Axisymmetric static and dynamic buckling of laminated thick truncated conical cap

Axisymmetric buckling analysis is presented for moderately thick laminated shallow, truncated conical caps under transverse load. Buckling under uniformly distributed loads and ring loads applied statically or as step function loads is considered. 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 mid-surface and the stress function Φ. The governing equations are solved by the orthogonal point collocation method. Truncated conical caps with a circular opening, which is either free or plugged by a rigid central mass, have been analysed for clamped and simple supports with movable and immovable edge conditions. Typical numerical results are presented illustrating the effect of various parameters.     

Surge effect during the water exit of an axisymmetric body traveling normal to a plane interface: experiments and BEM simulation

 Experiments are presented on the surge effect induced by cylindrical bodies piercing a free surface at constant velocity. The study covers the following ranges: 0.1 < Froude² < 7.3, 0.7 < Weber < 89, 46 < Reynolds < 6000; with the Goucher number (Go) evolving between 1.8 and 4.5. Free-surface profiles are compared with those issued from a boundary element method (BEM) simulation based on potential theory and axisymmetric flow configuration. The free-surface deformation is accurately predicted by the BEM approach for Reynolds numbers (Re) higher than 500. However, for Re less than about 200, computed results underestimate the interface elevation, except on the axis of symmetry, where they remain accurate. Finally, the magnitudes of the interface deformation predicted by the numerical simulation are provided over a wide range of Goucher numbers (0.001 < Go < 50) both for hemispherical and conical tips. Implications of these results for phase detection probes are discussed. Received: 18 December 2000/Accepted: 25 January 2001     

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.      

Dynamic analysis of 3-D beam elements including warping and shear deformation effects

In this paper the dynamic analysis of 3-D beam elements restrained at their edges by the most general linear torsional, transverse or longitudinal boundary conditions and subjected in arbitrarily distributed dynamic twisting, bending, transverse or longitudinal loading is presented. For the solution of the problem at hand, a boundary element method is developed for the construction of the 14 × 14 stiffness matrix and the corresponding nodal load vector of a member of an arbitrarily shaped simply or multiply connected cross section, taking into account both warping and shear deformation effects, which together with the respective mass and damping matrices lead to the formulation of the equation of motion. To account for shear deformations, the concept of shear deformation coefficients is used, defining these factors using a strain energy approach. Eight boundary value problems with respect to the variable along the bar angle of twist, to the primary warping function, to a fictitious function, to the beam transverse and longitudinal displacements and to two stress functions are formulated and solved employing a pure BEM approach that is only boundary discretization is used. Both free and forced transverse, longitudinal or torsional vibrations are considered, taking also into account effects of transverse, longitudinal, rotatory, torsional and warping inertia and damping resistance. Numerical examples are presented to illustrate the method and demonstrate its efficiency and accuracy. The influence of the warping effect especially in members of open form cross section is analyzed through examples demonstrating the importance of the inclusion of the warping degrees of freedom in the dynamic analysis of a space frame. Moreover, the discrepancy in the dynamic analysis of a member of a spatial structure arising from the ignorance of the shear deformation effect necessitates the inclusion of this additional effect, especially in thick walled cross section members.     

A Shear-Corrected Formulation for the Sandwich Twist Specimen

The sandwich plate twist test method involves torsion loading of a panel by application of concentrated loads at two diagonally opposite corners and supporting the panel at the other two corners. Compliance measured in this test can be used to extract the shear moduli of monolithic, composite and sandwich plates, and it may also be employed for determination of the twist stiffness, D ₆₆ . Previous studies of the plate twist specimen have shown that classical laminated plate theory does not adequately predict the compliance of sandwich panels with a low density/modulus core, as a result of transverse shear deformation. This work proposes a “shear-corrected” model for accurate prediction of the plate twist compliance by incorporation of the transverse shear stiffnesses of the core. This model was used to extract the transverse shear modulus of a range of low density PVC foam cores from the measured panel twist compliance. Good agreement with published PVC foam core shear modulus values was obtained.     

