Fatigue resistance of the deck plate in steel orthotropic deck structures

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Fatigue resistance of the deck plate in steel orthotropic deck structures ⁎

Johan Maljaarsa,b, , Eric Boneta, Richard J.M. Pijpersa a b

TNO, Unit of Buildings, Infrastructure and Maritime, The Netherlands Eindhoven University of Technology, Department of the Built Environment, The Netherlands

A R T IC LE I N F O

ABS TRA CT

Keywords: Probabilistic fracture mechanics Bridge deck Deck plate crack Rib-to-deck plate joint Fatigue

Steel orthotropic decks in bridges and ferries have shown to be prone to fatigue cracking. A typical fatigue crack observed in these structures initiates at the root of the weld between the deck plate and stringer and grows into the deck plate. Due to limited fatigue test data available and the deviating characteristics of this type of crack as compared to other types of fatigue cracks, the fatigue strength is uncertain. In this study, a linear elastic fracture mechanics model has been developed for this type of crack, providing insight into its fatigue performance. The model predicts a relatively high fatigue resistance which is close to the results of fatigue tests. The model further predicts a relatively long residual fatigue life after crack detection, providing large inspection intervals.

1. Introduction Steel orthotropic decks are applied in bridges and ferries. They consist of a deck plate stiffened by stringers and supported by crossbeams. In many cases the stringers are of trapezoidal shape and are running through openings in the crossbeam web. Steel orthotropic decks are prone to fatigue cracking due to passing wheels of heavy vehicles, [1]. In such a deck, a typical type of fatigue crack can initiate in the deck plate from the root of the weld between the stringer and deck plate, Fig. 1. This type of crack is reported for bridges in Japan [2], France [3], Belgium and The Netherlands [4]. Decks with a thin deck plate and a thin road pavement are especially sensitive to this type of crack. Fatigue test data are available for deck plate cracks. Test results for the deck plate in between crossbeams, Fig. 1a, are reported in [5–8]. In case of continuous stringers with crossbeams welded around, deck plate cracks are predominantly observed at the junction with the crossbeam, Fig. 1b, caused by the high stress concentration at that location. This crack location – which is the focus of this paper – has been tested in [9,10] and with slightly different geometry in [11]. Kolstein [10] combined the results of [9,10] and provided the fatigue reference strength at 2 · 106 cycles in MPa, which is further in this paper referred to as FAT class, with as subscript the survival probability. The FAT classes are provided in Table 1. These FAT classes are high when compared to other types of welded details for which fatigue test data are available. As a reference, standards and guidelines such as [12] and [13] provide FAT95% = 90 or 100 for weld toe cracks assessed with the hot-spot stress, whereas root cracks usually have a lower fatigue strength. The differences in FAT class may be due to important differences in geometry and loading between the deck plate geometry and more common types of fatigue prone details, Table 2. The main difference is that the crack initiates from the root of the weld between the stringer and the deck plate and grows into the deck plate, whereas usual fatigue cracks starting from the weld root grow through the weld. Another distinct characteristic of this deck plate crack is that the deck plate is loaded in bending with the initiation point

⁎

Corresponding author at: TNO, P.O. Box 155, 2600 AD Delft, The Netherlands. E-mail address: [email protected] (J. Maljaars).

https://doi.org/10.1016/j.engfracmech.2018.06.014 Received 23 March 2018; Received in revised form 1 June 2018; Accepted 11 June 2018 0013-7944/ © 2018 Elsevier Ltd. All rights reserved.

Please cite this article as: Maljaars, J., Engineering Fracture Mechanics (2018), https://doi.org/10.1016/j.engfracmech.2018.06.014

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Fig. 1. Geometry of the deck plate crack: (a) Deck plate crack between crossbeams; (b) Deck plate crack at crossbeam.

Table 1 Fatigue reference strength in MPa (FAT class) of the deck plate as determined in tests [10]. Location

No. of specimen

Stress parameter

FAT50%

FAT95%

FAT97.7%

Between crossbeams At crossbeam

21 (two series, [6,7]) 21 (two series, [9,10])

Nominal Hot-spot

183 214

146 167

140 158

Table 2 Comparison of deck plate detail and usual fatigue sensitive weld details for which test data are available. Aspect

Usual weld details tested in fatigue

Deck plate crack

Crack location

-Weld toe: through base plate -Weld root: through weld

Weld root, through base plate

Type of load

Tension (in fatigue tests). Sometimes bending with tension at the initiation location

Bending (in fatigue tests) with compression at the initiation location

Stress

Simple geometries – straightforward stress analysis

Complex geometry at the junction with the crossbeam – direct stress decreases with distance to crossbeam

