An efficient methodology for fatigue damage assessment of bridge details using modal superposition of stress intensity factors

International Journal of Fatigue 81 (2015) 61–77

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International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue

An efficient methodology for fatigue damage assessment of bridge details using modal superposition of stress intensity factors Carlos Albuquerque a,⇑, António L.L. Silva a,b, Abílio M.P. de Jesus a,b, Rui Calçada a a b

University of Porto – Faculty of Engineering, Rua Dr. Roberto Frias, 4200 465 Porto, Portugal INEGI, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal

a r t i c l e

i n f o

Article history: Received 13 January 2015 Received in revised form 30 June 2015 Accepted 3 July 2015 Available online 23 July 2015 Keywords: Fatigue crack propagation Modal stress intensity factor Computational efficiency Submodeling Railway bridge

a b s t r a c t The objective of this paper is to propose an accurate and computationally efficient method for the fatigue assessment of bridge details, using Fracture Mechanics and crack propagation laws. The proposed workflow benefits from the combination of finite element submodeling techniques and modal superposition method with the new concept of modal stress intensity factors. The new methodology was applied to the fatigue analysis of a complex bridge under real traffic conditions. The simulation of fatigue crack propagation in a critical detail of the structure was achieved with minimal computer resources and within a short time frame. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction The progressive damage of structures as a result of fatigue is a recurring problem in Civil Engineering. In the particular case of composite and steel railway bridges, fatigue is one of the main causes of severe damage [1–4]. The relevance of this topic is reflected by the increasing number of numerical and experimental fatigue damage assessment studies performed on railway bridges [5–12]. In this context, local models for fatigue analysis have been gaining popularity with respect to the classical nominal S–N approaches [13–16]. Fracture Mechanics and local/notch stress/strain-based fatigue approaches are becoming more frequent options to model fatigue crack initiation and propagation in bridge details. Fatigue of steel structures is a localized damage process requiring reliable models to compute local time-history data (e.g. notch stresses and strains, stress intensity factors for fatigue cracks) in order to permit trustworthy fatigue assessments, including residual life assessments. The local nature of the fatigue damage problem raises important challenges concerning the fatigue analysis of large structures using local fatigue models, which is the case of bridges subjected to an important diversity of traffic characteristics. Theoretically, a global computational model of the bridge is required, generally a finite element model. The critical location must be modeled in fine detail. That is sometimes performed directly in the global model or, ⇑ Corresponding author. http://dx.doi.org/10.1016/j.ijfatigue.2015.07.002 0142-1123/Ó 2015 Elsevier Ltd. All rights reserved.

alternatively, in a separate model, using sub-modeling techniques. That level of refinement at the details cannot be extended to the global structure [17]. This kind of modeling requirements corresponds to a multi-scale modeling problem [18], which inevitably leads to heavy computational models, demanding huge computational resources and time for the fatigue analyses. This situation could become impracticable, if a significant diversity of traffic has to be modeled and their dynamic effects to be accounted. The objective of this paper is to propose an accurate and computationally efficient method for the fatigue assessment of bridge details. Fracture Mechanics is adopted to support the numerical simulation of fatigue cracks. The local time-history data needed for the fatigue damage assessment is generated using the new concept of modal stress intensity factors and the modal superposition method. This procedure was presented previously by some of the co-authors of the present work and demonstrated for the computation of stress intensity factors for a simple structural problem of a simply supported beam [19]. In this paper the proposed procedure is applied to a complex problem, the Portuguese Alcácer do Sal Composite Railway Bridge, for which experimental data concerning both traffic and bridge local response are available [20]. A hypothetical scenario of fatigue residual life assessment, based on Fracture Mechanics concepts, is presented, to illustrate the efficiency of the proposed method. Finally, it is demonstrated that the modal stress intensity factors computation can be embedded in a long term monitoring system, allowing the real time assessment of fatigue cracks propagation.

