Random Geometric Imperfections Research Articles

Statistical Scaling in Localization-Induced Failures

Abstract The investigation of statistical scaling in localization-induced failures dates back to da Vinci's speculation on the length effect on rope strength in 1500 s. The early mathematical description of statistical scaling emerged with the birth of the extreme value statistics. The most commonly known mathematical model for statistical scaling is the Weibull size effect, which is a direct consequence of the infinite weakest-link model. However, abundant experimental observations on various localization-induced failures have shown that the Weibull size effect is inadequate. Over the last two decades, two mathematical models were developed to describe the statistical size effect in localization-induced failures. One is the finite weakest-link model, in which the random structural resistance is expressed as the minimum of a set of independent discrete random variables. The other is the level excursion model, a continuum description of the finite weakest-link model, in which the structural failure probability is calculated as the probability of the upcrossing of a random field over a barrier. This paper reviews the mathematical formulation of these two models and their applications to various engineering problems including the strength distributions of quasi-brittle structures, failure statistics of micro-electromechanical systems (MEMS) devices, breakdown statistics of high– k gate dielectrics, and probability distribution of buckling pressure of spherical shells containing random geometric imperfections. In addition, the implications of statistical scaling for the stochastic finite element simulations and the reliability-based structural design are discussed. In particular, the recent development of the size-dependent safety factors is reviewed.

Frequency mismatch analysis of hemispherical shell resonators with both radius and thickness imperfections

The hemispherical shell resonator (HSR) is the core element of the hemispherical resonator gyroscope (HRG), and its frequency mismatch is a key property influencing gyroscope accuracy. Investigating the mechanism of frequency mismatch is vital for improving the quality of HSRs and performance of HRGs. Midsurface radius imperfections and thickness imperfections are two principal causes of frequency mismatch, but their combined effects have rarely been discussed. This paper develops a model to comprehensively analyze the frequency mismatch of HSRs with both radius and thickness imperfections. The model derives a quantitative relation between the frequency mismatch and the two imperfections, and provides principles for evaluating dominant imperfections. To validate the model, we conduct experiments on a batch of 12 HSRs with random geometric imperfections. We apply the model in calculating the theoretical frequency mismatches of these HSRs, and the results agree well with the experimental data. The experiment also confirms that for macro-HSRs, thickness imperfections have a larger impact than radius imperfections, and the frequency mismatch is approximately linear to the fourth thickness harmonic. Our research can be a useful reference for the design and fabrication of HSRs and may open new possibilities for high-precision manufacture of HRGs.

Uncovering the dual role of dimensionless radius in buckling of spherical shells with random geometric imperfections

Localized deformation and randomly shaped imperfections are salient features of buckling-type instabilities in thin-walled load-bearing structures. However, it is generally agreed that their complex interactions in response to mechanical loading are not yet sufficiently understood, as evidenced by buckling-induced catastrophic failures which continue to today. This study investigates how the intimate coupling between localization mechanisms and geometric imperfections combine to determine the statistics of the pressure required to buckle (the illustrative example of) a hemispherical shell. The geometric imperfections, in the form of a surface, are defined by a random field generated over the nominally hemispherical shell geometry, and the probability distribution of the buckling pressure is computed via stochastic finite element analysis. Monte-Carlo simulations are performed for a wide range of the shell's radius to thickness ratio, as well as the correlation length of the spatial distribution of the imperfection. The results show that over this range, the buckling pressure is captured by the Weibull distribution. In addition, the analyses of the deformation patterns observed during the simulations provide insights into the effects of certain characteristic lengths on the local buckling that triggers global instability. In light of the simulation results, a probabilistic model is developed for the statistics of the buckling load that reveals how the dimensionless radius plays a dual role which remained hidden in previous deterministic analyses. The implications of the present model for reliability-based design of shell structures are discussed.

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.

