Kinematic Assumptions Research Articles

Large Amplitude Free Vibration Analysis of Nanobeams Based on Modified Couple Stress Theory

In this study, the large amplitude free vibration of nanobeams based on the modified couple stress theory was developed by using Total Lagrangian finite element formulation. In this study, Timoshenko beam theory has been used in free vibration analysis of nanobeams. Minimal kinematic assumptions have been used to model nanobeams. With this model, free vibration of nanobeams with small to large amplitude and with arbitrary boundary conditions can be analyzed. The numerical results obtained for free vibration based on the modified couple stress theory with small amplitude and the results obtained for free vibration with large amplitude without considering the modified couple stress theory are in good agreement with the similar results reported in previous research. Effects of the dimensionless length scale parameter, slenderness ratio, vibration amplitude and different boundary conditions on the nonlinear frequency ratio of nanobeams have been investigated. The results show the importance of considering nonlinear and size effects in the free vibration analysis of nanobeams with large amplitude.

Asymptotically consistent size-dependent plate models based on the couple-stress theory with micro-inertia

Several beam and plate models have been recently developed in the literature to accommodate for size-dependence. These are usually obtained starting from a generalized continuum theory (such as the couple-stress, strain-gradient or non-local theory or their modifications) and then deducing the governing equations through Hamilton’s principle and ingenuous kinematical assumptions. This approach, originated by Kirchhoff, usually fails to reproduce the dispersion features of the equivalent 3D theory. Besides, it produces a variety of models, in dependence of the different assumptions, such as Kirchhoff’s or Mindlin’s. In contrast, in this paper we adopt asymptotic reduction: moving from the couple-stress linear theory of elasticity with micro-inertia, we deduce new models for elongation and flexural deformation of microstructured plates. The resulting models are consistent, in the sense that they reproduce the dispersion features of the corresponding 3D body. Also, models are unique, for they may only differ by the order of the approximation. We find that microstructure especially affects inertia terms, which can be hardly captured by a-priori kinematical assumptions. For static flexural deformations, our results match those already obtained assuming plane cross-sections within the modified couple-stress theory. In fact, we show that couple-stress, reduced couple-stress and strain gradient theories all lead to equivalent results. Higher order models are also given, that describe the near first-cut-off behaviour and account for thickness deformations in the spirit of Timoshenko.

Higher order formulations for doubly-curved shell structures with a honeycomb core

Anisotropic doubly-curved shells reinforced with a honeycomb core are innovative structures for applications in civil, biomedical, and aerospace engineering. In this context, the homogenization technique represents one of the simplest way for analyzing such complex structures. A proper formulation must be capable to give accurate results for any cell configuration and/or curved shape. In the present work an innovative model is proposed, based on an Equivalent Single Layer (ESL) approach and higher order theories, for an accurate estimation of the vibrational response of plates, panels and shells, whose results are compared with predictions from a classical Finite Element Method (FEM). The work starts with a comparative study performed on aluminum sandwich plates with hexagonal, rectangular and re-entrant cells. Then, a sensitivity analysis evaluates the dynamic response of single- and doubly-curved panels with different cell typologies. The fundamental equations are tackled numerically by resorting to the 2D Generalized Differential Quadrature (GDQ) method. The influence of the kinematic assumptions throughout the thickness on the dynamic response of shells is investigated, accounting for different Representative Volume Element (RVE) deformation effects within the homogenized model. In all the analyses, cell units are analyzed by means of different geometric angles, thin and thick cores, as well as classic and double thickness vertical walls or commercial honeycomb cores.

An efficient isogeometric beam formulation for analysis of 2D non-prismatic beams

This study presents an efficient beam formulation for static and vibration analysis of 2D non-prismatic beams with the kinematic assumptions of the Timoshenko beam theory. Beams having complex configurations can be straightforwardly constructed by using B-spline basis functions, which are used for descriptions of the beam axis and the sectional height. This feature enables analysis of free-form non-prismatic beams with strong curvatures to be performed conveniently and accurately, and is not virtually found in any existing formulations that are related to non-prismatic beams. The concept of isogeometric analysis is adopted by discretizing the kinematic unknowns, i.e., the translational displacements of the beam axis and the sectional rotation, using B-spline basis functions. A procedure for the recovery of stress components is also presented. The accuracy and performance of the developed approach are assessed through several numerical examples which cover several types of beam, e.g., sinusoidal beam, tapered beam and Tschirnhausen's beam representing free-form beams. Furthermore, convergence tests are also performed to demonstrate convergence properties of the proposed formulation.

