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.     

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.     

An equivalent stiffness approach for modelling the behaviour of compression members according to Eurocode 3

The development of design procedures based on inelastic advanced analysis is a key consideration for future steel design codes. In advanced analysis the effect of imperfections has to be modelled in such a way that the incremental analysis fully captures this effect in the process of moment redistribution. In modelling the influence of imperfections on the behaviour of individual members of real structures, different approaches have been used to globally represent this effect in the overall analysis of structural systems. They are referred to as the initial bow imperfection approach or as the equivalent transverse load approach. When using the abovementioned approaches in analysis of multiple member structural systems, the designer is required to arrange the directions of bow imperfections or equivalent transverse loads in such a way that the imperfection arrangement leads to the least constrained solution, i.e. the lowest ultimate load predicted from all possible sets of member initial imperfection arrangements. Since there is still ongoing research on the development of simple application rules ensuring that the designer obtains a unique solution when choosing a certain set of member initial imperfections, there is at the same time interest in the development of alternative approaches to modelling the influence of member imperfections on the behaviour of structural systems. This paper provides the necessary background information as well as describes the formulation and modelling techniques used in the development of a new approach to modelling the influence of imperfections on the stability behaviour of structural components and systems. This new approach, called hereafter an equivalent stiffness approach, has an advantage over the previously described approaches since an imperfect member is treated as a hypothetically straight element, flexural and axial stiffnesses of which at each load level are predicted in a continuous fashion dependent upon the actual force and deformation states. This type of modelling does not require any explicit modelling of equivalent geometric imperfections or equivalent forces and their directions in advanced analysis; therefore also it does not require any buckling mode assessment. Moreover, the effects of strain hardening and section class may conveniently be included in modelling. Finally, European buckling curves are used to estimate the values of parameters of the developed model that can be immediately used in advanced analysis conducted according to Eurocode 3.     

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.     

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.     

Random imperfection method for stability analysis of a suspended dome

The traditional eigenvalue buckling mode method used for the stability analysis of suspended domes has been deemed unsafe and unreasonable. A random imperfection method is proposed in this paper that is suitable for the stability analysis of suspended domes. Using this method, the impact of initial imperfections, such as joint position deviation, eccentricity of members, variation in cable pretension and elastic modulus of materials, on the stability of nonlinear geometries as well as the stability of both nonlinear geometries and materials (i.e., ultimate bearing capacity) are studied. The analysis results using this method are compared to those obtained from the traditional eigenvalue buckling mode method. This study indicates that the random imperfection method can more reasonably demonstrate the impact of initial imperfections on the stability of suspended domes. Additionally, the shape and size of the initial imperfections applied by the random imperfection method are more reasonable, and the results are safer. The random imperfection method is successfully used in the stability analysis of a large-span suspended dome, and the results of this method are nearly equivalent to that of the model test, which verifies the method.     

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.     

Worst Multiple Perturbation Load Approach of stiffened shells with and without cutouts for improved knockdown factors

An optimization framework of determining the worst realistic imperfection was proposed by the present authors to study the reduction of the load-carrying capacity of unstiffened cylindrical shells. However, with regard to stiffened shells, especially when cutouts are included, the dimple combination pattern should be judged in a more rational manner. In this study, node coordinates are utilized to describe the position of each dimple-shape imperfection for Worst Multiple Perturbation Load Approach (WMPLA), which is an improvement of the MPLA using an optimization algorithm to find the application positions that will reduce the buckling load. Further, a novel method to determine the density of possible positions of dimple-shape imperfections is proposed based on eigenmode shape for stiffened shells without cutout. In addition, the effects of cutouts on the proposed method are investigated in detail. The effectiveness of the proposed method is demonstrated by comparison of several conventional methods to obtain improved knockdown factors (KDFs).     

