Dynamic Instability of Functionally Graded Shells Using Higher-Order Theory

This paper reports the dynamic instability behavior of functionally graded (FG) shells subjected to in-plane periodic load and temperature field using a higher-order shear deformation theory in conjunction with the finite-element approach. Properties of FG materials are assumed to be temperature dependent and graded in the thickness direction according to the power-law distribution in terms of volume fraction of the constituents. Five forms of shells considered in this investigation are singly curved cylindrical, doubly curved spherical, and hyperbolic paraboloid having two principal curvatures, doubly curved hypar having twist curvature only, and doubly curved conoid having one curvature and twist curvature. The boundaries of dynamic instability regions are obtained using Bolotin’s approach. The structural system is considered to be undamped. The correctness of the formulation is established by comparing the writers’ results with those of problems available in the published literature. Effects of material composition and geometrical parameters are studied on the dynamic instability characteristics of the aforementioned five forms of shells having practical applications in many engineering disciplines.     

Influence of Functionally Graded Material on Buckling of Skew Plates under Mechanical Loads

In this technical note, the critical buckling of simply supported functionally graded skew plate subjected to mechanical compressive loads is evaluated using first-order shear deformation theory in conjunction with the finite element approach. The material properties are assumed to vary in the thickness direction according to the power-law distribution in terms of volume fractions of the constituents. The effective material properties are estimated from the volume fractions and the properties of the constituents using the Mori–Tanaka homogenization method. The effects of aspect ratio, material gradient index, and skew angle on the critical buckling loads of functionally graded material plates are highlighted.     

Thermal Buckling Analysis of SMA Fiber-Reinforced Composite Plates Using Layerwise Model

Thermal buckling analysis of laminated smart composite plates subjected to uniform temperature distribution has been presented. Shape memory alloy (SMA) fibers whose material properties depend on temperature have been used as a smart material. A three-dimensional layerwise plate model has been employed in developing the system equations using variational approach. Finite-element method has been adopted for discretization of the laminate. Lagrangian interpolation functions have been used to approximate the displacement components along the thickness as well as in the in-plane direction. The actual variation of prebuckling stresses has been accounted for in the derivation of the geometric stiffness matrix of the laminates. An incremental load technique has been used in the analysis to take into account the nonlinearity in the material properties of the SMA arising due to their temperature dependence. The effects of thickness ratio, orthotropic ratio, fiber orientation, aspect ratio, stacking sequence and various boundary conditions on the critical buckling temperature have been examined in details. The results have been validated with those available in the literature.     

Postbuckling of Axially Loaded Functionally Graded Cylindrical Panels with Piezoelectric Actuators in Thermal Environments

A compressive postbuckling analysis is presented for a functionally graded cylindrical panel with piezoelectric actuators subjected to the combined action of mechanical, electrical, and thermal loads. The temperature field considered is assumed to be of uniform distribution over the panel surface and through the panel thickness and the electric field considers only the transverse component EZ. The material properties of the presently considered functionally graded materials (FGMs) are assumed to be temperature-dependent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents, whereas the material properties of the piezoelectric layers are assumed to be independent of the temperature and the electric field. The governing equations are based on a higher-order shear deformation theory with a von Kármán-Donnell-type of kinematic nonlinearity. A boundary layer theory for shell buckling is extended to the case of hybrid FGM cylindrical panels of finite length. The nonlinear prebuckling deformations and initial geometric imperfections of the panel are both taken into account. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the compressive postbuckling behavior of perfect and imperfect FGM cylindrical panels with fully covered piezoelectric actuators, under different sets of thermal and electrical loading conditions. The effects due to temperature rise, volume fraction distribution, applied voltages, panel geometric parameters, in-plane boundary conditions, as well as initial geometric imperfections are studied.     

Postbuckling of Pressure-Loaded Functionally Graded Cylindrical Panels in Thermal Environments

A postbuckling analysis is presented for a functionally graded cylindrical panel of finite length subjected to lateral pressure in thermal environments. Material properties are assumed to be temperature dependent, and graded in the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents. The governing equations of a functionally graded cylindrical panel are based on Reddy’s higher-order shear deformation shell theory with von Kármán–Donnell-type of kinematic nonlinearity and include thermal effects. The two straight edges of the panel are assumed to be simply supported and two curved edges are either simply supported or clamped. The nonlinear prebuckling deformations and initial geometric imperfections of the panel are both taken into account. A boundary layer theory of shell buckling, which includes the effects of nonlinear prebuckling deformations, large deflection in the postbuckling range, and initial geometric imperfections of the shell, is extended to the case of functionally graded cylindrical panels. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling behavior of simply supported, pressure-loaded, perfect and imperfect, functionally graded cylindrical panels with two constituent materials under different sets of thermal environments. The influences played by temperature rise, volume fraction distributions, transverse shear deformation, panel geometric parameters, as well as initial geometric imperfections, are studied.     