A new semi-analytical solution of bending, buckling and free vibration of functionally graded plates using scaled boundary finite element method

This paper is devoted to study the bending response, free vibration and mechanical buckling of functionally graded material (FGM) plates based on the scaled boundary finite element method (SBFEM) for the first time. On the basis of the three-dimensional (3D) theory of elasticity, the SBFEM governing equations of the FGM plate are derived from the principle of virtual work and solved analytically in the thickness direction to obtain the displacements, stresses, natural frequencies and critical buckling loads. A high order spectral element only with three degrees of freedom per node that is able to satisfy the high order continuity of the displacement fields while facilitating the numerical computation is applied. In the present method, only the bottom surface of plate structures needs to be discretized while numerical approximations in the thickness direction are no longer required, which leads to an accurate solution of the displacement in the thickness direction and a considerable reduction of the computational cost. Furthermore, the model strictly follows 3D theory of elasticity without employing any kinematic assumptions of plate theory, so that it is able to eliminate the shear locking problem for numerical simulations of FGM plates when the thickness becomes thinner. Accuracy and superior computational efficiency of the present formulations have been verified by comparing with the available analytical and numerical solutions achieved by other researchers.

A novel mixed finite element formulation based on the refined zigzag theory for the stress analysis of laminated composite plates

This study presents an accurate mixed variational formulation for the stress analysis of laminated composite plates based on Refined Zigzag Theory (RZT). A two-field variational concept based on the Hellinger-Reissner (HR) principle is employed associated with the kinematic assumptions of the RZT. The RZT provides a good mixture between the accuracy and computational efficiency for the thin and thick laminated composite structures without using the shear correction factors. A four-noded quadrilateral element and bi-linear shape functions are used for the discretization of the solution domain ensuring the C0-continuity. The main novelty of the present study is that the flexural behavior of the laminated composite plates is investigated based on RZT within the light of HR principle using monolithic approach for the first time. The proposed Mixed Finite Element (MFE) formulation assigns stress resultant type field variables in addition to the kinematic variables of the RZT. Therefore, the present approach, MRZT, paves the way of obtaining the stress resultants at each node directly from the solution of the system equations. Since the shear forces are obtained at each node, Equivalent (transformed) Section Principle (ESP) is utilized to achieve continuous through thickness transverse shear stress variations. In-plane strain components are calculated through the compliance matrix without resorting to the spatial derivatives of displacement components. The robustness and capability of the present approach are established through benchmark problems, and its applicability to challenging problems is demonstrated by modeling thick and highly heterogeneous plates, a delaminated plate and three-point bending tests.

Free vibrations of functionally graded material cylindrical shell closed with two spherical caps

ABSTRACT Free vibration response of a cylindrical shell closed with two hemispherical caps at the ends (hermit capsule) is analysed in this research. It is assumed that the system of joined shell is made from functionally graded materials (FGM). Properties of the shells are assumed to be graded through the thickness. Cylindrical and hemispherical shells are unified in thickness. To capture the effects of through-the-thickness shear deformations and rotary inertias, first order theory of shells is used. Donnell type of kinematic assumptions are adopted to establish the general equations of motion and the associated boundary and continuity conditions with the aid of Hamilton's principle. The resulting system of equations are discretised using the semi-analytical generalised differential quadrature (GDQ) method. After proving the efficiency and validity of the present method for the case of isotropic homogeneous joined shells, some parametric studies are carried out for the system of combined FGM hermetic capsule.

