Analysis of hyperbolic cooling towers with local imperfections

Two methods to analyse hyperbolic cooling towers with local imperfections are presented. One method relies on the finite element technique. For this a specialized finite-element program, which can model any arbitrary imperfections while retaining the advantage offered by the basically axisymmetric nature of the shell, was developed. The other method is an approximate procedure, which may be implemented with a purely axisymmetric analysis capability. The two methods are compared through numerical studies. A cooling tower shell with a bulge-type imperfection is examined under dead load and wind load conditions. It is concluded that the finite-element model presented is effective for the analysis of such shells, while the equivalent-load method may be adequate for some cases. Also, it is shown that both meridional and circumferential stress resultants may be radically influenced by a small bulge imperfection.     

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.     

Form of circumferential weld-induced imperfections for tapered-wall cylindrical shell structures

Initial geometric imperfections have a great effect on the buckling strength of thin-walled cylindrical shells under axial compression, and the circumferential weld-induced imperfection is usually the most deleterious imperfection form. Two axisymmetric imperfection forms proposed by Rotter and Teng have widely been employed in the buckling analysis of cylindrical shells. However, the applicability of the two forms for tapered-wall cylinders needs further study, since they are derived from the elastic bending theory for long thin-walled cylinders with a constant wall thickness. This paper presents a modified form of circumferential imperfection for tapered-wall cylinders. Finite element analyses are carried out by employing the trapezoidal strain field approach to model the welding process, and the obtained circumferential depression shapes are used to evaluate the availability of the modified imperfection form. It is shown that the modified imperfection form is reasonable for any wall thickness ratio between two adjacent strakes, and the most suitable shape function, which is very close to the FE results, can be obtained by giving suitable values of the roundness in the modified form.     

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.     

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.     

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.     

Geometric imperfections in plated structures subject to interaction between buckling modes

An analytical technique is described for expanding a measured geometric imperfection surface in the buckling modes of a structure. The surface is expanded longitudinally as well as transversely. In the transverse expansion, the constituent plates are treated as a plate assembly so that the imperfections of the individual plates can be interrelated. The technique is applied to a channel section which buckles interactively in an overall flexural mode in the plane of symmetry and a primary local buckling mode. The imperfection components of these modes are determined, as are the components of the secondary local buckling modes which may be triggered as a result of the interaction between the overall and local modes. The imperfection components are determined approximately from imperfection measurements taken at six points in each cross-section along the length of the column. The application of the technique to other cross-section types is mentioned.     

Computational modelling of geometric imperfections and buckling strength of cold-formed steel

Computational modelling of the buckling strength of cold-formed steel members as influenced by initial geometric imperfections is studied. The geometric imperfections are represented by the member eigenmode shapes. Along with the classical measure — the amplitude of imperfections, an energy measure defined by the square root of the elastic strain energy hypothetically required to distort the originally perfect structural element into the considered imperfect shape is used. Based on the measures, two approaches for the choice of the most unfavourable imperfections are suggested. Normalising imperfections by the amplitude, the energy measure is calculated as indicative parameter of imperfection significance. Vice versa, when adopting normalisation by the energy measure, the amplitude is used as a supporting parameter. The suggestions are illustrated on calculating the strength of an axially compressed steel lipped channel column with eigenmodes exhibiting local-distortional interactions. For eigenvalue and geometrically and materially non-linear strength calculations, the FEM codes MSC.NASTRAN and COSMOS/M are employed.     

The measurement of imperfections in cylindrical thin-walled members

The structural behaviour of thin-walled circular cylindrical members has been shown to be imperfection sensitive. However, only little information of the exact nature of imperfections in such members is available. In this paper a method of measuring imperfections in circular cylindrical members is described, the method is simple to implement in a laboratory environment while providing accurate measurements. Numerical methods to process the measurements into three-dimensional imperfection maps are also presented along with an algorithm to distinguish between significant imperfection patterns and measurement ‘noise’. Results from a recent research project where this method has been used illustrate the derivations in this paper.     

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.     

Shear resistance of longitudinally stiffened panels—Part 1: Tests and numerical analysis of imperfections

This paper deals with the results of four full-scale tests, numerical simulation of tests and initial geometric imperfection analysis for longitudinally stiffened panels in shear. The tests examine the influence of varying position and bending stiffness of one trapezoidal longitudinal stiffener on the panel shear resistance and its buckling behaviour. The stiffeners were designed such as to obtain both global and local buckling shapes. Numerical simulations (FEA), based on the test girder geometry, the measured initial geometric imperfections and elastic-plastic material characteristic from the tensile tests, demonstrate a very good agreement with the tests. The initial geometric imperfection study on different verified numerical models shows a limited sensitivity of the panel shear capacity to any kind of imperfection shape variation with amplitude at the allowable fabrication tolerances. Finally, the paper offers some ideas for modelling geometric imperfections with regard to the design or research demands.     

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.     

Simulation of geometric imperfections in cold-formed steel members using spectral representation approach

In this paper, a summary of the available imperfection measurements for cold-formed steel members is presented. Three methods to simulate imperfection fields are introduced: the first is the classical approach employing a superposition of eigenmode imperfections, but scaled to match peaks in the measured physical measurements. The second is a method based on the multi-dimensional spectral representation method, in which imperfections are considered as a two-dimensional random field and simulations are performed taking a spectra-based approach. The third is a novel combination of modal approaches and spectral representation that directly considers the frequency content of the imperfection field, but employs a spectral representation method driven by the cross-sectional eigenmode shapes to generate the imperfection fields. The effect of these different approaches on the simulated strength and collapse behavior of members is investigated using material and geometric nonlinear finite element collapse modeling. The third imperfection generation method, termed the 1D Modal Spectra Method, provides an intriguing new tool in the simulation of thin-walled members.     

