Combined use of SHMS and finite element strain data for assessing the fatigue reliability index of girder components in long-span cable-stayed bridge

The design of long-span cable-stayed bridge involves a large number of loads, geometric and material parameters, all of which can interact in a random fashion. It is desirable to have a total measure of the operational reliability and safety of the structural components. Based on the box-girder component strain history data of the Runyang Cable-stayed Bridge (RYCB) in China, a computer algorithm is developed to evaluate the fatigue damage that is assumed to occur in increments, according to a lognormal distribution. The corresponding probability density function is then found to obtain a fatigue reliability index β for ranking the integrity of the girders. Emphases are placed on the overall scheme of structural reliability evaluation such that the different fatigue damage criteria, probability density functions, and strain measurement techniques can be made. Finite element calculations are also used to provide strain data at locations that are not conducive for installing strain gauges, while the compatibility of measured and calculated data is made empirically. Each of the subroutine in the fatigue reliability algorithm can be altered for improvement. The flexibility allows up-dating the prediction as the monitored strains are changed by the environmental conditions. Preliminary results are first obtained to test the selected damage increments in relation to the probability function and fatigue damage criterion. Particular attention has been given to test the sensitivity of the combined governing parameters. The highly non-linear behavior of numerical calculations related to fatigue failure necessitates an in-depth understanding of the physical model. The condition under which fatigue damage accumulation is needed in contrast to the linear sum of fatigue cycles will be left for the future. Justification should be given to include the more complex issues. The aim here is to seek a simple, and yet reliable index that can account for the fatigue damage of box-girder of long cable-stayed bridges.     

Fatigue criteria for integrity assessment of long-span steel bridge with health monitoring

A structural health assessment (SHA) methodology is developed using data acquired from structural health monitoring (SHM) system installed on long-span bridges. A set of fatigue criteria has been proposed for pre-determining the global state of the bridge structure failure due to fatigue. This involves finding the threshold of fatigue initiation, below which the rate of fatigue damage may be undetectable under current technology or it is economically unfeasible. The state-of-art for large structures corresponds to the initiation of macro-cracks caused by the accumulation of damage generated by actual service loads for the case of bridges. In what follows, consideration is given to developing fatigue crack growth criterion based on the concept of the continuum damage mechanics (CDM). Fatigue accumulative is included in the model where a fatigue limit for multi-axial stress state is considered. The proposed criterion advocates the evolution of micro-crack growth up to the stage of macro-crack formulation. Considered are the loading histories that correspond to normal traffic loading for highways and railways, incidental or accident loadings such as those caused by typhoons and effective environmental loadings. The potential sites of damage are determined are discussed. The proposed criterion is applied to analyze the fatigue damage of the Tsing Ma Bridge with online strain history data acquired by the SHM system that is permanently installed in the bridge.     

Monitoring of the Alamillo cable-stayed bridge during construction

The Alamillo Bridge is one of the long-span bridges crossing the Guadalquivir River. It was built on the occasion of Expo '92 in 1992 in Sevilla, Spain. The bridge is a cable-stayed structure spanning 200 m without any intermediate supports. Its originality is the lack of back stays and the balancing of the front stays through the backward inclination of a massive pylon. This paper shows the importance of experimental in situ techniques when applied to unconventional civil engineering structures and how—with the help of an important amount of accurate instrumentation, monitoring the most important experimental variables—it was possible to build the bridge correctly, safely, and on schedule.     

Fatigue crack growth of cable steel wires in a suspension bridge: Multiscaling and mesoscopic fracture mechanics

