Simply-supported Boundary Conditions Research Articles

Three-dimensional vibration analysis of multilayered composite and functionally graded piezoelectric plates and shells

In the present paper, a coupled 3D exact electro-elastic shell model for Functionally Graded (FG) and composite piezoelectric structures is proposed. Primary variables of the electro-elastic model are the electric potential and the three displacement components. The model allows to evaluate the piezoelectric effect in terms of frequencies and vibration modes. Both closed and open circuit configurations are analyzed and compared. The 3D equilibrium equations and the 3D divergence electric displacement equation for spherical shells give the set of partial differential equations for the electro-elastic problem. The proposed model for spherical shells automatically degenerates into simpler models for plates and cylindrical shells via properly considerations on radii of curvature along in-plane directions. The orthogonal mixed curvilinear coordinates α, β and z are employed. The partial differential governing equations have constant coefficients considering fictitious layers and they are solved using the Navier harmonic form and the exponential matrix method. These features lead to an exact solution for simply-supported boundary conditions. Free vibration analyses are conducted and circular frequencies for the first three thickness vibration modes are computed. After a global assessment phase to verify the correctness of the developed model, new benchmarks are proposed: different thickness ratios and material configurations are investigated. The present work can be intended as a reference general solution for those scientists interested in the study of piezoelectric structures via 2D analytical and numerical formulations.

Thermal instability and vibration characteristics of laminated composite struts with Graphene reinforcements: An analysis of distribution patterns and geometrical imperfections

This study explores the effect of local buckling on the compressive performance of slender structural elements, particularly those with thin-walled sections. The phenomenon of local buckling significantly reduces the axial compressive stiffness, leading to a notable decrease in the load-bearing capacity of these elements. The main goal of this research is to examine how the post-buckling characteristics of polymeric composite channel section struts can be improved under thermal loading by incorporating multi-layer graphene reinforcements. The solution methodology incorporates the von Karman geometrical nonlinearity and is based on the layerwise third-order shear deformation theory (LW-TSDT). To ascertain the precision and computational performance of the results derived from LW-TSDT, a three-dimensional (3D) finite element model is created in ABAQUS for comparative evaluation. An extensive analysis of nonlinear thermal instability in perfect and geometrically imperfect FG-GRC laminated channel section struts is undertaken to discern the graphene distribution patterns that are most and least effective in elevating the critical buckling temperature and natural frequencies through pre- and post-buckling conditions. The comparative analysis indicates that employing the FG-X graphene distribution pattern across the thickness of the web and flanges in channel section struts leads to a projected increase of 12 % in the critical buckling temperature for clamped channel section struts, in contrast to those that adopt the FGO graphene distribution pattern. For cases with simply-supported boundary conditions, this increase is noted to be approximately 9 %. Moreover, findings confirm that incorporating an asymmetric graphene distribution pattern (FGV) or introducing geometrical imperfections in the flanges and web that generate a bending moment within the structure from the beginning of thermal loading effectively prevents the primary natural frequencies of FG-GRC channel section struts from declining to zero close to the critical buckling temperature. This is significantly different from scenarios involving perfectly structured and symmetrically reinforced graphene distribution patterns such as FGX.

Investigating forced vibration of an intelligent sandwich plate consisting of a metal foam core, stiff nanocomposite layers and active piezomagnetic face-sheets exposed to electro-magnetic potentials

The main purpose of the present article is to present a forced vibrational analysis for a new intelligent sandwich plate made up of various layers. The proposed intelligent structure contains a porous metal foam core, super stiff double-FG nanocomposite layers, and piezomagnetic actuators which will be actuated magnetically and electrically. For the metal foam core of the structure, three kinds of porosity patterns are considered and, in the double-FG nanocomposite layers, Carbon Nanotubes (CNTS) are dispersed through various FG patterns in an FG metal-ceramic matrix. The power-law function is used to form the FG metal-ceramic matrix of the nanocomposite layers. Then, first-order shear deformation theory (FSDT) is utilized to derive the governing equations, and the resonance and natural frequencies of the proposed sandwich structure with simply-supported boundary conditions are obtained employing Navier’s analytical solution method. Finally, the effects of various parameters such as the CNTs’ volume fraction, magnetic and electric potentials, porosity patterns, core-to-nanocomposite layers thickness ratio, power-law index, etc on the resonance behavior of the structure will be discussed based on tabular and graphical results. The results demonstrate that the occurrence of the resonance phenomenon in such a multi-layered structure remarkably depends on the mentioned parameters and can be somehow controlled by applying magnetic and electric potentials to the piezomagnetic actuators connected to the top and bottom of the structure.

