Reliability and Sensitivity Analysis of Steel Frames with Initial Random Imperfections

The effect of non-uniformity of axial loading on the buckling behaviour of shells with random imperfections

Loading conditions constrained wrinkling behaviors of thin-walled sheet/tube parts during metal forming

ABSTRACTSteel products which have various characteristic (e.g. high ductility, high yield stress, low yield stress and high weldability) are used as main structures by diversification of demanded application, and technology for steel manufacturing has developed. Because steel square tubular member is processed by cold‐forming and welding in the process of manufacturing, residual stress will exist in the products. Moreover, due to the cold‐forming process, geometrical initial imperfection, i.e., global and local, will exist. These initial imperfection are still remained even so the technology for steel manufacturing is developed. However, effect of the initial imperfection on elasto‐plastic behaviour of square hollow section columns under compressive axial force with bending moment is not clarified. In this paper, firstly, the effect of individual initial imperfection (e.g. initial global imperfection, initial local imperfection or residual stress) on elasto‐plastic performance is investigated independently; comparison between the experimental results are shown. It is shown that initial local imperfection has a significant impact on elasto‐plastic performance. Secondly, the effect of composite initial imperfections is investigated. The amplitude of the local imperfection is the most influential initial imperfection on the deformation capacity of the column, and the deformation capacity decreased with an increase of the amplitude of local imperfection. Validation of the numerical simulation were also performed and recommend values of imperfections are shown.

On the modelling of initial geometric imperfections of steel frames in advanced analysis

A new method for estimation of the statistical characteristics of geometric initial imperfections in rotational shells is proposed and applied to estimate the statistical properties of initial imperfections in hyperboloidal cooling tower shells. The method is based on the entropy maximum criteria, which are often used in information science for analysing random phenomena. From the present analysis, we have found that there exists a high possibility of causing geometric imperfections with relatively high harmonic numbers in the rotational direction. However, since there exist at present little available data for initial imperfections, the validity and applicability of results obtained in this paper for practical designs are still left unresolved. But in furure, if many measured imperfections in shell structures were assembled and analysed, it would be possible to correctly evaluate the estimated values and the method would have a possibility of application in practical designs.

This paper presents the comparison of reliability technique that employ Fourier series representations of random axisymmetric imperfections in axially compressed cylindrical shells with evaluations prescribed by ASME Section VIII, Division 2. The ultimate goal of the reliability type technique is to predict the buckling load associated with the axially compressed cylindrical shell. The representation of initial geometrical imperfections in the cylindrical shell requires the determination of appropriate Fourier coefficients. The buckling of cylindrical shells in any type of loading is sensitive to the form and amplitude of geometric imperfections present in the structure. Initial geometric imperfections have significant effect on the load carrying capacity of axisymmetrical cylindrical shells. Many deterministic and probabilistic techniques are there to predict shell behavior during buckling. Fourier decomposition is used to interpret imperfections as structural features can be easily related to the different components of imperfections. The mean vector and the variance-covariance matrix of Fourier coefficients are calculated from the simulated shell profiles. Recommendations for further use of Fourier coefficients through simulation by Monte Carlo Method are laid down. Large number of shells thus created can be used to calculate buckling stress for each shell. The probability of the ultimate buckling stress exceeding a predefined threshold stress can be calculated.

Geometrical imperfections are ubiquitous in load-bearing structures, including beams, columns, and shells. Fabrication processes of structural members most often create geometrical imperfections of random size and shape, which lead to non-deterministic load-carrying capacity. This study investigates the statistics of the buckling load of a beam with a random initial imperfection profile that rests on a nonlinear elastic foundation. The geometrical imperfection is represented by a zero-mean Gaussian random field, generated using the Karhunen–Loève expansion. The spatial distribution of the random imperfection is characterized by the probability distribution of the local imperfection magnitude and a spatial autocorrelation function. A finite-difference scheme is used to solve the governing equilibrium equation for a given initial imperfection profile, from which the buckling load is determined. Through a set of Monte Carlo simulations, the mean and variance of the buckling load are determined. The simulations reveal the influence of different length scales on the statistics of the buckling load, including the beam length and the autocorrelation length of the geometrical imperfection. The size effects predicted with the simplified model have implications for reliability-based structural design.

The initial geometric imperfection is recognized as substantially influential on the buckling behaviour and ultimate strength of stiffened plated structures. Conventionally, the initial geometric imperfection is defined in a deterministic manner, such as a hungry-horse mode, ARE mode or critical buckling mode with characteristic magnitudes. However, due to the uncertainty in manufacturing and in-service effects, the initial geometric imperfection should be better described as a random field. This paper aims to assess the effects of the stochastic nature of geometric imperfection on the ultimate strength of stiffened panel in compression. A probabilistically-based imperfection model developed by spectral representation method is applied to an orthogonally stiffened panel. A Monte-Carlo Simulation enhanced by a Latin-Hypercube Scheme (LHS) is utilized to sample the initial geometric imperfection. A series of nonlinear finite element analyses are completed to calculate the ultimate compressive strength of the case study model. With reference to the deterministic representations, the impact of the stochastic imperfection model is discussed. The present work may provide useful insights for the reliability-based ship structural assessment regarding the uncertainty due to geometric imperfection.