Intrinsically, fatigue failure problem is a typical multiscale problem because a fatigue failure process deals with the fatigue crack growth from microscale to macroscale that passes two different scales. Both the microscopic and macroscopic effects in geometry and material property would affect the fatigue behaviors of structural components. Classical continuum mechanics has inability to treat such a multiscale problem since it excludes the scale effect from the beginning by introducing the continuity and homogeneity assumptions which blot out the discontinuity and inhomogeneity of materials at the microscopic scale. The main obstacle here is the link between the microscopic and macroscopic scale. It has to divide a continuous fatigue process into two parts which are analyzed respectively by different approaches. The first is so called as the fatigue crack initiation period and the second as the fatigue crack propagation period. Now the problem can be solved by application of the mesoscopic fracture mechanics theories developed in the recent years which focus on the link between different scales such as nano-, micro- and macro-scale.On the physical background of the problem, a restraining stress zone that can describe the material damaging process from micro to macro is then introduced and a macro/micro dual scale edge crack model is thus established. The expression of the macro/micro dual scale strain energy density factor is obtained which serves as a governing quantity for the fatigue crack growth. A multiscaling formulation for the fatigue crack growth is systematically developed. This is a main contribution to the fundamental theories for fatigue problem in this work. There prevail three basic parameters μ^∗, σ^∗ and d^∗ in the proposed approach. They can take both the microscopic and macroscopic factors in geometry and material property into account. Note that μ^∗, σ^∗ and d^∗ stand respectively for the ratio of microscopic to macroscopic shear modulus, the ratio of restraining stress to applied stress and the ratio of microvoid size ahead of crack tip to the characteristic length of material microstructure.To illustrate the proposed multiscale approach, Hangzhou Jiangdong Bridge is selected to perform the numerical computations. The bridge locates at Hangzhou, the capital of Zhejiang Province of China. It is a self-anchored suspension bridge on the Qiantang River. The cables are made of 109 parallel steel wires in the diameter of 7 mm. Cable forces are calculated by finite element method in the service period with and without traffic load. Two parameters α and β are introduced to account for the additional tightening and loosening effects of cables in two different ways. The fatigue crack growth rate coefficient C₀ is determined from the fatigue experimental result. It can be concluded from numerical results that the size of initial microscopic defects is a dominant factor for the fatigue life of steel wires. In general, the tightening effect of cables would decrease the fatigue life while the loosening effect would impede the fatigue crack growth. However, the result can be reversed in some particular conditions. Moreover, the different evolution modes of three basic parameters μ^∗, σ^∗ and d^∗ actually have the different influences on the fatigue crack growth behavior of steel wires. Finally the methodology developed in this work can apply to all cracking-induced failure problems of polycrystal materials, not only fatigue, but also creep rupture and cracking under both static and dynamic load and so on.     

Plasticity induced fatigue crack growth retardation model for steel elements reinforced by composite patch

Fatigue tests on notched steel plates reinforced by composite patch showed that the application of carbon fiber reinforced polymers (CFRP) strips with pretension of the overlays prior to bonding. This resulted in a significant amount of additional fatigue life. In particular, the pre-tension produces a compressive field in the steel plate which reduces the stress ratio that enhances crack growth retardation. The fatigue crack propagation rate is postulated to be a function of the effective strain energy density factor range. Fatigue crack growth data showed that standard crack growth retardation model cannot be used to evaluate the minimum effective stress. Hence, an ad hoc plasticity model is introduced and validated using experimental results. The proposed technique is an extension of the well know Newman’s model. The bridging effect due to the reinforcing strips is analytically modeled in order to estimate the reduction of crack opening displacement and finally the magnification of the crack growth retardation. Numerical and experimental results match well and show a significant influence of the pre-tension level on the expected fatigue crack growth rate of a reinforced steel plate.     

Accumulative damage near crack tip for welded bridge members: fatigue life determination

The behavior of crack growth for the fatigue damage accumulation near tip where damage is most severe is analyzed. Fatigue life is assessed for the welded members of bridges under traffic loading. Two parts are considered. They consist of the development of a fatigue damage accumulation model for welded bridge members and a method for calculating the stress intensity factor that is needed for evaluating the fatigue life of welded bridge members with cracks. Based on the concept of continuum damage accumulation and fatigue and fatigue crack growth relations, results are obtained to describe the relationship between the cracking count rate and the effective stress intensity factor. Crack growth and fatigue life are found for two types of welded members assisted by using fatigue experimental results. The stress intensity factors are modified by correcting for the geometric shape of the welded members in order to reflect the influence of the weldment and geometry. This is accomplished via the stress intensity factor. The calculated and measured fatigue lives were generally in good agreement for the initial cracking conditions of two types of welded members widely used in steel bridges.     

Fatigue based three-dimensional structural design optimisation studies implementing the generalised Frost–Dugdale crack growth law