Artificial neural networks coupled with numerical approach for the stability prediction of non-uniform functionally graded microscale cylindrical structures

Functionally graded materials (FGMs) have garnered significant attention in research due to their potential for creating innovative materials with diverse applications. This study focuses on investigating the behavior of steel-concrete functionally graded structures under axial loading, specifically exploring nonlinear buckling. The research considers cylindrical structures with simply-supported boundary conditions and analyzes four different configurations with varying thickness. The governing equations are derived using the modified couple stress theory, incorporating assumptions of nonlinear von-Kármán theory coupled with classical beam theory. Numerical solutions are obtained using the general differential quadrature (GDQ) method. A notable aspect of this work is the use of Artificial Neural Networks (ANN) to predict nonlinear buckling forces, greatly enhancing the accuracy and efficiency of the predictions. This approach offers a novel method for analyzing concrete structures. The results are validated by comparing them with data from other published works, using dimensionless parameters for consistent comparisons. The findings provide a comprehensive analysis of various parameters, including wall thickness function and porosity, on the buckling stability of the cylindrical structure. Both numerical and ANN approaches are employed to report the effects. The study concludes that the uniform thickness cylinder exhibits the highest resistance to axial loading, resulting in buckling.

Vibration of cylindrical shells: Design criteria for transition from shell modes to beam modes

The dynamic behavior of the cylindrical shell can be predicted by more simplified beam models for a wide range of applications. The present paper deals with finding design conditions in which the cylindrical shell performs like a beam. Employing the Hamilton’s principle, the governing equations are obtained using both Donnell-Mushtari and Flugge shell theories and the analytical solution is obtained for long cylinders with simply-supported boundary conditions at both ends. Then, by equalizing the shell and beam vibration frequencies, the shell-to-beam transition conditions are obtained for both theories. To account for the effects of shear distortion and rotatory inertia of the shell, the finite element method is applied to find the best transition conditions with less approximating assumptions. Finally, the effects of boundary conditions on the transition parameters as well as the frequency response are studied. The obtained conditions simply define that if the shell can be assumed as a beam in any specific geometrical and material conditions.

Compact Finite Difference Scheme for Euler–Bernoulli Beam Equation with a Simply Supported Boundary Conditions

Compact Finite Difference Scheme for Euler–Bernoulli Beam Equation with a Simply Supported Boundary Conditions

Elastic Buckling of Prismatic Web Plate under Shear with Simply-Supported Boundary Conditions

This study aims to investigate the local elastic buckling behavior of simply-supported prismatic web plates under pure shear loading. Comprehensive finite element analysis is conducted to analyze the effects of various geometric parameters, such as tapering ratio, aspect ratio, and web slenderness, on the local elastic buckling behavior with simply-supported boundary conditions. An eigenvalue analysis is conducted to determine web plates’ natural frequencies and corresponding shape modes with varying geometric parameters. Particular attention is given to the effect of the slenderness ratio, since current formulas do not consider the impact of the slenderness ratio on the elastic shear buckling coefficient. A sensitivity analysis is conducted to examine the importance of the web slenderness ratio for estimating the critical buckling coefficient of a prismatic plate under pure shear loading. Finally, a formula of the elastic local critical buckling coefficient for a simply-supported prismatic web considering the web slenderness effect is proposed, which can be used in international codes.

Nonlinear coupled multi-mode vibrations of simply-supported cylindrical shells: Comparison studies

In spite of extensive studies on the nonlinear vibrations of cylindrical shells, the significant influences of some effective modes and the in-plane nonlinearity on the nonlinear coupled multi-mode vibrations are not clear. In this work, which is based on Donnell’s nonlinear shell theory and Amabili–Reddy’s third-order shear deformation theory, the nonlinear differential equations of motion of both the thin-walled and moderately thick cylindrical shells under the simply-supported boundary condition are established via Lagrange equations. These high-dimensional differential equations with the quadratic and cubic nonlinearities are solved by using an iteration procedure that is a combination of incremental harmonic balance method, pseudo-arclength method and extrapolation techniques. Turning and bifurcation points of the system are determined with the help of the direct method, the stability of solution of the frequency–response is examined by using the multi-variable Floquet theory. The numerical responses obtained by adopting two present shell theories are compared to investigate the influence of ignoring the in-plane nonlinearity on nonlinear vibrations. In the coupled multi-mode vibrations with respect to the fundamental mode m,n, apart from the regular modes i×m,j×ni=1,3,5,…,andj=0,1,2,3,… that include the axisymmetric and asymmetric modes, some irregular modes are taken into account to study the frequency–responses. Results show that present iteration procedure is efficient and successful to get the frequency–responses of the coupled multi-mode vibrations, and the influence of the irregular mode on the coupled multi-mode vibration is dependent on the relationship between the natural frequency of the irregular mode and that of the fundamental mode.

