Random Initial Imperfections Research Articles

Probabilistically modelled geometrical imperfections for reliability analysis of vertically loaded steel frames

The article explores the stochastic modelling of 3D steel frames under static vertical load, with a main focus on random initial geometrical imperfections. The random input parameters for initial geometrical imperfections are derived from the tolerance criteria in accordance with European standard. In order to derive statistical parameters from the corresponding standard tolerances, two methods noted as #RSS (random storey sway) and #RSP (random storey position) have been utilized and compared. Both of these methods are used along with the first-order reliability method (FORM) and the advanced finite element method (FEM) – geometrically and materially nonlinear imperfect analysis (GMNIA) to numerically analyse several geometries of steel frame structures. Ultimate resistances of these structures are monitored and basic sensitivity studies are conducted. The results are also compared with the deterministic approach of the European standard. The stochastic approach of the FORM to estimate the ultimate resistance of steel frame structures serves as a verification tool of the classic, in engineering praxis widely used, deterministic approach. This study provides useful provisions for the advanced numerical analyses of steel frames of various geometries. Additionally, the estimations of ultimate resistance by the deterministic European standard approach is verified for selected frame geometries.

Study on Global Imperfection of Modular integrated Construction Structures for Second‐order Analysis

AbstractThe global imperfection significantly influences the stability design of the Modular integrated Construction (MiC) buildings. Currently, the initial global imperfection shapes recommended for MiC structures are formulated based on steel frames whose members are continuously connected. In practice, the columns and beams in MiC structures are composed of the members of adjacent module units, which are discontinuously connected through the special joints. Ignorance of such discontinuous features in MiC structures leads to an inappropriate estimation of the structural resistance. This paper introduces a new initial imperfection shape for MiC structures considering out‐of‐verticality of module columns and eccentricity between modules, capturing the discontinuous feature. Random initial imperfection shapes were generated using Monte‐Carlo simulation based on tolerances in regulations. The proposed imperfection shape is adopted in the analysis of a typical MiC structure and the result is compared with results from existing global imperfections in design codes. This research provides a practical global imperfection for the second‐order analysis and design of steel MiC structures. Recommendation for basic value of the proposed imperfection shape is also provided.

Investigation of Intermediate-Height Horizontal Brace Forces under Horizontal and Vertical Loads including Random Initial Imperfections

In engineering practice, longitudinal brace systems for column-braced systems are designed to resist both horizontal and vertical loads. In previous experimental research on horizontal brace forces for column-braced systems of intermediate height, only vertical loads were considered. Hence, this paper presents a numerical simulation of numerous column-braced systems subjected to horizontal and vertical loads. In the numerical simulation, second-order analysis was adopted, and the Monte Carlo method was used to incorporate the randomness of initial imperfections in the horizontal brace and column. From the finite element (FE) analyses and probability model statistics, the normal probability density equation for intermediate-height horizontal brace forces under horizontal and vertical loads was obtained, and the corresponding design intermediate-height horizontal brace forces were determined and compared with those under vertical loads only. The results indicate that the design intermediate-height horizontal brace forces under horizontal and vertical loads are significantly greater than those under only vertical loads, and that the design intermediate-height horizontal brace forces under horizontal and vertical loads are also greater than the simple superposition results of horizontal loads and intermediate-height horizontal brace forces under only vertical loads.

Shear Buckling Analysis of Corrugated Web Steel Plate Girder with Random Material Properties

