Dynamic buckling of fiber composite shells under impulsive axial compression

The paper deals with dynamic buckling due to impulsive loading of thin-walled carbon fiber reinforced plastics (CFRP) shell structures under axial compression. The approach adopted is based on the equations of motion, which are numerically solved using a finite element code (ABAQUS/Explicit) and using numerical models validated by experimental static buckling tests. To study the influence of the load duration, the time history of impulsive loading is varied and the corresponding dynamic buckling loads are related to the quasi-static buckling loads. To analyse the sensitivity to geometric imperfections, the initial geometric imperfections, measured experimentally on the internal surface of real shells, are introduced in the numerical models. It is shown numerically that the initial geometric imperfections as well as the duration of the loading period have a great influence on the dynamic buckling of the shells. For short time duration, the dynamic buckling loads are larger than the static ones. By increasing the load duration, the dynamic buckling loads decrease quickly and get significantly smaller than the static loads. Since the common practice is to assume that dynamic bucking loads are higher than the static ones, which means that static design is safe, careful design is recommended. Indeed, taking the static buckling load as the design point for dynamic problems might be misleading.     

Simplified procedure for design of liquid-storage combined conical tanks

Steel vessels in the form of combined conical-cylindrical shells are commonly used for liquid storage in elevated water tanks. A number of such tanks collapsed in different places around the globe due to instability of the steel shells. An essential cause of those collapses is the lack of adequate design procedures for such structures. In this study, a simplified design approach is developed to ensure safety of hydrostatically loaded combined steel conical tanks against buckling. The study is conducted numerically using a non-linear finite element model that accounts for the effects of large deformations and geometric imperfections on the stability of combined conical tanks. The finite element results together with a non-linear regression analysis are used to develop magnification functions that relate the overall shell stresses to the membrane stresses which can be evaluated analytically. Numerical examples are presented to explain application of the suggested design approach.     

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.     

Behavior of stiffened liquid-filled conical tanks

Unreinforced steel conical-shaped containment vessels are frequently used in water tower applications. The failure of one of these structures in Fredericton, New Brunswick, Canada, several years ago, raises the question of whether there are adequate safety provisions for existing conical tanks. The aim of this investigation is to study the effect of welding rectangular-shaped longitudinal stiffeners to enhance the buckling capacity of existing conical tanks and to improve the design of new structures. The investigation is carried out numerically using an in-house developed shell element model that includes the effects of geometric and material non-linearities and accounts for geometric imperfections. The study focuses on two cases of tanks reinforced by longitudinal stiffeners in the lower region: the case of stiffeners free at their bottom edge, which would correspond to the retrofit of existing tanks; and the second having stiffeners anchored to the bottom slab of the tank, which can duplicate the situation of a new design. An extensive parametric study is conducted to assess the typical behavior of the two cases and to determine the critical imperfection shape that leads to the minimum buckling capacity of such type of stiffened shell structures. Finally, a comparison between the buckling capacity of unstiffened and longitudinally stiffened conical tanks that have the same volume of steel is conducted, revealing a major benefit of including stiffeners.     

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.     

Recommendations on imperfections in the design of plated structural elements of bridges

High-yield strength steel-plated structures represent competitive solutions when used in steel and steel–concrete composite bridges. Nevertheless, further modifications may still be introduced at the design stage in the case of slender sections, in order to minimize the number of their stiffeners and thereby economize on manufacturing costs. Eurocode 3 “Design of steel structures” specifies design methodologies for slender plates subjected to compression and for stiffeners. Moreover, the use of Finite Element Method (FEM) software is fast becoming an alternative analytical method for the design of complete structures or structural elements, as it offers a more realistic approach. This paper makes recommendations for FEM assessments of plated sections in bridges that take the initial imperfections, geometric imperfections and residual stresses of the sections into account, in order to arrive at realistic results.     

