Geometrically non-linear free and forced vibrations of simply supported circular cylindrical shells: A semi-analytical approach

The non-linear free and forced vibrations of simply supported thin circular cylindrical shells are investigated using Lagrange's equations and an improved transverse displacement expansion. The purpose of this approach was to provide engineers and designers with an easy method for determining the shell non-linear mode shapes, with their corresponding amplitude dependent non-linear frequencies. The Donnell non-linear shell theory has been used and the flexural deformations at large vibration amplitudes have been taken into account. The transverse displacement expansion has been made using two terms including both the driven and the axisymmetric modes, and satisfying the simply supported boundary conditions. The non-linear dynamic variational problem obtained by applying Lagrange's equations was then transformed into a static case by adopting the harmonic balance method. Minimisation of the energy functional with respect to the basic function contribution coefficients has led to a simple non-linear multi-modal equation, the solution of which gives in the case of a single mode assumption an expression for the non-linear frequencies which is much simpler than that derived from the non-linear partial differential equation obtained previously by several authors. Quantitative results based on the present approach have been computed and compared with experimental data. The good agreement found was very satisfactory, in comparison with previous old and recent theoretical approaches, based on sophisticated numerical methods, such as the finite element method (FEM), the method of normal forms (MNF), and analytical methods, such as the perturbation method.     

One-to-two global-local interaction in a cable-stayed beam observed through analytical, finite element and experimental models

The mechanism of cable end angle-variation induced oscillations in the non-linear interactions between beams and cables in stayed-systems is first explained by a proposed analytical model. It is then verified by both experimental and finite element models. The non-linear interaction maximizes its effects for cable oscillations when inherent quadratic coupling between local and global modes produces energy transfer from low to high frequency vibrations by means of a one-to-two global-local autoparametric resonance. The response of the analytical model is fully described using a continuation method applied directly to the reduced two degree of freedom discrete model showing that, for a selected one-to-two global-local resonant system, primary harmonic excitation of the global mode produces large oscillations of the local mode at twice the excitation frequency. Detailed comparisons between the responses of the analytical model, experimental results and finite element simulations show excellent agreement both in the qualitative behaviour and in the calculated/measured response amplitudes.     

Large oscillations of beams and columns including self-weight

Large-amplitude, in-plane beam vibration is investigated using numerical simulations and a perturbation analysis applied to the dynamic elastica model. The governing non-linear boundary value problem is described in terms of the arclength, and the beam is treated as inextensible. The self-weight of the beam is included in the equations. First, a finite difference numerical method is introduced. The system is discretized along the arclength, and second-order-accurate finite difference formulas are used to generate time series of large-amplitude motion of an upright cantilever. Secondly, a perturbation method (the method of multiple scales) is applied to obtain approximate solutions. An analytical backbone curve is generated, and the results are compared with those in the literature for various boundary conditions where the self-weight of the beam is neglected. The method is also used to characterize large-amplitude first-mode vibration of a cantilever with non-zero self-weight. The perturbation and finite difference results are compared for these cases and are seen to agree for a large range of vibration amplitudes. Finally, large-amplitude motion of a postbuckled, clamped–clamped beam is simulated for varying degrees of buckling and self-weight using the finite difference method, and backbone curves are obtained.     

薄膜褶皱的非线性屈曲有限元分析

有限元模拟已经成为目前薄膜褶皱预测的重要工具。本文采用ANSYS SHELL63单元对薄膜受剪情况下的褶皱形变进行了非线性屈曲分析。通过本征屈曲分析得到的模态模拟了薄膜的初始缺陷。利用本文的非线性有限元模型分析得到了薄膜褶皱的波长和幅度,并与理论分析结果进行了比较,结果相近。     

Efficient analysis of large-scale structural problems with geometrical non-linearity

The paper presents an efficient methodology for the analysis of large-scale structural problems with geometrical non-linearity. A finite element based tool is developed, taking advantage of the analytical formulation of the stiffness matrix of a beam element, which is explicitly separated in linear and non-linear terms. The methodology proposes the substitution of the typical Newton-type non-linear analysis procedure, by a series of incremental linear analyses and a set of ‘fictitious’ forces, replacing the non-linear effect. The proposed technique is demonstrated in several structural problems that exhibit geometrical non-linear behaviour, with satisfactory results. The method’s advantages on the analysis of large-scale non-linear problems are discussed, as well as the limitations and the further development that is required.     