Background: The non-linear finite element method with initial geometric imperfection is compulsory to capture the shear buckling behavior of the Corrugated Web Steel Plate Girder (CWSPG). These initial geometry imperfections can come from the slender structure that cannot maintain its perfect shape or lousy quality during the assembly process. Most researchers generate the initial geometry imperfection from the elastic buckling modes that may not represent the randomness in the geometric imperfection. Therefore, there is a need to develop a method to generate random initial geometry imperfection without carrying out elastic buckling modes from the analysis. Objectives: This paper investigated the shear buckling behavior of CWSPG using non-linear finite element analysis and proposed a method to generate the initial geometric imperfection using the random material imperfection. Methods: The random material properties for each meshed element follow a statistically random normal distribution of the material yield strength. The initial geometric imperfection is generated from the first second-order analysis with random material properties (using the in-house 3D-NLFEA package) to the point where the expected buckling shape is obtained. The final deformed geometry from the first second-order analysis is then scaled down to be used as the initial geometric imperfection. Results: The proposed method requires the scaling value such that the first buckling load from the available experimental test result and the one from the numerical model are at the same level. The proposed method was able to capture the shear buckling behavior of the CWSPG and was sensitive to the element’s size. The minimum size of the element required normalized with the element thickness was found to be less than four to maintain the accuracy for both the peak and residual load of the CWSPG specimen. Conclusion: The proposed method shows excellent agreement in predicting the peak load and the post-buckling behavior of the available test results. Therefore, the proposed method can be used as an alternative method to capture the buckling load of the CWSPG specimen.

Stochastic Buckling of Geometrically Imperfect Beams on Elastic Foundation

Abstract 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.

Design Forces of Horizontal Braces Unlocated at Middle of Columns considering Random Initial Geometric Imperfections

The horizontal bracing forces of column-bracing systems derived from past studies and current design codes were considered only located at middle of columns. Actually, the horizontal braces used to reduce the out-of-plane effective column lengths are frequently designed not to locate at middle of columns. In this paper, a large number of column-bracing systems with the horizontal braces unlocated at middle of columns were modelled and analyzed using the finite element method, in which the random initial geometric imperfections of both the columns and the horizontal braces unlocated at middle of columns were well considered by the Monte Carlo method. Based on the numerical calculations, parametric analysis, and probability statistics, the probability density function of the horizontal bracing forces was found, so that the corresponding design forces of horizontal braces unlocated at middle of columns were proposed which were compared with the design mid-height horizontal bracing forces in the previous study and the relevant codes. The results indicate that the design forces of the horizontal braces located at 0.6 column height are smaller than the mid-height horizontal bracing forces in the previous study while the design forces of horizontal braces located at 0.7 column height are larger than the mid-height horizontal bracing forces in the previous study. The proposed design forces of the horizontal braces located and unlocated at middle of columns are both smaller than the mid-height horizontal bracing forces stipulated in GB50017-2017, Eurocode 3-1992, and AS4100-1998. The above conclusions provide references for the engineering applications and further related code revisions.

Reliability calibration for the design of multiple-chord CFST trusses by advanced analysis

Concrete-filled steel tubular (CFST) truss with complex configurations are now widely used in practice, especially in large-scale bridge constructions. For such complex composite trusses, traditional structural analysis approaches are readily applied for the safety checks of individual members and connections, whilst inelastic analysis and reliability calibration are very limited. Recently, the development of advanced system-level analysis that takes into account various nonlinearities and uncertainties allows a rational reliability calibration for CFST trusses to assess their structural reliability. This paper aims to present reliability calibrations on three typical CFST trusses (two-chord, three-chord and four-chord) by advanced finite element models implemented with random uncertainties. The established finite element models are validated against experimental results. Afterwards, stochastic finite element analyses (FEA) are conducted taking into account the material and geometric nonlinearities, the random initial imperfections and the potential model errors. Using the obtained statistics of resistance, reliability calibration is then undertaken to calculate the reliability indexes of the three typical CFST trusses in respect of resistance factors under various load conditions, with the target reliability discussed in accordance with AASHTO and ANSI/AISC 360–16. The proposed novel computational approach and the reliability calibration contribute to both the practical design and the standard drafting for CFST trusses.

Reliability-based evaluation for concrete-filled steel tubular (CFST) truss under flexural loading

Concrete-filled steel tubular (CFST) truss is now widely used in many large-span constructions. In the past, relevant studies on the reliability of CFST structures mostly focused on structural components such as individual columns and beams. In current practices, a simplified method based on the ultimate axial strengths of the chords is commonly adopted to predict the flexural strength of a CFST truss, since there is no mature system-based design regulations nor reliability evaluations on this complex composite system. This paper aims to address this gap by advanced structural reliability analysis of CFST truss through numerical approach considering both the structural nonlinearities and random uncertainties. The finite element models are validated against experimental results. Afterwards, a mass finite element simulation, taking into account the material and geometric nonlinearities, random initial imperfections and potential finite element model errors, is conducted to generate the statistics of the flexural strengths of CFST trusses. Reliability analysis is then conducted to calculate reliability indexes in respect of different resistance factors under various load cases based on AASHTO. Finally, reliability-based evaluation on the safety level of CFST truss design according to mainstream design guidelines such as ANSI/AISC 360–16, Eurocode 4:2004 and JGJ/T D65–06–2015 are carried out. Results show that the current component-based strength prediction of CFST trusses meet the target reliability while the current regulations provide uniform reliability.

