Edge Supports Research Articles

Thermomechanical nonlinear in-plane analysis of fix-ended FGM shallow arches on nonlinear elastic foundation using two-step perturbation technique

An attempt is made in this research to analyse the nonlinear response of functionally graded material shallow arches with both edges clamped. The arch is resting on a three parameter nonlinear elastic foundation during deformation and is subjected to uniform lateral pressure and uniform temperature rise. Material properties are expressed according to a power law function and are assumed to be temperature dependent. The governing equilibrium equations of the arch are established with the aid of third order shear deformation curved beam theory of Reddy and von Karman type of strain–displacement relations. The obtained equations contain three coupled and nonlinear equations in terms of circumferential displacement, lateral displacement and cross section rotation. Considering the immovable type of edge supports, the equations are reduced to two new coupled and nonlinear equations. These equations are solved using the two step perturbation technique for the case of clamped boundary conditions. Explicit expressions are resulted which yield the deflected shape of the arch as a function of temperature elevation and uniform pressure. It is shown that the arch reveals the snap-through type of instability under certain conditions. The response of the arch is highly affected by the power law index, thermal environment, side to thickness ratio and stiffnesses of the foundation.

Elastic buckling analysis of polygonal thin sheets under compression

In this work, we have studied elastic buckling of regular hexagonal thin sheets made of homogeneous and isotropic materials under in-plane hydrostatic and uniaxial compression, with internal supports, translational and rotational elastic edge supports, and a combination of free, simple-support and clamped boundary conditions. An energy method called Rayleigh–Ritz was utilized and the kinematic hypotheses of the classical thin plate buckling theory were applied. Displacement conditions were applied to edges and internal supports by employing proper basic P-Ritz functions. This approach gives an approximate analytical response that was calculated with Maple software and validated by previous researches. Moreover, the results were compared to numerical results obtained from ANSYS software. The findings revealed that besides having the advantages of other analytical methods, the Rayleigh–Ritz method is easy to use and straightforward to apply different constraints on boundaries and internal points; however, increasing the number of edges and constraints adds significantly the burden of calculation.

Nonlinear bending of third-order shear deformable carbon nanotube/fiber/polymer multiscale laminated composite rectangular plates with different edge supports

The geometrically nonlinear bending behavior of carbon nanotube/fiber/polymer multiscale laminated composite (CNT-FPMLC) rectangular plates with various edge conditions subjected to the uniform transverse mechanical loading is investigated. Based on the Reddy’s third-order shear deformation plate theory and employing the von Karman hypotheses and fundamental lemma of calculus of variations, the governing equilibrium equations including the shear deformation effect and geometrical nonlinearity together with associated boundary conditions are developed. The fiber micromechanics and the Halpin-Tsai relations are employed to approximately calculate the material properties of multiscale composite. Also, the carbon nanotubes (CNTs) are assumed to be distributed uniformly and oriented arbitrarily through the epoxy resin matrix. For the large deflection analysis, first, the generalized differential quadrature (GDQ) method is used to discretize the differential governing equations and corresponding boundary conditions resulting in a set of nonlinear algebraic equations. Then, the pseudo-arclength continuation technique is utilized to numerically solve the resulting nonlinear parameterized equations and subsequently obtain the load-deflection curve of CNT-FPMLC rectangular plates with different edge supports. Several numerical results are provided to reveal the influences of the weight percentage of single-walled and multi-walled CNTs, CNT aspect ratio, volume fraction of fibers, length-to-thickness ratio of plate and boundary conditions on the nonlinear responses of the CNT-FPMLC plates.