Experimental studies of fatigue crack growth in aluminium alloys have shown that, at the low-to-mid stress intensity factor range, there is often a log-linear relationship between the crack length and the fatigue life. These observations have led to the development of the generalised Frost–Dugdale crack growth law, which allowed the accurate prediction of fatigue crack growth from Region I. For this research paper the ‘generalised Frost–Dugdale’ law was used to perform an optimisation study of 7050-T7451 Aluminium structures. The structural optimisation procedure proposed integrates geometrical modelling, structural analysis and optimization into one complete and automated computer-aided design process. The results from the structural optimisation study compared the ‘generalised Frost–Dugdale’ law and the traditional Paris law. Gradient-less, gradient-based optimisation algorithm and an enumeration scheme were considered in this investigation. The enumeration scheme takes advantage of a cluster computer architecture which enables a visualisation of the solution space allowing verification and validation of the optimisation algorithm. The results indicated that the optimal geometrical shape and predicted fatigue life depended on the crack sizes, structural geometry, boundary conditions and fatigue crack growth law. As a result, this procedure illustrates that for the design of light weight structures, a fatigue based optimisation used in conjunction with visualisation of the solution space may provide a viable design methodology. The importance of non-destructive inspection (NDI) and its role in determining optimal structural geometries is also revealed. Furthermore, the possibility of the application of the generalised Frost–Dugdale model in design optimisation has been demonstrated. This procedure has the potential to be applied to structures with complex structural configurations taking into account crack propagation in Region I.     

A boundary element modeling of fatigue crack growth in a plane elastic plate

This paper presents an extension of a boundary element method to fatigue growth analysis of mixed-mode cracked plane elastic bodies. The method consists of the non-singular displacement discontinuity element presented by Crouch and Starfield and the crack-tip displacement discontinuity element due to the author. In the boundary element implementation the left or the right crack-tip element is placed locally at the corresponding left or right crack tip on top of non-singular displacement discontinuity elements that cover the entire crack surface and the other boundaries. Crack growth is simulated with an incremental crack extension analysis based on the modified maximum strain energy density criterion. In numerical simulation, for each increment of crack extension, remeshing of existing boundaries is not required because of an intrinsic feature of the boundary element method. Crack growth is simulated by adding new boundary elements on the incremental crack extension to the previous crack boundaries. At the same time, the element characters of some related elements are adjusted according to the manner in which the boundary element method is implemented. Some numerical results of fatigue growth in a plane elastic plate with a center-inclined crack under uniaxial cyclic loading are given.     

Fatigue life prediction of out-of-plane gusset welded joints using strain energy density factor approach

Fatigue growth behavior of out-of-plane gusset welded joints is studied using the strain energy density factor approach. Fatigue tests on two types of specimens with curvatures of ρ = 0 and ρ = 30 were performed in order to estimate fatigue strength under tension. Fatigue crack growth analysis is carried out to show the effects of initial crack shape, initial crack length and stress ratio. Fatigue crack growth parameters were obtained from crack growth curves assuming constant crack shapes. The results of analysis for the assumed crack shapes agreed well with the experimental data. Fatigue propagation life of the ρ = 30 specimen was larger than that of the ρ = 0 specimen.     

Fatigue life prediction of in-plane gusset welded joints using strain energy density factor approach

Fatigue crack growth behavior of in-plane gusset welded joints is studied using the strain energy density factor approach. Fatigue tests were performed in order to estimate fatigue strength under tension. Fatigue crack growth analysis was carried out to show the effects of the initial crack shape, the initial crack length, and the stress ratio on the crack types of in-plane gusset welded joints. The assumed crack types were edge crack, semi-elliptical crack, and corner crack. Fatigue crack growth parameters were obtained from crack growth curves assuming constant crack shapes for the given crack types. The results of analysis for the assumed crack types agreed well with the experimental data. The fatigue life did not change as initial crack shape varied for a given initial crack length.     

Principle of least variance for dual scale reliability of structural systems

A R-integral is defined to account for the evolution of the root functions from Ideomechanics. They can be identified with, though not limited to, the fatigue crack length or velocity. The choice was dictated by the available validated data for relating accelerated testing to real time life expectancy. The key issue is to show that there exists a time range of high reliability for the crack length and velocity that correspond to the least variance of the time dependent R-integrals. Excluded from the high reliability time range are the initial time span where the lower scale defects are predominant and the time when the macrocrack approaches instability at relatively high velocity. What remains is the time span for micro-macro cracking. The linear sum (ls) and root mean square (rms) average are used to delineate two different types of variance. The former yields a higher reliability in comparison with that for the latter. The results support the scale range established empirically by in-service health monitoring for the crack length and velocity. The principle of least variance can be extended to multiscale reliability analysis and assessment for multi-component and multi-function systems.     

Fatigue growth prediction of elliptical cracks in welded joint structure: Hybrid and energy density approach

A hybrid weight function approach (HWFM) is presented for the fatigue life prediction of infinite body and welded joint structure containing elliptical cracks. A self-containing computer code has been developed for this purpose. Numerical computations were first conducted on cracked infinite body showing a physical fact, that the elliptical shape of the crack becomes circular during its evolution. The prediction of the fatigue crack growth shows that the present results are in perfect concordance with those reported in the literature. Then, numerical tests were carried out on two types of specimens of welded joint structure. The present results were compared to the experimental and predicted ones of other authors, demonstrating that the hybridization method is a powerful numerical technique, and that the SEDF approach (using the Sih’s law) is more valid for the critical cases of welded joints than the SIF approach (using the Paris law). A parametric study has been conducted on the stress ratio “R” showing that the fatigue life to failure decreases with the increase of “R”.     

