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.     

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.     

残余应力对局部轴压焊接圆柱壳承载力的影响

焊接钢圆柱薄壳广泛应用于钢筒仓和钢油罐结构中,屈曲通常是该结构的设计控制条件,圆柱薄壳的屈曲在大多数荷载工况下对焊接几何缺陷十分敏感.现有研究标明,焊接残余应力可少量提高均匀轴压圆柱壳的稳定承载力,但对于局部轴压荷载下圆柱薄壳中残余应力的效应,相关的研究很少.采用施加收缩应变法,建立了分别考虑焊接几何缺陷以及考虑或不考虑焊接残余应力焊接圆柱薄壳的数值分析模型,研究了含有周向焊缝、竖向焊缝以及砌砖式焊缝(patterned welds)的局部轴压焊接圆柱壳屈曲行为,通过比较考虑/不考虑残余应力圆柱薄壳的计算结果,得到残余应力对局部轴压圆柱壳承载力的影响.     

对称双圆弧柱壳的非线性屈曲

对于潜艇艇体耐压壳结构,屈曲特性在设计中被广泛关注。针对一种新型潜艇耐压艇体结构-对称双圆弧环肋柱壳,推导了相应的弹塑性失稳系数。采用非线性大挠度理论,给出了静水压力作用下含初始缺陷的对称双圆弧环肋柱壳大挠度弹塑性屈曲临界压力计算式。讨论了开口角、周向相当波数和初始几何缺陷对临界压力的影响。计算结果表明,开口角对结构弹塑性屈曲的临界压力影响很小,而周向相当波数是影响临界压力的主要因素。     

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.     

圆弧焊接薄壁圆筒残余应力场的分析研究

由于圆弧焊接的薄壁圆筒在高科技工程,如航空机器、压力容器和核工程中的广泛应用,对焊接导致的如焊接变形和残余应力等缺陷的研究就显得非常重要。介绍了一个计算程序,用以分析低碳钢薄壁圆筒在圆弧焊接中的温度分布以及随后将产生的残余应力场。基于数值模拟的参数研究用于分析焊接所产生的残余应力中的关键参数的影响。考虑了低碳钢、焊丝和双椭圆热源模型的基于温度的热机械性能。通过全尺寸焊接试验中的瞬时温度分布和残余应力场证实了有限元模拟方法的精确性。其目的在于提供数据以证实薄壁圆筒现有焊接工艺的有效性,从而将预加应力导致的结构在使用期间失效的可能性降至最低。     

Reliable and accurate prediction of the experimental buckling of thin-walled cylindrical shell under an axial load

Reliable and accurate method of the experimental buckling prediction of thin-walled cylindrical shell under an eccentric load is presented. The experimental arrangement and specimens are discussed in detail, including the measurement of the geometric imperfections of the specimen's surface using a coordinate measuring machine. Different FE models, in terms of complexity, are used to simulate the experiment arrangement in an attempt to get a good agreement with the experimental buckling loads and study the effect of measured initial geometric imperfections, load eccentricity, load eccentricity position along the shell's circumferential direction and different experimental arrangement that influence the boundary conditions. It has been demonstrated that FE models with simplified rigid support conditions overestimate the prediction of the experimental buckling load even though these models included the effects of the measured initial geometric imperfections and load eccentricity. By contrast, FE models with realistically modeled support conditions achieved the best result. The average deviation −1.59% from the experimental buckling loads was achieved using the FE model simulating the mounting devices as elastic bodies and with surface-to-surface contact interaction behavior on the support. The presented work also demonstrated the strong influence of the eccentric load position along the imperfect shell's circumferential direction on the buckling of the thin-walled shell.     

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.     

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.     

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 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.     

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.     

On the modelling of different types of imperfections in silo shells

The assessment of imperfections is most important for determining the load-bearing capacity of a thin-walled shell structure. Different ways of modelling imperfections are discussed in this paper and steel silo shells are used as an application. Buckling tests were performed on different types of model shell - standard quality and high quality with reduced heat input during welding. For the numerical studies two different approaches were used as well: an axisymmetric approach with substitute geometric imperfections and an FEM approach, where the nodal coordinates were derived from surveying the specimen. It was found that there is considerable gain in the buckling strength due to the presence of the granular solid. The larger the initial imperfections in the shell the greater the gain in strength compared to the empty cylinder. The modelling of the uneveness of the edges with uneven dead loading is also discussed.     

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.     