No. cycles between through-thickness crack and failure

Limited

Significant

Inspection

Weld toe: possible Weld root: difficult unless the crack is large

Difficult; no access from above (road pavement), no access from below (stringer wall and crossbeam)

experiencing compression due to wheel loading. The cracks experienced in fatigue tests are believed to be influenced by residual welding stresses, which result into internal tension at the crack initiation point. In practice, yielding of the deck plate due to loading by extremely heavy axles, may further contribute to residual tensile stresses or reduce residual compression stresses. Accurate stress determination is performed for the deck plate at the weld toe and weld root in [14], indicating a relatively high effective notch stress at the weld root. The FAT class of this detail is relatively uncertain due to limited test data available and the deviating characteristics of this type of fatigue crack, as compared to other types. In addition, the fatigue tests are a simplified representation of reality because, deck plate road pavement was not applied, a fixed load position was applied, and the crossbeam was vertically supported over the entire length. A lack of access to the crack location makes fatigue inspection of the deck plates difficult. Practical experience indicates that small cracks can best be detected from above the deck with the time of flight diffraction (TOFD) technique, but only after removal of the road pavement. Consequently, an inspection is best planned when the road pavement is to be replaced. The question is, if the 2

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resulting inspection interval is sufficient from a structural perspective, i.e. are cracks detected before instable crack growth or failure takes place? Several crack types in orthotropic decks are analysed in [15] using linear elastic fracture mechanics (FM) and it shows that FM can be applied to characterise fatigue cracks in orthotropic bridge decks. In [16,17], the deck plate crack was characterised using an approximate FM model based on a modified version of a FM model for weld toe cracks in [18]. The current paper provides a FM assessment based on finite element (FE) models of the actual deck plate geometry. This assessment provides insight into the characteristics of the fatigue performance of this type of crack, including the relatively high FAT class and long life between the attainment of a surface breaking crack and a critical crack subjected to instable crack growth. This assessment also provides a FAT class that can be used in determining an inspection interval. 2. Considered geometries This study considers the location at the crossbeam because it is most sensitive to fatigue damage. The decisive dimensions of the deck for this type of crack are those that determine the bending stiffness of the deck plate between the stringer walls: - The centre-to-centre distance of the stringer walls is taken as 300 mm. This is typical for orthotropic bridge decks applied in Europe and it has been applied in the tests [6,9–11] that are used as reference in this study. - The decks that have proven to be sensitive to fatigue have a relatively thin deck plate, in the order of 10 or 12 mm. The tests in [6,9–11] have been carried out with 12 mm thick deck plates. For proper comparison, the same thickness is applied in the current study. - Two extreme cases have been considered for the road pavement. The first case is a bare deck plate without road pavement. This case is equal to the condition in the tests and it is representative for ferries, for bridge decks with a very thin top layer of low stiffness (e.g. 3 mm epoxy) as applied on movable bridges, and for bridge decks with asphalt at temperatures above 25 °C when the asphalt stiffness is negligibly low. In addition, a case is defined with a thick and stiff road pavement consisting of 3 mm epoxy and 72 mm high strength concrete. This combination is applied in renovation projects, [19]. In practice, many bridge decks have an asphalt layer. At moderate temperatures, this road pavement constitutes a case in between the two extremes. The hot-spot stresses have been determined for an asphalt layer in this research, but not the stress intensity factors. Table 3 gives an overview of the road pavement considered and the constitutive properties. - The other dimensions of the deck are taken equal as in the tests, Table 4 and Fig. 2. These are representative for orthotropic bridge decks applied in Europe. The welds between stringer and deck plate are partial penetration welds and all other welds are fillet Table 3 Road pavement cases considered. Case

Description (E = Young’s modulus, ν = Poisson’s ratio)

Lay-out

Case 1

12 mm bare steel plate (E = 210000 MPa, ν = 0.3)

Case 2

72 mm high-strength concrete (E = 45000 MPa, ν = 0.4) 3 mm epoxy (E = 1150 MPa, ν = 0.3) 12 mm steel (E = 210000 MPa, ν = 0.3) Fully coupled – i.e. full composite working – at interfaces

Case 3

90 mm asphalt at 15 °C (E = 4000 MPa, ν = 0.3) 12 mm steel (E = 210000 MPa, ν = 0.3) Fully decoupled at interface

Table 4 Dimensions of the orthotropic deck, the load patch, and the considered cracks. Variable