Features of long-term health monitored strains of a bridge with wavelet analysis

This paper analyses the five years' monitored strains collected from a long-term health monitoring system installed on a bridge with wavelet transform. In the analysis, the monitored strains are pre-processed, features of the monitored data are summarized briefly. The influences of the base functions on the results of wavelet analysis are studied simultaneously. The results show that the db wavelet is a good mother wavelet function in the analysis, and the order N should be larger than 20, but less than 46 in decomposing the monitored strains of the bridge. According to the strain variation features of concrete bridge, the proper decomposition level is 4 in the wavelet multi-resolution analysis. With the present method, the strains caused by random loads and daily sunlight can be accurately extracted from the monitored strains. The decomposed components of the monitored strains show that the amplitudes of the strains caused by random loads, daily sunlight, and annual temperature effect, are about 5 με, 25 με, and 50 με respectively. The structural response under random load is smaller than the other parts.     

Optimal placement of sensors for sub-surface fatigue crack monitoring

This paper discusses the scattering of stress waves by defects in representative aircraft structures with multi-layered construction and geometry variation. The approaches for determining and enhancing the probability of detection of non-surface-penetrating defects in such structures as well as minimising the contributions of multi-layered construction and geometry variation to false indications are presented. The results demonstrate the importance of selecting the appropriate frequency and location of the sensor in monitoring sub-surface defects on these structures. The findings suggested that a computer solution of the problem may be required to determine the optimal combination of frequency and sensor location. This study suggests the possibility of incorporating structural health monitoring into the design of future structures which will constitute a significant leap in the current knowledge base of structural health monitoring.     

Non-probabilistic reliability method and reliability-based optimal LQR design for vibration control of structures with uncertain-but-bounded parameters

Uncertainty is inherent and unavoidable in almost all engineering systems. It is of essential significance to deal with uncertainties by means of reliability approach and to achieve a reasonable balance between reliability against uncertainties and system performance in the control design of uncertain systems. Nevertheless, reliability methods which can be used directly for analysis and synthesis of active control of structures in the presence of uncertainties remain to be developed, especially in non-probabilistic uncertainty situations. In the present paper, the issue of vibration con- trol of uncertain structures using linear quadratic regulator （LQR） approach is studied from the viewpoint of reliabil- ity. An efficient non-probabilistic robust reliability method for LQR-based static output feedback robust control of un- certain structures is presented by treating bounded uncertain parameters as interval variables. The optimal vibration con- troller design for uncertain structures is carried out by solv- ing a robust reliability-based optimization problem with the objective to minimize the quadratic performance index. The controller obtained may possess optimum performance un- der the condition that the controlled structure is robustly re- liable with respect to admissible uncertainties. The proposed method provides an essential basis for achieving a balance between robustness and performance in controller design ot uncertain structures. The presented formulations are in the framework of linear matrix inequality and can be carried out conveniently. Two numerical examples are provided to illustrate the effectiveness and feasibility of the present method.     

Correlation of crack growth rate and stress ratio for fatigue damage containing very high cycle fatigue regime

A model is proposed to correlate the crack growth rate and stress ratio containing very high cycle fatigue regime. The model is verified by the experimental data in literature. Then a formula is derived for the effect of mean stress on fatigue strength, and it is used to estimate the fatigue strength of a bearing steel in very high cycle fatigue regime at different stress ratios. The estimated results are also compared with those by Goodman formula.     

Crack growth rate and cracking velocity in fatigue for a class of ceramics

Fatigue crack growth rate data and cracking velocity data are studied for a class of ceramics including SiC, TiB₂, Si₃N₄, ZrO₂ and Al₂O₃. Both sine and square wave cyclic loading are combined such that the data could be converted to cracking velocity for a given frequency of cyclic load. An effective stress intensity factor range is defined and used in an relation for computing the crack growth rate and cracking velocity. As for the metal alloys, the data for ceramics also fall into three regimes identified with near-threshold, stable growth and rapid crack extension, except that the slope of the da/dN (the crack growth rate) curves for ceramics are steeper in comparison with that for metals. Reported are the empirical constants in the relations for the crack growth rate and the cracking velocity for a variety of ceramics.