关于圆柱壳几何缺陷分析中傅立叶级数选取的探讨

圆柱壳屈曲一般对壳壁上微小几何缺陷的型式和幅值均十分敏感.为了能将缺陷的不同分量和圆柱壳的结构特征联系起来以及研究缺陷各分量对壳屈曲强度的影响,缺陷通常采用傅立叶级数分解.然而,大多数先前的研究选取不适当的傅立叶级数得到不正确的结果.本文首先考察傅立叶级数的数学描述基础,进而讨论不同傅立叶级数在描述不同型式几何缺陷的表现,从而得出如何选取适当的傅立叶级数用来描述圆柱壳几何缺陷的结论.采用这些适当的傅立叶级数,能更好地了解圆柱壳几何缺陷的特征分量以及这些分量对壳体屈曲强度的影响.     

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.     

Wind pressures and buckling of cylindrical steel tanks with a dome roof

An experimental/computational strategy is used in this paper to evaluate the buckling behavior of steel tanks with a dome roof under exposure to wind. First, wind tunnel experiments using small scale rigid models were carried out, from which pressure distributions due to wind on the cylindrical part and on the roof were obtained. Second, a computational model of the structure (using the pressures obtained in the experiments) was used to evaluate buckling loads and modes and to study the imperfection sensitivity of the tanks. The computational tools used were bifurcation buckling analysis (eigenvalue analysis) and geometrical nonlinear analysis (step-by-step incremental analysis). Geometric imperfections and changes in the buckling results due to reductions in the thickness were also included in the study to investigate reductions in the buckling strength of the shell. For the geometries considered, the results show low imperfection sensitivity of the tanks and buckling loads associated with wind speeds 45% higher than those specified by the ASCE 7-02 standard.     

Robust design of composite cylindrical shells under axial compression — Simulation and validation

Thin-walled shell structures like circular cylindrical shells are prone to buckling. Imperfections, which are defined as deviations from perfect shape and perfect loading distributions, can reduce the buckling load drastically compared to that of the perfect shell. Design criteria monographs like NASA-SP 8007 recommend that the buckling load of the perfect shell shall be reduced by using a knock-down factor. The existing knock-down factors are very conservative and do not account for the structural behaviour of composite shells. To determine an improved knock-down factor, several authors consider realistic shapes of shells in numerical simulations using probabilistic methods. Each manufacturing process causes a specific imperfection pattern; hence for this probabilistic approach a large number of test data is needed, which is often not available. Motivated by this lack of data, a new deterministic approach is presented for determining the lower bound of the buckling load of thin-walled cylindrical composite shells, which is derived from phenomenological test data. For the present test series, a single pre-buckle is induced by a radial perturbation load, before the axial displacement controlled loading starts. The deformations are measured using the prototype of a high-speed optical measurement system with a frequency up to 3680 Hz. The observed structural behaviour leads to a new reasonable lower bound of the buckling load. Based on test results, the numerical model is validated and the shell design is optimized by virtual testing. The results of test and numerical analysis indicate that this new approach has the potential to provide an improved and less conservative shell design in order to reduce weight and cost of thin-walled shell structures made from composite material.     

Parametric study on buckling behaviour of dented short carbon steel cylindrical shell subjected to uniform axial compression

Generally, thin cylindrical shells are susceptible for geometrical imperfections like non-circularity, non-cylindricity, dents, swellings, etc. All these geometrical imperfections decrease the static buckling strength of thin cylindrical shells, but in this paper only effect of a dent on strength of a short (Lc/Rc∼1, Rc/t=117, 175, 280) cylindrical shell is considered for analysis. The dent is modeled on the FE surface of perfect cylindrical shell for different angles of inclination and sizes at half the height of cylindrical shell. The cylindrical shells with a dent are analyzed using non-linear static buckling analysis. From the results it is found that in case of shorter dents, size and angle of inclination of dents do not have much effect on static buckling strength of thin cylindrical shells, whereas in the case of long dents, size and angle of inclination of dents have significant effect. But both short and long dents reduce the static buckling strength drastically. It is also found that the reduction in buckling strength of thin cylindrical shell with a dent of same size and orientation increases with increase in shell thickness.     

Effects of openings of the buckling of cylindrical shells subjected to axial compression

Experimental and numerical methods are used to study the stability problem of cylindrical shells with cut-outs. The paper presents parametric research of the shape (square, rectangular, circular), the dimensions (axial and circumferential sizes, diameter) of the hole. The effect of the location and the number of the holes are also studied. The analysis indicates that the critical load is sensitive to the opening angle or circumferential size of the hole. The function (critical load-opening angle) is linear for large openings and independent of the geometrical imperfections of the shell. However for small openings, it is necessary to take into account the coupling between the initial geometrical imperfections and the openings. The linear approach does not fit because of the importance of the evolution of the displacements near the openings. These results will be used for the development of European rules.