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2. Theoretical background This section presents the theoretical background supporting the methodology proposed in this paper. In Section 2.1, the modal superposition method and its benefits for the dynamic analysis of railway bridges are briefly presented. The adopted fatigue model, which is based on Linear Elastic Fracture Mechanics, is detailed in Section 2.2. Finally, Section 2.3 describes how the modal superposition of stress intensity factors can be performed. 2.1. Dynamic analysis using modal superposition When trains pass over a certain railway bridge, they originate a dynamic response/vibration of the structure. If the traffic loads are known, the dynamic response can be simulated numerically using a finite element model of the bridge. Trains are frequently simulated as moving loads, representing the actions of the axles over the structure. The positions of the axles of the trains, at certain time step, may be computed based on their initial position and on train velocity. Once the axles position is defined, the corresponding axle loads are converted into equivalent vertical nodal forces, applied to the nodes of the modeled rails. The position of the train axles and the corresponding nodal forces need to be updated at each time step. Two common numerical methods employed to perform the dynamic analysis are the direct integration and the modal superposition methods [21]. For finite element models with a high number (N) of degrees-of-freedom, the direct integration method becomes highly time and resources consuming, as it implies solving Eq. (1) directly, without additional assumptions:

€ ðtÞ þ D  uðtÞ _ Mu þ K  uðtÞ ¼ FðtÞ

ð1Þ

In Eq. (1) M, D and K are the mass, damping and stiffness matrices, respectively, and have a N  N dimension, each. FðtÞ is the N  1 vector of the nodal forces acting in the structure for a certain time-step and uðtÞ is the N  1 vector of displacements associated to each degree-of-freedom at the same time step. On the other hand, the modal superposition method is computationally more efficient since the global dynamic response of the structure can be accurately reproduced superimposing the individual modal responses of a limited number (J) of modes of vibration. This approach can be applied to the case of flexible structures with elastic behaviour, such as composite railway bridges. Using modal superposition, the full N  N system of simultaneous equations is converted into N uncoupled single-variable equations (Eq. (2)) that can be solved independently:

Y€ j ðtÞ þ 2xj  nj  Y_ j ðtÞ þ x2j  Y j ðtÞ ¼ f j ðtÞ

current study, the proposed methodology could also be extended to elastic local/notch approaches which would allow to model fatigue crack initiation. Moreover, in the present investigation the Linear Elastic Fracture Mechanics (LEFM) principles were followed since the plasticity effects at crack tip are negligible. In the linear elastic domain, the stress fields around the crack tip are expressed by the stress intensity factor (SIF), K [22]. The fatigue crack propagation rate, on the other hand, is related with the range of K, which is commonly represented as DK. A variety of fatigue crack propagation laws are available in the literature, with more or less complexity depending on the number of variables considered [23]. In the current work, the well-known Paris Law (Eq. (3)) is adopted [24]:

da ¼ C  DK m dN

ð3Þ

where da/dN is the fatigue crack propagation rate and C and m are material-dependent constants. In spite of its known limitations, Paris Law has been widely used (e.g. in design codes) due to its simplicity. It should be stressed that this relation is only valid for the fatigue crack propagation in Region II (see Fig. 1). Near threshold (Region I) and near unstable crack propagation (Region III) regimes are not accounted by the Paris relation. Complex structures, such as railway bridges, under complex loading, such as railway traffic, can be subjected to mixed mode (I + II + III) crack growth conditions. Nevertheless, a preliminary analysis performed in the detail of the case study used in this paper showed that, in this particular case, Mode III has a residual contribution to the fatigue crack propagation. Therefore, Mode III was disregarded and mixed mode (I + II) crack propagation was assumed in the current work. Mode I (opening mode) is originated by stresses normal to the crack plane (see Fig. 2(a)). Mode 2 (shear or sliding mode) results from in-plane shear with the displacement between the crack faces being in the plane of the crack and perpendicular to the crack front (see Fig. 2(b)). In the case of mixed mode fatigue crack propagation, the Paris relation may be applied with an equivalent stress intensity factor range. For the computation of the equivalent stress intensity factor, the relation proposed by Tanaka was selected [25]:

K eq ðtÞ ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 K 4I ðtÞ þ 8  K 4II ðtÞ

where KI is the stress intensity factor component originated by Mode I and KII is the stress intensity factor component originated by Mode II loading conditions.

ð2Þ

where Y j is the modal coordinate, xj is the natural frequency, nj is the modal damping ratio and f j represents the modal forces, for the jth mode of vibration. The efficiency of the modal superposition technique arises not only from the uncoupling of the system of equations but also from the fact that the total number of modes of vibration needed to accurately reproduce the dynamic response of the flexible structure, J, is usually much lower than the total number of degrees of freedom (J  N). 2.2. Fatigue model The proposed methodology is applied with a fatigue model in order to track the evolution of the damage at the investigated detail (welded detail). Since welded joints can present initial cracks, their fatigue behaviour is usually well characterized by Fracture Mechanics principles [16]. Therefore, the initiation phase is disregarded in this paper. Nevertheless, in the context of the

ð4Þ

Fig. 1. Typical fatigue crack growth behaviour.