Non-linear and linear evolution of perturbation in stochastic basic flows

In this paper, we examine the non-linear and linear evolutions of perturbation in stochastic basic flows with two-dimensional quasi-geostrophic equations on a sphere. As the analytic solutions for the considered quasi-geostrophic equations are not available, the Fourier finite volume element method is used to perform numerical simulation. It is found that, the non-linear and linear evolutions of perturbation in stochastic basic flow will be consistent for a short period of time and small stochastic fluctuations when they are consistent in the deterministic basic flow. However, the tangent linear model will fail to approximate the original non-linear model when the time period is considerably long and stochastic fluctuation becomes large. Moreover, the global energy decays faster for stochastic basic flow with stronger fluctuations.     

Multistable composite plates with piecewise variation of lay-up in the planform

The non-linear out-of-plane displacements of partially unsymmetric laminates are modelled using both finite element analysis and an analytical method. Attention is focused on the effects that thermal stresses have on the potential multiple shapes of a composite structure. The paper extends previous analytical models which could only take into account “free-free” boundary conditions. The shape functions that model the out-of-plane displacements are modified to include variations of the curvatures within the domain. The new analytical formulation is compared with literature and finite element analysis for a square plate and then it is tested for laminates with piecewise variation of lay-up in the planform. The results are validated against finite element analysis and experimental tests and a good correlation is obtained. Finally, a parametric study is made on the effect of changing the fibres orientation and the laminate thickness. The results confirm that it is possible to introduce bi-stable composites within structures to obtain systems that are both flexible and stiff depending on the loading environment.     

A transonic quasi-3D analysis for gas turbine engines including split-flow capability for turbofans

A numerical approximation is taken to the solution of the complex flows existing in gas turbine engines with transonic blading. The quasi-3D approach decouples the problem into through-flow and blade-to-blade solutions. An industrially practical finite element through-flow solution is developed and for blade-to-blade solutions a transonic finite areas method is utilized. The finite element code developed is capable of operating in an analysis or a design mode. In both modes a dynamic relaxation factor is employed and considerable reduction in solution time can be achieved. Comparisons to streamline curvature methods are carried out for simple analytical and complex industrial problems.     

Non-linear in-plane analysis and buckling of pinned–fixed shallow arches subjected to a central concentrated load

This paper is concerned with an analytical study of the non-linear elastic in-plane behaviour and buckling of pinned–fixed shallow circular arches that are subjected to a central concentrated radial load. Because the boundary conditions provided by the pinned support and fixed support of a pinned–fixed arch are quite different from those of a pinned–pinned or a fixed–fixed arch, the non-linear behaviour of a pinned–fixed arch is more complicated than that of its pinned–pinned or fixed–fixed counterpart. Analytical solutions for the non-linear equilibrium path for shallow pinned–fixed circular arches are derived. The non-linear equilibrium path for a pinned–fixed arch may have one or three unstable equilibrium paths and may include two or four limit points. This is different from pinned–pinned and fixed–fixed arches that have only two limit points. The number of limit points in the non-linear equilibrium path of a pinned–fixed arch depends on the slenderness and the included angle of the arch. The switches in terms of an arch geometry parameter, which is introduced in the paper, are derived for distinguishing between arches with two limit points and those with four limit points and for distinguishing between a pinned–fixed arch and a beam curved in-elevation. It is also shown that a pinned–fixed arch under a central concentrated load can buckle in a limit point mode, but cannot buckle in a bifurcation mode. This contrasts with the buckling behaviour of pinned–pinned or fixed–fixed arches under a central concentrated load, which may buckle both in a bifurcation mode and in a limit point mode. An analytical solution for the limit point buckling load of shallow pinned–fixed circular arches is also derived. Comparisons with finite element results show that the analytical solutions can accurately predict the non-linear buckling and postbuckling behaviour of shallow pinned–fixed arches. Although the solutions are derived for shallow pinned–fixed arches, comparisons with the finite element results demonstrate that they can also provide reasonable predictions for the buckling load of deep pinned–fixed arches under a central concentrated load.     

Fuzzy variational principle and its applications

Linear and non-linear peaky fuzzy numbers and their arithmetic operations are constructed for the analysis of engineering structures with fuzzy characteristic quantities. Fuzziness of the corresponding quantities is consistently incorporated into the functional of the total potential energy. A set of deterministic recursive equations is obtained as the alternative expressions of the fuzzy variational principle by means of the second-order perturbation technique. The fuzzy Ritz method and the fuzzy finite element method are presented as the applications of the fuzzy variational principle. Accordingly, the roundabout procedures frequently used in the formulations of the fuzzy finite element method are avoided. A benchmark problem of a bending beam with fuzzy Young's modulus under fuzzy external loading is solved by the developed fuzzy numerical methods. Numerical examples show that results determined by these two fuzzy methods are both little conservative, and are in good agreement with those obtained by the analytical method. Moreover, the fuzzy Ritz method or the fuzzy finite element method can provide more valuable information than the conventional deterministic methods.     