Vibration response of perforated thick beam under moving load

This manuscript aims to investigate the dynamic responses of thick Timoshenko perforated beams under a moving load using Ritz method for the first time. The beam is perforated by equally spaced square holes arranged in array form. The perforated geometrical parameters of beam are presented as a function of full beam, hole number throughout the cross section, and filling ratio. The governing equation is derived by using the Lagrange procedure using the kinematic assumption of Timoshenko shear deformation theory to include the rotation of the thickness effect. Therefore, the transverse-shear strain and rotary inertia are considered through analysis. Ritz method is exploited to solve the governing equation and transform it to ordinary differential equation in time domain. Newmark average acceleration method is adopted to solve the dynamic equation of motion to obtain the time response. The effects of moving load parameter, filling ratio, and the number of holes throughout the cross section on the dynamic responses of the perforated beam are investigated and analyzed. The proposed model can be used in many civil and mechanical perforated structures to reduce the vibration amplitudes.

Thermomechanical analysis of large deflection in shear deformable FG‐CNT reinforced composite beams using perturbation technique

Present investigation deals with the geometrically nonlinear bending response of functionally graded carbon nanotube reinforced composite (FG‐CNTRC) beams. It is assumed that beam is resting on a three‐parameter elastic foundation. Uniform thermal field is also included into the formulation and properties are considered as temperature dependent. Distribution of CNTs in the beam may be uniform or functionally graded according to linear distribution of volume fraction of CNTs across the thickness. The equivalent properties of the composite media are obtained with the aid of the refined rule of mixtures approach which is calibrated with the available data based on the molecular dynamics simulations. The governing equations of the beam are obtained based on first‐order, third‐order, and sinusoidal theories. To capture the large deformations, the von Kármán type of kinematic assumptions is included. Two types of boundary conditions are included which are immovable pinned and immovable clamped. The established equations are reformulated and transferred into dimensionless presentation. After that, using the powerful two step perturbation technique, closed form expressions are provided to trace the nonlinear load‐deflection paths of the beam. Results of this study are compared with the available data in the open literature. After that, novel results are provided to discuss the effects of boundary conditions, temperature change, foundation stiffness, slenderness ratio, CNT volume fraction, and distribution profiles. It is shown that graded pattern of CNTs and volume fraction of CNTs are two important factors on the geometrically nonlinear response of the CNTRC beam.

Assessment of negative Poisson’s ratio effect on the postbuckling of pressure-loaded FG-CNTRC laminated cylindrical shells

This paper examines the postbuckling behaviors of pressure-loaded laminated cylindrical shells made of carbon nanotube reinforced composite (CNTRC) under thermal environmental conditions. The shell has negative in-plane or out-of-plane effective Poisson’s ratio (EPR). The thermo-mechanical properties of CNTRCs are temperature dependent. The CNT volume fraction of the shell is in a piece-wise functionally graded (FG) distribution across the shell thickness. The extended rule of mixture (EROM) model is applied to determine the thermo-mechanical properties of CNTRCs. Based on the framework of the Reddy's third order shear deformation shell theory and the von Kármán-type of kinematic assumptions, the governing partial differential equations for pressure-loaded CNTRC laminated cylindrical shells are formulated. The thermal environmental condition effect is also included. The postbuckling solution for pressure-loaded FG-CNTRC laminated cylindrical shell is obtained in the asymptotic sense by means of a singular perturbation technique in associate with a two-step perturbation approach. Numerical investigations are carried out for the (10/60/10)S shell including in-plane NPR effect, and (20/-20/20)S and (70/-70/70)S shells including out-of-plane NPR effect. Assessment is presented through numerical comparisons.

High performance model for buckling of functionally graded sandwich beams using a new semi-analytical method

In this paper, a new semi-analytical approach based on the scaled boundary finite element method (SBFEM) is proposed to solve buckling problem of functionally gradient material (FGM) sandwich beams. Material properties of each individual layer are assumed to be continuously graded along the thickness according to a power law function with respect to the volume fractions. Based on the layer-wise theory, the two-dimensional constitutive model is directly performed in the proposed formulations without any kinematic assumptions of plate theory. The buckling governing equations of FGM sandwich beams based on the SBFEM are derived using the weighted residual method and solved by the means of the eigenvalue analysis. The advantage of the present approach is that only the longitudinal dimension of the beam structure needs to be discretized using one-dimensional higher-order spectral element so that it can be dealt with as a one-dimensional mechanical system while maintaining the analytical characteristics in the thickness direction, which possesses a key feature to make structure modelling more effective and accuracy. To evaluate the validity of proposed formulations, a series of numerical examples involving the element convergence and parametric analysis are carried out and the results are compared with existing solutions available in open literature. The numerical studies confirm the accuracy and adaptability of the presented method for the buckling analysis of FGM sandwich plates.

