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.     

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.     

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.

     

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.     

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.     

Imperfections in steel girder webs with and without perforations under patch loading

In this paper, the problem of stability of web plates with imperfections, subjected to patch load, is studied. The aim is to give some insights about the best way to take into account real imperfections in non-linear stability analyses of plates with and without perforations. In this context, the study is developed on the basis of the measured imperfections or numerically deriving the deformed model (theoretical imperfections). The influence of the patch load length, out-of-plane imperfection amplitude, dimension and position of the hole on stability behaviour and buckling strength are studied comparing some theoretical deformed configurations corresponding to different modal shapes. The results obtained with a three-dimensional model of the whole real beam with stiffeners, with experimentally measured imperfections, and each corresponding single web panel are compared and discussed obtaining some insights about the accuracy of the simplified (and conservative) model of the single panel.The main insights of this work are as follows. The deformed shape, corresponding to the first buckling mode, can be assumed as the initial configuration of the panels with and without holes to study post-critical behaviour until ultimate condition. The shape of the imperfection does not severely change the critical buckling stress. A longer patch load reduces the ultimate stress in the panel. An initial imperfection amplitude of less than 1% of the height of the panel does not reduce the ultimate load by more than about 5%.     

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.     

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.     

Imperfection sensitivity of plate girder webs subjected to patch loading

This paper is aimed at studying the influence of initial geometric imperfections on the postbuckling behavior of longitudinally stiffened plate girder webs subjected to patch loading. A sensitivity analysis is conducted herein using two approaches (deterministic and probabilistic) in order to investigate the effect of varying imperfection shape and amplitude on both, the postbuckling response and ultimate strength of plate girders under patch loading. This sensitivity analysis is performed by means of nonlinear finite element analysis. At first, the initial shape imperfections are modeled using the buckling mode shapes resulting from an eigenvalue buckling analysis. Thereafter, the amplitude of the buckling shapes for the various modes is varied, and then introduced in the nonlinear analysis. The results show the influence of these modes and amplitudes on the resistance to patch loading.     

Finite Element simulation of dynamic buckling of cylinders subjected to periodic shear

This paper deals with the buckling of cylindrical shells under a dynamic shear load. The aim of our study is to compare static buckling load and buckling load during a sweep frequency excitation. First, we describe the special experimental device and the two Finite Element codes used in this study. In a second part static tests and corresponding Finite Element calculations are presented in order to have a reference buckling load and to understand the effect of initial imperfections. Then, a vibration analysis is performed in order to investigate the effect of geometric imperfection and a preload. In the last part, we discuss dynamic results. When we reach the first eigen frequency, the buckling load drops and the buckling deformations increase due to a parametric resonance. There is a coupling between vibration and buckling modes.     

Control of axi-symmetric vibrations of cylindrical shells using active constrained layer damping

Distributed-parameter modeling of thin cylindrical shells which are fully treated with active constrained layer damping (ACLD) is presented. Hamilton's principle is utilized to develop the shell/ACLD model as well as the associated boundary conditions. A globally stable boundary control strategy is developed to damp out the vibration of the shell/ACLD system. The devised boundary controller is compatible with the operating nature of the ACLD treatments where the strain induced, in the active constraining layer, generates a control force acting at the boundary of the treated shell. As the boundary control strategy is based on a distributed-parameter model of the shell/ACLD system, the classical spillover problems resulting from using “truncated” finite element models is eliminated. Also, such an approach makes the boundary controller capable of controlling all the modes of vibration of the shell/ACLD and guarantees that the total energy norm of the system is continuously decreasing with time. Numerical examples are presented to demonstrate the effectiveness of the ACLD in damping out the vibration of cylindrical shells. Such effectiveness is determined for different control gains and compared with the performance of conventional passive constrained layer damping (PCLD). The results obtained demonstrate the high damping characteristics of the boundary controller particularly over broad frequency bands.     

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。     

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.     

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.     

A numerical imperfection sensitivity study of cold-formed thin-walled tubular steel columns at uniform elevated temperatures

A numerical study is carried out on cold-formed rectangular hollow section columns to evaluate the sensitivity of column failure strength to initial imperfections, stress–strain relationships and to assess the existing design methods. It is shown that the magnitude of initial local buckling imperfection has a significant effect on the ultimate strength of short columns where failure is predominantly local buckling. Its effect on long columns is relatively small. Similarly the magnitude of initial global imperfection has more influence on the ultimate strength of a long column, whose failure is governed by global buckling, than on short columns, where local buckling controls. The shape of the stress–strain relationship of cold-formed steel will have noticeable effect on the column failure load. Current design methods, for high temperatures in ENV1993-1-2 and for ambient temperature in ENV1993-1-3, can provide a valid basis of calculation but modification will be necessary, depending on the exact model of stress–strain relationship of cold-formed steel at elevated temperatures.     

Iterative Fourier decomposition of imperfection measurements at non-uniformly distributed sampling points

Buckling of cylindrical shells subject to axial compression is acutely sensitive to the form and amplitude of geometric imperfections present in the structure. As a result, many attempts have been made to measure geometric imperfections in cylindrical shells both in laboratory specimens and less frequently in full-scale structures. The imperfections are generally interpreted using the well-known method of Fourier decomposition, so that the different components of imperfections can be more easily related to structural features such as positions of welds and their effects on buckling strength better understood. A common situation in imperfection measurements on full-scale shell structures is that some parts of the structure are not accessible, due to the presence of accessories such as service ladders and pipes. As a result, a measurement grid with non-uniform intervals is generally employed in imperfection surveys on full-scale structures. This paper first shows that when results from such surveys are interpreted using the traditional Fourier decomposition method, the resulting Fourier series cannot provide an accurate representation of the discrete measurement data due to the non-uniform distribution of sampling points. The paper then presents an iterative Fourier decomposition method which overcomes this problem. The theoretical background of the proposed method is detailed, followed by a numerical demonstration of the effectiveness of the method.     

初始几何缺陷对网壳结构线性动力反应的影响初探

自振频率的密集分布和不可避免的初始几何缺陷,将导致网壳结构出现模态局部化现象,而模态局部化将导致结构振动呈现局部化。通过矩阵摄动理论分析并给出了模态局部化现象发生的诱因,讨论并确定了初始几何缺陷的大小和分布模式。以一个单层球面网壳为例,进行了同一地面运动激励下,50个包含不同初始几何缺陷网壳结构的线性动力反应分析,结果表明,初始几何缺陷对网壳结构线性动力反应的影响是不可忽略的,建议在网壳结构的抗震计算中予以重视。     

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.     

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.