Structural behaviour and reliability of CFST trusses with random initial imperfections

Concrete-filled steel tubular (CFST) truss is a type of composite structure with CFST chords and hollow tubular braces. CFST trusses have been increasingly used in large-scale structures such as towers, bridge girders, piers and arch ribs. The compression and flexural behaviour of CFST trusses are greatly improved compared to hollow tubular trusses due to the concrete infill in chords. With the complex configuration, nonlinear material interaction and sophisticated construction process, initial imperfections may largely affect the strength and stability of a CFST truss structure. Relevant studies in the past mostly focused on one single type of imperfection with assumed magnitude in CFST member whilst in reality a variety of combinations of steel and concrete imperfection exist. This paper develops a nonlinear analysis formwork with the reliability analysis of CFST truss with random imperfections based on on-site measurement data and Monte-Carlo (MC) simulations. Advanced nonlinear finite element analysis (FEA) that can account for the material interaction and confinement in CFST trusses is established and validated by reported test data. The flexural behaviour of CFST trusses with deterministic and probabilistic initial imperfections are evaluated and compared, based on which the reliabilities and related system resistance factors in regards to random initial imperfections are proposed.

Buckling reliability evaluation of a clamped beam with random imperfections subjected to axial impact using probability density evolution method

ABSTRACTBased on the generalized probability density evolution theory, the joint probability density function of impact responses and random factors, that is, random initial imperfection and load intensity for a clamped beam, is derived. And a Total Variation Diminishing (TVD)-based numerical scheme is employed to obtain the numerical solutions. With given threshold values, the dynamic reliability is assessed in terms of a bilateral displacement criterion. Two numerical examples are then presented to demonstrate the proposed method. One is a clamped beam with random initial perfections subjected to a deterministic impulse load, and another is a clamped beam with a deterministic initial imperfection subjected to impact loading with a random amplitude. The predicted dynamic reliability is in good agreement with the result by Monte Carlo method.

Study on mid-height horizontal bracing forces considering random initial geometric imperfections

In order to investigate the different mid-height horizontal bracing forces of column–bracing system between pin-ended column base and fixed-ended column base, a large number of column–bracing systems with pin-ended column base and fixed-ended column base have been modeled and analyzed using finite element method, in which the random combination of the initial geometric imperfections between columns and braces was well considered by the Monte Carlo method. Based on the above comparative study, the probability density function of mid-height horizontal bracing forces was found through probability statistics and the design bracing forces were also obtained. It is founded that the buckling mode of columns for pin-ended column base is three half-waves of bending while the buckling mode of columns for fixed-ended column base is two half-waves of bending, so that the ultimate load-carrying capacity and the mid-height horizontal bracing forces of column–bracing systems with pin-ended column base are higher than those of column–bracing systems with fixed-ended column base, and the relative high ultimate load-carrying capacity of the former more significantly increases its mid-height horizontal bracing forces. The results also indicate that random combination of the initial geometric imperfections between columns and braces leads to the randomness of mid-height horizontal bracing forces in compression or in tension, so that the design bracing forces can be reasonably reduced which are smaller than those stipulated in GB50017-2003, Eurocode3-1992 and AS4100-1998. Moreover, practical design formulas of mid-height horizontal bracing forces are proposed.

Stochastic stability analysis of steel tubes with random initial imperfections

In this paper, the effect of initial geometric imperfections on the buckling load of steel tubes (relatively thick cylindrical shells) under axial load and lateral pressure is investigated. The geometric imperfections are modeled as a 2D-1V non-homogeneous Gaussian stochastic field simulated using the spectral representation method. The evolutionary power spectrum of the non-homogeneous field is derived from available experimental measurements using the recently proposed method of separation. For the determination of the limit load variability of the tubes, a stochastic formulation based on Monte Carlo simulation is implemented. It is shown that the imperfections can lead to a substantial reduction of the buckling load and thus should be taken into account via a realistic description through stochastic field modeling.