Pseudostatic Stability Analysis of Rock Slopes Using Variational Method

In this paper, a rectilinear slope of homogenous rock is analyzed by using the variational method. The strength of the rock medium is assumed to be governed by the Hoek–Brown failure criterion. Although the variational method is developed within the framework of limit equilibrium method, however, neither the shape of the critical slip surface (kinematical assumption) nor the distribution of normal stresses along the slip surface (statical assumption) is required to assume. The aim of this paper is to obtain critical slip surface and the corresponding critical factor of safety (Fs) for different combinations of (i) slope geometries (slope angle, β and height of slope, h), (ii) rock properties (Geological Strength Index, GSI, non-dimensional strength parameter for intact rock, mi, disturbance factor, Df, and uniaxial compressive strength of intact rock, σci) and (iii) seismic loading conditions (coefficient of horizontal seismic acceleration, kh). The magnitude of Fs is evaluated uniquely by implementing certain necessary constraints on the slip and the slope surfaces and optimizing the functionals generated from resisting and driving moments. The failure surfaces drawn for several cases provide an insightful observation of the influence of different (a) material properties, (b) slope geometries and (c) seismic conditions.

Free vibration of a cracked FG microbeam embedded in an elastic matrix and exposed to magnetic field in a thermal environment

A mathematical model is developed, though this article, to investigate a vibrational behaviour of functionally graded (FG) cracked microbeam rested on elastic foundation and exposed to thermal and magnetic fields. The model includes a size scale effect and temperature dependent material properties, for the first time. The crack is modelled as a rotating spring, that is connecting the two parts of the microbeam at the crack’s position. The equation of motion of the FG microbeam is obtained by using the Euler-Bernoulli beam theory for kinematic assumption and nonlocal elasticity theory for size-dependency effects. The transverse Lorentz force induced from the magnetic field is derived using Maxwell's equations. By adding the effects of thermal loading and foundation parameters on the cracked micro beam, the motion equation of the entire system is obtained using the Hamilton’s principle and then solved with a Navier type solution method. Eight constraints equations are used to derived the frequency equation, which are boundary conditions at the end points and the displacement, slope, bending moment and transverse force continuity in the section where the crack is located. The resulting system of equations is solved sequentially, and natural frequencies and vibration modes of the cracked microbeam are obtained. The model is verified with previous published work. Numerical results are presented to illustrate influences of temperature, material composition, foundation parameters and magnetic field on the dynamics of the cracked FG microbeam.

Nonlinear geometry effects investigation on free vibrations of beams using shear deformation theory

This article presents an analytical procedure to predict the beams nonlinear natural frequencies using the first-order shear deformation theory. The nonlinear kinematics assumptions include the moderately large deformation of the mid-plane stretching and transverse deflection that are defined by using the von Kármán relations. The strains are small and Hooke’s law is used as the constitutive equation. A coupled nonlinear longitudinal-transverse set of motion equations are derived utilizing Hamilton’s principle. The equations are solved analytically using the powerful multiple scales method of the perturbation technique. At first, the linear natural frequencies and mode shapes are determined. Then the nonlinear frequencies which contain the corrections on the linear frequencies are calculated. The corrected parts of the nonlinear frequencies are functions of the axial and transverse amplitudes of vibrations. The influences of the axial load and aspect ratio on the linear and nonlinear frequencies are studied too. The results of the special cases are compared with the available results in the literature and the finite element analysis. The results show the noticeable effect of the axial amplitude as well as the transverse amplitude of vibrations on the nonlinear frequencies.

Nitsche-based models for the unilateral contact of plates

This paper aims to present different Nitsche-based models for the unilateral contact of plate structures. Our analysis is based on the consideration of Nitsche’s method on a 3D structure with kinematic assumptions of thin or thick plate theories. This approach is compared to that of Gustafsson, Stenberg and Videman which consists of Nitsche’s method applied directly on a 2D plate model. To simplify the presentation, we focus on the contact of an elastic plate with a rigid obstacle. The different approaches are compared numerically in terms of reliability compared to the 3D elastic model.