Initial imperfections can cause evident influence on the bearing capacity of lattice shells. In this article, the member initial imperfections including the initial bending and residual stress are considered by the Marshall model which is a phenomenon model based on the hysteresis loop of the real structural imperfect member under cyclic loading. The geometrical initial imperfection of lattice shells resulting from the structure assembly deviation is represented by the normally distributed random variable. The bearing capacities of lattice shells with initial imperfections are assumed to be distributed normally, and then this assumption is proved correct by the nonparametric test. The expressions of the mean value and variance estimators of the normally distributed structural bearing capacities are derived by the maximum likelihood method. A new method to calculate the design bearing capacity of the initial imperfect lattice shell at a uniform probability level is proposed, and its error expression at certain confidence level is derived. Numerical analysis indicates that some members of the structure at the ultimate limit state get buckled as a result of the member initial imperfections. The structure bearing capacity can be markedly overestimated if the member initial imperfections are neglected. The nonlinear design bearing capacity of the lattice shell at a uniform probability level cannot be acquired by the methods suggested by the structure design code.

Geometric imperfections have an important influence on the buckling response of structural components. This paper describes an experimental technique for determining imperfections in long (5.7 m) structural members using a series of overlapping measurements. Measurements were performed on 31 austenitic stainless steel sections formed from three different production routes: hot-rolling, cold-rolling and press-braking. Spectral analysis was carried out on the imperfections to obtain information on the periodic nature of the profiles. Two series were used to model the profile firstly the orthogonal cosine and sine functions in a classic Fourier transform and secondly a half sine series. Results were compared to the relevant tolerance standards. Simple predictive tools for both local and global imperfections have been developed to enable representative geometric imperfections to be incorporated into numerical models and design methods.

Capacity of liquid steel conical tanks under hydrodynamic pressure due to horizontal ground excitations

This research studies the first part of the failure of a compression member structure due to buckling. This unstable equilibrium collapse, exposes brittle failure which occurs suddenly and therefore should be avoided wherever possible. Unavoidable geometric imperfections due to structural fabrication, will weaken the structure against buckling. The behavior of bar under compression will be closely examined by taking a set of geometric imperfection data synthesized from previously available from the measurement of conical shells. Therefore, the two-dimensional surface imperfection is converted into several one-dimensional imperfection with some probability properties. In order to obtain a comparison tool for different type of imperfections, Fourier analysis is used to convert the imperfection into coefficients of trigonometric function. By examining the coefficients, geometric imperfection patterns introduced by a certain fabrication process are able to be identified. The study successfully demonstrates the applicability of Fourier analysis in representing inherent geometric imperfections as an initial step for conducting probabilistic buckling analysis. Fourier analysis has shown its capability to simultaneously characterize imperfections in two crucial parameters - the magnitude and configuration of the imperfection.

Structural behaviour and reliability of CFST trusses with random initial imperfections

Results from a numerical study of the buckling response of a thin-walled compressionloaded isotropic circular cylindrical shell with initial geometric and loading imperfections are used to determine a lower bound buckling load estimate suitable for preliminary design. The lower bound prediction techniques presented herein include an imperfection caused by a lateral perturbation load, an imperfection in the shape of a single stress-free dimple (similar to the lateral pertubation imperfection), and a distributed load imperfection that induces a nonuniform load in the shell. The ABAQUS finite element code is used for the analyses. Responses of the cylinders for selected imperfection amplitudes and imperfection types are considered, and the effect of each imperfection is compared to the response of a geometrically perfect cylinder. The results indicate that compression-loaded shells subjected to a lateral perturbation load or a single dimple imperfection, and a nonuniform load imperfection, exhibit similar buckling behavior and lower bound trends and the predicted lower bounds are much less conservative than the corresponding design recommendation NASA SP-8007 for the design of buckling-critical shells. In addition, the lateral perturbation technique and the distributed load imperfection produce response characteristics that are physically meaningful and can be validated via laboratory testing.

Mean values as well as confidence intervals for the critical load of axially loaded cylindrical shells containing general random initial imperfections are obtained. The results display trends that have been observed experimentally and provide insight into previously proposed design criteria. It has been determined that nonaxisymmetric initial imperfections play a vital role in the determination of the critical load statistics and that this role varies with the shell radius-to-thickness ratio. The work should provide a sound theoretical basis for the design of randomly imperfect cylindrical shells.