Seismic analyses of single-layer dome structures with random geometrical imperfections under stochastic ground motions

In this paper, seismic analyses of single-layer dome structures are carried out, with a particular emphasis on the incorporation of randomness in both geometrical imperfections and ground motions. First, the paper introduces models for random geometrical imperfections and stochastic ground motions, employing the Stochastic Imperfection Mode Superposition Method (SIMSM) for imperfections and utilizing the Spectral Representation Method (SRM) for ground motions. These models are designed to comprehensively account for the inherent uncertainties in real-world structural behavior. Subsequently, the Probability Density Evolution Method (PDEM) is applied to investigate the probabilistic response and seismic reliability of dome structures, which allows for a detailed examination of key performance indicators, such as displacement, plastic ratio, and strength damage, under the influence of random factors. Both deterministic and stochastic analyses are conducted to gain insights into the impact of randomness in geometrical imperfections and ground motions. The computational results unveil a crucial finding: an unfavorable distribution of imperfections can significantly compromise the dynamic performance and safety of single-layer dome structures. Furthermore, the paper delves into parametric analyses, with a specific focus on imperfection amplitude, the rise-span ratio and roof load as pivotal parameters. These analyses provide valuable insights into the sensitivity of the structural response to variations in these critical design factors.

Creating Geometric Imperfections in Thin-Walled Structures Using Acoustic Excitation

Abstract Thermomechanical buckling of slender and thin-walled structural components happens without warning and can lead to catastrophic failure. Similar phenomena are observed during plasmolysis (contraction of a plant cell’s protoplast) and rupture of viral capsids. Analytical formulas derived from stability analyses of elastic plates and shells that do not account for the effects of random geometric imperfections introduced during the manufacturing process or biological growth may vastly over-estimate buckling capacity. To ensure structural safety, the formulas must therefore be combined with empirical data to define “knockdown factors” which are in turn used to establish safety factors. Towards improved understanding of the role of imperfections on mechanical response, ingenious methods have been used to fabricate and test near-perfectly hemispherical shells and those containing dimple-like defects. However, a method of inducing imperfections in the form of randomly shaped surfaces remains elusive. We introduce a protocol for realizing such imperfect shells and measuring the pressure required to buckle them. Silicone is poured onto an elastomeric mold under an acoustic excitation, which can be either random sound, or if desired the same as the modal frequency of the mold. Illustrative micro-computed-tomography images and buckling pressure experiments of a nearly perfect shell and an imperfect one show that the method is effective in introducing randomly shaped imperfections of significant magnitudes. This proof-of-concept study demonstrates that the experimental results when combined with computational simulations can lead to improved understanding of stochastic buckling phenomena.

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.

Reliable crack detection in a rotor system with uncertainties via advanced simulation models based on kriging and Polynomial Chaos Expansion

In this study the authors propose to take into account the nonlinear effects induced by the presence of a transverse crack to carry out vibratory monitoring and detect transverse cracks in rotating systems subject to model uncertainties. More precisely, we focus more particularly on the global complexity of the nonlinear dynamic behaviour of cracked rotors and the evolution of their harmonic components as a function of the parameters of a transverse breathing crack (its position and depth) when numerous uncertainties are considered. These random uncertainties correspond to random geometric imperfections (two disc thicknesses), random material properties (Young modulus and material density) and boundary conditions uncertainty (two bearing stiffnesses). The objective of the present work is to identify robust indicators capable of determining the presence of a crack and its status even though numerous uncertainties are present.To conduct such a study, an advanced surrogate modelling technique based on kriging and Polynomial Chaos Expansion (PCE) is proposed for the prediction of both the critical speeds and the harmonic components n× during passage through sub-critical resonances. An extensive study to ensure the validation of the surrogate models and a relevant choice of both the parametric and random Design of Experiments (i.e. kriging DoE and PCE DoE) is proposed. The proposed methodology is applied on a flexible rotor with a transverse breathing crack and subjected to random geometric imperfections and fluctuations in material properties of the rotor system.

Advanced computational technique based on kriging and Polynomial Chaos Expansion for structural stability of mechanical systems with uncertainties

In this paper, a numerical strategy based on the combination of the kriging approach and the Polynomial Chaos Expansion (PCE) is proposed for the prediction of buckling loads due to random geometric imperfections and fluctuations in material properties of a mechanical system. The original computational approach is applied on a beam simply supported at both ends by rigid supports and by one punctual spring whose location and stiffness vary. The beam is subjected to a deterministic axial compression load. The PCE-kriging meta-modelling approach is employed to efficiently perform a parametric analysis with random geometrical and material properties. The approach proved to be computationally efficient in terms of number of model evaluations and in terms of computational time to predict accurately the buckling loads of a beam system. It is demonstrated that the buckling loads are substantially impacted not only by both the location and the stiffness of the spring, but also by the random parameters.