Effect of the order of the finite element and selection of the constrained modes in deformable body dynamics

Finite elements with different orders can be used in the analysis of constrained deformable bodies that undergo large rigid body displacements. The constrained mode shapes resulting from the use of finite elements with different orders differ in the way the stiffness of the body bending and extension are defined. The constrained modes also depend on the selection of the boundary conditions. Using the same type of finite element, different sets of boundary conditions lead to different sets of constrained modes. In this investigation, the effect of the order of the element as well as the selection of the constrained mode shapes is examined numerically. To this end, the constant strain three node triangular element and the quadratic six node triangular element are used. The results obtained using the three node triangular element are compared with the higher order six node triangular element. The equations of motion for the three and six node triangular elements are formulated from assumed linear and quadratic displacement fields, respectively. Both assumed displacement fields can describe large rigid body translational and rotational displacements. Consequently, the dynamic formulation presented in this investigation can also be used in the large deformation analysis. Using the finite element displacement field, the mass, stiffness, and inertia invariants of the three and six-node triangular elements are formulated. Standard finite element assembly techniques are used to formulate the differential equations of motion for mechanical systems consisting of interconnected deformable bodies. Using a multibody four bar mechanism, numerical results of the different elements and their respective performance are presented. These results indicate that the three node triangular element does not perform well in bending modes of deformation.     

变厚度中厚板和中厚壳的大挠度分析

采用摄动有限元法分析了变厚度中厚板和中厚壳的大挠度问题。文中借助虚功原理导出了这类板壳的一般非线性有限元方程，同时利用摄动展开求得了逐级摄动有限元的递推算式。算例表明，摄动有限元法分析变厚度中厚板壳问题同样能获得效率高精度好的结果。     

基于mbS模式的有限单元法

mbS模式及其有限元法是在固体和结构分析模型中引入薄膜、弯曲和剪切理论,且采用纯拉压、纯弯和纯剪单元进行分析的数值方法。在时空系中剖分物质单元和时间单元上构造以指数函数和贝塞尔函数为插入函数且按Lagrange插值条件的薄膜、弯曲和剪切等基本位移函数,由此得到更加完备和耦合的固体和结构实体单元的变形模式,根据能量泛函变分原理得到静动力有限元基本方程的一致格式。研究表明,mbS模式及其有限元法可用于梁柱和板壳等结构的静动力分析及屈曲分析。     

Necking of biaxially stretched elastic-plastic circular plates

The necking of an elastic-plastic circular plate under uniform radial tensile loading is investigated both within the framework of the three-dimensional theory and within the context of the plane-stress approximation. Attention is restricted to axisymmetric deformations of the plate. The material behavior is described by two different constitutive laws. One is a finite-strain version of the simplest flow-theory of plasticity and the other is a finite-strain generalization of the simplest deformationtheory, which is employed as a simple model of a solid with a vertex on its yield surface. For an initially uniform plate made of an incompressible material, bifurcation from the uniformly stretched state is studied analytically. The regimes of stress and moduli where the governing axisymmetric three-dimensional equations are elliptic, parabolic or hyperbolic are identified. The plane-stress local-necking mode emerges as the appropriate limiting mode from the bifurcation modes available in the elliptic regime. In the elliptic regime, the main qualitative features of the bifurcation behavior are revealed by the plane-stress analysis, although three-dimensional effects delay the onset of necking somewhat. For the deformation theory employed here, the first bifurcation modes are encountered in the parabolic regime if the hardening-rate is sufficiently high. These bifurcations are not revealed by a plane-stress analysis. For a plate with an initial inhomogeneity, the growth of an imperfection is studied by a perturbation method, by a plane-stress analysis of localized necking, and by numerical computations within the framework of the three-dimensional theory. When bifurcation of the corresponding perfect plate takes place in the elliptic regime, the finite element results show that the plane-stress analysis gives reasonably good agreement with the numerical results. When bifurcation of the corresponding perfect plate first occurs in the parabolic regime, then a bifurcation of the imperfect plate is encountered, that is, the finite element stiffness matrix ceases to be positive definite.     

Non-linear vibrations of free-edge thin spherical shells: modal interaction rules and 1:1:2 internal resonance

This paper is devoted to the derivation and the analysis of vibrations of shallow spherical shell subjected to large amplitude transverse displacement. The analog for thin shallow shells of von Kármán’s theory for large deflection of plates is used. The validity range of the approximations is assessed by comparing the analytical modal analysis with a numerical solution. The specific case of a free edge is considered. The governing partial differential equations are expanded onto the natural modes of vibration of the shell. The problem is replaced by an infinite set of coupled second-order differential equations with quadratic and cubic non-linear terms. Analytical expressions of the non-linear coefficients are derived and a number of them are found to vanish, as a consequence of the symmetry of revolution of the structure. Then, for all the possible internal resonances, a number of rules are deduced, thus predicting the activation of the energy exchanges between the involved modes. Finally, a specific mode coupling due to a 1:1:2 internal resonance between two companion modes and an axisymmetric mode is studied.     