Resonant vibration and heating of ring plates with piezoactuators under electromechanical loading and shear deformation

The bending vibration and dissipative heating of a viscoelastic isotropic ring plate with piezoceramic actuators under electromechanical loading and shear deformation are studied by solving a coupled problem. The temperature dependence of the complex characteristics of the passive and piezoactive materials is taken into account. The nonlinear problem of thermoviscoelasticity is solved by time stepping with discrete orthogonalization used at each iteration to integrate the equations of elasticity and using an explicit finite-difference scheme to solve the heat-conduction equation with a nonlinear heat source. The effect of shear deformation, fixation conditions for the plate, the geometry of the piezoactuators, and the dissipative-heating temperature on the active damping of the forced vibration of a circular plate subject to uniform transverse monoharmonic compression is studied Translated from Prikladnaya Mekhanika, Vol. 45, No. 2, pp. 124–132, February 2009.     

Effects of transverse shear on the nonlinear bending of rectangular plates laminated of bimodular composite materials

This paper investigates the application of Dynamic-Relaxation (DR) method to the problems of nonlinear bending of rectangular plates laminated of bimodular composite materials. The classical lamination theory and a shear deformation theory of layered composite plates, taking account of large rotations (in the von Karman sense) are employed separately to analyze the subject. It has been found here that the estimation of the fictitious densities which control the convergence and numerical stability of nonlinear DR solution considering transverse shear effect still needs to be further investigated. In this paper, a procedure to calculate fictitious densities has been presented; hence the numerical stability of this topic has been ensured. In this paper the main steps of solving the nonlinear bending of bimodular composite laminates by means of DR method are outlined. The numerical results are given for simply supported, two-layer cross-ply rectangular plates made of mildly bimodular material (Boron-Epoxy (B-E)) and highly bimodular materials (Aramid-Rubber (A-R) and Polyester-Rubber (P-R)) under sinusoidally distributed and uniformly distributed transverse loads. The results obtained have been compared with linear results and those obtained for laminates fabricated from conventional composite materials, the elastic moduli of which are identical with the tensile moduli of the bimodular materials. In addition, the effect of transverse shear deformation on the nondimensionalized center deflection has been studied.The main contents of this paper were presented at the International Symposium of Composite Materials and Structures (June 1986, Beijing).The authors thank Prof. Zhou Li for his guidance.     

Bending of a fiber-reinforced viscoelastic composite plate resting on elastic foundations

Composite structures on an elastic foundation are being widely used in engineering applications. Bending response of inhomogeneous viscoelastic plate as a composite structure on a two-parameter (Pasternak’s type) elastic foundation is investigated. The formulations are based on sinusoidal shear deformation plate theory. Trigonometric terms are used in the present theory for the displacements in addition to the initial terms of a power series through the thickness. The transverse shear correction factors are not needed because a correct representation of the transverse shear strain is given. The interaction between the plate and the foundation is included in the formulation with a two-parameter Pasternak’s model. The effective moduli and Illyushin’s approximation methods are used to derive the viscoelastic solution. The effects played by foundation stiffness, plate aspect ratio, and other parameters are presented.     

Two new refined shear displacement models for functionally graded sandwich plates

Two refined displacement models, RSDT1 and RSDT2, are developed for a bending analysis of functionally graded sandwich plates. Unlike any other theory, the number of unknown functions involved is only four, as against five in case of other shear deformation theories. The developed models are variationally consistent, have strong similarity with classical plate theory in many aspects, do not require shear correction factor, and give rise to transverse shear stress variation such that the transverse shear stresses vary parabolically across the thickness satisfying shear stress-free surface conditions. The accuracy of the analysis presented is demonstrated by comparing the results with solutions derived from other higher-order models. The functionally graded layers are assumed to have isotropic, two-constituent material distribution through the thickness, and the modulus of elasticity, Poisson’s ratio of the faces, and thermal expansion coefficients are assumed to vary according to a power-law distribution in terms of the volume fractions of the constituents. The core layer is still homogeneous and made of an isotropic ceramic material. Numerical results for deflections and stresses of functionally graded metal–ceramic plates are investigated. It can be concluded that the proposed models are accurate and simple in solving the bending behavior of functionally graded plates.     