A numerical and experimental investigation on composite stiffened panels into post-buckling

The structural behaviour of composite stiffened flat panels under axial compression is here investigated up to collapse. The panel configuration is designed to buckle once the limit load is reached and to work in post-buckling until the ultimate load. The design phase is based on the use of four different kinds of finite element analyses: eigenvalue, non-linear static with modified Riks’ method and both implicit and explicit dynamic analyses. Once the final configuration is identified, two specimens are manufactured. The initial geometrical imperfections are measured and analyzed, then axial compression tests are performed until collapse. As foreseen by the numerical analyses, experimental results prove the ability of the panels designed to work in the post-buckling field until collapse which takes place due to the failure of the stiffener blades. Finally, the measured initial imperfections are included in the model significantly increasing the numerical–experimental correlation.     

Axially and transversely loaded Timoshenko and laced built-up columns with arbitrary supports

The two main effects that should be taken into account in the design of built-up columns are the shearing effect due to their reduced shear rigidity and the interaction between global and local buckling modes in the presence of imperfections. EC3 considers such columns as Timoshenko members for accounting for shear deformation and provides guidance for the calculation of their collapse load under axial compression only for the simply supported case.In the present work an approximate analytical procedure for the calculation of the maximum 2nd order bending moment along imperfect Timoshenko members with arbitrary rotational and translational boundary conditions under combined axial and lateral external loading is formulated. The procedure is verified by means of geometrically nonlinear numerical analyses with the use of shear deformable beam elements. This method is then used for the evaluation of the maximum actions due to combined axial and lateral loading along laced built-up columns with various end conditions accounting also for initial imperfections. Incorporating the 2nd order bending moments in an interaction equation specifically formulated for laced built-up columns, the collapse load is calculated. Analytically obtained results based on the proposed method are compared with the ones found from geometrically and materially nonlinear imperfection analyses (GMNIA) of laced built-up columns modeled using either beam or shell elements. The accuracy of the analytical results is found to be very satisfactory. Due to its simplicity and accuracy the proposed method is a useful tool for the design of laced built-up columns in structural engineering practice.     

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。     

考虑缺陷的板片空间结构稳定分析

在工程中,一致缺陷模态法是对空间结构进行缺陷稳定分析的主要方法,但该法并不适用于板片空间结构的缺陷稳定分析。鉴于此,提出一种改进的方法,即利用结构屈曲前后两个邻近状态的位移之差来模拟结构的初始几何缺陷。由随机缺陷模态法得到的改进方法的可靠度以及与模型试验结果的对比可以证明,改进方法是可行的。     

板片空间结构的改进一致缺陷模态法

目前工程上对结构进行缺陷稳定分析的主要方法是一致缺陷模态法。而利用该法对板片空间结构进行缺陷稳定分析得到的并不是结构的最小临界荷载。为此提出了改进的方法,即通过N-R法来引入结构的初始几何缺陷。为了检验改进方法的可行性,采用一个单层模型试验加以验证并且通过随机缺陷模态法得到了改进方法的可靠度。通过以上的检验,改进方法是可行的。这对板片空间结构的设计和研究有着重要的意义。     

Measurement and assessment of geometric imperfections in thin-walled panels

Thin-walled members may be subject to performance limitations arising through local or distortional buckling of slender elements comprising the cross-section of the member, or overall buckling of the member. The effects of structural instability may be aggravated by the presence of geometric imperfections in these elements. An investigation is presented into methods of measuring and assessing geometric imperfections in cold-rolled thin-walled steel panels. These methods can be used to characterise the geometry of prismatic thin-walled members that exhibit performance sensitivity due to geometric imperfections. The measurement procedures investigated include close-range photogrammetry, precise optical levelling, and the use of a co-ordinate measurement machine. The assessment procedure comprises a least-squares spectral decomposition of the measurements to characterise the imperfections existent in the panels under investigation, and estimates of the precision of the derived Fourier coefficients are used to inter-compare the three measurement procedures. The investigation has demonstrated that statistically significant imperfections may exist in thin-walled members at short and medium wavelengths, leading to a reduction in the load carrying capacity. Both optical levelling and the co-ordinate measurement machine technique can yield desirable results, but for high precision work, use of the co-ordinate measurement machine is recommended.     