Buckling and post buckling of thin-walled composite columns with intermediate-stiffened open cross-section under axial compression

The thin-walled composite columns with an open cross-section reinforced by intermediate stiffener under axial compression have been considered. The finite element method is employed to study the buckling behaviour of the thin-walled composite column. Eigenvalue analyses are carried out first to predict the buckling load and buckling mode shapes of the column, and then the geometric nonlinear analyses are performed to investigate the nonlinear buckling properties and post-buckling behaviour of the thin-walled structures. The type of angle ply symmetric laminate is used. The investigation is performed over several values of ply arrangement angle and various values of stiffener parameter. The numerical results show a significant effect of the intermediate stiffeners and composite ply angle on loading capacity and buckling behaviour of the thin-walled composite column. The research provides insight into the thin-walled structure and composite laminate, which is employed to enhance the loading capacity of thin-walled composite structures.     

A semi-analytical approach for linear and non-linear analysis of unstiffened laminated composite cylinders and cones under axial,torsion and pressure loads

A semi-analytical model for the non-linear analysis of simply supported, unstiffened laminated composite cylinders and cones using the Ritz method and the Classical Laminated Plate Theory is proposed. A matrix notation is used to formulate the problem using Donnell׳s and Sanders׳ non-linear equations. The approximation functions proposed are capable to simulate the elephant׳s foot effect, a common phenomenon and a common failure mode for cylindrical and conical structures under axial compression. Axial, torsion and pressure loads can be applied individually or combined, and solutions for linear static, linear buckling and non-linear buckling analyses are presented and verified using a commercial finite element software. The presented non-linear buckling analyses used perturbation loads to create the initial geometric imperfections, showing the capability of the method for arbitrary imperfection patterns. The linear stiffness matrices are integrated analytically and for the conical structures an approximation is proposed to overcome the non-integrable expressions.     

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.     

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.     

GBT formulation to analyse the buckling behaviour of isotropic conical shells

This paper presents a Generalised Beam Theory (GBT) formulation developed to analyse the elastic buckling behaviour of isotropic conical shells with constant thickness under axial compression. The GBT approach provides a general solution for 1st and 2nd order analysis using bar elements capable of describing the global and local deformations. The kinematic relations of the theory of thin shells are greatly simplified by the assumptions of null membrane shear strain and transverse extension before they are used for the evaluation of the strain energy variation. Because of the cross-section variation specific to conical shells, the mechanical and geometric properties are no longer constant along the bar axis as it is the case of cylinders and prismatic thin-walled structures. This formulation is validated by comparison between GBT results and values obtained by means of shell finite element analyses.     

Local buckling of steel plates in concrete-filled thin-walled steel tubular beam-columns

The availability of high strength steels and concrete leads to the use of thin steel plates in concrete-filled steel tubular beam-columns. However, the use of thin steel plates in composite beam-columns gives a rise to local buckling that would appreciably reduce the strength and ductility performance of the members. This paper studies the critical local and post-local buckling behavior of steel plates in concrete-filled thin-walled steel tubular beam-columns by using the finite element analysis method. Geometric and material nonlinear analyses are performed to investigate the critical local and post-local buckling strengths of steel plates under compression and in-plane bending. Initial geometric imperfections and residual stresses presented in steel plates, material yielding and strain hardening are taken into account in the nonlinear analysis. Based on the results obtained from the nonlinear finite element analyses, a set of design formulas are proposed for determining the critical local buckling and ultimate strengths of steel plates in concrete-filled steel tubular beam-columns. In addition, effective width formulas are developed for the ultimate strength design of clamped steel plates under non-uniform compression. The accuracy of the proposed design formulas is established by comparisons with available solutions. The proposed design formulas can be used directly in the design of composite beam-columns and adopted in the advanced analysis of concrete-filled thin-walled steel tubular beam-columns to account for local buckling effects.     

