Analysis and load carrying behaviour of imperfection sensitive shells

This paper deals with the numerical evaluation of the limit loads and of the imperfection sensitivity of shells. A semi-analytic treatment of the field equations governing the nonlinear static behaviour of shells of revolution as well as a novel approach for general shells to construct the relationship between the limit load and amplitudes and shapes of the ”worst” imperfections in a direct way are covered. Applying these methods a large number of results for thin shells are analyzed investigating the influence of the material behaviour, the geometry and the boundary conditions. Included are also combinations of shells and high imperfection sensitive shells. For several types of shells the load carrying behaviour and their imperfection sensitivity is discussed. W. Wunderlich and U. Albertin 2 1 PERFECT AND IMPERFECT SHELLS Shells under compressive loading investigated under the assumption of perfect properties may be considered to be optimal structures. Their load carrying capacity is significantly larger compared to shells which show deviations in geometry, material behaviour, loading and boundary conditions. As actual structures exhibit such imperfections it is necessary to investigate the sensitivity of shells with respect to different shapes and magnitudes of the imperfections. Unfortunately, comparatively little quantitative information exists about the initial imperfections in actual structures, and there are still no generally accepted standard procedures for characterizing them for analytical or numerical purposes. In most cases the amount of reliable experimental data is not sufficient, or the specimens as well as the manner in which the experiments are carried out often differ from the actual service conditions. Individual test results alone are usually not sufficient to allow specific predictions of the failure risk of a given structure. One possibility to improve this situation is to perform systematic numerical simulations which take into account all relevant details of a given shell‘s geometry, loading, boundary conditions, and material properties as well as the shape and magnitude of its initial imperfections. This approach allows to obtain detailed and comprehensive information on the load carrying behaviour which may then be reduced to diagrams or explicit design formulas. Classical numerical concepts of the load carrying capacity of imperfect structures focus on the model of a perfect shell configuration and on the analytical estimation of unstable, postcritical equilibrium paths. This was first demonstrated by Koiter, whose postbuckling theory describes the nonlinear static load carrying behaviour of structures in the initial stages of buckling. The asymptotic postbuckling analysis yields information on the shape of the initial postbuckling path, the stability of the respective equilibrium states, and the way in which initial geometric imperfections (in the shape of the bifurcation mode) influence the structure‘s load carrying behaviour. Particularly, Koiter‘s theory shows that buckling will be catastrophic for negative initial slope of the perfect structure‘s postbuckling path, and, as a result, initial unfavourable imperfections will lead to a reduction in load carrying capacity. This approach has certain restrictions as the results are evaluated by linearisation around the bifurcation point of the perfect shell. For the numerical simulation of the load carrying behaviour of imperfect shells it is commonly assumed that the initial geometric imperfections have the shape of the lowest bifurcation mode of the respective shell. This procedure gives sufficient results for the practical evaluation of the load carrying capacity. In the cases of high imperfection sensitive shells multi-mode-buckling may lead to high sensitive solutions, however, and the lowest bifurcation mode is not always the ”worst” imperfection shape. Recently, a specific concept employing finite element procedures was proposed which directly evaluates the ”worst” imperfection shape and which bases on the analysis of the imperfect shell space. This approach yields more detailed informations about the load carrying behaviour of shells and on the shape of the initial geometric imperfections. W. Wunderlich and U. Albertin 3 2 SCOPE OF NUMERICAL SIMULATIONS In addition to the description of numerical methods used for the nonlinear analysis of imperfect shells a variety of results is given which have been obtained by extensive numerical investigation of the nonlinear elastic and elastic-plastic load carrying behaviour and of the imperfection sensitivity of different shell structures. Together with some new results they are collected from a number of previous papers of the first author and his coworkers. In the meantime the number of results are sufficient to summarize them and to draw conclusions from comparisons. Especially, elastic and elastic-plastic material behaviour of torispherical and spherical shells under external pressure is considered to show the differences in load carrying behaviour and imperfection sensitivity. As boundary conditions also influence the limit loads conical shells subjected to uniform axial loading are investigated to evaluate the influence of the semi-angle β and of the boundary conditions on the load carrying behaviour. Furthermore, the paper deals with combinations of shells to demonstrate their complex buckling behaviour. Particularly, toriconical shells under internal and external pressure are calculated to observe the different influences of the cylinder, cone and knuckle. In dealing with imperfect toriconical shells it is observed that initial imperfections in form of a single buckle yield smaller limit loads than imperfections in the shape of the lowest bifurcation mode. In this aspect high imperfection sensitive shells were investigated. Special high imperfection sensitive shells are cylindrical shells under axial loads and externally pressurized spherical shells. Their load carrying behaviour is difficult to describe as multi-mode-buckling leads to highly sensitive solutions. Although most shells exhibit similar load carrying behaviour for different materials or vary boundary conditions the buckling behaviour of shells remains quite complex and allows only limited generalizations. Specially, combinations of different shells and high imperfection sensitive shells demonstrate new aspects in the evaluation of the load carrying behaviour and imperfection sensitivity. 3 NUMERICAL MODELLING 3.1 Semi-analytic treatment of shells of revolution In the first section of this chapter a problem-oriented method for the nonlinear elastic and elastic-plastic analysis of shells of revolution is briefly summarized. Details may be found in a variety of papers in the literature. The numerical analysis of most of the shells considered is based on a semi-analytic treatment of the field equations. This method is proven for static and dynamic nonlinear analysis of general shells of revolution and leads to important advantages in efficiency and accuracy compared with a common finite element analysis. As a starting point of this approach the governing nonlinear differential equations are formulated in such a way that the forces and displacements are chosen as unknowns and only first derivatives with respect to the meridional coordinate appear. Thus, a set of eight nonlinear partial differential equations with four displacement quantities and four stress resultants as unknowns is obtained. The linearized incremental equations, which typically W. Wunderlich and U. Albertin 4 appear in the course of an incremental or iterative Newton-Raphson solution procedure may be written compactly in the form ) , ( ∂ ∂ θ s s z . s s , s ( ) , ( + ) , ( ) , , ∂ ∂ = Τ 0 θ θ θ z z E z A (1) In Equ. (1) the variables s and θ denote the meridional coordinate and the circumferential angle of an orthogonal convected coordinate system, respectively. The vector z={v1, v2, v3, φ2, s21, n22, s2, m22} contains the increments of the three displacement components v1, v2, v3, the meridional angle of rotation φ2, the inplane modified shear force s21, the meridional stress resultants n22 and m22 and the modified transverse shear force s2. The components of matrix A depend on the meridional coordinate s, on the geometric parameters of the shell, the components of the state vector of the present configuration denoted by z, derivatives with respect to the circumferential coordinate θ, and, in the case of elastic-plastic material behaviour integrals through the thickness or the current tangent moduli, here denoted by ET. The dependence of z(s,θ) on the angle θ is approximated by Fourier series in the form ) ( = ) 0 s s z z θ , ( ( ) ( ) . s ) n ( n ∑ = Ψ + N