Dynamic instability of reinforced hollow cylinders in a confined underwater environment

An experimental investigation was conducted to understand the collapse mechanisms of internally ring stiffened aluminum cylinders under uniform hydrostatic loading in a confined underwater environment. The implosion of ring stiffened cylinders was studied using a combination of high-speed photography and 3D Digital Image Correlation (DIC). The results show that as stiffener thickness is decreased, the collapse of the structure transitions from two segments collapsing in mode III with the stiffener serving as boundary separating the segments to one uniform mode II collapse where the ring stiffener collapses with the structure. Furthermore, in the cases with two collapses, the behavior of the structure can be described as approaching that of two cylinders divided by a simply-supported boundary condition at the location of the stiffener. The ring thickness also affects the collapse pressure of the cylinder, radial velocity during collapse at the location of the ring stiffener, and dwell time between the collapse of two sections in cases with two collapses.

Investigation of energy-absorbing connections for application in mass timber structures

This study presents the results of an experimental program investigating the use of steel energy-absorbing connections (EAC) for the purpose of improving the energy dissipation capabilities of timber assemblies. The performance of the EACs is investigated both experimentally and numerically. A total of thirty-six specimens, representing nine configurations, including angle and circular hollow-steel sections, were tested under quasi-static and dynamic loading. The results of the study showed that connection configurations with well-defined plateaus were capable of dissipating a significant amount of energy and as such can be used in design for blast mitigation. Scalability of resistance based on the connection's depth was shown to be achievable experimentally. The welding configurations were shown to impact the behaviour significantly, and thus the details used in manufacturing must be identical to those assumed in design. The resistance curves of the EAC configurations investigated were reasonably predicted using Finite Element models. Analysis with single degree of freedom utilizing resistance curves obtained experimentally or from the finite element model mimicked the observed behaviour of the EACs. An example of the application of EACs in a mass timber assembly is demonstrated. This example showed that a 247 % increase in the energy absorption of the system might be obtained when compared to glulam members with idealized simply-supported boundary conditions.

Parametric resonance for pipes conveying fluid in thermal environment

Parametric resonance is a kind of specific fluid-induced vibration for pipes conveying fluid. This paper focuses on revealing the qualitative characteristics of parametric resonances of the pipe with pulsating fluid speed in thermal environment, and compares the differences between the resonance characteristics in subcritical and supercritical regions. According to the generalized Hamilton's principle, the partial-differential-integral governing equation of a straight pipe is established. Under simply-supported boundary conditions, the non-trivial equilibrium configuration is obtained analytically. The governing equation of a supercritical pipe is derived by coordinate substitution on the basis of the new equilibrium configuration. The approximate analytical solution of parametric resonance for a pipe conveying fluid in a thermal environment is obtained by using the direct multi-scale method. The approximate analytical solution is verified to be reliable by using the Runge-Kutta method. The stability bounds of the parameters inducing parametric resonance are investigated via the introduced analytical method. The results show that the influence of temperature increment on sub-harmonic resonance is non-monotonic. In the subcritical region, when the temperature increment increases, the unstable region widens. This means that the system is more prone to parametric resonance, and the response amplitude decreases. In the supercritical region, when the temperature increment increases, the unstable bandwidth decreases, and the response amplitude increases. When viscous damping is increased or the average velocity is decreased, the sub-critical instability region decreases, and the supercritical instability region increases. With the increase in the pulsation velocity, the unstable bandwidth of subcritical and supercritical regions will increase, and parametric resonance is more likely to occur.

Behavior of unrestrained hot-rolled steel I-beams exposed to localized fire: An experimental study