Dimension

Deck plate thickness Stringer wall thickness Centre-to-centre distance of stringer walls Throat of stringer-to-deck plate weld, including penetration Lack of penetration of-stringer-to-deck plate weld Throat of cross beam web-to-deck plate weld (fillet weld) Stringer height Load patch length Load patch width Crack depth of the semi-elliptical cracks (a) Aspect ratio of the semi-elliptical cracks (a/c) Semi crack length of the through-thickness cracks (c)

12 mm 6 mm 300 mm 5 mm 1.9 mm 4 mm 325 mm 320 mm 270 mm 0.15–11.5 mm 0.1–1 35–125 mm

3

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Fig. 2. Dimensions of the orthotropic deck and crossbeam [mm].

welds with a throat of 4 mm. Two load patches are applied, representing the contact areas of an axle with two tires. The load patch dimensions are taken equal to that of rear axles of heavy vehicles in fatigue load model 4 in a European standard [20]. The same load patch dimensions were applied in the tests. A uniform pressure is applied on each patch. In the tests, the fluctuating load is applied at a fixed position centred over a trough and a crossbeam whereas, in practice, wheels are rolling over the deck plate in direction x, Fig. 1. Both load conditions are modelled. In the tests, the crossbeam bottom flange was supported in vertical direction over its full length. In addition to this condition, a case is considered in the current study with a crossbeam that is hinge supported at its edges. This case is applied to study the possible influence of the stress range imposed to the deck plate by global bending of the crossbeam. The global geometry represents an actual bridge with crossbeams spaced at 3.645 m, a crossbeam span (centre-to-centre distance of main girders) of 23.8 m, and load patches at 14.35 m and 16.50 m from the crossbeam end. 3. Finite element model The commercially available FE analysis software Abaqus 2014 [21] is used to create the models for deriving stresses and stress intensity factors (SIF). The models represent a steel orthotropic bridge deck including three crossbeams. Symmetry conditions are applied at the centre crossbeam in the models where loads are centred over this crossbeam, Fig. 3, and the full length geometry with 5 stringers is used in the models in which loads are applied away from the crossbeam and the crossbeam bottom flange is fully supported. Each model is constructed with 8-noded brick elements of type C3D8R. The element sizes vary between 75 mm away from the point of interest to 0.01 mm near the crack tip of the smallest crack modelled. Tie constraints, tying two separate surfaces together so that there is no relative motion between them, are used for a multi-part mesh refinement. This modelling technique is demonstrated in Fig. 4. Smaller geometry models without tie constrains are constructed and compared with the full model, to make sure that the ties were applied sufficiently far away from the point of interest as to not influence the stress nor the stress intensity factor (SIF). The cracks are modelled by duplicating nodes in the deck plate allowing the surfaces to separate. Fig. 5 shows the mesh details of two cracks. As an example, Fig. 6 shows the stresses and deformations of one of the full models. Note that the load causes the crack surfaces to interfere. If residual stresses are not considered – as in the model – the maximum interference at the plate surface of very deep cracks is approximately 3 μm per kN load and it is almost zero for small cracks. Apart from plasticity induced crack closure that is always present near the crack front, crack flank contact away from the crack front is not expected for small (e.g. 1 mm deep) cracks because of the residual tensile stress. If, for deep cracks, the crack flanks make contact away from the crack front, Smith and Smith [22] demonstrate that wear and rubbing of the flanks take place because of mode II and mode III deformations that are observed in 4

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Fig. 3. Symmetry conditions and load locations applied in the model with load patch centred over the crossbeam: (a) Full model; (b) Selection, indicating the stringers.

Fig. 4. Model used to calculate Case 2 hot spot stress.

5

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Fig. 5. Mesh details at crack front: (a) semi-elliptical crack with a = 0.15 mm and a/c = 1; (b) through thickness crack with c = 125 mm, d/c = 0.5 (dimensions a, c and d explained in Fig. 1).

Fig. 6. Stresses (contours) and deformations of the case with a bare steel plate and a crack with a = 3 mm, a/c = 0.5: (a) Deformation scale factor = 5000; (b) Close-up of fig. a, deformation scale factor = 5000; (c) Close-up of fig. b, deformation scale factor = −1000.

the current model. This implies that a significant reduction of the effective SIF is also not expected from crack flank contact away from the crack front for larger cracks. The contour integral method is applied to extract the SIF. The modelling approach is validated by modelling three extensively studied geometries in literature and comparing the SIF: - A plate with a penny-shaped crack. The SIF differed 1% between the current model and [23]. - A plate with a semi-elliptical crack. The SIF differed 2% between the current model and [23]. - A block with a straight crack front. The SIF differed 2% between the current model and [24]. 4. Fracture mechanics model 4.1. Hot-spot stress The stress was measured in the tests in [9] for the case of a fully supported crossbeam and a steel deck plate only. As a first validation step, this measured stress is compared to the stress extracted from the FE model for a geometry without crack. The maximum measured stress for a 49 kN tire load, averaged over the specimens, was 198 MPa at the deck plate bottom surface at a distance approximately 4.0 mm away from the root – the exact distance was not provided. The FE model (of the geometry without cracks) predicts the same stress value. The structural hot-spot stress σhs is intended to include all stress raising effects of a structural detail excluding that due to the local weld profile itself, [13]. Note that σhs is usually considered for the weld toe only, but it can be applied to the root for this specific case 6