Stability of a multilayered composite conical shell under uniform external pressure

The stability of equilibrium of a layered composite circular conical truncated shell loaded with uniform external pressure is investigated. A parametric analysis of the critical pressure intensities is carried out with allowance for the transverse shear, the moment character of the subcritical state of equilibrium, and the subcritical strains. Kemerovo State University, Kemerovo 650043. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 4, pp. 198–207, July–August, 1999.     

The effect of transverse pressure on the stability of a plate

The stability problem of a centrally compressed infinite plate is solved with allowance for the transverse normal deformation caused by uniform load for various boundary conditions at the edges. The linearized nonlinear equations of elastic deformation of thin plates taking into account transverse shear and transverse normal deformation are used. The obtained critical loads are compared with existing solutions.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 2, pp. 170–178, March–April, 2005.     

Three-dimensional supersonic turbulent flows past conical bodies

The results of the numerical simulation of supersonic three-dimensional flow past sharp-nosed cones with circular and elliptic cross-sections in the turbulent shock-layer flow regime are presented. The calculations are performed in the local conical approximation using the system of Reynolds equations and the differential one-equation turbulence model. The numerical solutions are obtained by means of an implicit constant-direction finite-difference scheme. The emphasis is placed on the investigation of the transverse flow separation and the flow features associated with the turbulent flow regime. St.Petersburg. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 99–105, January–February, 2000. The study was carried out with the support of the Russian Foundation for Basic Research (project No. 99-01-00735).     

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.     

Analytical solutions for single- and multi-span functionally graded plates in cylindrical bending

A recently developed plate theory using the concept of shape function of the transverse coordinate parameter is extended to determine the stress distribution in an orthotropic functionally graded plate subjected to cylindrical bending. The transfer matrix method is presented to derive the shape function. The equations governing the plate deformation are then solved analytically using the transfer matrix method for arbitrary boundary conditions. For a simply supported functionally graded plate, a comparison of the present solution with the exact elasticity solution, the first- and third-order shear deformation plate theories is presented and discussed. It is demonstrated that the present method yields more accurate stresses than the first- and third-order shear deformation theories. The effect of boundary conditions and inhomogeneity of material on the displacements and stresses in functionally graded plates are investigated. A multi-span functionally graded plate with arbitrary boundary conditions is further considered to demonstrate the efficiency of the present method.     

Vibration analysis of rotating twisted and open conical shells

Based on a non-linear strain–displacement relationship of a non-rotating twisted and open conical shell on thin shell theory, a numerical method for free vibration of a rotating twisted and open conical shell is presented by the energy method, where the effect of rotation is considered as initial deformation and initial stress resultants which are obtained by the principle of virtual work for steady deformation due to rotation, then an energy equilibrium of equation for vibration of a twisted and open conical shell with the initial conditions is also given by the principle of virtual work. In the two numerical processes, the Rayleigh–Ritz procedure is used and the two in-plane and a transverse displacement functions are assumed to be algebraic polynomials in two elements. The effects of characteristic parameters with respect to rotation and geometry such as an angular velocity and a radius of rotating disc, a setting angle, a twist angle, curvature and a tapered ratio of cross-section on vibration performance of rotating twisted and open conical shells are studied by the present method.     

