Finite element techniques for the analysis of cooling tower shells with geometric imperfections

Two finite element methods for analysing geometrically imperfect cooling tower shells are presented. In the first the geometry of the imperfection is modelled by the elements; in the second the imperfection is represented by an equivalent load on the shell. Axisymmetric and general shell elements have been considered.Results are given which show that the first approximation to the equivalent load is sufficiently accurate and that it is possible to represent local imperfections by axisymmetric imperfections which require less computation. It is also shown that axisymmetric elements should be used wherever possible, because of their greater efficiency, following the geometry of an axisymmetric imperfection but representing local imperfections by equivalent loads.     

Stresses in thin spherical shells with imperfections. Part I: Influence of axisymmetric imperfections

A finite element model of axisymmetric geometry is used to obtain stress and moment fields in the region of an imperfection in thin, spherical shells. In Part I the studies are restricted to axisymmetric imperfections with a cosine variation along the meridian. Parametric studies are carried out to identify the main parameters controlling the response for internal pressure and gravity load. The results show that the behaviour of the shell is similar to the imperfect cylindrical shell with the same radius as in the spherical shell.     

Stresses in thin spherical shells with imperfections. Part II: Influence of local imperfections

The changes in stress resultants in thin spherical shells, associated with a local imperfection introducing curvature errors in all directions, are investigated. An axisymmetric finite element model of the shell and imperfection is employed to carry out the linear elastic analysis. Parametric studies have been performed, to identify the main parameters controlling the response, for the case of internal pressure. The results are compared with those obtained in Part I for axisymmetric imperfections, and bounds for maximum elastic stress resultants are established to cover the possibility of both local and axisymmetric imperfections.     

Buckling analysis of a cooling tower shell with measured and theoretically-modelled imperfections

Geometrical imperfections were measured using photogrammetric techniques on an existing reinforced concrete cooling tower shell. The imperfections, related to the radii of such a real shell, were used as input data to create a real shape of the cooling tower. Numerical analysis was carried out for three models: (P) perfect shell of revolution, (M) shell with measured imperfections, (T) shell with a theoretical imperfection corresponding to the primary buckling mode under dead load. The buckling analysis was related to the linearized eigenvalue problem of elastic shells. The shell midsurface was approximated by eight-node quadrilateral isoparametric finite elements. Computations were carried out using the ANKA computer code. Critical values of the load parameter enable confirmation of a partial correlation between existing imperfections and buckling modes under dead load. The most disadvantageous direction of the wind load application on the real shell was found, in order to evaluate the decrease in the load-carrying capacity of the cooling tower shell against buckling. Theoretically modelled imperfections give rather unrealistic values of buckling loads of the real shell.     

Coupling Effect of Nodal Deviation and Member Imperfection on Load-Carrying Capacity of Single-Layer Reticulated Shell

Single-layer reticulated shell is sensitive to imperfections. To clarify the effect of member imperfection, nodal deviation and their couplings on load-carrying capacity of reticulated shell, the equivalent load method (ELM) is developed in the present study to establish single-layer reticulated shell with random member imperfection, and its realization procedures in FEM package are well-elaborated. The main conclusions are summarized as follows: the proposed ELM is of high efficiency to form member imperfection in space structures. For reticulated shell only with member imperfection, the limit load is more or less influenced by member imperfection. With the increase of amplitude of member bow imperfection, limit load gradually decreases. Load-carrying capacity of reticulated shell with larger amplitude of bowed member is more sensitive to bending direction of bowed member than the one with smaller amplitude. Load-carrying capacity of the reticulated shell considered is extremely sensitive to nodal deviation. For reticulated shell with nodal deviation and member imperfection, the effect of member imperfection on load-carrying capacity is closely related to the magnitude of nodal deviation. Load-carrying capacity of shell with smaller nodal deviation is obviously affected by member imperfection, while not sensitive to member imperfection for reticulated shell with larger one. Member imperfection can be neglected when nodal deviation is large enough. Load-carrying capacity of reticulated shell is also influenced by bending angle of bowed member. On the whole, load-carrying capacity of reticulated shell with larger amplitude of member imperfection is more sensitive to bending direction than the reticulated shell with smaller one.

     

单层网壳结构稳定性分析的改进随机缺陷法

初始几何缺陷对网壳结构稳定性的影响是结构设计中的一个关键问题.对一致缺陷模态法和随机缺陷法的优缺点进行理论分析,提出了改进随机缺陷法,它弥补了随机缺陷法人工计算量大的缺点,也回答了设计临界荷载和一致缺陷模态法得到的临界荷载的可靠性问题.     