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Table 5 Hot spot stress parameters for a patch load of 1 kN. Road pavement

Bare steel plate

Crossbeam supporta

Full

Simple

Full

Simple

Full

Simple

σhs0 g|h for fixed load pos. g|h for rolling load

−4.61 MPa 0.64|29.2 mm 0.67|24.5 mm

−4.95 MPa

−0.43 MPa 0.55|21.3 mm 0.58|17.9 mm

−0.70 MPa

−1.47 MPa 0.66|24.3 mm 0.68|21.0 mm

−1.55 MPa

a

Steel & concrete

Steel & asphalt

‘Full’ means bottom flange of crossbeam is vertically supported along its entire length. ‘Simple’ means crossbeam is hinged supported at its ends.

because the crack grows into the deck plate instead of the weld. Following the definition applied in the tests in [10], σhs is determined by linear extrapolation to the root of the weld from the stress in the deck plate bending direction z (Fig. 1) at the deck plate bottom surface at 0.4 t and 1.0 t distance away from that root, where t is the deck plate thickness. The FE model indicates that σhs reduces as a function of the distance to the crossbeam in wheel rolling direction x because of the stiffening effect of the crossbeam web. In addition, it depends on the location of the load patch in x direction. The hot-spot stress is fit with the function:

σhs (x ) = σhs0 θ (x ) θ (x ) = g + (1−g ) cos

( ) πx h+x

(1)

where σhs0 is the maximum hot-spot stress (at the crossbeam location, x = 0) and g and h are fit parameters for the range 0 ≤ x ≤ 125 mm. Table 5 lists the values of these parameters for a load patch of 1 kN (pressure 11600 Pa) and the hot spot stress is visualised in Fig. 7. In all cases, the fit has a coefficient of determination of R2 = 0.99. The table and figure demonstrate that global bending of the crossbeam can increase the hot-spot stress in the deck plate. The increase depends on the crossbeam dimensions and meeds to be assessed per structure. The table and figure further reveal the important contribution of the road pavement to the stress and thus to the fatigue performance. By adding the asphalt layer with a temperature of 15 °C or the concrete road pavement, the maximum hot-spot stress reduces to approximately 30% or 10%, respectively, of that of a bare steel plate. These large reductions are attributed to a combination of a stiffer deck and spread of the load patch pressure in the road pavement. Fig. 7 shows that the load position centred over the crossbeam, as applied in the tests, gives the largest maximum hot-spot stress σhs0, but the envelope of the hot-spot stress away from the crossbeam is larger for a load rolling over the deck. 4.2. Surface cracks The SIF is evaluated along the crack front for 26 discrete surface cracks, with the ranges of dimension listed in Table 4. The loss of inverse square-root singularity generally observed for the J-integral method, [25], affects the results at the crack surface tip element. Therefore, this element is ignored and the adjacent element is taken as the free surface crack tip. The error introduced through this approximation is negligible because of the dense mesh chosen.

Fig. 7. Hot-spot stresses for a fully supported crossbeam and a patch load of 1 kN: (a) stress extraction locations and direction; (b) Hot-spot stress for a fixed load position (as in the tests); (c) Envelope of hot-spot stress for a rolling load (as in reality). 7

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Fig. 8. SIF along the crack front for two cracks.

A crack has an optimal shape if the SIF is constant along the crack front. The crack aspect ratio – i.e. ratio a/c – giving the optimal shape is selected hindsight. With the predefined semi-elliptical shape, the SIF is almost constant along 85% of the crack front for relative crack depths a/t ≤ 0.75 (where a is the crack depth, Fig. 1) but it increases close to the crack surface tip, see the black curve in Fig. 8 for an example. However, the SIF near the crack surface tip is subject to uncertainty anyway because of weld imperfections present along the entire weld seam. The SIF appears to be almost independent of the crack size for 0.04 ≤ a/t ≤ 0.75 and it even decreases with crack size for a/t > 0.75, whereas in many other geometries the SIF increases with crack size. This difference is attributed to the surface tip experiencing a decreasing hot-spot stress (Fig. 7) for larger cracks. Loading in bending attributes to a decreasing stress at the deepest point. For large cracks, i.e. a/t > 0.75, the SIF was non-constant along the crack front irrespective of the selected crack aspect ratio – see the grey curve in Fig. 8 for an example – indicating that a semi-elliptical crack does not reflect the optimal crack shape. The SIF at the deepest point and at the surface tip of the crack, Ka and Kc, respectively, are decomposed into the following components:

K a = Mka Mba σhs0 πa K c = Mkc Mbc θc σhs0 πa

(2)

where θc = θ(x = c) using Eq. (1), c is the semi crack length (Fig. 1), Mba and Mbc are the geometric correction factors for semielliptical surface cracks in a flat plate loaded in bending and Mka and Mkc are the additional geometric correction factors for the deck plate geometry of Fig. 1b. Factors Mba and Mbc are taken from [23] in their validity range a/t ≤ 0.75. For deeper cracks, modified factors are taken from [26]. The unknown factors Mka and Mkc in Eq. (2) are then determined by substituting the SIFs from the FE model for Ka and Kc. A curve is fit through the numerically obtained values for Mka and Mkc. Factors Mka and Mkc appear to be relatively insensitive for non-optimal aspect ratios because the aspect ratio is already accounted for in factors Mba and Mbc in Eq. (2). For a maximum error in Mka and Mkc of 5%, the results with different aspect ratios indicate that the maximum difference of the SIF along the crack front that can be allowed is 25%. Thus, Mka and Mkc of the evaluated crack sizes with less than 25% deviation of the SIF along the crack front are considered in the fit. For deep cracks, a > 0.75 t, the deviation in SIF along the crack front is larger than 25% for semi-elliptical cracks irrespective of the aspect ratio. Consequently, the relationships derived for Mka and Mkc are uncertain for deep cracks. The smallest crack considered has a depth of 0.15 mm and an aspect ratio of 1.

Fig. 9. Geometry correction factors: (a) factor Mka; (b) factor Mkc; (c) product Mtcθc. 8

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Table 6 Fit parameters of Mk■ (Eq. (3)) and Mtc (Eq. (8)) based on the FE analyses. Case

pa

qa

ra

sa

ua

pc

qc

rc

sc

uc

pt

qt

rt

st

Bare steel Steel&Concr.

1.0 1.0

2.0 3.8

−30 −11

−0.5 −0.7

−6 −6

0.9 1.7

4.0 7.5

−7 −7

0.8 1.0

−8 −10

0.50 0.75

0.13 0.30

105 mm 110 mm

0 mm 20 mm

This aspect ratio also provides a deviation in SIF larger than 25% along the crack front, but is considered in the Mka and Mkc relationships because it reflects the initial crack size, as will be explained in the next section. The black curve in Fig. 9a gives factor Mka as a function of relative crack depth for the case of a bare steel deck plate and a fully supported crossbeam. The uncertain part of the relationship is indicated with a dashed curve and the coefficient of determination is indicated in the graph. As an illustration of results, some of the numerically obtained values are indicated with dots in the figure. The fit function is of shape:

a t −a ⎞ Mk■ = p■ + q■exp ⎛r■ ⎞ + s■exp ⎛u■ t ⎠ ⎝ t⎠ ⎝

(3)

where subscript ■ is ‘a’ or ‘c’ indicating deepest points or surface points, respectively, and the second row of Table 6gives the values of fit parameters pa, qa, sa and ua. Factor Mka is large in case of small cracks (a/t < 0.2) when compared to cracks initiating at a weld toe in a straight plate geometry, e.g. [27], indicating the severity of the deck plate geometry. For deeper cracks – roughly between 0.4 < a/t < 0.7 – factor Mka is close to 1, implying that the plate and geometry of the deck plate crack do not significantly influence the SIF. For deep cracks – a/t > 0.75 – factor Mka is smaller than one, which is attributed to the decreasing stress at the crack surface tip for larger cracks. This influences the optimal aspect ratio and thus also Mka. The case with concrete road pavement has been considered with a fully supported crossbeam and with a hinged crossbeam and the resulting Mka factors are equal despite the difference in hot-spot stress (Table 5). This implies that the fatigue resistance is equal for these two cases. The grey curve in Fig. 9a results for the case of a concrete road pavement. As for the bare steel plate, the results for the concrete road pavement are uncertain for deep cracks (a/t > 0.75). The fit parameters are provided in the third row of Table 6. Comparing the cases with a concrete road pavement to the bare steel plate in Fig. 9a, it appears that the correction factor with concrete is larger than that of a bare steel plate for small crack sizes (a/t ≤ 0.4). This is attributed to the fact that the neutral axis, for the case with concrete road pavement, is located above the centreline of the steel deck plate, so that the original steel deck plate is not loaded in bending only, but in a combination of membrane and bending stress, which generally results into larger SIF values when compared to bending only, e.g. [18]. Evaluation of factor Mkc is less straightforward than that of Mka because of the observed increase of the SIF near the crack surface tip. Two different approaches are therefore followed: - The reference approach is selecting Mkc one node away from the singularity point along the crack front for each considered crack size. - The alternative approach, applied to determine the influence on the fatigue strength, is selecting Mkc at an ellipse angle φ = 5°for each considered crack size. Fig. 9b provides the resulting fit functions and Table 6 lists the fit parameters for the reference strategy. 4.3. Through thickness cracks The remaining ligament above the crack reduces with crack growth and eventually plastic break-through takes place when the ligament is no longer able to bear the load. In agreement with [28], local plastic failure is considered and the crack is assumed surface-breaking – so that the crack depth, a, instantaneously increases to the deck plate thickness, t – when the following condition is met:

σref σf

Uf > 1

(4)

where σf is the flow stress defined as the average of the yield stress, σy, and the ultimate tensile strength σu (Eq. (5)), Uf is a model uncertainty factor, and σref is the reference stress for which the model in [29] is adopted:

σf = 0.5(σy + σu )

σref =

2 σnom 3 (1 − α eff )2

α eff =

πca 2t (2c + t )

(5)

(6) 9

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Based on a comparison between the local plastic failure model and 26 tests on simple specimens including cracks and loaded in bending, reported in [29], Uf is characterized by a mean of 0.78 and a standard deviation of 0.12. The nominal stress σnom in Eq. (6) should not contain the stress raising effects of the welds and the crossbeam. Consequently, the nominal stress can be determined for a deck plate supported by stringer walls. This results in σnom/F = 1.8 MPa/ kN for the bare steel deck plate case, where F is the load applied on the patch. Because of the uncertain structural performance of the road pavement once the deck plate fails, the road pavement is conservatively not considered in the local plastic failure model: the nominal stress is determined as if there is a bare steel deck plate only and it is assumed that a deck plate crack with top side length 2d (Fig. 1) extends in the road pavement. The tests indicated that a surface breaking crack grows further in length direction through fatigue. Semi-elliptical throughthickness cracks according to Fig. 5b are therefore evaluated. The SIF predicted at the surface point, on the underside of the deck plate, Ktc, appears to be insensitive for the aspect ratio of these cracks: the variation in Ktc is only 3% for cracks with 0.15 ≤ d/ c ≤ 0.95, where d and c are the semi crack lengths at the top side and the bottom side of the deck plate, respectively, see Fig. 1. The SIF is therefore specified as a function of c only:

Ktc = Mtc θc σhs0 πc

(7)

c + rt ⎤ ⎞ Mtc = pt + qt ⎛⎜1−cos ⎡π ⎟ ⎥⎠ ⎢ st ⎦ ⎣ ⎝

(8)

Table 6 provides the values of parameters pt, qt and rt. The relation is valid for 35 mm ≤ c ≤ 125 mm, being the smallest and largest crack lengths analysed. Larger cracks have not been analysed because the SIF values in shear modes II and III increase for increasing crack length and they may become too dominant for c > 125 mm. Fig. 9(c) provides the product Mtcθc. Despite Mtc being larger, this product is smaller for the concrete road pavement case as compared to the case of a bare steel plate. This is attributed to the concrete transferring a larger portion of the load as the crack reduces the steel plate stiffness. 5. Fatigue resistance 5.1. Simulation of the tests The Paris equation (Eq. (9)) is used to simulate the number of cycles, N, required for a crack to grow from an initial size, with dimensions a0 and c0, to a final size. The final size is either a surface breaking crack or a through-thickness crack with maximum semi length of c = 125 mm: da dN dc dN

( (1−

ΔK 0 n a Us

= Ca (ΔK a Us )m 1− ΔK =

Cc (ΔK c Us )m

) )