The influence of the fabrication process on the buckling of thin-walled steel box sections

Steel box sections are usually fabricated from flat plates which are welded at the corners. The welding process can introduce residual stresses and geometric imperfections into the sections which can influence their strength. For some thin-walled sections, large periodic geometric imperfections have been observed in manufactured sections. Subsequent investigations have indicated that the imperfections are in fact buckling deformations i.e. the box section has buckled due to welding residual stresses prior to any application of external load. The welding procedure and the behaviour of the box sections under load has been modelled using a finite element analysis that accounts for both geometric and material non-linearities. Tests have been carried out on box sections with a range of width to thickness ratios for the plate elements. Modelling has been shown to give good correlation with the test results. The conditions for buckling to take place as a result of the welding process have been established. A design method has been proposed.     

Future directions and challenges in shell stability analysis

Recent advances in structural analysis and design technology for buckling-critical shell structures are discussed. These advances include a hierarchical analysis strategy that includes analyses that range from classical analysis methods to high-fidelity nonlinear finite element analysis methods, reliability based design methods, the development of imperfection data bases, and the identification of traditional and nontraditional initial imperfections for composite shell structures. When used judiciously, these advances provide the basis for a viable alternative to the traditional and conservative lower-bound design philosophy of the 1960s. These advances also help answer the question of why, after so many years of concentrated research effort to understand the behavior of buckling-critical thin-walled shells, one has not been able to improve on this conservative lower- bound design philosophy in the past.     

On polymorphic uncertainty modeling in shell buckling

Buckling is typically the governing failure mode of thin-walled shells. In particular, geometric and material imperfections have a major influence on the buckling behavior. Small variations of imperfections have large effects on the load-bearing behavior. However, the design of shells is still characterized by a deterministic way of thinking, in which uncertainties have not yet been sufficiently considered. Even in probabilistic approaches, false assumptions are often generated due to the small amount of experimental data. The focus of this paper is an appropriate uncertainty quantification based on the available data. Therefore, the concept of polymorphic uncertainty modeling is presented on axially loaded shells with different types of imperfections. Finally, an idea for a novel design concept for shells based on a fuzzy-valued safety level is introduced. The paper is intended to initiate a rethinking of the methodology for the numerical design of shells with an appropriate uncertainty quantification.     

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.     

Stub column tests of thin-walled complex section with intermediate stiffeners

A series of stub column tests on complex sections with intermediate stiffeners is presented in this paper. Initial geometric imperfections and material properties of the test specimens were measured. It is shown that the intermediate stiffeners could effectively enhance the local buckling stress of thin-walled sections. The test strengths are compared with the design strengths calculated using the direct strength method in the North American Specification and Australian/New Zealand Standard for cold-formed steel structures. It is shown that the direct strength method using finite strip method to obtain the buckling stresses is very conservative. Therefore, finite element method was used to predict the elastic buckling stresses. It is shown that the design strengths calculated using direct strength method based on the buckling stresses obtained from finite element analysis results generally agree with the test results well.     

Probabilistic Buckling Analysis of Plates and Shells

For many years, a significant amount of research has been directed towards experimental modelling of thin-walled plates and shells, as well as towards the development of analytical and numerical methods to improve their design against buckling. This paper presents methodologies for probabilistic buckling analysis and reliability assessment of such structural components and examines the link between probabilistic and deterministic studies. In particular, the effect of manufacturing variabilities, such as initial geometric imperfections and residual stresses, on elastoplastic buckling response is investigated through parametric reliability studies of plate panels and cylinders under axial compression.     

带加劲肋的复杂截面薄壁柱的试验研究

进行了带加劲肋的复杂截面管柱的一系列试验研究。测量了试件的初始几何缺陷和材料特性。加劲肋能有效增强薄壁构件的局部稳定性。将试验强度与采用北美、澳大利亚、新西兰规范中针对冷弯型钢结构的直接强度法计算的设计强度进行对比,结果表明,采用有线条法确定屈曲荷载的直接强度法非常保守。因此,采用有限元法确定弹性屈曲荷载,结果表明,采用此直接强度法计算的设计强度与试验结果吻合较好。     

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.