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contains a crack of dimension an, the corresponding stress intensity factor, Ksta, through the crack front, is:

K sta ðtÞ ¼ C n

pffiffiffiffiffiffiffiffi pan  rsta

ð9Þ

If the same structure is subjected to a dynamic loading, the corresponding stress field, variable in time, is rdyn(t). Assuming that during each dynamic loading event the crack remains with approximately constant dimensions, an, then:

K dyn ðtÞ ¼ C n

pffiffiffiffiffiffiffiffi pan  rdyn ðtÞ

ð10Þ

Considering the contribution of both the static and dynamic loadings and considering that the static load is usually constant over time, the total stress intensity factor at the crack front becomes:

K total ðtÞ ¼ K sta þ K dyn ðtÞ

ð11Þ

K⁄total

Nevertheless, is not yet the final total stress intensity factor, as it requires a ‘‘signal’’ correction to be performed. This correction is detailed in later paragraphs. By applying the modal superposition method, rdyn(t) can be expressed as:

rdyn ðtÞ ¼

X

rj  Y j ðtÞ

ð12Þ

j

Fig. 2. Crack loading modes intervening in mixed mode (I + II) crack propagation. (a) Mode I (opening mode); (b) Mode II (sliding mode).

In Eq. (12), the j subscript refers to the number of each mode of vibration, rj is the nominal stress in the jth mode shape and Yj(t) is the modal coordinate of the jth mode of vibration [21]. Replacing Eq. (12) into Eq. (10), the SIF due to the dynamic loading becomes:

K dyn ðtÞ ¼ C n

X pffiffiffiffiffiffiffiffi X pffiffiffiffiffiffiffiffi X pan  rj  Y j ðtÞ ¼ C n pan  rj  Y j ðtÞ ¼ K j  Y j ðtÞ j

For a variable amplitude loading, the rainflow method [26] can be applied in order to extract the histogram of equivalent stress intensity factor ranges (DKeq,i, ni). The fatigue crack propagation corresponding to that histogram can be computed by the direct integration of the Paris Law:

Da ¼

X ni  C  DK m eq;i

ð5Þ

i

j

j

ð13Þ where Kj is the modal stress intensity factor, or in other words, the stress intensity factor obtained with the configuration of the jth mode shape. Finally, the total stress intensity factor can be expressed as:

K total ðtÞ ¼ K sta þ

X K j  Y j ðtÞ

ð14Þ

j

In order to reproduce the crack propagation path, a crack branching criterion was adopted. The first step consisted in the computation of the kink angle time history, hðtÞ, using the Maximum Tangential Stress (MTS) criterion [27]:

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 0 2 3  K ðtÞ þ K 4I ðtÞ þ 8  K 2I ðtÞ  K 2II ðtÞ II A hðtÞ ¼ cos1 @ K 2I ðtÞ þ 9  K 2II ðtÞ

ð6Þ

For an histogram of stress intensity factors, e.g. corresponding to the passage of a train, an equivalent kink angle is computed using the weighted average function described by Ref. [28]:

h ¼

Pn

da i¼1 dN ðDK eq;i Þ  hi ðK I ; K II Þ Pn da i¼1 dN ðDK eq;i Þ

ð7Þ

As detailed in Ref. [19], the computed K total ðtÞ can have a negative sign during some periods of time. In the case of Mode I (opening mode) that situation would correspond to the crack faces overlapping each other due to compression, which is not physically acceptable. In reality, crack faces do not interpenetrate. Instead, as crack closes, the minimum allowable Ktotal is zero, which corresponds to a completely closed crack. Therefore, the final expression for Mode I stress intensity factor computation becomes:

X X 8 K I;sta þ K I;j  Y j ðtÞ ( K I;sta þ K I;j  Y j ðtÞ P 0 > > < j j X K I ðtÞ ¼ > 0 ( K I;sta þ K I;j  Y j ðtÞ < 0 > : j

If mixed mode crack propagation is assumed, besides KI also KII needs to be computed. That is done, in a similar way, by applying Eq. (16):

2.3. Modal superposition of stress intensity factors

K II ðtÞ ¼ K II;sta þ As described in Ref. [19], the utilization of the modal superposition method allows the computation of stress intensity factor time histories, K total ðtÞ, with a low computational cost. K can be defined, in general terms, as:

KðtÞ ¼ C

pffiffiffiffiffiffi pa  r

ð15Þ

ð8Þ

where C is a function of the crack dimensions and of the geometry of the structure, r is the remote stress acting on the detail and a is the crack dimension. Static loading acting on the structure (e.g. self-weight) induces a certain stress state on it, rsta. If the structure

X K II;j  Y j ðtÞ

ð16Þ

j

In the case of Mode II, negative values of stress intensity factor are allowed, as they do not represent overlapping of crack faces. On the contrary, symmetrical values of KII correspond to opposite directions of relative movement between the faces of the crack. As can be observed, in order to obtain the time histories of stress intensity factors (Mode I and/or Mode II), the requirement is to compute the modal stress intensity factors for an adequate number of vibration modes, Kj, plus the stress intensity factor regarding to the static load (e.g. self-weight), Ksta.

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3. Proposed workflow for residual fatigue life assessment of bridge details Based on the fatigue model and on the methodology described in previous section, a workflow was developed, in order to assess the fatigue crack propagation behaviour in critical details of bridges. The workflow is divided in two main steps. The first step (Fig. 3) consists in some pre-processing and requires the following inputs:  global numerical model of the bridge;  numerical model of the detail(s) under analysis;  traffic information, such as axle loads, axle’s spacing and trains’ speed. The traffic scenario may come from monitoring systems or from other sources such as standards. The modal analysis performed on the global numerical model of the structure allows the computation of the modal displacement fields, Uj , and dynamic properties, xj and mj. The modal damping ratios, nj, can be estimated based on standards or field measurements. On the other hand, the displacement field originated by the static loading, Usta , is obtained after a static analysis of the same model. The above mentioned displacement fields are then extrapolated to the boundary nodes of the numerical model of the detail, through a submodeling process. The boundary nodal displacements associated to the static loading are stored in a file designated BDCOsta (BDCO is a short form standing for ‘‘boundary conditions’’). Additionally, the displacements associated to each jth mode of vibration are stored in the BDCOj file. Finally, the traffic data available in the monitoring system database (trains’ speed, axle loads and axles spacing) is combined with the dynamic properties (Uj , nj, xj and mj) of the global model in order to obtain the modal coordinates time histories, Y j ðtÞ. That is achieved by solving Eq. (2) for each traffic event and for each mode of vibration. Therefore, the outputs of the first step of the workflow are:  the time history of the modal coordinates for each mode of vibration of the structure and for each traffic event considered;

 the nodal displacements to be applied to the boundaries of the local model of the detail in order to replicate the static and modal displacement fields. One of the major advantages of the proposed procedure is that these outputs only need to be computed once. They then become the only inputs needed for the second step of the workflow (see Fig. 4), which consists of the crack propagation simulation. The simulation starts with an initial crack in the local model of the detail. Applying the previously stored boundary conditions (BDCOsta and BDCOj), the static and modal stress intensity factors, Ksta and Kj are computed, using an appropriate numerical method (such as virtual crack closure technique (VCCT) [29] or Displacement Extrapolation (DE) [30]). For each train stored in the monitoring system database the corresponding modal coordinates are loaded allowing to compute KI(t) and KII(t), by application of Eqs. (15) and (16), respectively. Equivalent stress intensity factor, Keq(t) (Eq. (4)), and the kink angle h (Eq. (6)) are subsequently computed. At this point, if the maximum observed value of Keq(t) exceeds the material toughness, KIc, that means the crack would propagate in unstable way. Therefore the simulation stops and current crack length is considered the final crack length before failure of the detail. Otherwise, if the maximum observed value of Keq(t) does not exceed the adopted material toughness, the crack propagation associated to the train passage is computed. The rainflow method is applied to the Keq(t) time history, to obtain the histogram of stress intensity factor ranges. Then, Eqs. (5) and (7) are applied to compute the fatigue crack growth length and direction corresponding to that train passage, Da. The contribution of all trains is summed up to obtain a final crack increment, Dan. For computational reasons, it is useful to adopt constant crack length increments, Dainc, between iterations. In that case, the crack increment corresponding to a loading block needs to be scaled by Bn = Dainc/Dan, where Bn represents the number of loading blocks needed, in order to achieve the Dainc increment. After computing the crack increment for all the trains in the monitoring system database, the total crack length and direction is updated in the local model and the process is repeated. In the next section, the application of the proposed workflow to a case study is described.

Fig. 3. Workflow 1st step: pre-processing of the input data.

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Fig. 4. Workflow 2nd step: crack propagation simulation.