夹层梁总体屈曲及皱曲的有限元计算

皱曲是夹层结构的一种短波屈曲模式,通常发生于夹心较厚或夹心刚度较低的情况。由于模型规模的限制,在常规有限元建模时通常将夹层板模拟为二维板单元,这种方法忽略了面板和夹心在厚度方向上的相互作用,无法计算出皱曲模式。针对上述问题,本文首先介绍了一个计算夹层结构总体屈曲和皱曲的统一理论,并将此理论的计算结果作为理论解。为了同时...     

Wave induced thermal boundary layers in a compressible fluid: analysis and numerical simulations

The response of a semi-infinite compressible fluid to a step-wise change in temperature of its boundary is investigated analytically and numerically. Numerical results of the boundary layer structure are compared with Clarke’s analytical solution for a gas with thermal conductivity proportional to temperature. To avoid unwanted numerical dissipation in the numerical analysis, the space-time conservation element and solution element (CESE) method has been adopted to solve the unsteady 1-D Navier-Stokes equations. Good agreement between analytical and numerical results has been found for the development of the thermal boundary layer on a long time scale. Weak shock waves and expansion waves induced by the thermal boundary layer due to its compressibility, are observed in the numerical simulation. Finally, the numerical method has been applied to the reflection of a non-linear expansion wave and to a shock wave from an isothermal wall, thereby illustrating the effect of the boundary layer on the external flow field.     

GBT pre-buckling and buckling analyses of thin-walled members under axial and transverse loads

This paper presents an analytical approach for pre-buckling and buckling analyses of thin-walled members implemented within the framework of the Generalised Beam Theory (GBT). With the proposed GBT cross-sectional analysis, the set of deformation modes used in the analysis is represented by the dynamic modes obtained for an unrestrained frame representing the cross-section. In this manner, it is possible to account for the deformability of the cross-section in both pre-buckling and buckling analyses. Different loading conditions, including both axial and transverse arrangements, are considered in the applications to highlight under which circumstances the use of the GBT deformation modes is required for an adequate representation of the pre-buckling and buckling response. The numerical results have been validated against those determined using a shell element model developed in the finite element software ABAQUS.     

An aeroelastic model for composite rotor blades with straight and swept tips. Part I: Aeroelastic stability in hover

An analytical model for predicting the aeroelastic behavior of composite rotor blades with straight and swept tips is presented. The blade is modeled by beam type finite elements along the elastic axis. A single finite element is used to model the swept tip. The non-linear equations of motion for the finite element model are derived using Hamilton's principle and based on a moderate deflection theory and accounts for: arbitrary cross-sectional shape, pretwist, generally anisotropic material behavior, transverse shears and out-of-plane warping. Numerical results illustrating the effects of tip sweep, anhedral and composite ply orientation on blade aeroelastic behavior are presented. It is shown that composite ply orientation has a substantial effect on blade stability. At low thrust conditions, certain ply orientations can cause instability in the lag mode. The flap-torsion coupling associated with tip sweep can also induce aeroelastic instability in the blade. This instability can be removed by appropriate ply orientation in the composite construction.     

Mode I stress intensity factor solutions for spot welds in lap-shear specimens

The analytical solutions of the mode I stress intensity factor for spot welds in lap-shear specimens are investigated based on the classical Kirchhoff plate theory for linear elastic materials. First, closed-form solutions for an infinite plate containing a rigid inclusion under counter bending conditions are derived. The development of the closed-form solutions is then used as a guide to develop approximate closed-form solutions for a finite square plate containing a rigid inclusion under counter bending conditions. Based on the J integral, the closed-form solutions are used to develop the analytical solutions of the mode I stress intensity factor for spot welds in lap-shear specimens of large and finite sizes. The analytical solutions of the mode I stress intensity factor based on the solutions for infinite and finite square plates with an inclusion are compared with the results of the three-dimensional finite element computations of lap-shear specimens with various ratios of the specimen half width to the nugget radius. The results indicate that the mode I stress intensity factor solution based on the finite square plate model with an inclusion agrees well with the computational results for lap-shear specimens for the ratio of the half specimen width to the nugget radius between 4 and 15. Finally, a set of the closed-form stress intensity factor solutions for lap-shear specimens at the critical locations are proposed for future applications.