ΔK 0 n ΔK a Us

(9)

where Δ is the range operator, Ca, Cc, m, ΔK0 and n are material dependent parameters that can be determined in crack growth tests and Us is an uncertainty factor related to the derivation of Ka and Kc. The mean value of Us is 1. Based on Section 4, the standard deviation of Us is small for a/t ≤ 0.75 and large for a/t > 0.75, see Table 7. In order to obtain an as accurate as possible representation of the tests, spare material from the specimens in [9] is obtained and used for tensile tests and crack growth tests. An assessment in [30] indicates that the residual stress near the root of a fillet weld – representative for the surface tip – is approximately equal to 0.6σy. The crossbeam acts as a welded attachment at the deepest point. A residual stress of approximately 0.8σy was measured at the end of a welded attachment in [31,32]. The average hot-spot stress applied in the fatigue tests is −0.7σy. Consequently, the effective stress ratio in the material is expected to be close to zero 0 both for deepest point and surface tip. The crack growth tests are carried out at stress ratio R = 0.1. Table 7 provides the resulting values of σy and σu, and distributions of Ca and Cc. The initial crack of an as-welded detail to be used in a fracture mechanics analysis has been calibrated in a number of studies. For Table 7 Distributions of variables for simulating test results (units N, mm). Variable

a0

a0/c0

σy

σu

log10 Ca

log10 Cc

m

ΔK0

n

Uf

Us (a/t ≤ 0.75)

Average St.dev. Distr.a Source a b c

0.15 0.66 L [33]

0.62 0.40 L [33]

396 0 D

549 0 D

−13.00 0.179 N

−12.52 0.093 N

3 0 D –b

170 34 L [34]c

D = deterministic, N = normal distribution, L = lognormal distribution. Predefined value, used in fitting Ca and Cc. Average value of similar steels reported in [34], standard deviation of ΔK0 taken from [28]. 10

0.5 0 D [34]c

0.78 0.12 N [29]

1 0.05 N Section 4 based

(a/t > 0.75) 1 0.15 N estimate

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Fig. 10. Crack size as function of number of cycles for a simulation at Δσhs0 = 283 MPa for a bare steel deck: (a) Crack depth, a; (b) Semi crack width, c.

weld toes, this initial defect is well established and often taken as a0 = 0.15 mm and a0/c0 = 1. However, experience is lacking for the current crack type. As it grows into the deck plate and not through the weld, the initial defect for weld toes is probably better representing the actual condition than the defect of a usual weld root. The distribution of the initial crack size in [33] is used here, see Table 7, and its sensitivity will be determined hereafter. Fig. 10 provides the crack size development resulting from the model for the case with a bare steel plate, a fully supported crossbeam, average values of the variables of Table 7 and a hot-spot stress range of 283 MPa. The load ratio applied in the tests was R = 0.06, resulting in a nominal stress in Eq. (6) of σnom = F/(1 − R) = −0.43Δσhs0. The crack grows with relatively constant rate up to a certain depth, from which crack propagation is almost absent in depth direction. The crack continues to grow in length direction up to 8 · 105 cycles where local plastic failure is predicted and the crack is surface breaking. From there onwards, a relatively constant crack growth rate is predicted in length direction. As a result of the fast initial crack growth, the prediction appears to be insensitive to the initial crack size: For a constant aspect ratio, the number of cycles to the final crack size varies only 5% for an initial crack depth ranging between 0.05 mm ≤ a0 ≤ 0.50 mm. The dots in Fig. 11 represent the results from available tests on deck plates with t = 12 mm at the crossbeam location. The horizontal axis provides the number of cycles at the detection of a surface breaking crack. Three types of data are distinguished: ‘failure’ data, where a surface breaking crack was detected (before significant crack growth of the crack in the through-thickness stage took place), ‘runout’ data, where a crack was not detected and ‘early stop’ data, where a crack was detected – either by crack size measurement or by strain gauges – but that was stopped before the crack was surface breaking. The results of the fracture mechanics model, with average values of variables (Table 7), is indicated with a straight black line in the figure. Note that none of the variables in Table 7 is calibrated with the tests. Two linear regression methods using the equation log10 N = α + β log10 Δσ are employed on the test data for a comparison with the model prediction: The regression equation is fit using the least squares method, considering failure data only. This gives the continuous grey line in the figure. At the stress range tested with highest confidence −Δσhs0 = 283 MPa – the difference in number of cycles between the model and the linear regression is less than 1%. However, there is a difference in slope of the curves: slope parameter β in the regression equation is equal to β = −4.4 in the model and β = −2.8 in the tests. The fact that the predicted −β is larger than m in Eq. (9) is due to the nominal stress in the plastic failure model being proportional to the stress range; a lower stress range thus also gives a larger final crack size. The regression equation considering failure data and ‘early stop’ data. The regression is fit using the maximum likelihood method considering the ‘early stop’ data as right-censored, with a likelihood equal to one minus the cumulative density function of the normal distribution centred on the predicted value, using the method described in [35]. Employing this regression, it is assumed that the ‘early stop’ data would have grown into a surface breaking flaw, had the test continued. It gives the dashed grey line in the figure. The slopes of this regression and the model are in good agreement: The model predicts a 4% lower FAT50% class than the average

