•General nonlocal theory with two parameters is used for the porous FG nanobeam.•The nonlocal parameters cause softening and hardening behavior.•Reddy’s beam theory is used to model higher-order ...transverse shear strains.•Material variations in the length and thickness directions are investigated.•The porosity volume fraction and the length to thickness ratio are varied.
A comprehensive vibrational analysis of bi-directional functionally graded (2D-FG) rotating nanobeams with porosities is studied for the first time. The beam is modeled based on general nonlocal theory (GNT) where the beam governing equations are derived depending on two different nonlocal parameters. Unlike Eringen’s conventional form of nonlocal theory, the general nonlocal theory can reveal both hardening and softening behaviors of the material. Here, the attenuation functions are altered in both transverse and longitudinal directions of 2D-FG nanobeam. This feature, which has a significant effect on the vibrational characteristics, has not been considered in previous studies. Moreover, to estimate the effects of the higher-order transverse shear strains on the vibration of the nanobeam, Reddy’s beam theory (RBT), which includes higher-order shear deformation, is employed. The material properties of the 2D-FG rotating nanobeam vary both in the length and thickness directions according to a power law. The generalized differential quadrature method (GDQM) is used to predict the vibration response. Also, the effects of material variation along the length and thickness directions, the rotating velocity of the nanobeam, the porosity volume fraction and the length to thickness ratio of the rotating nanobeam are illustrated and discussed in detail. The investigations performed in this study expose new phenomena for the vibration of nanobeams.
This paper investigates the linear free vibration of nanocomposite beams reinforced by single-walled carbon nanotubes (SWCNTs). Two types of CNT reinforced beams, namely uniformly distributed CNT ...reinforced (UD-CNT) beams and functionally graded CNT reinforced (FG-CNT) beams, are considered. It is assumed that the SWCNTs are aligned along the beam axial direction and the distribution of the SWCNTs may vary through the thickness of the beam. The virtual strain and kinetic energies of the FG-CNT composite beam are obtained using the classic variational method of Hamilton’s principle and then solved by the p-Ritz method. Vibration frequency parameters for the FG-CNT beams based on the first order and third order beam theories are presented and the effects of CNT filler volume fraction, distribution, beam span to depth ratio and end support conditions on the free vibration characteristics of the beams are discussed. Comparison studies for UD-CNT and FG-CNT beams based on the first order and the third order beam theories are also performed and the differences in vibration frequencies between these two theories are highlighted.
Compared to rigid robotic actuators, soft actuators based on fiber-reinforced elastomeric actuators have great potential for use in exploring complex terrain, medical operations, and flexible capture ...of targets. However, it is still challenging to model the mechanical behavior of soft actuators because of the large deformation and nonlinear characteristics of elastic materials. This article proposes an analytical model to predict the shape of a soft actuator to better describe the relationship between its deformation and input pressure, gravity, and other external forces. We first derive the relationship between soft actuator deformation and pressure using volume maximization and the principle of virtual work. We further use Euler-Bernoulli beam theory to investigate the influence of self-gravity and external forces on the soft actuator configuration. The model is verified by fabricating a soft actuator prototype via mold casting. Finally, we perform a series of experiments to evaluate the accuracy of our proposed model. The longest total time to solve the model is 0.035 s. Experimental results show that the model's maximum error rate is between 2.89% and 9.75%. This indicates that the model can effectively predicts the deformation of soft actuators considering gravity and other external forces.
The nonlocal continuum theory, either integral or differential form, has been formulated, evolved and widely used in our era to explain size effect phenomena in micro– and nano– structures. In the ...case of Euler Bernoulli beam theory (EBBT), the nonlocal integral form produces energy consistent formulas, unlike the nonlocal differential form revealing inconsistencies and giving rise to paradoxes. In this work, our overall research objective mainly focuses on nonlocal integral elasticity analysis of beams by employing a normalized symmetric kernel. This kernel corresponds to a finite domain (and will be called modified kernel from now on), that satisfies all the properties of a probability density function as well as successfully handles the physical inconsistencies of classic types of kernel. Our concern is to investigate the static response of a beam with various types of loading and boundary conditions (BCs) by making use the modified kernel and the kernel corresponding to the two phase nonlocal integral (TPNI) model. Carrying out numerical methods to our problems, the deducing results appear more flexible behavior than those of classic-local and the common nonlocal differential, respectively. What is more, a comparison is made between the results of the aforementioned kernels to demonstrate the advantages of the modified kernel. Unlike the common nonlocal differential form, the nonlocal integral forms do not give rise to paradoxes. Moreover, it is critical that the TPNI model does not raise paradoxes as has been observed in other publications presented in the literature.