Collapse of axially compressed cylindrical shells with random imperfections

The establishment of an International Imperfection Data Bank is discussed. Characteristic initial imperfection distributions associated with different fabrication techniques are shown.

Using a first-order, second-moment analysis, a stochastic method is presented, whereby the stability of isotropic, orthotropic and anisotropic nominally circular cylindrical shells under axial compression, external pressure and/or torsion possessing general nonsymmetric random initial imperfections can be evaluated. Results of measurements of initial imperfections are represented in Fourier series and the Fourier coefficients are used to construct the second-order statistical properties needed. The computation of the buckling loads is done with standard computer codes and includes a rigorous satisfaction of the specified boundary conditions.

It is shown that the proposed stochastic approach provides a means to combine the latest theoretical findings with the practical experiences spanning about 75 years in an optimal manner via the advanced computational facilities currently available.     

The single perturbation load approach applied to imperfection sensitive conical composite structures

The importance of taking into account geometric imperfections for cylindrical and conical thin-walled structures prone to buckling had been already recognized by the first authors dealing with new formulations. Nowadays, the analysts still use empirically based lower-bound methods such as the NASA SP-8007 guideline to calculate the required knock-down factors (KDFs), which does include important mechanical properties of laminated composite materials, such as the stacking sequence. New design approaches that allow taking full advantage of composite materials are required.The single perturbation load approach (SPLA), a new deterministic approach first proposed by Hühne, will be investigated with unstiffened composite conical structures varying the geometry, lamina and layup. The SPLA׳s capability for predicting KDF is compared with the NASA approach. The SPLA was applied to the geometrically perfect structures and to the structure with geometric imperfections of two types, mid-surface imperfections and thickness imperfections. The study contributes to the European Union (EU) project DESICOS, whose aim is to develop less conservative design guidelines for imperfection sensitive thin-walled structures.     

The development of shell imperfection measurement techniques

The development of techniques for measuring geometric imperfections is traced from the sixties to the present day. First, the early laboratory experiments that initiated the concepts and developed the basic techniques are outlined. Imperfection measurements of large and full-scale shells are then considered, including those performed in industry. Modern, more sophisticated, semi-automated measurement techniques as well as imperfection data banks are discussed. Finally, a design methodology which includes the correlation of realistic imperfections with manufacturing techniques is outlined, and ways are indicated of inclusion of the relevant imperfection data into design methods.     

Buckling and postbuckling of cylindrical shells with one end pinned and the other end free

Using finite element analysis, this paper examines the linear bifurcation buckling loads, and nonlinear collapse loads, of cylindrical shells with one end pinned and the other end free, under a variety of axial and pressure load combinations. The pinned end is formulated so as to provide no axial restraint. For the bifurcation analysis, loads are related back to the classical solutions for cylinder buckling loads, to explain the very low values found for this set of boundary conditions.The nonlinear analysis includes both imperfections and material plasticity. In this analysis, it is found that cylindrical shells with pinned-free boundary conditions are notably imperfection insensitive, and for a range of geometries are able to reach collapse loads significantly greater than their bifurcation load. For other geometries, collapse loads very close to the bifurcation load are found. This unusual imperfection insensitivity for a cylindrical shell is explained in terms of the large flexibility engendered by the pinned-free boundary conditions and the oval buckling mode.     

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.     

Development of a new equivalent analysis technique to determine the ultimate load in an electric transmission tower

The current design practice of an electric transmission tower is based on allowable stress design method in Korea. However, it is difficult to find reasons for the collapse of a transmission tower designed with this approach because the collapse tends to occur as a result of secondary large deformations. The main influential factors of nonlinear behavior are residual stresses, initial imperfections, and the end restraints on members. In this study, the necessity of the new analysis technique is examined through a comparison of elastic analysis and inelastic analysis. A new equivalent analytical technique (EAT) is proposed as a means of minimizing the complexity. Furthermore, the reliability of the proposed method is validated by comparing the computed ultimate load of the transmission tower with that of nonlinear finite element analysis.