4. Application of the proposed procedure to a case study 4.1. The bridge of the new railway crossing of river Sado The case study used to employ and validate the proposed methodology is the bridge of the new railway crossing of river Sado, which is located in the Lisboa–Algarve railway connection, in Portugal (Fig. 5). This composite bowstring bridge has 3 continuous spans (160 m each) and a total length of 480 m. Each span is suspended by an arch made of steel, through 18 tension rods (Fig. 6). The deck, which is prepared to host two tracks, is formed by a concrete slab laid over a trapezoidal steel box (Fig. 7). At each tension rod-to-deck connection, the deck has a steel diaphragm and two diagonal strings that transfer the suspension loads from the tension rods to the deck (Fig. 7). Diagonals are 600 mm wide and have a thickness of 35 mm.

The application of fatigue assessment methods present in the Eurocode 3 [35] are challenging for a complex structure, such as the present bridge, and complex loading histories, in particular if we are interested in a damage tolerant assessment method. The standard global S–N fatigue assessment procedures do not allow a clear picture of the actual fatigue damage evolution (e.g. crack size and path) in critical bridges details which difficult residual life calculations. Also the details categorization may be not simple and consensual when we are dealing with complex welded details. Some of the bridge’s most critical details, concerning fatigue damage, are not well captured in the detail categories present in that standard. That is the case of the top connection of the diagonals present at each tension rod-to-deck connection (Fig. 8). The classification of this welded detail according to Eurocode 3 is not straightforward, since the local weld features of this detail (Fig. 9(a)) are not well captured in any of the fatigue recommended detail categories, for nominal stresses. In this detail we have load

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Fig. 5. Bridge of the new railway crossing of the river Sado. (a) Location [31]; (b) overview (adapted from [32]).

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Fig. 6. Side elevation of the 2nd span of the bridge [33].

Fig. 7. Cross section of the deck [33].

carrying fillet welds, for which the Eurocode 3 recommends that the fillet weld should finish 10 mm from the plate end (see Fig. 9(b)). Instead, it continues its path through the gusset thickness generating a double stress concentration at the diagonal (slot radius notch stress concentration plus weld stress concentration (Fig. 9(a)). The structural detail shown above was the one chosen to apply the proposed methodology. In particular, the diaphragm analysed was the Diaphragm 51 (see Fig. 10).

4.2. Monitoring system Due to the paramount relevance of this structure, a monitoring system was installed on it. This monitoring system allowed the real time traffic characterization and structural behaviour evaluation [20]. The structural behaviour was assessed by means of strain gauges installed at the diagonals of Diaphragms 51 and 54 (Fig. 10). At Diaphragm 51, in particular, 5 local strain gauges were

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Fig. 8. Critical detail to fatigue damage. (a) Cross-section of the deck; (b) critical detail.

installed at the top of the diagonal in order to better assess the stress field near the detail (Fig. 11). The traffic was characterized using shear strain gauges welded to the rails (axle loads estimate) and instrumented rail pad sensors (axles spacing and trains speed quantification). Each train crossing the bridge is totally characterized (train speed and direction, axle loads and axles spacing), through routines developed for that purpose, in MATLAB, and the corresponding information stored in a database also in MATLAB format. The information is sent, through a 3G connection, to a server at the Faculty of Engineering of University of Porto. Therefore, an extensive database, including real strain measurements and trains characteristics is available and was used in the present work. 4.3. The global numerical model of the bridge The global numerical model of the bridge was developed using the ANSYS software and its Parametric Design Language (APDL). The concrete slab and the steel box girder were modeled using linear 4-noded shell elements (SHELL63 of ANSYS library) while most of the diaphragms and diagonals, the arches and the tension rods were modeled with linear beam elements (BEAM44 of ANSYS library) (see Fig. 12(a)). The connection between the concrete slab and the upper flanges of the steel box was achieved by means of multipoint constraint procedures, assuming rigid connections (MPC184 of ANSYS library). The Diaphragms 51 and 54 and corresponding diagonals were modeled with a fine mesh of shell elements, with maximum dimension of 0.1 m, in order to facilitate the comparison of the numerical results with the strain gauge measurements captured with the permanent monitoring system (see Fig. 12(b)). The remaining deck was modeled with a mesh size of 2 m, as per the results of a sensitivity analysis.

Fig. 9. Schematic of local weld features of critical detail. (a) Actual weld connection; (b) Eurocode 3 detail category.