Anomalous predictions of pressure drop and heat transfer in ducts of arbitrary cross-section with modified power law fluids

 The apparent viscosities of purely viscous non-Newtonian fluids are shear rate dependent. At low shear rates, many of such fluids exhibit Newtonian behaviour while at higher shear rates non-Newtonian, power law characteristics exist. Between these two ranges, the fluid's viscous properties are neither Newtonian or power law. Utilizing an apparent viscosity constitutive equation called the “Modified Power Law” which accounts for the above behavior, solutions have been obtained for forced convection flows. A shear rate similarity parameter is identified which specifies both the shear rate range for a given fluid and set of operating conditions and the appropriate solution for that range. The results of numerical solutions for the friction factor–Reynolds number product and for the Nusselt number as a function of a dimensionless shear rate parameter have been presented for forced fully developed laminer duct flows of different cross-sections with modified power law fluids. Experimental data is also presented showing the suitability of the “Modified Power Law” constitutive equation to represent the apparent viscosity of various polymer solutions. Received on 21 August 2000     

On the use of solid- and shell-type finite elements in creep-damage predictions of thinwalled structures

Summary  Solid- and shell-type finite elements available for plasticity and creep analysis are applied to the creep-damage prediction of a thinwalled pipe bend under uniform internal pressure. Conventional creep-damage material model with scalar damage parameter is used. Based on the comparative numerical study, performed using solid and shell elements, the applicability frame of the shell concept is discussed. Particularly, if a dependence on the stress state is included in the material model, the cross-section assumptions of the first-order shear deformation theory should be refined. The possibilities to modify the through-thickness approximations are demonstrated on the beam equations. The first-order shear-deformation beam theory is discussed in detail. It is shown that if the damage evolution significantly differs for tensile and compressive stresses, the classical parabolic transverse shear-stress distribution and the shear-correction coefficient have to be modified within time-step simulations. Received 30 January 2000; accepted for publication 30 May 2000     

Buckling and postbuckling of moderately thick plates

This paper gives the basic differential equations for finite deflections of elastic plates according to Reissner’s approximate stress distributions. The buckling and postbuckling problems of elastic rectangular plates, including the effect of transverse shear deformation, are solved and discussed, by using perturbation method suggested in ref. [8]. The postbuckling equilibrium paths of perfect and imperfect moderately thick rectangular plates are presented and compared with the results based on thin plate theory.     

The effect of shear deformation on the post-buckling behavior of imperfect nonlinear viscoelastic columns

Summary The post-buckling behavior of imperfect columns made of nonlinear viscoelastic materials is investigated, taking into account the effect of shear deformation. The material is modeled according to the Leaderman representation of nonlinear viscoelasticity. Solutions are developed, within the elastica and the shear deformation theories, in order to calculate the growth in time of the total deflection. The numerical results establish the importance of the shear and the nonlinear viscoelasticity effects, and of the h/ℓ ratio in the column post-buckling behavior. Accepted for publication 11 November 1996     

Nonlinear dynamic analysis of Timoshenko beams by BEM. Part II: applications and validation

In this two-part contribution, a boundary element method is developed for the nonlinear dynamic analysis of beams of arbitrary doubly symmetric simply or multiply connected constant cross section, undergoing moderate large displacements and small deformations under general boundary conditions, taking into account the effects of shear deformation and rotary inertia. In Part I the governing equations of the aforementioned problem have been derived, leading to the formulation of five boundary value problems with respect to the transverse displacements, to the axial displacement and to two stress functions. These problems are numerically solved using the Analog Equation Method, a BEM based method. In this Part II, numerical examples are worked out to illustrate the efficiency, the accuracy and the range of applications of the developed method. Thus, the results obtained from the proposed method are presented as compared with those from both analytical and numerical research efforts from the literature. More specifically, the shear deformation effect in nonlinear free vibration analysis, the influence of geometric nonlinearities in forced vibration analysis, the shear deformation effect in nonlinear forced vibration analysis, the nonlinear dynamic analysis of Timoshenko beams subjected to arbitrary axial and transverse in both directions loading, the free vibration analysis of Timoshenko beams with very flexible boundary conditions and the stability under axial loading (Mathieu problem) are presented and discussed through examples of practical interest.