Buckling analysis of cylindrical shells with random geometric imperfections

In this paper the effect of random geometric imperfections on the limit loads of isotropic, thin-walled, cylindrical shells under deterministic axial compression is presented. Therefore, a concept for the numerical prediction of the large scatter in the limit load observed in experiments using direct Monte Carlo simulation technique in context with the Finite Element method is introduced. Geometric imperfections are modeled as a two dimensional, Gaussian stochastic process with prescribed second moment characteristics based on a data bank of measured imperfections. (The initial imperfection data bank at the Delft University of Technology, Part 1. Technical Report LR-290, Department of Aerospace Engineering, Delft University of Technology). In order to generate realizations of geometric imperfections, the estimated covariance kernel is decomposed into an orthogonal series in terms of eigenfunctions with corresponding uncorrelated Gaussian random variables, known as the Karhunen-Loéve expansion. For the determination of the limit load a geometrically non-linear static analysis is carried out using the general purpose code STAGS (STructural Analysis of General Shells, user manual, LMSC P032594, version 3.0, Lockheed Martin Missiles and Space Co., Inc., Palo Alto, CA, USA). As a result of the direct Monte Carlo simulation, second moment characteristics of the limit load are presented. The numerically predicted statistics of the limit load coincide reasonably well with the actual observations, particularly in view of the limited data available, which is reflected in the statistical estimators.     

Geometrically nonlinear vibration of laminated composite cylindrical thin shells with non-continuous elastic boundary conditions

The geometrically nonlinear forced vibration response of non-continuous elastic-supported laminated composite thin cylindrical shells is investigated in this paper. Two kinds of non-continuous elastic supports are simulated by using artificial springs, which are point and arc constraints, respectively. By using a set of Chebyshev polynomials as the admissible displacement function, the nonlinear differential equation of motion of the shell subjected to periodic radial point loading is obtained through the Lagrange equations, in which the geometric nonlinearity is considered by using Donnell’s nonlinear shell theory. Then, these equations are solved by using the numerical method to obtain nonlinear amplitude–frequency response curves. The numerical results illustrate the effects of spring stiffness and constraint range on the nonlinear forced vibration of points-supported and arcs-supported laminated composite cylindrical shells. The results reveal that the geometric nonlinearity of the shell can be changed by adjusting the values of support stiffness and distribution areas of support, and the values of circumferential and radial stiffness have a more significant influence on amplitude–frequency response than the axial and torsional stiffness.

     

Nonlinear strain components of general shells with initial geometric imperfections

On the basis of nonlinear strain component formulations of three-dimensional continuum, this paper has derived the nonlinear strain component formulations of shells with initial geometric imperfections. The derivation is not confined to a special shell, therefore they possess general properties. These formulations provide the theoretical basis of the strain analysis for geometric nonlinear problems of shells with initial geometric imperfections     

Stress analysis of hyperbolic shells of revolution with non-axisymmetrical geometric imperfections

In analyzing hyperbolic shells of revolution with non-axisymmeteric imperfections, an approximate method based on simulating the effect of imperfections by the application of fictitious normal pressure loading on the perfect shell is investigated. In the analysis of a shell of revolution with a bulge-type imperfection under non-axisymmetric loads, an efficient algorithm of applying the method is developed: the effect of individual curvature errors on stress resultants and couples are separately considered, while the interactions among various curvature errors are properly treated in the analysis by an iterative procedure. This algorithm avoids repeated analyses for non-axisymmetric loads and may be implemented with a purely axisymmetric analysis capability.A hyperbolic cooling tower shell with a bulge-type imperfection is analyzed under dead load and wind load conditions by the equivalent load method. A direct analysis of the imperfect shell is also made by a specialized finite element program. Through numerical studies, the accuracy and applicability of the equivalent load method are examined.     

Thermomechanical postbuckling of composite laminated cylindrical shells with local geometric imperfections

The effect of local geometric imperfections on the buckling and postbuckling of composite laminated cylindrical shells subjected to combined axial compression and uniform temperature loading was investigated. The two cases of compressive postbuckling of initially heated shells and of thermal postbuckling of initially compressed shells are considered. The formulations are based on a boundary layer theory of shell buckling, which includes the effects of the nonlinear prebuckling deformation, the nonlinear large deflection in the postbuckling range and the initial geometric imperfection of the shell. The analysis uses a singular perturbation technique to determine buckling loads and postbuckling equilibrium paths. Numerical examples are presented that relate to the performances of cross-ply laminated cylindrical shells with or without initial local imperfections, from which results for isotropic cylindrical shells follow as a limiting case. Typical results are presented in dimensionless graphical form for different parameters and loading conditions.     

Effect of multiple localized geometric imperfections on stability of thin axisymmetric cylindrical shells under axial compression

Stability of imperfect elastic cylindrical shells which are subjected to uniform axial compression is analyzed by using the finite element method. Multiple interacting localized axisymmetric initial geometric imperfections, having either triangular or wavelet shapes, were considered. The effect of a single localized geometric imperfection was analyzed in order to assess the most adverse configuration in terms of shell aspect ratios. Then two or three geometric imperfections of a given shape and which were uniformly distributed along the shell length were introduced to quantify their global effect on the shell buckling strength. It was shown that with two or three interacting geometric imperfections further reduction of the buckling load is obtained. In the ranges of parameters that were investigated, the imperfection wavelength was found to be the major factor influencing shell stability; it is followed by the imperfection amplitude, then by the interval distance separating the localized imperfections. In a wide range of parameters this last factor was recognized to have almost no effect on buckling stresses.     