The behaviour of hot-rolled steel I-section beams was studied experimentally and numerically in uniform temperatures in the past, but only limited experimental data are available for localized fire tests. The present paper reports ten experimental tests for short to medium-range laterally unrestrained I-beams of Class 1 section with a non-dimensional slenderness ratio in the range of 0.74 to 1.1. The localized fire behaviour of unrestrained I-beams is experimentally studied with a high thermal zone having a temperature range of 595 to 1200 °C. Three-point loading is provided considering simply-supported boundary conditions with pre-load and peak load (ultimate load) in the range of 0.26 to 0.76 of the design load capacity of a beam member at ambient temperature. The two types of fire scenarios are considered (a) fire concentration at the middle of the span and (b) fire concentration near the support of the beam under steady and transient experimental conditions. The geometrical imperfection is measured using an industrial blue-light optical scanner. A numerical validation study was also conducted using the robust FE modelling software ABAQUS. The experimental study concluded that failure mode can depend on fire location, load ratios, and span length. In the case of localized fire at the middle of the span, lateral torsional buckling happened, and when the fire was near support, local buckling happened at higher load ratios. However, local buckling happened in the beam for lower to mid-load ratios. It is observed that the steady and transient fire tests provided a negligible effect on the response of the beam member. The experimental and numerical results are in good agreement, with a mean percentage variation of 1.5 %, with a maximum value of up to 4.67 %. The study provides quite a simple experimental setup that can easily be replicated for different types of members that enhance the fire safety study of structure. Further, a parametric study is conducted under simply-supported and varying end restraints condition at supports.

Analytical solution for the bending problem of micropolar plates based on the symplectic approach

Abstract An analytical solution for the bending problem of micropolar plates is derived based on the symplectic approach. By applying Legendre's transformation, we obtain the Hamiltonian canonical equation for the bending problem of a micropolar plate. According to the method of separation of variables, the homogenous Hamiltonian canonical equation can be transformed into an eigenvalue problem of the Hamiltonian operator matrix. We derive the eigensolutions of the eigenvalue problem for the simply-supported, free and clamped boundary conditions at the two opposite sides. Based on the adjoint symplectic orthogonal relation of the eigensolutions, the solution of the bending problem of micropolar plate is expressed as a series expansion of eigensolutions. Numerical results confirm the validity of the present approach for the bending problem of micropolar plates under various boundary conditions and demonstrate the capability of the proposed approach to capture the size-dependent behavior of micropolar plates.

Vibration analysis of piezoelectric semiconductor beams with size-dependent damping characteristic

In this paper, the vibration analysis of the piezoelectric semiconductor (PS) beam with size-dependent damping characteristic is conducted based on the nonlocal PS model. The nonlocal constitutive relations including the stress, electric displacement as well as current density of electrons are achieved. The one-dimensional vibration governing equations are obtained by integrating the three-dimensional governing equations along the cross section. Then, the natural frequency and damping characteristic are given by solving the governing equations of the PS beam with simply-supported boundary conditions. Through the numerical results, the effect of the nonlocal parameter, initial electron concentration, length to radius ratio and different scales on the vibration behavior of the system is demonstrated. The main innovation of the manuscript is that the peak value of the damping shifts to the smaller initial electron concentration because of the nonlocal effect, and the damping characteristics of the PS beam at nanoscale, micrometer scale and millimeter scale are obviously different.

A general electroelastic analysis of piezoelectric shells based on levy-type solution and eigenvalue-eigenvector method

Eigenvalue-Eigenvector approach as well as Levy type solution are used for electroelastic analysis of a doubly curved shell made of piezoelectric material based on a shear deformable model and piezoelasticity relations. The electroelastic governing equations are derived using virtual work principle. The solution is proposed for a Levy type boundary conditions with two simply-supported boundary conditions and two clamped ones. After derivation of the governing equations, a solution satisfying two simply supported boundary conditions is assumed to arrive a system of ordinary differential equations. The latest governing equations are solved using Eigenvalue-Eigenvector method to satisfy clamped-clamped boundary conditions. The distribution of displacements, rotations, electric potential, strain and stress is presented along the planar coordinate. Accuracy of the proposed solution is justified through comparison with results of previous papers.

Flügge shell theory and solution for the vibration analysis of a thin elastic cylindrical panel of varying curvature

In this paper, the free vibration behavior of isotropic and orthotropic cylindrical shell panels with variable curvature subjected to certain ends boundary conditions is analyzed by the transfer matrix technique and solved by the Romberg integration approach. Flügge’s shell theory is employed to model the governing three-dimensional equations of vibration and Fourier’s approach is used to deform the displacement fields as trigonometric functions in the longitudinal direction. As a result of a semi-analytical procedure is conducted, the coupled governing equations of panel problem can be written in a matrix partial differential equation of first order with variable coefficients and reduced from the two-dimensional problem to one-dimensional one which is solved numerically as an initial-value problem. The proposed model is applied to get the natural frequencies and mode shapes for the vibration behavior of two-side simply-supported cylindrical panels with simply-supported, free-free, clamped-free and clamped-clamped ends boundary conditions lying at their curved edges. The sensitivity of the vibration behavior to the variety of curvature, panel parameters and material orthotropy of shell is studied.