The numerical model was calibrated, based on the results of an Ambient Vibration Test and a load test performed in the bridge [34]. The main parameter affecting the global response of the bridge was the Young Modulus of the concrete, Ec, forming the concrete slab. This parameter was calibrated in order to minimize the differences between the experimental natural frequencies, mode shapes and traffic-induced deformations and those simulated with the numerical model. The calibration process led to an optimum value of Ec = 43 GPa. After calibration of the global numerical model, a modal analysis was performed and the corresponding modal properties (Uj , nj, xj and mj) were computed and exported to MATLAB. In MATLAB, those modal properties are combined with the traffic information already stored in the database, F(t), and the dynamic response of the structure is simulated, using the modal superposition technique (Eq. (2)) and adopting a total of 1500 modes of vibration. The number of modes of vibration used

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Fig. 10. Location of Diaphragms 51 and 54.

Fig. 11. Location of strain gauges at Diaphragm 51. (a) Global and local strain gauges – schematics; (b) local strain gauges – location and labeling.

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Fig. 12. Global numerical modal of the bridge. (a) Deck, tension rods and arch; (b) Diaphragm 51 and corresponding diagonals.

in the analysis (1500) was defined based on a sensitivity analysis on the simulated dynamic response of the bridge. An increasing number of modes of vibration was considered, until the dynamic response stabilized and no visible changes in response are observed as extra modes of vibration are added. Additionally, the good agreement observed between the numerical simulation and the experimental records obtained with strain gauges installed in the bridge [35] confirm also the adequacy of the number of modes of vibration employed. A modal damping ratio of 0.5% was adopted for all the modes of vibration. This is the reference value proposed in Eurocode 1 [36] for the case of steel and composite bridges with spans longer than 20 m. At this stage, the modal coordinates, Y j ðtÞ, are computed and stored. 4.4. The numerical model of the critical detail and shell-to-solid submodeling A local finite element model of the investigated welded detail was also built. Its geometry was defined using the commercial software SOLIDWORKS and then exported to the finite element software ANSYSÒ. All the properties of the finite element model are defined in a parametric format, using ANSYS APDL language. The parametric code allows the inclusion of a fatigue crack. If a crack is considered, the code allows the definition of its geometry, i.e. the crack path. An uncracked geometry was first modeled aiming the validation of the submodeling process and the submodel itself, comparing numerical results to experimental data from the local strain gauges

placed near the welded joint (see Fig. 11). Furthermore, the uncracked finite element model allowed the confirmation of the potential fatigue cracking location at the weld toe. Once the potential crack location is defined, the APDL routine developed allows the implementation of the initial crack explicitly in the finite element model of the welded detail. Moreover, the APDL routine was also used to update the fatigue crack path after each iteration of the workflow. Fig. 13 shows the local finite element model with an initial crack. The bulk of the welded detail was modeled using tetrahedral quadratic finite elements. However, at the crack tip, a refined region was modeled with hexahedral quadratic finite elements, in order to accurately assess the stress intensity factors. The transition between the refined and coarse regions is achieved by means of pyramidal finite elements. In order to be able to compute accurately the stress/strains at the local finite element model of the detail, the boundary conditions resulting from the global finite element model of the bridge must be interpolated to the whole set of boundary nodes of the local finite element model. This is achieved using the shell-to-solid submodeling procedure as available in ANSYS [37]. In order to allow this procedure work properly, the shell planes of the global model should fit the mid thickness of the plates of the local model. That was achieved in this case as illustrated in Fig. 14. By applying the shell-to-solid submodeling procedure, the computation of the static and modal stress intensity factors in the fine numerical models of the critical details, built with volume elements, can be uncoupled from the analysis of the global numerical