Probabilistic analysis of secant piles with random geometric imperfections

Probabilistic analysis of secant piles with random geometric imperfections

Seismic performance of imperfect unanchored tanks

Liquid storage tanks play a significant role in storing hazardous liquids in process industries. Hence, their proper seismic performance is considerably important to prevent the catastrophic consequences of tank failure. This study focuses on seismic safety assessment of oil storage tanks. To this end, four unanchored tanks of various height-to-diameter ratios (H/D) were considered and their elastic and inelastic buckling failures were studied, as well as their uplifting behaviour. To this end, incremental dynamic analysis was employed in an Abaqus platform using the lateral seismic excitations. Different patterns of random geometric imperfections of various amplitudes were considered in the tank models. The Budiansky–Roth procedure was followed to calculate the buckling capacities of tank shells. The results of this study revealed that the imperfection has a small but notable effect on buckling capacity of tank shells, but its effect on tank uplift is negligible.

Big influence of small random imperfections in origami-based metamaterials

Origami structures demonstrate great theoretical potential for creating metamaterials with exotic properties. However, there is a lack of understanding of how imperfections influence the mechanical behaviour of origami-based metamaterials, which, in practice, are inevitable. For conventional materials, imperfection plays a profound role in shaping their behaviour. Thus, this paper investigates the influence of small random geometric imperfections on the nonlinear compressive response of the representative Miura-ori, which serves as the basic pattern for many metamaterial designs. Experiments and numerical simulations are used to demonstrate quantitatively how geometric imperfections hinder the foldability of the Miura-ori, but on the other hand, increase its compressive stiffness. This leads to the discovery that the residual of an origami foldability constraint, given by the Kawasaki theorem, correlates with the increase of stiffness of imperfect origami-based metamaterials. This observation might be generalizable to other flat-foldable patterns, in which we address deviations from the zero residual of the perfect pattern; and to non-flat-foldable patterns, in which we would address deviations from a finite residual.

Effect of random geometric imperfections on the water-tightness of diaphragm wall

Diaphragm walls are commonly used as cut-off walls to prevent water from entering dewatered excavation pits, landfills and downstream water. However, due to the geometrical imperfections such as the positioning error of grab during trenching, small seepage passages through the diaphragm wall may occasionally occur, which will undermine its water-tightness and lead to significant leakages. This study therefore used a reliability-based approach to quantitatively estimate the effect of geometrical imperfections on the water-tightness of diaphragm walls using a three-dimensional discretised algorithm (TDA). This algorithm enables a rigorous examination of penetration of seepage passages and it further quantitatively estimates the flow rate through the diaphragm wall under certain circumstances. Finally, detailed design procedures and design graphs were provided, which may facilitate the choice of a rational grab model and construction parameters in the design of diaphragm walls.

Modelling of Random Geometrical Imperfections and Reliability Calculations for Thin Cylindrical Shell Subjected to Lateral Pressure

It is well known that it is very difficult to manufacture perfect thin cylindrical shell. Initial geometrical imperfections existing in the shell structure is one of the main determining factor for load bearing capacity of thin cylindrical shell under uniform lateral pressure. As these imperfections are random, the strength of same size cylindrical shell will also random and a statistical method can be preferred to find the allowable load of these shell structures and therefore a In this work the cylindrical shell of size R/t = 228, L/R = 2 and t=1mm is taken for study. The random geometrical imperfections are modeled by linearly adding the first 10 eigen mode shapes using 2kfullfactorial design matrix of DoE. By adopting this method 1024 FE random imperfect cylindrical shell models are generated with tolerance limit of ± 1 mm. Nonlinear static FE analysis of ANSYS is used to find the buckling strength of these 1024 models. FE results of 1024 models are used to predict the reliability based on MVFOSM method.