Numerical study on the nonlinear resonant dynamics of carbon nanotube/fiber/polymer multiscale laminated composite rectangular plates with various boundary conditions

This work deals with the numerical investigation of the geometrically nonlinear resonant dynamics of carbon nanotube/fiber/polymer multiscale laminated composite (CNT-FPMLC) rectangular plates with different boundary conditions. It is assumed that a uniform distributed harmonic excitation load in the transverse direction is applied to the CNT-FPMLC plates. The material properties of multiscale composite are estimated by means of the fiber micromechanics and Halpin–Tsai relations. Furthermore, it is assumed that the carbon nanotubes (CNTs) are distributed uniformly and oriented arbitrarily through the epoxy resin matrix. Based upon the Mindlin plate theory and using the von Kármán hypotheses, the governing equations of motion for the in-plane and out-of-plane motions including the effects of geometric nonlinearity, rotary inertia and shear deformation are achieved by means of the Hamilton's principle. In the solution process, the nonlinear partial differential equations of motions and associated boundary conditions are discretized via the generalized differential quadrature (GDQ) and afterward converted into a Duffing-type nonlinear time-varying set of ordinary differential equations via a numerical Galerkin approach. Then, the time periodic discretization method and the pseudo-arc length continuation technique are employed to solve the obtained equations in order to achieve the frequency–response curves associated with nonlinear free and forced resonances for the CNT-FPMLC rectangular plates with various edge supports. Finally, the influences of important design parameters including the weight percentage of single-walled and multi-walled CNTs, volume fraction of fibers, CNT aspect ratio, plate geometry and boundary conditions on the nonlinear resonant dynamics and linear natural frequencies of CNT-FPMLC rectangular plate are investigated in the numerical results.

A new analytical solution and novel energy formulations for non-linear eccentric impact analysis of composite multi-layer/sandwich plates resting on point supports

In the present research, an analytical solution based on a new idea of superposition of two kinematic descriptions is presented for dynamic response analysis of a multi-layer/sandwich composite plate with point supports subjected to an eccentric low-velocity impact. Direct and virtual-work-based novel energy formulations are proposed for the problem that take into account the potential energy of the indentation region. The nonlinear governing equations of motions are found based on minimization of the total potential energy of the whole mass-plate system, including work of the inertia forces, employing Ritz technique and transformation of the time-dependent nonlinear system of governing equations to a non-linear algebraic one through a novel concept. In contrast to the available researches, influence of the lower layers on the stiffness of the contact region is incorporated. Time-dependent responses of a sandwich composite plate with simply supported edges are compared with those of a plate resting on point supports. Verification of the results has been accomplished based on results of ABAQUS computer code. In the results section, the significant effects of the point supports (in comparison to the complete edge supports), initial velocity of the indenter, aspect ratio of the plate, and material properties of the layers on time histories of the contact force and lateral deflection of the plate are investigated.

Geometrically nonlinear resonance of higher-order shear deformable functionally graded carbon-nanotube-reinforced composite annular sector plates excited by harmonic transverse loading

This article presents an attempt to study the nonlinear resonance of functionally graded carbon-nanotube-reinforced composite (FG-CNTRC) annular sector plates excited by a uniformly distributed harmonic transverse load. To this purpose, first, the extended rule of mixture including the efficiency parameters is employed to approximately obtain the effective material properties of FG-CNTRC annular sector plates. Then, the focus is on presenting the weak form of discretized mathematical formulation of governing equations based on the variational differential quadrature (VDQ) method and Hamilton’s principle. The geometric nonlinearity and shear deformation effects are considered based on the von Karman assumptions and Reddy’s third-order shear deformation plate theory, respectively. The discretization process is performed via the generalized differential quadrature (GDQ) method together with numerical differential and integral operators. Then, an efficient multi-step numerical scheme is used to obtain the nonlinear dynamic behavior of the FG-CNTRC annular sector plates near their primary resonance as the frequency-response curve. The accuracy of the present results is first verified and then a parametric study is presented to show the impacts of CNT volume fraction, CNT distribution pattern, geometry of annular sector plate and sector angle on the nonlinear frequency-response curve of FG-CNTRC annular sector plates with different edge supports.