Nonlinear Winkler-based Frame Element with Inclusion of Shear-Flexure Interaction Effect for Analysis of Non-Ductile RC Members on Foundation

Non-ductile reinforced concrete (RC) members are common in the existing RC frame buildings with the old seismic code (lightly and inadequately detailed transverse reinforcement) and may suffer shear failure or flexure-shear failure. To investigate the failure behaviors of those RC structures, performance-based numerical models are needed. Thus, a new fiber frame element on Winkler-based foundation including the interaction effects between shear and flexure was developed to analyze non-ductile RC frames resting on foundation, in this study. The proposed element is formulated for implementation in displacement-based finite element formulation under the kinematic assumptions of Timoshenko beam theory. As a result, axial and flexural mechanisms are automatically coupled through the fiber-section model, while shear and flexural actions interact via the UCSD shear-strength model within the framework of modified Mergos-Kappos interaction procedure to evaluate sectional shear force and shear stiffness within the shear constitutive law. Therefore, the presented model is simple, but able to capture several salient behaviors of non-ductile RC frames resting on foundation, including interaction between shear and flexure, soil-structure interaction, degradation of shear strength due to inelastic flexural deformation, and shear failure. Those features and efficiency of the proposed model are demonstrated by two numerical simulations in this work.

Nonlinear analysis of thermal-mechanical coupling bending of clamped FG porous curved micro-tubes

A nonlinear analysis on the thermal-mechanical coupling bending behavior of functionally graded porous curved micro-tubes is performed in this research. The case of curved micro-tubes with doubly-clamped boundary conditions resting on nonlinear elastic foundation is considered. Based on the modified couple stress theory, curved micro-tubes under uniformly distributed transverse pressure and uniform temperature rise are analyzed. It is assumed that the material properties of the curved micro-tube are temperature-dependent and graded across the radius of cross-section. The governing equations of the curved micro-tube are established within the framework of higher-order shear deformation theory and the von Kármán kinematic assumptions. These equations are reformulated for the case of curved micro-tubes with even distributed porosities based on the uncoupled thermoelasticity theory. The system of nonlinear differential equations governing the equilibrium position of curved micro-tubes is solved using the two-step perturbation technique. The analytical solutions are given in this study to obtain the large deflection in curved micro-tubes as an explicit function of the thermal/mechanical load. The influences of porosity volume fraction, power law index, microstructural length scale parameter, foundation stiffness, and geometrical parameters on the thermomechanical deflection of curved micro-tubes are investigated.

Free vibration of FGM conical–spherical shells

Natural frequencies of a conical–spherical functionally graded material (FGM) shell are obtained in this study. It is assumed that the conical and spherical shell components have identical thickness. The system of joined shell is made from FGMs, where properties of the shell are graded through the thickness direction. The first order shear deformation theory of shells is used to investigate the effects of shear strains and rotary inertia. The Donnel type of kinematic assumptions are adopted to establish the general equations of motion and the associated boundary and continuity conditions with the aid of Hamilton’s principle. The resulting system of equations are discretized using the semi-analytical generalized differential quadrature (GDQ) method. Considering various types of boundary conditions for the shell ends and intersection continuity conditions, an eigenvalue problem is established to examine the vibration frequencies. After proving the efficiency and validity of the present method for the case of thin isotropic homogeneous joined shells with the data of conventional finite element software, parametric studies are carried out for the system of combined moderately thick conical–spherical joined shells made of FGMs and various types of end supports.

A Thermodynamically Consistent Formulation for Dynamic Response of Thermoviscoelastic Plate/Shell Based on Classical Continuum Mechanics (CCM)