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model of the structure, also reducing the total computation cost [19]. The virtual crack closure technique (VCCT), as described by Krueger [29], as well as the Displacement Extrapolation (DE) method were implemented in APDL for stress intensity factors computation. 4.5. Fatigue model assumptions The implementation of the adopted fatigue model required the definition of the appropriate Fracture Mechanics parameters. The Paris Law material constants, C and m, were determined by fatigue crack propagation tests conducted using compact tension (CT) specimens built on the same material used in the bridge and with similar thickness. Results were published in reference [38]. In order to bound the fatigue crack propagation domain, the pffiffiffiffiffiffiffiffiffi material toughness was considered K C ¼ 1434 MPa mm. This value was based on the maximum stress intensity factors measured at failure, during a set of fatigue tests performed on CT specimens of the same material [39]. Fatigue crack propagation was simulated until Keq reached KC. The adopted initial crack length, ai, was 15 mm. Modern welded fabrication assures through inspection that cracks and larger discontinuities are not incorporated into the new welded joints. Small crack-like discontinuities may occur at the intersection of the fusion line of the weld and the plate surface at the weld toe but they are too small for detection. One may postulate the initial defects using the maximum detectable defect assured by the inspection technique. Also, assumptions about the initial shape of defects and their distribution will be required but not available for this bridge detail. For thick plates, as the current case, it is not plausible that initial defects are through thickness and multiple initial cracks are also very likely to occur. Only after a certain operational period, that complex crack pattern will coalesce, leading to through thickness cracks, which was the base geometry of the fatigue model selected in this paper. In this paper, a residual fatigue life calculation is followed which is an approach consistent with data from periodic bridge inspections. For very large structures, such as bridges, the more economic inspection procedures are based on direct visual inspection but they only allow the detection of macro through thickness cracks. The visual inspection is also restricted by the paint layer applied in the structure. The crack

Fig. 14. Shell-to-solid sub-modeling: fit of the local model on the global model. (a) Front view; (b) side view; (c) top view.

Fig. 13. Local finite element model.

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Fig. 15. Strains at the location of the local SG: experimental vs numerical. (a) Local SG 1; (b) Local SG 2; (c) Local SG 3; (d) Local SG 4; (e) Local SG 5.

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Fig. 16. Fatigue crack initiation spot. (a) Numerical simulation; (b) tested experimental joints [30].

size of 15 mm was therefore proposed as a crack detectable by direct visual inspection. Cracks of that size were not actually detected on the current bridge detail under investigation since the bridge is new. However, inspection routines carried out on other welded bridges have detected cracks of similar, or even higher, length [40]. At the end of each iteration, the crack length is incremented by 5 mm (Dainc). As the traffic loading block originates a much lower crack progression, Dan, the equivalent number of traffic loading blocks, Bn, has to be computed. Table 1 summarizes the Fracture Mechanics parameters considered in the analysis. 5. Results 5.1. Experimental validation In order to validate the adopted submodeling procedure, the experimental strain gauge data available was compared with the numerical simulation results obtained at the same locations, in the uncracked submodel. The numerical simulation was performed in MATLAB using the modal superposition technique. An example of that comparison, for a freight train (Fig. 19(c)), is shown in Fig. 15. The strains are plotted as a function of the position of the 1st axle of the train. A very good agreement can be observed between the experimental and the numerical results, which confirms the adequacy of the submodeling techniques employed. After validation of the submodeling procedure, the uncracked local model was used to determine the most probable crack initiation spot. For several train passage simulations, the maximum normal stresses at the diagonal yielded at the same location (Fig. 16(a)). This location was coincident with the crack initiation location observed at the experimental tests performed in small-scale replicas of this welded joint [30] (Fig. 16(b)). The

Fig. 17. Computation of K(t): VCCT vs. DE methods. (a) KI(t); (b) KII(t).

critical location shows the superposition of two stress concentration contributions: the hole and weld toe effects. This location was then adopted for the initial crack in the local model. It must be stressed that even if experimental observations pointed to the formation of 2 symmetrical fatigue cracks (Fig. 16(b)), only one fatigue crack was modeled. That allowed reducing the complexity of the model without jeopardizing the validation of the proposed methodology. 5.2. Comparison of stress intensity factor’s computation techniques Different numerical methods are available for the numerical computation of stress intensity factors. In the current work, two different methods were used for cross-validation: the virtual crack

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Fig. 18. Fatigue crack propagation path.

Table 1 Adopted parameters for fatigue analysis. c

m

a

a

9.93e14 [40]

3.14 [40]

KIC

ai

Dainc

pffiffiffiffiffiffiffiffiffi MPa mm 1434 [41]

mm 15

mm 5

a Paris Law parameters set for da/dN in mm/cycle and stress intensity factors in pffiffiffiffiffiffiffiffiffi MPa mm.