On the energy absorption of the reinforced sandwich curved beam equipped with piezoelectric layers through ANN and MCS analyses

The aim of this research is to investigate the impact of a viscoelastic foundation on the vibration of a curved beam structure with clamped and simply-supported boundary conditions. The structure considered is a micro-scale laminate composite beam made up of two piezoelectric face layers and a carbon nanotube-reinforced composite core. The study employs non-classical elasticity theory and nonlocal strain gradient theory to account for both nano-scale and microscale effects. The equations of motion are derived using the energy terms of the beam and a variational principle, with boundary conditions assumed to be clamped at one end and simply supported at the other end. A semi-analytical method of general differential quadrature method is used to solve the equations due to their nonlinear terms. To capture the effects of various parameters on the in-plane vibration of sandwich arches, an artificial neural network is designed and implemented. The study analyzes the impact of nonlocal and gradient parameters, geometrical aspect ratios, and substrate constants of the structure on the natural frequency and amplitude of vibration of the laminate beam. Results show that increasing nonlocal and gradient parameters have different effects on the amplitude and frequency of vibration of the beam.

On the pressure–deflection relations and instability of carbon-based composite nonlinear pipes

In this paper, for the first time, the nonlinear buckling of curved pipes is investigated for different distributions of single-walled carbon nanotubes (SWCNTs) in the matrix. The curved pipe is subjected to uniformly distributed transverse pressure and experiences uniform temperature rise. Additionally, the pipe is in one-sided contact with an elastic nonlinear medium. Defined thickness-dependent and temperature-dependent material properties of SWCNTs are taken from the results of molecular dynamics (MD) simulations, well-established in the literature on the subject. Five technologically justified types of CNT distributions are considered. Shen’s rule of the mixture is applied to determine the effective properties of CNT-based composites. To avoid determining shear correction factor, the higher-order shear deformation theory is used. The nonlinear components of the Green–Lagrange strain tensor and the principle of virtual displacements are applied to derive the nonlinear system of partial differential equations of motion of the curved pipe. The solution to formulated boundary value problem is obtained by the perturbation technique for both simply-supported and clamped-clamped boundary conditions. The pressure–deflection curves of the curved pipe are obtained and discussed. The investigation of snap-buckling intensity and upper/lower limit load of the curved pipe is presented and new insights for its nonlinear stability are presented. For instance, it is shown that the V-type of CNT distribution profile affects on lower deflection of the curved pipe while the higher deflection exists for the Λ-type profile.

A novel higher-order refined zigzag theory for static bending analysis in sandwich composite beam

Sandwich composite beams are widely used in variety fields. A typical sandwich composite beam includes a soft core covered by two stiff face layers. For such beam structures, large stiffness difference between two adjacent layers can result in shear deformations and zigzag displacement phenomenon. In this research, a novel higher-order refined zigzag theory (HRZT) is presented for solving the static bending problems of a sandwich composite beam with a soft core. The HRZT is derived based on the refined zigzag theory (RZT) by adding the higher-order zigzag terms. Unlike RZT, the kinematics assumption of HRZT can obtain a continuous shear stress distribution across the thickness. The governing equations of HRZT are derived by variational principle and the general solutions of which are derived in exact forms. The HRZT, RZT and FEM with commercial software are used to solve the static bending responses of sandwich composite beams with cantilevered and simply-supported boundary conditions. By comparing the FEM results, both the displacements and shear stresses calculated by HRZT are verified with high accuracy. In the present study, it is shown that HRZT is able to preserve the advantages of the RZT on the modelling of the zigzag displacement, but also improves the shear stress distributions that are continuous across the interface between two layers.

Exact solution of the longitudinally-graded rectangular plate under mixed instantaneous shock loading

This study follows a novel path toward analysis of the thermoelastic reaction of the longitudinally-graded graphene-nanoplates enriched composite (LG-GNPEC) rectangular-shaped plates exposed to the mixed-form instantaneous shock. Three-dimensional form of the linear thermoelasticity theory is picked as the base for establishing the governing equations. Lord-Shulman theorem and energy balance relations are employed to determine the time-dependency of the system’s behavior. Due to simply-supported boundary conditions, solving the governing equations through an analytical solution on the bases of Navier’s approach is performed. Accuracy of the applied methods is verified by comparing the obtained results with those presented in the published studies. Results of this research can be directly considered in the analysis of the thermoelastic behavior of the advanced composite materials exposed to the mixed-form thermomechanical shocks.