Probabilistic investigation on defective jet-grouted cut-off wall with random geometric imperfections

This study examines the effect of geometric imperfections arising from variations in both the diameter and orientation of jet-grouted columns on the water-tightness of a jet-grouted cut-off wall. A three-dimensional discretised algorithm is used to detect and measure the discontinuities in the wall. A statistical evaluation of the gross flow rate through the cut-off wall with random flow-through pathways is carried out using Monte Carlo simulations. Based on the statistical results, a dimensionless design chart is proposed which can help engineers to devise safe and economical wall designs through trade-offs among various parameters such as target diameter, column spacing and number of rows.

On the robustness of spider capture silk’s adhesion

Through evolution, spider orb webs have acquired an outstanding ability to capture flying preys. In this work, we aim to understand the robustness of spider adhesion by analyzing the effects of randomness in the adhesion strengths and inter-spacings of glue droplets along a spider orb web capture silk on its load-bearing and energy absorption capacities when adhered on a substrate. It is found that the spider silk exhibits a high level of robustness in the sense that both its total adhesion strength and energy absorption capacity are insensitive to random mechanical and geometric imperfections. Specifically, randomness in the spatial distributions of glue droplets is shown not to deteriorate the adhesion property, rather it can slightly enhance the load-bearing and energy absorption capacities of the silk. This finding may help us understand the principle of spider adhesion and guide the design of spider-inspired adhesion and capture devices.

Stochastic Modeling of Geometric Imperfections in Aboveground Storage Tanks for Probabilistic Buckling Capacity Estimation

AbstractThis study develops a methodology to model random geometric imperfections in welded cylindrical aboveground storage tanks (ASTs). Precise modeling of the imperfections in ASTs is important for quantitative assessment of buckling strength of ASTs under various external loads, such as wind and storm surge, because buckling phenomenon is highly sensitive to stochastic imperfections. Even though buckling of ASTs due to extreme winds during hurricanes has been reported as a common failure mode, existing literature lacks models for random geometric imperfections to facilitate probabilistic buckling capacity estimation of ASTs under such external loadings. Current studies on buckling behavior of ASTs use deterministic imperfection models, neglecting the inherent uncertainties present in the imperfections and the response of ASTs. Usually, studies use eigen or buckling mode shapes to depict imperfections which are most detrimental to the buckling strength of ASTs; furthermore, the amplitude of these imper...

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

Steel structural members and frames always indicate imperfections to various degrees. These include initial out-of-straightness and initial out-of-plumb due to manufacturing and erection tolerances. In general, the shape and magnitude of geometric imperfections may have a significant influence on the response of a structure, and hence need to be modelled accurately when determining the load carrying capacity of a steel frame by advanced structural analysis. Most conveniently, geometric imperfections can be introduced in structural models as scaled eigenmodes obtained a priori from an elastic buckling analysis. However, it remains unanswered how many eigenmodes need to be incorporated and how to choose the scaling factors of each mode.This paper presents a study of how the strength of steel frames varies with the number and magnitudes of eigenmodes. Frames with random geometric imperfections are produced using the statistics of measurements of out-of-plumb and member imperfections, and analysed using advanced geometric and material nonlinear analysis. The imperfections are then resolved into eigenmodes and a second set of advanced analysis is carried out using a finite number of modes to represent the imperfections. Conclusions are drawn about the appropriate number and magnitudes of eigenmodes to use in advanced structural analyses of steel frames.

Experimental determination of critical loads in thin-walled bars with Z-section subjected to warping torsion

Critical loads were determined experimentally from the condition of the local buckling of thin-walled bars with Z-section subjected to warping torsion. The experimental investigations were carried out using simply supported models, loaded with a concentrated torsional moment at the mid-span. A method of determining the so-called “local ordered deflection interval” was developed. In this interval, the modeled deflection of the component plates (walls) of a thin-walled bar with random wall geometrical imperfections is compliant with the local buckling mode. The “local ordered deflection interval” makes it possible to adjust the known experimental methods so that they could be used to determine critical torsional loads and local critical bimoments. Experimental investigations showed the occurrence of two local critical bimoments in bars with Z-section. The bimoments are different in their absolute values, depending on the sense (sign) of the torsional load. Experimentally determined critical loads were compared with theoretical results.

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.