Nonlinear Buckling of Composite Shells of Revolution

By adopting the energy method, a method of calculating the stability of the rotational composite shell is presented that takes into account the influence of nonlinear prebuckling deformations and stresses on the buckling of the shell. The relationships between the prebuckling deformations and strains are calculated by nonlinear Karman equations. The numerical method is used to calculate the energy of the whole system. The nonlinear equation is solved by combining the gradient method and the amended Newton iterative method. A computer program is also developed. Examples are given to demonstrate the accuracy of the method presented in this paper.     

Dynamic Axial-Moment Buckling of Linear Beam Systems by Power Series Stiffness

The paper applies the power series method to find the dynamic stiffness for the dynamics axial-moment buckling analyses of linear framed structures. Since the formulation is exact in classical sense, one element is good enough for the entire beam. The dynamic stiffness thus obtained can be decomposed into the stiffness, mass and initial stress matrices at a particular frequency, a particular axial force and a particular initial moment. The given axial force and moment can be nonuniformly distributed. The interaction diagrams in classical loading conditions of uniform moment, moment due to concentrated and distributed lateral force are given explicitly. The effects of warping rigidity, torsion rigidity, axial tension and compression are investigated in detail. The static and dynamic interaction buckling of a two-section I-beam structure is studied. Finally, we conclude that the three dimensional interaction diagram of the dynamic biaxial moment buckling can be obtained simply by rotating the three dimensional interaction diagram of the dynamic mono-axial moment buckling about the frequency axis if the bimoments are appropriately scaled. It is shown that application for non-uniform section is not suitable due to convergent problem. The method is very efficient that many interaction diagrams are produced for the first time.     

Spatial Rotation Kinematics and Flexural–Torsional Buckling

This paper aims to clarify the intricacies of spatial rotation kinematics as applied to three-dimensional (3D) stability analysis of metal framed structures with minimal mathematical abstraction. In particular, it discusses the ability of the kinematic relationships traditionally used for a spatial Euler–Bernoulli beam element, which are expressed in terms of transverse displacement derivatives, to detect the flexural–torsional instability of a cantilever and of an L-shaped frame. The distinction between transverse displacement derivatives and vectorial rotations is illustrated graphically. The paper also discusses the symmetry and asymmetry of tangent stiffness matrices derived for 3D beam elements, and the concepts of semitangential moments and semitangential rotations. Finally, the fact that the so-called vectorial rotations are independent mathematical variables are pointed out.     

Thermal Stresses in Spherical Shells

This paper presents an iterative finite-difference technique for the analysis of axisymmetric spherical shells with variable wall thickness. The formulation is based on thin elastic shell theory. One-dimensional finite-difference points are used to discretize the shell into strips along the meridian, and an iterative technique is employed to determine the normal and meridional displacements. The stress resultants and bending stresses are then evaluated. Unlike existing analytical and finite-difference techniques, the proposed method is applicable with ease to any variation in rigidity along the meridian, to general loading conditions, and to steep and shallow shells. Results are presented and compared with those of the finite-element method.     

Thermal Scale Modeling by FEM and Test

Scale model approach in thermal testing of aircraft components has been proposed in this study. The idea of thermal scaling has been validated experimentally. Quite often, it may not be possible to carry out experimental validation of design analysis at high temperatures on prototype structures at the design stage since exact simulation of thermal environments may not be possible. Under such conditions, the designer may resort to study the structural behavior using scale model testing at a different temperature where the material does not change phase at the scaled temperature.     