杆件初始缺陷对单层凯威特球面网壳地震响应影响研究

为研究杆件初始缺陷对单层凯威特球面网壳地震响应的影响,利用OpenSEES有限元程序,采用多段梁法模拟杆件初始缺陷,给出了OpenSEES模拟空间网格结构圆管杆件滞回的多段梁法建模参数合理取值,基于GB 50017—2003《钢结构设计规范》中压杆稳定系数拟合了圆管等效偏心率和正则化长细比的关系式。考虑杆件初弯曲方向随机分布,建立了不同参数单层凯威特球面网壳模型,对网壳增量动力分析的最大位移进行了比较。结果表明：当网壳处于弹性状态时,杆件缺陷对其地震响应的影响可以忽略;当网壳进入塑性状态后,杆件缺陷对其地震响应的影响不可忽略,不同杆件初弯曲方向的网壳地震响应离散程度随地震动强度提高基本呈增大趋势;地震动强度较大时,考虑杆件缺陷和结构整体缺陷的单层网壳地震响应和仅考虑整体缺陷的单层网壳地震响应差异较大,两者最大位移的相对大小并无规律性,其关键影响因素为地震作用下塑性杆件分布的离散程度。     

The effect of geometric imperfections on the vibrations of anisotropic cylindrical shells

Analytical–numerical models to analyse the flexural vibration behaviour of anisotropic cylindrical shells are presented. The two models (denoted as Level-1 and Level-2 Analysis) have different levels of complexity and can be used to study the influence of important parameters, such as geometric imperfections, static loading, and boundary conditions. A specific anisotropic shell is used in the calculations in this paper. The influence of the imperfection shape and amplitude on the natural frequency is investigated for this shell via both the Level-1 and the Level-2 Analysis. Imperfections with the shape of the “lowest vibration mode” give a decrease of the natural frequency with increasing imperfection amplitude. The results of the Level-2 Analysis for the effect of imperfections on the natural frequency are in reasonable agreement with Finite Element calculations.     

Buckling of elastic cylindrical shells considering the effect of localized axisymmetric imperfections

The effect of localized axisymmetric initial imperfections on the critical load of elastic cylindrical shells subjected to axial compression is studied through analytical modeling. Some classical results regarding sensitivity of shell buckling strength with respect to distributed defects having axisymmetric or asymmetric forms are recalled. Special emphasis is placed after that on the more severe case of localized defects satisfying axial symmetry by displaying an analytical solution to the Von Kármán–Donnell shell equations under specific boundary conditions. The obtained results show that the critical load varies very much with the geometrical parameters of the localized defect. These variations are not monotonic in general. They indicate, however, a clear reduction of the shell critical load for some defects recognized as the most hazardous isolated ones. Reduction of the critical load is found to reach a level which is up to two times lower than that predicted by general distributed defects.     

Imperfection sensitivity of elastic and elastic-plastic torispherical pressure vessel heads

The paper deals with the results of a systematic numerical investigation of the nonlinear elastic and elastic plastic load-carrying behaviour and imperfection sensitivity of torispherical pressure vessel heads under uniform external pressure. In particular, the presentation focuses on the qualitative and quantitative influence of the radius-to-thickness ratio R/t, the yield stress σ₀ and the magnitude of initial geometric imperfections (in the shape of the elastic bifurcation mode) on the elastic-plastic load-carrying behaviour. It is found that thinner shells are more sensitive to the value of the yield stress and the magnitude of initial geometric imperfections, but their load-carrying capacity, relative to the elastic bifurcation pressure, may also be significantly higher than that of thicker shells.     

Probability analysis of double layer barrel vaults considering the effect of initial curvature and length imperfections simultaneously

Load carrying capacity of reticulated space structures majorly depend on the structures’ imperfections. Imperfections in initial curvature, length, and residual stress of members are all innately random and can affect the load-bearing capacity of the members and consequently that of the structure. The present study investigated the effect of the probability distribution of initial curvature imperfection and lack of fit of members on the load-bearing capacity of double-layer barrel vault space structures with different types of support. A random number was first assigned to each member using gamma and normal distributions for initial curvature and member length imperfections, respectively. Afterwards, the ultimate bearing capacity and the collapse behavior of the structure was determined using nonlinear finite-element analysis in OpenSees software and finally structures reliability was acquired. The results demonstrate that the collapse behavior of doable-layer barrel vault space structures is sensitive to the random distribution of initial imperfections.     

Stresses and pressures in thin-walled structures with damage and imperfections

This paper is a sequel to a recent book of the author, in which stress concentrations and redistributions due to structural imperfections in thin-walled structures were reviewed. Here the emphasis is on other complementary aspects. Only shape damage and imperfections are considered in the paper, while intrinsic imperfections are out of the scope discussed. The mechanics of generation of damage is studied for several cases, including a relative in-plane displacement that induces an out-of-plane distortion; impact of an object on the structure; impact of the structure on a rigid surface; damage due to local buckling; and removal of part of the structure. Possible mechanisms that lead to the generation of imperfections include errors in construction; influence of misfits and misalignments; welding; and design-imperfections. Practical cases in which imperfections are measured and data banks of such information is also briefly reviewed. The field of mechanics of stress redistributions, and the modeling of distortional damage using numerical tools, are discussed with reference to previous publications by the author. New problems of changes in the pressures due to modifications in the shape are highlighted, with reference to the discharge of silos. Finally, several areas are identified in which future developments are expected for the next five to ten years.