Buckling load variability of cylindrical shells with stochastic imperfections

In this paper, the effect of random initial geometric, material and thickness imperfections on the buckling load of isotropic cylindrical shells is investigated. To this purpose, a stochastic spatial variability of the elastic modulus as well as of the thickness of the shell is introduced in addition to the random initial geometric deviations of the shell structure from its perfect geometry. The modulus of elasticity and the shell thickness are described by two-dimensional univariate (2D-1V) homogeneous non-Gaussian translation stochastic fields. The initial geometric imperfections are described as a 2D-1V homogeneous Gaussian stochastic field. A numerical example is presented examining the influence of the non-Gaussian assumption on the variability of the buckling load. In addition, useful conclusions are derived concerning the effect of the various marginal probability density functions as well as of the spectral densities of the involved stochastic fields on the buckling behaviour of shells, as a result of a detailed sensitivity analysis.

Buckling analysis of imperfect shells with stochastic non-Gaussian material and thickness properties

In this paper, the effect of material and thickness spatial variation on the buckling load of isotropic shells with random initial geometric imperfections is investigated. To this purpose, a random spatial variability of the elastic modulus as well as of the thickness of the shell is introduced in addition to the random initial geometric deviations of the shell structure from its perfect geometry. The main novelty of this paper compared to previous works is that a non-Gaussian assumption is made for the distribution of the two aforementioned uncertain parameters i.e. the modulus of elasticity and the shell thickness which are described by two-dimensional uni-variate (2D-1V) homogeneous non-Gaussian stochastic fields. The initial geometric imperfections are described as a 2D-1V Gaussian non-homogeneous stochastic field with properties derived from corresponding experimental measurements. Numerical examples are presented focusing on the influence of the non-Gaussian assumption on the variability of the buckling load, which is calculated by means of the Monte Carlo Simulation method. It is shown that the choice of the marginal probability distribution for the description of the material and thickness variability is crucial since it affects significantly the statistics of the buckling load of imperfection sensitive shell-type structures.

Optimization of shell buckling incorporating Karhunen-Loève-based geometrical imperfections

The optimization of shell buckling is performed considering peak normal force and absorbed internal energy in the presence of geometrical imperfections implemented through Karhunen-Loeve expansions. Initially, the mass of a shell is minimized in the presence of random initial imperfections by allowing cutouts in the material, subject to constraints on the average peak force and average internal energy. Then, robustness is considered by minimizing the coefficient of variation of the normal peak force while constraining the average peak force and average internal energy. LS-OPT® is used both to generate an experimental design and to perform a Monte Carlo simulation (96 runs) using LS-DYNA® at each of the experimental design points. The effect of imperfections when minimizing the mass is not large, but when considering robustness, however, the optimal design has a substantially increased hole size and increased shell thickness, resulting in a heavier design with maximal robustness within the constraints.

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

In the present paper, the effect of random non-uniform axial loading on the buckling behaviour of isotropic thin-walled imperfect cylindrical shells is investigated. Random initial (out-of-plane) geometric imperfections, thickness and material property variability, together with a non-uniform stochastic axial loading are incorporated into a cost-effective non-linear stochastic finite element analysis using the non-linear TRIC shell element. For this purpose, the concept of an initial ‘imperfect’ structure is introduced involving not only deviations of the shell structure from its perfect geometry but also a spatial variability of the modulus of elasticity as well as of the thickness of the shell. The initial imperfections as well as the axial loading are modeled as stochastic fields with statistical properties that are either based on an available data bank of measured initial imperfections or assumed, in cases where no experimental data is available. Based on these simulation features, a simple and realistic approach is proposed for the estimation of the variability (scatter) of the limit loads by means of a brute-force Monte Carlo Simulation procedure. In addition, ‘worst case’ buckling scenarios are identified by means of a sensitivity analysis with respect to assumed parameters used for the description of stochastic fields that are not supported by corresponding experimental measurements. In addition it is shown that in the context of such sensitivity analysis, modeling of the non-uniformity of the axial loading is, from a computational point of view, fully equivalent to modeling the geometric boundary imperfections. The numerical tests performed demonstrate the significant role that the random varying axial loading plays on the buckling behaviour of imperfection sensitive structures like the axially compressed thin-walled cylinder considered in this study.