单层组合加肋拱支网壳的稳定性

介绍了某相贯单层组合加肋拱支网壳结构在竖向荷载及风荷载作用下的整体稳定性。采用有限元软件MIDAS,在两种荷载工况下分别进行了此网壳结构的特征值屈曲、几何非线性屈曲分析,初始缺陷对网壳结构的整体稳定性影响分析。结果表明:网壳结构对初始缺陷较为敏感,初始缺陷会明显降低结构稳定性;不同的荷载工况下,失稳的情况亦不同且网壳结构稳定临界系数随着荷载的增大而降低。     

钢结构几何缺陷的直接分析方法

本文介绍了结构几何缺陷的几种处理方法,导出了一种显式的缺陷分析模型,将构件初始缺陷直接引入到单元位移插值函数中,可以同时考虑缺陷以及单元横向荷载引起的二阶效应。分析表明对于构件数较少且缺陷敏感型结构,局部缺陷的影响显著;而对于构件数较多的复杂结构系统,局部缺陷的影响是局部的。给出的几个数值算例表明了本文模型的高效性和准确性,计算结果可直接用于设计,而无需进行构件验算和长度系数的计算。     

The shape of circumferential weld-induced imperfections in thin-walled steel silos and tanks

The strength of thin-walled cylindrical shell structures is highly dependent on the nature and magnitude of imperfections. Most importantly, circumferential imperfections have been reported to have an especially detrimental effect on the buckling resistance of these shells under axial load. Due to the manufacturing techniques commonly used during the erection of steel silos and tanks, specific types of imperfections are introduced into these structures, among them circumferential weld-induced imperfections between strakes of steel plates. The shape of such a localised circumferential imperfection has been shown to have a great influence on the degree of strength loss of thin-walled cylindrical shell structures. The results of a survey of imperfections in an existing silo at a location in Port Kembla, Australia in combination with linear elastic shell bending theory was used to develop and calibrate a shape function which accurately describes the geometric features of circumferential weld imperfections. The proposed shape function is the first function to combine shell theory with actual field imperfection measurements. It is a continuous function and incorporates all the necessary features to represent the geometry of a circumferential weld-induced imperfection. It was found that after filtering out the effects of overall imperfections three parameters governed the shape of the surveyed imperfections: the depth, the wavelength and the roundness.     

Effect of size and orientation of a centrally located dent on the ultimate strength of a thin square steel plate under axial compression

Thin steel plates are widely used in many structural applications because of its high load carrying capacity with less weight. The load carrying capacity of thin plates mainly depends on the imperfections present in them. Dent is one of the common geometrical imperfections present in thin shell structures which may be formed due to mechanical damage caused by accidental loading or impact. In this work, influence of various dent parameters (dent length, dent width, dent depth and angle of orientation of the dent) on the ultimate strength of a thin square plate with a centrally located dent is studied using nonlinear static finite-element analysis, under uni-axial compressive loading with simply supported boundary conditions.     

A simplified approach to the analysis of geometrically imperfect cooling tower shells

A simplified method for analysing the elastic stresses in geometrically imperfect cooling tower shells is outlined. The method is based on the replacement of the imperfection by an appropriate additional normal pressure. Illustrations with respect to both axi-symmetric and non axi-symmetric imperfections are presented and the validity of the method is established by comparisons with the direct analysis for the axi-symmetric case. Further simplifications are considered and a possible basis for codefied tolerances on meridional imperfections is developed. Some extensions to the method are suggested, particularly for the case of non axi-symmetric geometric imperfections.     

Combined axial and pressure buckling of end supported shells of revolution

A reduced stiffness theoretical analysis of the imperfection sensitive elastic buckling for end supported shells of revolution is extended to the case of arbitrary combinations of axial and radial pressure loading. Depending upon the shell and loading parameters, the potential reductions in load capacity due to imperfections are shown to involve two distinct forms of post-buckling loss of stiffness. Lower bounds in each of these regimes are provided by appropriate reduced stiffness models, and shown by comparisons with available test data to be reliable even for relatively perfect test models. By attributing reductions in load carrying capacity to weakened end support conditions, it is suggested that past interpretations of these tests may have underestimated the deleterious effects of initial imperfections.     

The conception of quasi-collapse-affine imperfections: A new approach to unfavourable imperfections of thin-walled shell structures

Despite of the intensive research effort of the last decades there are considerable gaps of knowledge concerning the imperfection sensitivity of steel shell structures, even with regard to the basic buckling cases. It is explained in the presented paper why the most unfavourable imperfection pattern does not exist for shell structures but only different unfavourable patterns depending on the imperfection amplitude. This amplitude-depending pattern cannot be determined with certainty because of the substantial influence of the material non-linearity and because of the numerous post-buckling paths which cross each other. However, the method of quasi-collapse-affine imperfections allows a reasonable approximation to the most unfavourable imperfection pattern. The basic thoughts of this concept are presented. The application of the concept to slender wind-loaded shells illustrates its capability.     

