Axisymmetric buckling of laminated, moderately thick shallow conical cap

Summary  Axisymmetric buckling and postbuckling analysis is presented for a moderately thick, laminated shallow conical cap under static transverse load. Marguerre-type, first-order shear deformation shallow-shell theory is formulated in terms of transverse deflection w, the rotation ψ of the normal to the midsurface and the stress function Φ. The governing equations are solved by the orthogonal point-collocation method. Clamped conical caps and simple supports with movable and immovable edge conditions are considered. Typical numerical results are presented, illustrating the effect of various parameters. The dependence of the effect of the shear deformation on the thickness parameter, boundary conditions, ratio of Young's moduli and cap height is investigated. Received 10 February 2000; accepted for publication 3 May 2000     

An efficient higher order zigzag theory for composite and sandwich beams subjected to thermal loading

A new efficient higher order zigzag theory is presented for thermal stress analysis of laminated beams under thermal loads, with modification of the third order zigzag model by inclusion of the explicit contribution of the thermal expansion coefficient α₃ in the approximation of the transverse displacement w. The thermal field is approximated as piecewise linear across the thickness. The displacement field is expressed in terms of the thermal field and only three primary displacement variables by satisfying exactly the conditions of zero transverse shear stress at the top and the bottom and its continuity at the layer interfaces. The governing equations are derived using the principle of virtual work. Fourier series solutions are obtained for simply-supported beams. Comparison with the exact thermo-elasticity solution for thermal stress analysis under two kinds of thermal loads establishes that the present zigzag theory is generally very accurate and superior to the existing zigzag theory for composite and sandwich beams.     

A second order solution of the non-linear equations of conical shells

A second order asymptotic solution to the Donnell type non-linear equations of elastic homogeneous conical shells is presented. Closed form solutions for the displacements and stress resultants including twelve constants of integration are obtained by considering the lateral displacement to be of the order of kt where k is a geometric parameter and t is the shell thickness. Terms of order up tot/r in comparison to unity are omitted. The solutions are valid for asymmetric slowly varying edge or surface loads.     

Experimental investigation of the buckling instability of monocoque shells

Buckling of a series of thin-metal, shallow spherical shells under a uniform hydrostatic pressure has been investigated. Stress and deformation histories, as well as the critical buckling pressure and the post-buckling behavior, have been determined. Comparisons with theoretical analyses for buckling of spherical caps are given. Results are presented for an initial phase of a stability study of truncated conical sections which have been subjected to combinations of axial load and lateral pressure. A series of roll-formed and butt-welded, truncated aluminum conical shells with a 75-deg base angle have been tested. Buckle modes for axial-load condition alone, laterial-pressure load alone and combinations of these loading conditions are described. Interaction curves for the conditions investigated are given.     

Development of a multiaxial fatigue theory by considering constraint effects on small mixed-mode cracks

The effects of the transverse strain (the normal strain in the crack-line direction) on the near-tip fields of small shallow surface cracks (Case A cracks) in power-law hardening materials are investigated by finite element analyses. The small Case A cracks are under plane stress, general yielding, and mixed mode I and II conditions. Constant effective stress contours representing the intense straining zones near the tip, deformed crack-tip profiles and near-tip mode mixity factors are presented for different transverse strains in the crack-line direction. Based on the concept of characterization of fatigue crack growth by the cyclic J-integral, the effects of the transverse strain on J are investigated. The results suggest that the fatigue life prediction based on multiaxial fatigue theories and the critical plane approach should include the constraint effects due to the transverse strain. Consequently, the concept of constant fatigue life contour on the Γ-plane in multiaxial fatigue theories is generalized to the constant fatigue life surface in the Γ-space where the shear strain and the two normal strains are the three axes. Finally, a damage parameter as a function of the shear strain and the two normal strains is proposed for evaluation of fatigue damage under multiaxial loading conditions.     

A Donnell type theory for finite deflection of stiffened thin conical shells composed of composite materials

A Donnell type theory is developed for finite deflection of closely stiffened truncatedlaminated composite conical shells under arbitrary loads by using the variational calculusand smeared-stiffener theory.The most general bending-stretching coupling and the effectof eccentricity of stiffeners are considered.The equilibrium equations,boundary conditionsand the equation of compatibility are derived.The new equations.of the mixed-type ofstiffened laminated composite conical shells are obtained in terms of the transversedeflection and stress function.The simplified equations are also given for some commonlyencountered cases.     

Nonlinear static and dynamic analysis for laminated,annular, spherical caps of moderate thickness

Based on Timoshenko-Mindlin kinematic hypothesis, the shallow shell theory is extended to include the transverse shear deformation for the nonlinear axisymmetric dynamic analysis of the symmetric cross-ply shallow spherical shell. Using the orthogonal point collocation method and the Newmark scheme, an iterative solution is formulated. The numerical results for the nonlinear static and dynamic responses and dynamic buckling of these shallow spherical shells with circular holes under uniformly distributed static or dynamic normal impact loads are presented and compared with available data.     