Optimum design of shell structures with random geometric, material and thickness imperfections

The optimum design of isotropic shell structures with random initial geometric, material and thickness imperfections is investigated in this paper and a robust and efficient methodology is presented for treating such problems. For this purpose, the concept of an initial “imperfect” structure is introduced involving not only geometric deviations of the shell structure from its perfect geometry but also a spatial variability of the modulus of elasticity as well as of the thickness of the shell. An efficient reliability-based design optimization (RBDO) formulation is proposed. The objective function is considered to be the weight of the structure while both deterministic and probabilistic constraints are taken into account. The overall probability of failure is taken as the global probabilistic constraint for the optimization procedure. Numerical results are presented for a cylindrical panel, demonstrating the efficiency as well as the applicability of the proposed methodology in obtaining rational optimum designs of imperfect shell-type structures.

The effect of material and thickness variability on the buckling load of shells with random initial imperfections

The effect of material and thickness imperfections on the buckling load of isotropic shells is investigated in this paper. For this purpose, the concept of an initial ‘imperfect’ structure is introduced involving not only geometric deviations of the shell structure from its perfect geometry but also a spatial variability of the modulus of elasticity as well as the thickness of the shell. The initial geometric imperfections are described as a two-dimensional uni-variate (2D-1V) stochastic field with statistical properties that are either based on an available data bank of measured initial imperfections or assumed, in cases where no experimental data is available. In order to describe the non-homogeneous characteristics of the initial imperfections, the spectral representation method is used in conjunction with an autoregressive moving average model with evolutionary power spectra based on a statistical analysis of the experimentally measured imperfections. In cases where no experimental results is available, the initial imperfections are assumed to be homogeneous and their impact on the buckling load is investigated on the basis of ‘worst’-case scenarios with respect to the correlation length parameters of the stochastic fields. The elastic modulus and the shell thickness are described as 2D-1V non-correlated homogeneous stochastic fields, while the stochastic stiffness matrix of the shell elements is formulated using the local average method. The Monte Carlo Simulation method is used to calculate the variability of the buckling load, while for the determination of the limit load of the shell, a stochastic formulation of the elastoplastic and geometrically non-linear TRIC facet triangular shell element is implemented.

Finite-Element Analysis of Cylindrical Panels with Random Initial Imperfections

Stochastic finite-element analysis of shells is performed using the spectral representation method for the description of the random fields in conjunction with the local average method for the formulation of the stochastic stiffness matrix of the elements. A stochastic formulation of the nonlinear triangular composites facet triangular shell element is implemented for the stability analysis of cylindrical panels with random initial imperfections. The imperfections are described as a two-dimensional univariate homogeneous stochastic field. The elastic modulus and the shell thickness are also described as two-dimensional uni-variate homogeneous stochastic fields. The variability of the limit load of the cylindrical panel is then computed using the Monte Carlo simulation. Useful conclusions for the buckling behavior of cylindrical panels with random initial imperfections are derived from the numerical tests presented in this paper. These tests also demonstrate the applicability of the proposed methodology in realistic problems.

Strongly non-linear stochastic response of a system with random initial imperfections

Strongly non-linear responses of structures with random imperfections of Gaussian type in their geometrical form under slowly increasing loading are investigated. Large displacements as a source of non-linearities are taken into account. Imperfections are considered as stochastic functions of space coordinates. The finite element method is supposed as a basis. A stochastic version of the arc length method of stochastic non-linear algebraic systems in linearised formulation is proposed. A closed interaction between deterministic and stochastic parts of response is demonstrated. Several numerical tests of theoretical results on simple Mieses frames modelling imperfect shallow shells have been carried out. Analytical and numerical results make it possible to demonstrate non-conventional properties typical for a randomly imperfect structure with a tendency to various types of snap-through effects.