Ersatzimperfektionen für den numerischen Beulsicherheitsnachweis stählerner Schalentragwerke – State of the Art

Equivalent Geometric Imperfections for the Numerical Buckling Strength Verification of Steel Shell Structures – State of the Art. Steel shell structures are very imperfection‐sensitive. Therefore, the inevitable deviations from the nominal data of the resistance parameters have to be included in a numerical calculation of the load‐bearing capacity. Because steel shell structures are unique, representative statistical data about the arising imperfections during manufacturing, transport and erection are missing. Therefore, many imperfection assumptions of the codes are based on engineering considerations. The new Eurocode for steel shell structures allows a numerical buckling strength verification in explicit consideration of the effect of imperfections. The assumed imperfections are fundamental for the numerical buckling strength verification, because they have to cover the influence of all accidental imperfections of the structure in a consistent manner. In the contribution, an overview is given on the fundamental imperfection assumptions within the framework of the Eurocode. Still existing knowledge gaps are discussed at the example of the circular cylindrical shell. Hints for application are given.     

Influence of geometric imperfections on the load capacity of orthotropic stiffened and composite shells of revolution with arbitrary meridians and boundary conditions

A stiffness matrix for an element of a shell of revolution has been derived, considering arbitrary load distributions and initial geometric imperfections. This element-stiffness matrix is based on the transfer-matrix method and describes the whole section of a shell of revolution between two rings in modal coordinates (a so-called super-element). The modal coordinates here are circumferential Fourier members, thus reducing the partial differential equations to ordinary ones.

Several stability analyses investigating the sensitivity of composite shells to different geometric imperfection shapes were carried out. The influence of the load distribution and boundary conditions in combination with geometric imperfections was analysed by different modellings of a hypothetical Jupe Avant shell of the ARIANE 5 rocket.     

Effects of imperfections of the buckling response of composite shells

The results of an experimental and analytical study of the effects of initial imperfections on the buckling response and failure of unstiffened thin-walled compression-loaded graphite-epoxy cylindrical shells are presented. The shells considered in the study have six different shell-wall laminates two different shell-radius-to-thickness ratios. The shell-wall laminates include four different orthotropic laminates and two different quasi-isotropic laminates. The shell-radius-to-thickness ratios includes shell-radius-to-thickness ratios equal to 100 and 200. The numerical results include the effects of traditional and nontraditional initial imperfections and selected shell parameter uncertainties. The traditional imperfections include the geometric shell-wall mid-surface imperfections that are commonly discussed in the literature on thin shell buckling. The nontraditional imperfections include shell-wall thickness variations, local shell-wall ply-gaps associated with the fabrication process, shell-end geometric imperfections, nonuniform applied end loads, and variations in the boundary conditions including the effects of elastic boundary conditions. The cylinder parameter uncertainties considered include uncertainties in geometric imperfection measurements, lamina fiber volume fraction, fiber and matrix properties, boundary conditions, and applied end load distribution. Results that include the effects of these traditional and nontraditional imperfections and uncertainties on the nonlinear response characteristics, buckling loads and failure of the shells are presented. The analysis procedure includes a nonlinear static analysis that predicts the stable response characteristics of the shells, and a nonlinear transient analysis that predicts the unstable response characteristics. In addition, a common failure analysis is used to predict material failures in the shells.     

Inelastic stability of conical tanks

The aim of this investigation is to study the effect of different imperfection shapes on the inelastic stability of liquid-filled conical tanks and to determine the critical imperfection shape that would lead to the minimum inelastic limit load. The study is carried out numerically using a self-developed shell element used to simulate a number of conical tanks having an imperfection shape in the form of Fourier series of equal coefficients. The Fourier analysis of the buckling modes indicates that the existence of axisymmetric imperfection will lead to the critical inelastic limit load for conical tanks.     

Effects of geometric imperfections on stress distributions in cooling towers

The effect of geometric imperfections on the static stress distributions in cooling towers is investigated. The antisymmetric imperfections as well as the axisymmetric ones are assumed to be localized band imperfections at some height in a tower. Analyses are carried out for the self weight, seismic lateral load and wind load. Hoop stress and meridional bending moment are strongly influenced and the quantitative effects are summarized in simple figures.     

Influence of loading imperfections on the stability of an axially compressed cylindrical shell

An important phenomenon is presented which has not been given sufficient attention in the stability of shell structures: the buckling of a circular cylinder subjected to a set of equally distributed discrete axial concentrated loads is studied. It is shown that the critical load of a circular cylinder under axial compression is very sensitive to imperfection of the applied loads as well as to initial geometric imperfections and the boundary conditions.     

Static buckling analysis of thin cylindrical shell with centrally located dent under uniform lateral pressure

One of the common failure modes of thin cylindrical shell subjected to external pressure is buckling. The buckling pressure of these shell structures are dominantly affected by the geometrical imperfections present in the cylindrical shell which are very difficult to alleviate during manufacturing process. Dent is one of the common geometrical imperfections present in thin shell structures which may be formed due to mechanical damage caused by accidental loading or impact. In this work, influence of various dent parameters (dent length, dent width, dent depth and angle of orientation of the dent) on the critical buckling pressure of thin cylindrical shells with a centrally located dent is studied using non-linear static finite-element analysis of ANSYS under external pressure with simply supported boundary conditions at the top and bottom edges of the thin cylindrical shell.