超音速等离子喷涂制备梯度功能热障涂层的特点

在国内最新研制成功的超音速等离子喷涂系统基础上,开发了国内外独创的“双通道、双温区”超音速等离子喷涂工艺,将高熔点陶瓷粉末和低熔点合金粉末分别送入等离子焰流的不同温度区,保证两种粉末分别达到各自的熔点,再均匀混合后以超音速喷射到基体形成涂层,克服了传统的“金属/陶瓷混合法”制备梯度热障涂层时,低熔点金属相过熔而引起的氧化、脆化问题。制备出的Ce-YSZ/NiCoCrAlY梯度功能热障涂层(FG-TBCs)组织结构均匀致密,陶瓷与金属组份呈连续梯度分布,显示出良好的韧性。在自行开发的多功能旋转式热震试验机上对约1 mm厚的涂层经1 200℃加热、淬水,200周次热震试验后,涂层表面仅出现了细微网状裂纹,裂纹垂直向下扩展最深350μm,止于梯度层中塑性金属区,未发现任何涂层剥落现象。     

Temporal Thermal Behavior and Damage Simulations of FRP Deck

The finite-element method (FEM) has been employed to study the structural behavior of the fiber-reinforced polymer (FRP) bridge deck. The numerical results were verified with the field-test results provided by New York State Department of Transportation. Fully coupled thermal-stress analyses were conducted using the FEM to predict the failure mechanisms and the “fire resistance limit” of the superstructure under extreme thermal loading conditions. Furthermore, damage simulations of the FRP deck as a result of snow and ice plowing process were performed to investigate any possibility of bridge failure after damage occurs. Thermal simulations showed that FRP bridge decks are highly sensitive to the effect of elevated temperatures. The FRP deck approached the fire resistance limit at early stages of the fire incident under all cases of fire scenarios. The damage simulations due to the snow plowing showed minimal possibility of bridge failure to take place under the worst-case damage scenario when the top 5 mm of the FRP deck surface was removed. The results of both phases of simulations provide an insight into the safety and the reliability of the FRP systems after the stipulated damage scenarios were considered. Moreover, this paper provides discussions concerning the recommended immediate actions necessary to repair the damaged region of FRP deck panels and possible use of the bridge after the damage incident.     

Wrinkling and Edge Buckling in Orthotropic Sandwich Beams

A sandwich beam buckling problem is studied here using two-dimensional elasticity to model the beam constituents. The global and local instability of such a beam with orthotropic constituents under various boundary conditions are investigated. The face sheet and the core are assumed to be linear elastic orthotropic continua. General buckling deformation modes of the sandwich beam subjected to uniaxial compressive loading are considered. The appropriate incremental stress and conjugate incremental finite-strain measure for the instability problem of the sandwich beam, and the corresponding constitutive model are addressed. It is shown that a sandwich beam having a core with a negligible stiffness compared to the face sheets is prone to fail by edge buckling. The present analysis is compared with several previous analytical studies and corresponding experimental results. Finite-element analyses are carried out for comparison against the theoretical predictions. The formulation used in the finite-element code is discussed in relation to the formulation adopted in the theoretical derivation.     

Approximate Derivation of Critical Buckling Load

This paper proposes an approximate derivation for the critical buckling load of a column, based on the application of a uniformly loaded beam's midspan moment and deflection to the buckled column's rotational equilibrium. The curvature of a pin-ended member, when it buckles under axial load, is similar to the curvature assumed by the same member when it deflects under a uniformly distributed load applied transversely along its entire length. Euler's famous equation for critical buckling load is based, of course, on the former assumption, in which the deflected column assumes the shape of a sine curve. However, dividing a uniformly loaded beam's midspan moment by its deflection provides a conservative result for the critical buckling load, within 3% of Euler's value, that can be derived solely on the basis of these commonly used beam equations.     

Dynamic Stability of Laminated Composite Curved Panels with Cutouts

The present investigation deals with the dynamic stability behavior of laminated composite curved panels with cutouts subjected to in-plane static and periodic compressive loads, analyzed using the finite element method. A generalized shear deformable Sanders’ theory with tracers is used in this study. Numerical results obtained for vibration and buckling of composite panels with cutouts compare well with literature. The principal dynamic instability region of composite perforated panels is obtained using Bolotin’s approach. The study reveals that curved panels with cutouts depict higher stiffness with the addition of curvatures. The laminated hyperbolic paraboloid panel shows the highest stiffness with the onset of instability at higher excitation frequencies. The effect of curvature in laminated composite curved panels is reduced with an increase in size of the cutout. The principal instability regions are influenced by the lamination parameters. Thus, the laminate construction, coupled with cutout geometry, can be used to the advantages of tailoring during design of composite structures for practical applications.     