Implementation of numerical approximations in studying vibration of functionally graded beams

In this investigation, a brief review on three efficient computational techniques viz. Finite Element Method, Differential Quadrature Method and Rayleigh–Ritz Method along with their mathematical formulation to study free vibration of thin Functionally Graded (FG) beams subject to various classical boundary supports have been presented. The deformation of FG beam is based on the framework of classical beam theory. Three different FG beam constituents assumed in this study are Al/Al $$_{2}$$ O $$_{3}$$ , Al/ZrO $$_2$$ and SUS304/Si $$_3$$ N $$_4$$ , in which the first component is meant for the metal constituent and the second for ceramic constituent respectively. The material properties of FG beam are assumed to vary continuously along thickness direction in a power-law form. The objective is to outline exemplary works carried out by various researchers on the concerned problem and also to find the effect of volume fraction of FG constituents on natural frequencies. The natural frequencies of different FG beams under four sets of classical edge supports have been evaluated along with two-dimensional mode shapes after finding the convergence with reference to concerned numerical methods and validation with available literature.

Experimentally Investigating Annealed Glazing Response to Long-Duration Blast

This paper investigates the response of annealed (float) glazing during long-duration blast as a function of glazing thickness, area, aspect ratio and edge support conditions. With positive phase durations in excess of 100ms, long-duration blasts produce substantial impulse and pronounced dynamic pressures. Transient dynamic response of annealed glazing during these events is a complex phenomenon dependent upon explosive proximity, structural arrangement and material properties. In particular, breakage time and initial fracture location are dictated by maximum principal stress exceedence at randomly distributed micro-flaws. As part of a larger research study, twelve full-scale air-blast trials employing 24 annealed glazing panels were conducted with ~14kPa peak static overpressure and ~110ms positive phase duration. Results are reported, where it is shown that notionally elastic edge supports can prevent glazing breakage versus rigidly clamped arrangements when suitable panel dimensions are employed. Fragmentation modes are also demonstrated to be a strong function of edge conditions with elastically supported panels producing large, angular fragments. In contrast, rigid arrangements are shown to induce localised impulsive stress transmission at clamped edges, leading to significant cracking and small fragments. Substantially different fragment masses and geometries demonstrate the need to accurately quantify edge supports when appraising fragment hazard. Quantification of peak panel deflection, breakage time and applied breakage impulse is then presented where results indicate the influence of edge supports and aspect ratio on glazing response to be dependent on proximity to the threshold area for a particular thickness.

A unified nonlocal nonlinear higher-order shear deformable plate model for postbuckling analysis of piezoelectric-piezomagnetic rectangular nanoplates with various edge supports

By considering the small-scale effect based on the nonlocal elasticity theory, the nonlinear postbuckling of thick and moderately thick rectangular piezoelectric-piezomagnetic nanoplates with various edge supports subjected to the magneto-electro-thermo-mechanical loading is investigated. For this objective, a unified nonlinear higher-order shear deformable plate model is proposed. By adopting the nonlocal theory to capture the small-scale effect and utilizing a generalized displacement field to consider the influence of transverse shear deformation, unified size-dependent nonlinear governing equations and related boundary conditions are derived based on the virtual work principle in conjunction with von Kármán geometric nonlinearity. By choosing appropriate shape functions, the developed plate model can be reduced to the size-dependent Kirchhoff, Mindlin, Reddy, parabolic, trigonometric, hyperbolic and exponential shear deformable plate models. The nonlinear governing equations and boundary conditions are discretized using the generalized differential quadrature method first. Then, the pseudo arc-length continuation technique is used to solve the discretized equations and obtain the secondary equilibrium path of nanoplate in the postbuckling regime. In addition to providing significant guidelines for accurate prediction of the stability conditions nanoplates, extracting various plate models on the basis of any existing shear deformable plate theory becomes readily attainable by utilizing the proposed unified plate model.

Nonlocal nonlinear first-order shear deformable beam model for postbuckling analysis of magneto-electro-thermo elastic nanobeams

In this study, the size-dependent postbuckling behavior of magneto-electro-thermo-elastic (METE) nanobeams with different edge supports is investigated. Based on the nonlocal first-order shear deformation beam theory and considering the von Karman hypothesis, a size-dependent nonlinear METE nanobeam model is developed, in which the effects of small scale parameter and thermo-electro-magnetic-mechanical loadings are incorporated. A numerical solution procedure based on the generalized differential quadrature (GDQ) and pseudo arc-length continuation methods is utilized to describe the size-dependent postbuckling behavior of METE nanobeams under various boundary conditions. The effects of different parameters such as nonlocal parameter, external electric voltage, external magnetic potential and temperature rise on the postbuckling path of METE nanobeams are explored. The results indicate that increasing the non-dimensional nonlocal parameter, imposed positive voltage, negative magnetic potential and temperature rise decrease the critical buckling load and post-buckling load-carrying capacity of METE nanobeams, whilean increase in the negative voltage, positive magnetic potential lead to a considerable increase of critical buckling load as well as postbuckling strength of the METE nanobeams.