Buckling analysis of composite laminated beams is developed based on modified couple stress theory. By applying principle of minimum potential energy and considering two different beam theories, ...i.e., Euler–Bernoulli and Timoshenko beam theories, governing equations, boundary and initial conditions are derive for micro composite laminated beams. By using the new curvature tensor and modified couple stress theory, the size effects are captured unlike the classical continuum theories. A model of simply supported composite laminated beams is considered. These models can cover the classical composite beam theories and isotropic beam theories. Governing equations are solved by utilizing the Fourier series expansions. Comparison between results obtained by present study and those obtained by literature reveals that they are in good agreement. Some numerical results are presented to study the effect of material length scale parameter, beam thickness and length of beam on the micro composite laminated beam behavior. In addition, to investigate the effect of lamination on the buckling analysis of micro composite laminated beams, two types of lamination, i.e., 0,90,0 and 90,0,90 are considered.
Investigations of wave and vibration properties of single- or multi-walled carbon nanotubes based on nonlocal beam models have been reported recently. However, there are numerous inconsistencies in ...the handling of the governing equations, applied forces, and boundary conditions based on some of the reported nonlocal beam models. In this paper, the consistent equations of motion for the nonlocal Euler and Timoshenko beam models are provided, and some issues on the nonlocal beam theories are discussed. The models are then applied to the studies of wave properties of single- and double-walled nanotubes. The wave and vibration properties of the nanotubes based on the presented nonlocal beam equations are studied, and scale effects are discussed.
This paper presents an analytical refined beam model taking into account the axial, the shear and the transverse normal stress distributions. Based on the equilibrium equations in local form and an ...iterative procedure for the solution of the bipotential equation, the beam differential equations for the axial and the transverse motion are derived. From a mathematical point of view the formal structure of the obtained differential equations is identical to the Timoshenko beam equations but with increased accuracy. This is achieved by splitting the ansatzfunctions for the displacement field into three weight-averaged degrees of freedom (for the axial and the transverse deflection and the rotation) and residual deflections, so-called higher order terms which depend on the state of stress. It is shown that the shear traction condition is always fulfilled and even the normal traction condition may hold. Furthermore an analogy to the Timoshenko beam equations is given and it is shown that the shear correction coefficient depends on Poisson’s ratio. Finally analytical results of the refined beam theory are compared to several analytical beam theories and two-dimensional finite element results for statically determinate and indeterminate beam configurations.
Reissner mixed variational principle is employed for establishment of the nonlinear differential and boundary conditions of dynamic equilibrium governing the flexure of beams when the effects of true ...shear stresses are included. Based on the Reissner mixed variational principle, the nonlinear size-dependent model of the Reissner nano-beam is derived in the framework of the nonlocal strain gradient elasticity theory. Furthermore, the closed form analytical solutions for the geometrically nonlinear flexural equations are derived and compared to the nonlinear flexural results of the Timoshenko size-dependent beam theory. The profound differences in the assumptions and formulations between the Timoshenko and the Reissner beam theory are also comprehensively discussed. The Reissner beam model is shown not to be a first-order shear deformation theory while comprising the influences of the true transverse shearing stress and the applied normal stress. Moreover, it is exhibited that the linear and nonlinear deflections obtained based on the Reissner beam theory are consistently lower than their Timoshenko counterparts for various gradient theories of elasticity.
The program FlϵX (flexural ϵ for xtals) has been developed for a quick, easy and accurate evaluation of the maximum deformation reached in flexible crystals from a simple optical microscope picture. ...The program takes advantage of computer vision libraries to find the contours of a bent crystal and fit these to semicircles. It can then calculate the theoretical maximum deformation along its long axis using equations from the Euler–Bernoulli beam theory.
A program has been developed to assist researchers interested in bending crystals to evaluate crystal deformation from a single optical image.
•Stochastic B-spline wavelet on the interval based finite element method for beams.•Formulations, based on both Euler-Bernoulli and Timoshenko beam theory are given.•Wavelet scaling functions are ...utilized for discretization of random field.•Perturbation method is suggested for solving of stochastic boundary value problem.
The current paper presents the formulation of stochastic B-spline wavelet on the interval (BSWI) based wavelet finite element method (WFEM) for beams wherein, the spatial variation of modulus of elasticity is modelled as a homogeneous random field. Stochastic beam element formulations based on both Euler-Bernoulli beam theory and Timoshenko beam theory are proposed. BSWI scaling functions are used for the discretization of the random field and the response statistics are obtained using the perturbation approach. Numerical examples are solved and the results from perturbation approach are compared with that obtained from Monte Carlo simulation (MCS). A parametric study is also done to understand the effect of different coefficient of variation (CV) values and correlation length parameters on the response statistics. The study concludes that the proposed BSWI WFEM based perturbation approach for beams produce accurate response statistics for values of CV less than 15%. A comparative study is carried out between the results obtained from the proposed stochastic WFEM with stochastic finite element method (SFEM) wherein the random field discretization is done using Lagrange shape functions. Furthermore, normalized computational times for the execution of perturbation approach and MCS based on WFEM are evaluated and compared with those obtained for SFEM.
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