Flexural-Torsional Buckling of Cantilever Strip Beam-Columns with Linearly Varying Depth

In this paper, one investigates the elastic flexural-torsional buckling of linearly tapered cantilever strip beam-columns acted by axial and transversal point loads applied at the tip. For prismatic and wedge-shaped members, the governing differential equation is integrated in closed form by means of confluent hypergeometric functions. For general tapered members (0<(hmax?hmin)/hmax<1), the solution to the boundary value problem is obtained in the form of a Frobenius’ series, which is shown to converge in the interior of the domain and at the boundary if and only if 0<(hmax?hmin)/hmax<1/2. Therefore, for 1/2?(hmax?hmin)/hmax<1 the Frobenius’ series solution cannot be used to establish the characteristic equation for the cantilever beam-columns; the problem is then solved numerically by means of a collocation procedure. Some of the analytical solutions (buckling loads) were compared with the results of shell finite-element analyses and an excellent agreement was found in all cases, thus validating the mathematical model and confirming the correctness of the analytical results. The paper closes with a discussion on the convexity of the stability domain (in the load parameter space) and the accuracy of approximations based on Dunkerley-type theorems.     

Thermal Postbuckling of Uniform Columns on Elastic Foundation—Intuitive Solution

Finite element and Rayleigh-Ritz methods have been effectively used to evaluate the thermal postbuckling behavior of columns with immovable ends in the axial direction. However, these methods do suffer from problems like large computational effort or complex algebra because of the nonlinear nature of the problem. A simple, intuitive method is proposed herein to predict the postbuckling load carrying capacity of uniform columns on elastic foundation. The present method requires the knowledge of only the linear thermal buckling load parameter and the tension developed in the column. The present method, when applied to simply supported and clamped columns gives exactly the same results as those obtained by the Rayleigh-Ritz method.     

Buckling Analysis of Nonuniform and Axially Graded Columns with Varying Flexural Rigidity

In this paper, we present a novel analytic approach to solve the buckling instability of Euler-Bernoulli columns with arbitrarily axial nonhomogeneity and/or varying cross section. For various columns including pinned-pinned columns, clamped columns, and cantilevered columns, the governing differential equation for buckling of columns with varying flexural rigidity is reduced to a Fredholm integral equation. Critical buckling load can be exactly determined by requiring that the resulting integral equation has a nontrivial solution. The effectiveness of the method is confirmed by comparing our results with existing closed-form solutions and numerical results. Flexural rigidity may take a majority of functions including polynomials, trigonometric and exponential functions, etc. Examples are given to illustrate the enhancement of the load-carrying capacity of tapered columns for admissible shape profiles with constant volume or weight, and the proposed method is of benefit to optimum design of columns against buckling in engineering applications. This method can be further extended to treat free vibration of nonuniform beams with axially variable material properties.     

Novel Formulation to Study Thermal Postbuckling of Circular Plates with Edges Elastically Restrained against Rotation

A novel formulation is used to study the thermal postbuckling behavior of circular plates, with the edges supported to not have lateral deflection and elastically restrained against rotation. The elastic restraint is mathematically represented by an elastic rotational spring. The circular plate is subjected to a uniform edge compressive radial load, developed because of a uniform temperature rise. The formulation is on the basis of on the radial tensile load developed in the plate because of the large deflections of the plate with edges immovable in the plane normal to the edge and the linear buckling load corresponding to the uniform edge radial compressive load. The developed radial tensile load is obtained by using Berger’s approximation. The numerical results obtained from the present investigation in terms of the ratios of the postbuckling to the buckling loads for several rotational spring stiffness values compare well with those obtained by using the versatile finite-element analysis.     

Numerical/Experimental Correlation of a Plasma Wind Tunnel Test on a UHTC-Made Nose Cap of a Reentry Vehicle

The nose cap demonstrator named Nose_2 has been tested for the second time in the plasma wind tunnel (PWT) facility which is part of the sharp hot structure (SHS) technology project, focused on the assessment of the applicability of ultrahigh temperature ceramics (UHTC) to the fabrication of high performance vehicles and SHS for reusable launch vehicles. In this paper the FEM based thermal analyses, carried out for the rebuilding of this PWT test, are presented. Experimental data measured in the PWT have been compared with numerical ones in order to validate the FEM model and to help in interpreting the experimental test itself. The knowledge on the physical phenomenon under investigation has been greatly improved, thanks to the synergy between numerical and experimental activities. In particular, a qualitative study of the modeling of the tip-dome interface has been performed in order to estimate the thermal contact resistance that heat flux encounters in passing through the demonstrator. The correlation between numerical and experimental temperature curves has been found to be satisfactory for both internal and surface temperature distribution, and the FEM model was found reliable in reproducing the thermal behavior of the nose cap.