Are measurements of geometric imperfections of plates and shells useful?

This paper refers to the question whether it would be advantageous to make numerous measurements of geometric imperfections of shells. Measurements are presented of the geometric imperfections of sixteen flat plates of box columns and are compared with the effective imperfections of those plates during postbuckling; the two are not found to agree always. Therefore, the paper wishes to raise the question whether, for a program of imperfection measurements to be useful, it should not concentrate on effective imperfections.     

Studies of nonlinear theories for thin shells

In order to formulate the equations for the study here, the Fourier expansions upon the system of orthonormal polynomials areused.It may be considerably convenient to obtain the expressions of displacements as well as stresses directly from the solutions.Based on the principle of virtual work the equilibrium equations of various orders are formulated. In particular, the system of third-order is given in detail, thus providing the reference for accuracy analysis of lower-order equations. A theorem about the differentiation of Legendre series term by term is proved as the basis of mathematical analysis. Therefore the functions used are specified and the analysis rendered is no longer a formal one.The analysis will show that the Kirchhoff-Love’s theory is merely of the first-order and the theory which includes the transverse deformation but keeps the normal straight is essentially of the first order, too.     

Composite shells with interlaminar imperfections

Summary In this contribution the effect of interlaminar initial imperfections on a composite shell behavior is investigated. The constitutive equations for shells with initial interlaminar bonding imperfections are obtained.

---

Verbundstoffschalen mit Zwischenschichtdefekten

Übersicht In diesem Beitrag wird der Einfluß von anfänglichen Zwischenschichtdefekten auf das Verhalten einer Verbundstoffschale untersucht und die Materialgesetze für solche Schalen werden aufgestellt.

---

---

The main theses of this paper have been presented on EUROMECH 292, Sept. 1992     

Effects of uni-directional geometric imperfections on vibrations of pressurized shallow spherical shells

This paper deals with the effects of initial geometric uni-directional imperfections on vibrations of a pressurized spherical shell or spherical cap. The analysis is based upon shallow shell theory. Frequency vs applied pressure interaction curves are plotted for various values of the imperfection amplitude. Imperfections are shown to have a severe effect in reducing the natural frequencies similar to that demonstrated in the buckling behavior of spherical shells.     

Stability investigations of shells with imperfections

Kiev Structural Engineering Institute. Translated from Prikladnaya Mekhanika, Vol. 26, No. 4, pp. 49–56, April, 1990.     

Geometrically nonlinear bending of thin-walled shells and plates under creep-damage conditions

Summary A phenomenological constitutive model for characterization of creep and damage processes in metals is applied to the simulation of mechanical behaviour of thin-walled shells and plates. Basic equations of the shell theory are formulated with geometrical nonlinearities at finite time-dependent deflections of shells and plates in moderate bending. Numerical solutions of initial/boundary-value problems have been obtained for rectangular thin plates (two-dimensional case) and axisymmetrically loaded shells of revolution (one-dimensional case). Based on the numerical examples for the two problems, the influence of geometrical nonlinearities on the creep deformation and damage evolution in shells and plates is discussed. Accepted for publication 30 October 1996     

Effect of geometric imperfections on non-linear stability of circular cylindrical shells conveying fluid

Circular cylindrical shells conveying incompressible flow are addressed in this study; they lose stability by divergence when the flow velocity reaches a critical value. The divergence is strongly subcritical, becoming supercritical for larger amplitudes. Therefore the shell, if perturbed from the initial configuration, has severe deformations causing failure much before the critical velocity predicted by the linear threshold. Both Donnell's non-linear theory retaining in-plane displacements and the Sanders-Koiter non-linear theory are used for the shell. The fluid is modelled by potential flow theory but the effect of steady viscous forces is taken into account. Geometric imperfections are introduced and fully studied. Non-classical boundary conditions are used to simulate the conditions of experimental tests in a water tunnel. Comparison of numerical and experimental results is performed.     

考虑周长约束的圆柱薄壳初始几何缺陷随机场建模方法

利用随机场对圆柱薄壳结构的初始几何缺陷进行建模,并据此建立了一种用于含初始几何缺陷轴压圆柱薄壳屈曲分析的随机分析方法。首先,指出已有将圆柱薄壳初始几何缺陷表征为二维高斯随机场的方法会导致与实际不相符的初始几何缺陷,如圆柱周长显著增大或缩小的几何缺陷。其次,提出一种考虑周长不变约束的随机场建模方法,以剔除与实际不相符的随机几何缺陷。最后,基于所建立的初始几何缺陷随机场模型,利用非干涉多项式混沌展开法进行圆柱薄壳的随机屈曲分析,给出临界屈曲载荷的概率分布。数值试验结果表明,基于随机场理论的初始几何缺陷建模方法可有效刻画几何缺陷对结构承载能力的影响,而提出的约束随机场建模方法又能有效减小结果的分散性。