Imperfection sensitivity of axially compressed stringer reinforced cylindrical sandwich panels

This paper presents some numerical results of the effects of several nondimensional parameters on the buckling and initial post buckling behaviors of shallow sandwich panels under axial compression. Results are presented that show these effects due to transverse shearing resistance of the core material, different face-sheet thicknesses, and different core thicknesses. Further effects on the buckling and initial postbuckling behaviors of sandwich panels are presented due to the torsional resistance of longitudinal edge stiffeners.The results show that the range of flatness parameter, δ/d, for which sandwich panels remain imperfection-insensitive increases with increases in transverse shearing resistance of the core material and with larger core thicknesses. These results also indicate that this range of δ/d is smallest when the face-sheet thicknesses are equal. Finally, as in the case of homogeneous panels, torsional resistance of the longitudinal edge stiffeners has the effect of making the sandwich panel less imperfection-sensitive.     

Nonlinear natural frequency of shallow conical shells with variable thickness

IntroductionTheplatesandtheshellswithvariablethicknessarewidelyusedinengineering .Theproblemaboutstaticshasbeenstudiedbymanyscholars;therearemanyRefs .[1 -4 ]inthisfield .Papersaboutnonlineardynamicsaremuchless[5 ,6 ].Inthispaper,selectingthemaximumamplitudeinthecenterofshallowconicalshellswithvariablethicknessasperturbationparameter,thenonlinearnaturalfrequencyofshallowconicalshellswithvariablethicknessisobtainedbymethodgiveninRef.[7] .Thenonlinearnaturalfrequencyisnotonlyconnectedwiththeva…     

Elasticity solutions for functionally graded annular plates subject to biharmonic loads

Based on England’s expansion formula for displacements, the elastic field in a transversely isotropic functionally graded annular plate subjected to biharmonic transverse forces on its top surface is investigated using the complex variables method. The material parameters are assumed to vary along the thickness direction in an arbitrary fashion. The problem is converted to determine the expressions of four analytic functions α (ζ), β (ζ), ? (ζ) and ψ (ζ) under certain boundary conditions. A series of simple and practical biharmonic loads are presented. The four analytic functions are constructed carefully in a biconnected annular region corresponding to the presented loads, which guarantee the single-valuedness of the mid-plane displacements of the plate. The unknown constants contained in the analytic functions can be determined from the boundary conditions that are similar to those in the plane elasticity as well as those in the classical plate theory. Numerical examples show that the material gradient index and boundary conditions have a significant influence on the elastic field.     

Investigation of the effect of nozzle shape on supersonic/hypersonic impactors designed for size discrimination of nanoparticles

In this study the flow field and the nanoparticle collection efficiency of supersonic/hypersonic impactors with different nozzle shapes were studied using a computational modeling approach. The aim of this study was to develop a nozzle design for supersonic/hypersonic impactors with the smallest possible cut-off size d₅₀ and rather sharp collection efficiency curves. The simulation results show that the changes in the angle and width of a converging nozzle do not alter the cut-off size of the impactor; however, using a conical Laval nozzle with an L/D_n ratio less than or equal to 2 reduced d₅₀. The effect of using a cap as a focuser in the nozzle of a supersonic/hypersonic impactor was also investigated. The results show that adding a cap in front of the nozzle had a noticeable effect on decreasing the cut-off size of the impactor. Both flat disks and conical caps were examined, and it was observed that the nozzle with the conical cap had a lower cut-off size.     

Non-symmetrical large deformation of a shallow thin conical shell

I.IntroductionItisimportanttoresearchnon-symmetricalquestionsofshallowcollicalshellsintheoryoronapplication.Asonekindofpressurevessel'sparts,shallowconicalshellsareverycommonlyusedillellgineerillgpractice,becausethedifficultyofmanul\lctul.illgthemis'small.AlthotlghwelookupmanyChineseandtbreignperiodicalswhicharctlblctobefound,wehavenotyeth'ulldarticlesanddocumentsfornon-symmetricalandnolllinearquestionsofshitllowconictllshells.Oval'rccelltyears,ProlbssorWangXinzhiandhiscolleagueshavedonealot…     

The exact solution for the general bending problems of conical shells on the elastic foundation

The general bending problem of conical shells on the elastic foundation (Winkler Medium) is not solved. In this paper, the displacement solution method for this problem is presented. From the governing differential equations in displacement form of conical shell and by introducing a displacement function U(s,θ), the differential equations are changed into an eight-order soluble partial differential equation about the displacement function U(s,θ) in which the coefficients are variable. At the same time, the expressions of the displacement and internal force components of the shell are also given by the displacement function U(s θ). As special cases of this paper, the displacement function introduced by V.S. Vlasov in circular cylindrical shell^[5], the basic equation of the cylindrical shell on the elastic foundation and that of the circular plates on the elastic foundation are directly derived.Under the arbitrary loads and boundary conditions, the general bending problem of the conical shell on the elastic foundation is reduced to find the displacement function U(s,θ).The general solution of the eight-order differential equation is obtained in series form. For the symmetric bending deformation of the conical shell on the elastic foundation, which has been widely usedinpractice,the detailed numerical results and boundary influence coefficients for edge loads have been obtained. These results have important meaning in analysis of conical shell combination construction on the elastic foundation,and provide a valuable judgement for the numerical solution accuracy of some of the same type of the existing problem.     