This paper presents a thermodynamically consistent and kinematic assumption free formulation for dynamics of thermoviscoelastic plates/shells based on the conservation and balance laws of classical continuum mechanics (CCM) in which dissipation mechanism has been incorporated through ordered rate constitutive theory for deviatoric stress tensor. In this paper, we consider small deformation, small strain. The conservation and balance laws of CCM in [Formula: see text] in Lagrangian description using Cauchy stress tensor ([Formula: see text]) and linearized Green strain tensor ([Formula: see text]) constitute the mathematical model. The constitutive theory for the deviatoric Cauchy stress ([Formula: see text]) is derived using conjugate pairs in entropy inequality in conjunction with representation theorem. The argument tensors of [Formula: see text] are [Formula: see text] and rates of [Formula: see text] up to order [Formula: see text]. This yields a constitutive theory with dissipation mechanism based on rates of strain up to order [Formula: see text]. Constitutive theory for heat vector is also derived using the conjugate pairs in the entropy inequality and representation theorem. Finite element method is used to obtain solutions of the initial value problems descried by the balance of linear momenta (BLM), energy equation and the constitutive theories. The shell element geometry is described by the middle surface and the nodal vectors at the middle surface defining bottom and top surfaces of the element. The local approximation for the displacement field is [Formula: see text] - version hierarchical in the plane of the element as well as in the transverse direction. A space-time decoupled finite element formulation using Galerkin Method with Weak Form (GM/WF) in space is constructed for BLM as well as energy equation, both resulting in ordinary differential equations (ODEs) in time. The ordinary differential equations (ODEs) in time resulting from the finite element formulation of BLM are used to study: (i) natural undamped modes of vibration (ii) the transient dynamic response using the ODEs in time recast in modal basis: (a) using Rayleigh damping (b) using the ordered rate damping proposed in this paper. Time response is calculated using modal damping based on Rayleigh damping as well as using the proposed ordered rate damping mechanism. Model problem studies are presented to demonstrate: (1) accuracy of the natural frequencies obtained from the present formulation for thin and thick plates/shells (in which shear deformation is significant) and the results are compared with the currently used plate formulations (2) accuracy of damped transient response using proposed damping mechanism is compared with time response using Rayleigh damping (3) it is shown that Rayleigh damping has no physical basis and leads to spurious stationary states. The proposed damping yields accurate stationary states that are in exact agreement with the solution of corresponding BVP. A single formulation presented in this paper remains valid and accurate for very thin as well as very thick plates/shells and correctly simulates 3D state of deformation regardless of plate/shell thickness and is free of shear locking problems as well as need for shear corrections. When obtaining the time response, solution for an increment of time alternates between the solution of BLM followed by the solution of the energy equation. Details are presented in the paper.

Thermodynamically consistent thermoelastic plate and shell formulations for small deformation and small strain using non-classical continuum mechanics incorporating internal rotations

The work presented in a recent paper by the authors [35] for a thermodynamically consistent and kinematic assumption free plate and shell formulation for small deformation and small strain based on the conservation and balance laws of classical continuum mechanics (CCM) is extended here for non-classical continuum mechanics (NCCM). This formulation incorporates additional physics due to internal rotations that arise due to the deformation gradient tensor. This physics is neglected in CCM, hence is absent in the plate and shell formulation of reference [35]. Consideration of this new physics requires modifications of the current balance laws as well as consideration of a new balance law “balance of moment of moments” (BMM) [2, 3]. Cauchy stress tensor becomes non-symmetric. Cauchy moment tensor is conjugate to the symmetric part of the rotation gradient tensor which exists now due to new physics. Balance of angular momenta yields additional three differential equations as part of the mathematical model. The new balance law (BMM) establishes symmetry of the Cauchy moment tensor. The new physics considered here exists in all deforming solid continua as it is due to the deformation gradient tensor, but is ignored in CCM. The consequence of this new physics is additional stiffness, hence additional strain energy storage and change in the time history of displacements and stress field compared to formulations based on CCM. The basic mathematical model for the plate and shell deformation consists of conservation and balance laws in $$\mathbb {R}^3$$ based on NCCM incorporating internal rotations. The associated finite element formulations for obtaining the solution of the mathematical model consists of : (i) geometry of the plate or shell described by the flat or curved middle surface (as done conventionally) and nodal vectors locating the top and bottom faces of the plate/shell (ii) the displacement field approximation that is p-version hierarchical in the plane as well as in the transverse direction (iii) integral form is constructed using Galerkin Method with Weak Form (GM/WF) and the corresponding element equations. The formulation presented here remains valid and accurate for thin as well as thick plate/shell and naturally reduces to the formulation of reference [35] based on classical continuum mechanics. Model problem studies and comparisons with the studies based on CCM formulation [35] will be presented in a follow up paper.