfest beam, a stationary variational framework is established. The boundary-value problem of dynamic equilibrium along with the constitutive model is appropriately integrated into a single functional. Various generalized elasticity theories of gradient type are restored as particular cases of the developed mixture unified gradient theory. The flexural wave propagation is formulated within the context of the introduced size-dependent elasticity theory and the propagation characteristics of flexural waves are analytically addressed. The phase velocity of propagating waves in CNTs is inversely reconstructed and compared with the numerical simulation results. A viable approach to inversely determine the characteristic length-scale parameters associated with the generalized continuum theory is proposed. A comprehensive numerical study is performed to demonstrate the wave dispersion features in a Timoshenko–Ehrenfest nanobeam. Based on the presented wave propagation response and ensuing numerical illustrations, original benchmark for numerical analysis is detected.

Analytical Solutions of Viscoelastic Nonlocal Timoshenko Beams

Abstract: A consistent nonlocal viscoelastic beam model is proposed in this paper. Specifically, a Timoshenko bending problem, where size- and time-dependent effects cannot be neglected, is investigated. In order to inspect scale phenomena, a stress-driven nonlocal formulation is used, whereas to simulate time-dependent effects, fractional linear viscoelasticity is considered. These two approaches are adopted to develop a new Timoshenko bending model. Analytical solutions and application samples of the proposed formulation are presented. Moreover, in order to show influences of viscoelastic and size effects on mechanical response, parametric analyses are provided. The contributed results can be useful for the design and optimization of small-scale devices exhibiting flexural behaviour.