Extended meshfree analysis of transverse and inplane loading of a laminated anisotropic plate of general planform geometry

An extended meshfree method is presented for the analysis of a laminated anisotropic plate under elastostatic loading. The plate may be of any planform shape with its thickness profile composed of perfectly bonded uniform thickness layers of distinct anisotropic materials. Both transverse and inplane loads are considered using a first order shear deformation theory for flexural behavior and generalized plane stress for the membrane behavior. In this extended meshfree method, a rectangular domain is initially considered with the plate of arbitrary geometry inscribed within it. A particular solution in the form of an analytic generalized Navier solution (a compound double Fourier series) is used to capture the response due to the loading within the rectangular domain. Then, a homogeneous solution by meshfree analysis is added to treat the augmented boundary conditions on the actual contour of the plate. These augmented conditions are composed of the prescribed values and that of the particular solution evaluated around the plate’s contour.Concentrated transverse and inplane loads in the form of uniform loads over a very small patch are considered with this generalized Navier solution representation. When a meshfree portion is added to account for the boundary conditions, such solutions constitute the Green’s functions for the plate. The viability of these double Fourier series representations is shown by the convergence rates for the kinematic and force/moment fields. An additional example of a two layer ±30° angleply circular plate is given to illustrate the capability of this extended meshfree method.     

Nonlinear free vibration of orthotropic shallow shells of revolution under the static loads

I.Intr0ducti0nThenon1inearvibrationprob1emsofshellsofrevolutionarealwaysofgreatdifficultyandofgreatvaluetostudyfortheircomplexityinmathematicsandmechanicsaswellasinwideapplications.ManyinvestigatorshavemaderesearchontheseinoneWayoranother,butfewinvolvedth…     

考虑横法向热应变的C~0型Reddy板理论和三角形板单元

考虑横法向热变形,建议了C0型Reddy理论,并用于分析复合材料层合/夹层板热膨胀问题。虽然考虑了横法向热应变,但不增加额外的位移变量。此理论位移场不含有横向位移一阶导数,构造有限元时仅需C0插值函数。基于这一模型,运用虚位移原理推导了复合材料板平衡方程以及构造了6节点三角形板单元,并分析了简支复合材料层合/夹层板的热膨胀问题。数值结果表明,建立的模型能准确分析复合材料层合/夹层板热膨胀问题,而忽略横法向热应变的理论分析热膨胀问题误差较大。     

Nonlinear Axisymmetric Static and Transient Response of Orthotropic Annular Shallow Spherical Caps

ABSTRACT

This investigation deals with the nonlinear axisymmetric static and transient response of orthotropic annular shallow spherical caps with a free edge hole and with a rigid central mass, subjected to uniformly distributed load and a central load. The dynamic analogue of Marguerre equations, in terms of normal displacement w and stress function Ψ, are employed. An orthogonal point collocation method is used for spatial discretization and a Newmark-β scheme is used for time integration. Static load, step function load, and sinusoidal pulse load are considered. The influence of annular ratio on the response is studied for isotropic (β = 1) and orthotropic (β = 3) clamped caps, with a rise to thickness ratio of 1.5.     

Non-linear axisymmetric static analysis of orthotropic thin annular plates

This investigation deals with the geometrically non-linear axisymmetric static analysis of elastic orthotropic thin annular plates subjected to uniformly distributed and ring loads. Non-linear differential equations in terms of the transverse displacement w and stress function ψ have been employed. Both w and ψ are expanded in finite power series and the orthogonal point collocation method has been used to obtain discretised algebraic equations from the governing differential equations. Results have been presented for annular plates with and without a rigid plug in the hole; simply supported, clamped and free outer edge conditions; three orthotropic parameters β; two annular ratios and two loading conditions. Results have also been presented illustrating the effect of an elastic rotational constraint at the edge and the effect of a prescribed inplane displacement of the edge.     

功能梯度截顶圆锥壳的热弹性弯曲精确解

研究了功能梯度材料截顶圆锥壳在横向机械载荷与非均匀热载荷同时作用下的变形问题. 基于经典线性壳体理论推导出了以横向剪力和中面转角为基本未知量的功能梯度薄圆锥壳轴对称变形的混合型控制方程. 假设功能梯度圆锥壳的材料性质为沿厚度方向按照幂函数连续变化的形式. 然后采用解析方法求解,得到了问题的精确解. 分别就两端简支和两端固支边界条件,给出了圆锥壳的变形随其载荷、材料参数等变化的特征关系曲线,重点分析和讨论了载荷参数与材料梯度变化参数对变形的影响.