Shear buckling of FG-CNT reinforced composite plates using Chebyshev-Ritz method

Abstract Shear buckling response of carbon nanotube reinforced composite (CNTRC) rectangular plates in thermal environment is investigated in this research. Distribution of CNTs across the plate thickness may be uniform or graded via a mid-plane symmetric pattern. Properties of the CNTRC plate are obtained using the modified rule of mixtures approach by introduction of efficiency parameters. First order plate theory is adopted to construct the basic equations of the plate. A two-dimensional Ritz formulation with Chebyshev basis polynomials is implemented to obtain the elastic and geometric stiffness matrices of the plate. The proposed solution method may be used for arbitrary in-plane and out-of-plane edge supports. After performing convergence and comparison studies to show the effectiveness and accuracy of the proposed method, parametric studies are conducted to examine the influences of boundary conditions, volume fraction of CNTs, graded pattern of CNTs, thermal environment and plate geometry. It is shown that, shear buckling capacity of the plate may be enhanced through functionally graded distribution of CNTs. Besides, enrichment of matrix with CNTs enhances the shear buckling loads of FG-CNTRC plates.

The influence of structural arrangement on long-duration blast response of annealed glazing

This paper investigates the influence of structural arrangement on long-duration blast loaded annealed glazing via variable thickness, area, aspect ratio and edge support conditions. Initially, the findings of eighteen full-scale air-blast trials employing 33 annealed glazing panels are reported where it is demonstrated that fracture mode and fragmentation are a strong function of edge supports. Rigidly clamped edges are shown to induce localised stress transmission, producing significant cracking and small fragments. In contrast, elastic edges are shown to produce large, angular fragments, demonstrating the importance of accurately modelling edge conditions when analysing fragment hazard. Quantification of peak centre panel deflection and breakage time is then presented where variable results indicate the influence of edge supports and aspect ratio to be dependent on proximity to the threshold area as a function of glazing thickness. An initial Applied Element Method (AEM) analysis is then employed to model the influence of structural arrangement on long-duration blast-loaded annealed glazing. AEM models are shown to reasonably predict glazing fragmentation behaviour, breakage time and peak panel deflection at the moment of breakage. Thus indicating AEM's potential suitability to provide a predictive capacity for annealed glazing response during long-duration blast.

Size-dependent modeling of the free vibration characteristics of postbuckled third-order shear deformable rectangular nanoplates based on the surface stress elasticity theory

The present study deals with the nonlinear postbuckling and free vibration of third-order shear deformable rectangular nanoplate with various edge supports in the pre- and post-buckling regimes incorporating the surface effects. The Gurtin–Murdoch surface stress elasticity theory in conjunction with the third-order shear deformation plate theory is used for the size-dependent mathematical modeling of the nanoplates. The von Kármán-type kinematic nonlinearity is used to consider the nonlinear behavior of the nanoplate subjected to the in-plane loadings. The normal stress is assumed to be changed cubically through the thickness direction of nanoplate to satisfy the equilibrium conditions between on the surfaces and bulk layers. The size-dependent coupled in-plane and out-of-plane governing differential equations of motion and corresponding boundary conditions are derived by means of an energy method based on Hamilton's principle. The generalized differential quadrature (GDQ) method and pseudo-arc length continuation are employed to obtain the postbuckling load-deflection curves of nanoplates with various edge supports. A time-dependent small disturbance around the buckled configuration is considered to analyze the free vibration of postbuckled nanoplates. The effects of thickness and surface parameters on the postbuckling path and free vibration characteristics of nanoplates in the pre- and post-buckling regimes are studied. Also, a comparison is made between the results based upon the surface stress elasticity and classical continuum theories so as to show the significance of surface effects.