closure technique (VCCT) [29] and the Displacement Extrapolation (DE) method [41]. The stress intensity factors were computed in ANSYS using both methods for the same freight train presented before and for an initial crack dimension of 15 mm. The results are presented in Fig. 17. The difference between both methods is less than 10%. Therefore, the DE method was adopted as the reference and is the basis for the numerical results presented in the rest of this paper. From the results presented in Fig. 17, it can also be observed that for the initial crack, Mode I stress intensity factor, KI, is one order of magnitude higher than Mode II stress intensity factor, KII. 5.3. Residual fatigue life computation Once all the different techniques employed were validated, the entire workflow was run (Figs. 3 and 4), in order to assess the residual fatigue life of the structure, taking into account a postulated initial defect of 15 mm. The crack propagation occurred, as expected, in a direction perpendicular to the principal stresses, which underlined the importance of considering a crack branching criteria. The crack propagation stopped at iteration number 33, when the maximum computed stress intensity factor reached the adopted material toughness. The entire crack propagation path (a = 175 mm) is illustrated in Fig. 18. The crack propagation simulation workflow, comprising blocks of 565 trains, 33 crack increments and 1500 modes of vibration, was completed in approximately 3.5 days, confirming the high computational efficiency of the proposed methodology. A personal computer with a 3.2 GHz i7 processor and 24 GB RAM memory was used. Each traffic load block includes 565 trains corresponding to a total traffic load of 0.52 million t. The equivalent annual traffic volume is 3.1 million t/year. The trains are freight and passenger trains (Fig. 19), with a variety of weights, speeds and lengths, as illustrated in Fig. 20. The evolution of the total crack length as a function of the cumulative traffic volume is shown in Fig. 21. The approximately

Fig. 19. Types of trains crossing the bridge.

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Fig. 20. Trains’ characteristics. (a) Trains’ load per unit length; (b) trains’ speed; (c) trains’ length.

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it should be noted that the fatigue traffic mixes present in the standards usually assume higher traffic volumes. In the Eurocodes, for instance, an annual traffic volume of 25 MM t is considered as reference. Therefore, a scenario of 25 MM t per year was also included in the analysis. In that case, the simulated crack would take approximately 12 years to propagate (Fig. 22). It should also be noted that these results were achieved with a highly conservative assumption on the initial crack dimension (15 mm). The hypothetical crack initiation period and/or propagation period up to that 15 mm crack dimension are therefore disregarded. The employment of the Paris Law is also considered a conservative assumption for the range of stress intensity factors observed in this analysis, since no crack propagation threshold DKth was adopted. 6. Conclusions

Fig. 21. Crack propagation length as a function of cumulative traffic.

linear increase of the crack length means that there was not observed an increasing equivalent stress intensity factor as the crack size increased. This could be explained by stress/load redistribution due the damage spread which does not allow the acceleration of the crack growth rates. This means that the structure shows some degree of tolerance to the damage. It should be noted that the changes in crack propagation rate and in the crack path observed over time are affected by the length of the finite crack increments adopted in the workflow (Dainc = 5 mm). Smaller crack increments would provide more accurate results, but with associated costs in terms of computational time. If the current traffic volumes on the bridge were kept stable in the remaining life of the bridge, the hypothetical crack would propagate during approximately 95 years (Fig. 22). Nevertheless,

This paper presents a new efficient procedure for the fatigue assessment of railway bridges. This procedure was validated by application to a case study, the bridge of the new railway crossing of river Sado. The simulation of the fatigue crack propagation in a critical detail of the bridge was achieved using minimal computational resources. Shell-to-solid submodeling confirmed to be an effective way to address multiple-scale structural problems, such as localized fatigue crack propagation in large structures. The main conclusions withdrawn from the current work are as follow: – Modal superposition of stress intensity factors confirms to be an adequate and efficient method when multiple and complex load histories are considered, such as multiple traffic events on bridges. – For the current case study and critical detail considered, the assumption of mixed mode fatigue crack propagation confirmed to have impact on crack propagation path. – The proposed methodology allows the quick assessment of different traffic scenarios, e.g. the consideration of an increase of annual traffic volume over time. Finally, it should be underlined that the simulation can be optimized, in the future, by the implementation of parallel computing, further increasing the computational efficiency. Acknowledgements The present work has been funded by the Portuguese Foundation for Science and Technology (FCT), in the context of the Research Project Advanced methodologies for the assessment of the dynamic behaviour of high speed railway bridges (FCOMP-01-0124-FEDER-007195), and by the European Commission, in the context of the Research Project FADLESS Fatigue damage control and assessment for railway bridges (RFSR-CT-2009-00027). FCT scholarships SFRH/BD/47545/2008 and SFRH/BD/72434/2010 are acknowledged. The authors wish also to thank the collaboration and support provided by REFER, LNEC and Teixeira Duarte. References

Fig. 22. Crack propagation length vs time.

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