Surface effect on the large amplitude periodic forced vibration of first-order shear deformable rectangular nanoplates with various edge supports

Surface stress and surface inertia effects may play a significant role in the mechanical characteristics of nanostructures with a high surface to volume ratio. The objective of this study is to present a comprehensive study on the surface stress and surface inertia effects on the large amplitude periodic forced vibration of first-order shear deformable rectangular nanoplates. To this end, the Gurtin–Murdoch theory, first-order shear deformation theory (FSDT) and Hamilton׳s principle are employed to develop a non-classical continuum plate model capable of taking the surface stress and surface inertia effects and also the rotary and in-plane inertias into account. To solve numerically the geometrically nonlinear forced vibration of nanoplates with different boundary conditions, the generalized differential quadrature (GDQ) method, numerical Galerkin scheme, periodic time differential operators and pseudo arc-length continuation method are employed. The effects of parameters such as thickness, surface residual stress, surface elasticity, surface mass density, length-to-thickness ratio, width-to-thickness ratio and boundary conditions on the nonlinear forced vibration of rectangular nanoplates are fully investigated. The results demonstrate that surface effects on the nonlinear frequency response of aluminum (Al) nanoplate are more prominent in comparison with the silicon (Si) nanoplate.

Enhanced thermal buckling of laminated composite cylindrical shells with shape memory alloy

Thermal bifurcation behavior of cross-ply laminated composite cylindrical shells embedded with shape memory alloy fibers is investigated. Properties of the constituents are assumed to be temperature-dependent. Donnell's kinematic assumptions accompanied with the von-Karman type of geometrical non-linearity are used to derive the governing equations of the shell. Furthermore, the one-dimensional constitutive law of Brinson is used to predict the behavior of shape memory alloy fibers through the heating process. Governing equilibrium equations are established by employing the static version of virtual displacements principle. Linear membrane pre-buckling analysis is performed to extract the pre-buckling deformations of the shell. Applying the well-known adjacent equilibrium criterion to the pre-buckling state of the shell, stability equations are derived. The governing equations are solved via a semi-analytical solution employing the exact trigonometric function in circumferential direction and the harmonic differential quadrature method in the longitudinal direction. Numerical results cover various cases of edge supports, cross-ply lamination, shape memory alloy fibers volume fraction and shape memory alloy fiber pre-strain. It is shown that, proper usage of shape memory alloy fibers results in considerable delay of the thermal bifurcation type of buckling.

Non-Linear Thermal Stability Analysis of SMA Wire-Embedded Hybrid Laminated Composite Timoshenko Beams on Non-Linear Hardening Elastic Foundation

In the present study, non-linear thermal post-buckling analysis of hybrid laminated composite Timoshenko beams embedded with the shape memory alloy (SMA) wires resting on a non-linear hardening elastic foundation were studied. Mechanical properties of composite media are considered temperature-dependent. The theory of Timoshenko beams and von Kármán's strain-displacement relations are applied simultaneously in virtual work principles to derive the system of non-linear equilibrium equations. Various types of boundary conditions such as clamped, simply supported, and rolled edges were studied for edge supports. Generalized Differential Quadrature Method (GDQM) was utilized to discrete the equilibrium equations in space domain. Different types of lay-ups, such as symmetric and asymmetric, were considered. Post-buckling paths are depicted for different values of non-linear elastic foundation parameters, volume fractions, pre-strains of the SMA fibers, and boundary conditions. The one-dimensional thermo-mechanical constitutive law suggested by Brinson [1] is applied to model the SMA wires. Numerical results make it possible to recognize that increases in the volume fraction and pre-strain of SMA lead to a dramatic enhancement in thermal buckling and post-buckling capacity of the beam. Pre-buckling, buckling, and post-buckling behavior of the beam are totally different and this is due to variations among critical buckling, austenite start and finish temperatures. Due to the recovery stress of SMA wires, particular consequences are shown.

Free vibration of functionally graded thin elliptic plates with various edge supports

In this article, free vibration of functionally graded (FG) elliptic plates subjected to various classical boundary conditions has been investigated. Literature review reveals no study has been performed based on functionally graded elliptic plates till date. The mechanical kinematic relations are considered based on classical plate theory. Rayleigh-Ritz technique is used to obtain the generalized eigenvalue problem. The material properties of the FG plate are assumed to vary along thickness direction of the constituents according to power-law form. Trial functions denoting the displacement components are expressed in simple algebraic polynomial forms which can handle any edge support. The objective is to study the effect of geometric configurations and gradation of constituent volume fractions on the natural frequencies. New results for frequency parameters are incorporated after performing a test of convergence. A comparison study is carried out with existing literature for validation in special cases. Three-dimensional mode shapes for circular and elliptic FG plates are also presented with various boundary conditions at the edges.

Surface effects on the free vibration behavior of postbuckled circular higher-order shear deformable nanoplates including geometrical nonlinearity

This investigation deals with the free vibration characteristics of circular higher-order shear deformable nanoplates around the postbuckling configuration incorporating surface effects. Using the Gurtin–Murdoch elasticity theory, a size-dependent higher-order shear deformable plate model is developed which takes account all surface effects including surface elasticity, surface stress and surface density. Geometrical nonlinearity is considered based on the von Karman type nonlinear strain–displacement relationships. Also, in order to satisfy the balance conditions between bulk and surfaces of nanoplate, it is assumed that the normal stress is distributed cubically through the thickness of nanoplate. Hamilton׳s principle is utilized to derive non-classical governing differential equations of motion and related boundary conditions. Afterwards, an efficient numerical methodology based on a generalized differential quadrature (GDQ) method is employed to solve numerically the problem so as to discretize the governing partial differential equations along various edge supports using Chebyshev–Gauss–Lobatto grid points and pseudo arc-length continuation technique. A comparison between the results of present non-classical model and those of the classical plate theory is conducted. It is demonstrated that in contrast to the prebuckling domain, for a specified value of axial load in the postbuckling domain, increasing the plate thickness leads to higher frequencies.

Thermal buckling of temperature-dependent FGM conical shells with arbitrary edge supports

Bifurcation behavior of heated conical shell made of a through-the-thickness functionally graded material is investigated in the present research. Properties of the shell are obtained based on a power law form across the thickness. Temperature dependency of the constituents is also taken into account. The heat conduction equation of the shell is solved based on an iterative generalized differential quadrature method (GDQM). General nonlinear equilibrium equations and the associated boundary conditions are obtained using the virtual displacement principle in the Donnell sense. The prebuckling solution of the shell is obtained under the assumption of linear membrane deformations. The stability equations are extracted via the concept of the adjacent equilibrium criterion. A semi-analytical solution employing the GDQM and trigonometric expansion is implemented to solve the stability equations. Numerical results of the present research are compared and validated with the known available data through the open literature. Some parametric studies are conducted to investigate the influences of various involved parameters, such as the cone semi-vertex angle, boundary conditions, power law index of composition rule, length to thickness ratio, and the radius to thickness ratio.

An experimental study on the fatigue performance of CFRP and BFRP repaired aluminium plates

Cracked aluminium plates were repaired with multidirectional carbon/epoxy and boron/epoxy reinforcements. Aircraft repair conditions were simulated by using a steel bonding rig that constrained the effective thermal expansion to the level measured from real locally heated aircraft structures. Residual thermal stresses were varied using the rig and by varying the curing and testing temperatures. The repairs were tested using a variable amplitude loading (R=−0.14) with anti-buckling edge supports. The residual thermal stresses had a rectilinear and distinct effect on the crack growth rate. The structural support against bending had a significant effect on the fatigue life of the repairs. The fatigue life improvement factor with the centre-cracked single-sided repairs was greatest with the wet-laminated carbon/epoxy repairs having a stiffness ratio of 1.04. With edge-cracked specimens the longest life was achieved with the double-sided boron/epoxy repairs having a stiffness ratio of 0.65. The cracked aluminium plate 2024-T3 clad had the longest fatigue life, followed by the S07-1020 and 7075-T6 clad aluminium grades.