Fig. 11. S-N curves of the test and the model prediction for the bare steel deck, fully supported crossbeam, and a surface breaking crack. 11

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regression line and the prediction at any stress range is within the two-sided 50% confidence interval of the regression. If the alternative approach is adopted – i.e. selecting Mkc at φ = 5°, Section 4.2 – the model predicts a 2% higher FAT50% class than the average regression line. Other simulated cases also demonstrate a small influence of the Mkc selection criterion on the fatigue resistance. Simulations are carried out with the fracture mechanics model using the distributions of variables according to Table 7. The 5% and 95% survival fractions are determined with the first order reliability method (FORM) and are indicated with dashed black lines in Fig. 11. The standard deviation of Log10N obtained with the model is 0.18, whereas it is 0.21 for the linear regression line. Crack size measurements using TOFD were applied in one, ‘early stop’ test in [11] carried out at Δσhs0 = 184 MPa. Fig. 12 provides the measured crack sizes (bullets) and the prediction with the model (curves). A very good agreement is obtained: both test and model show an initially high crack growth rate in depth direction after which it reduces (model) or the crack even stops growing (test, the difference may be due to uncertainty in residual stress level for deep cracks and – related to this – ΔK0). In length direction, the crack grows with approximately constant rate both in the test and in the model. The length of the crack at the top side of the deck plate, 2d, is measured as a function of the number of cycles in [9] for throughthickness cracks. The model is not able to predict this length and hence the growth rate cannot be compared. However, a constant growth rate with crack size is reported in [9] for the top side of the crack and this agrees with the model prediction at the bottom side, Fig. 12. 5.2. Fatigue resistance of deck plates in real structures For determining the design fatigue resistance of deck plates of real orthotropic decks, some distributions are modified in order to represent the range of material properties and loads experienced in practice. The yield and tensile strength distributions representing steel grade S355 are taken from [36]. The Paris equations and the distributions of its variables are taken from [28]: da dN dc dN

min(A1 (ΔK a Us )m1, A2 (ΔK a Us )m2) =⎧ ⎨ ⎩0 min(A1 (ΔK c Us )m1, A2 (ΔK c Us )m2) =⎧ ⎨ ⎩0

for for

ΔK a > ΔK 0 ΔK a ⩽ ΔK 0

for for

ΔK c > ΔK 0 ΔK c ⩽ ΔK 0

(10)

Two sets of variables (A1, m1, A2, m2, ΔK0) are applied; one for stress ratio R < 0.5 representing the ‘normal’ situation and one for R ≥ 0.5 representing high residual stresses caused by significant weld repairs of an existing deck plate with cracks. In a real deck, axles of heavy vehicles cause crack growth but local plastic failure is due to very heavy (upper tail) axles passing with lower frequency. Hence, the nominal stress in Eq. (7) is not directly related to the stress range. The nominal stress applied in the model is determined from the highest load on a small patch provided in the ‘set of frequent lorries’ in a European standard [20]. The patch load of F = 70 kN (axle load of 2F = 140 kN) provides a nominal stress of σnom = 1.8 F = 127 MPa. Axle load measurements indicate that this load is frequently exceeded, [37]. For this reason, a relatively large coefficient of variation of 0.25 is considered for the maximum stress. The stress reduction function θ for rolling loads is used. Table 8 provides the distributions of the variables. The model now predicts a slope parameter β close to −3, the difference with the simulations of the tests being caused by the independency of the stress range and the maximum nominal stress for real decks. Fig. 13 gives the resulting FAT95% class from the FORM simulations for a deck with bare steel plate and for a deck with concrete road pavement (left-hand set of columns). The class is related to the hot-spot stress. A significant difference in FAT95% class results between a surface breaking crack and a final crack with c = 125 mm especially for the concrete road pavement case, demonstrating the significant remaining life after the attainment of a surface breaking crack. The figure provides a lower FAT95% class for a surface breaking crack if concrete road pavement is added but an almost equal FAT95% class for a final crack with c = 125 mm, as compared to the bare steel plate.

Fig. 12. Predicted (model - curve) and measured (TOFD - bullets) crack size for a test in [11] with Δσhs0 = 184 MPa, bare steel deck, fully supported crossbeam. 12

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Table 8 Distributions of variables for simulating decks in real structures (units N, mm). Variable

a0 a0/c0

σy

σu

log10 A1 R < 0.5|R ≥ 0.5

log10 A2 R < 0.5|R ≥ 0.5

m1 R < 0.5|R ≥ 0.5

m2

ΔK0 R < 0.5|R ≥ 0.5

Uf Us

σmax

Average St.dev. Distr.a Source

See Table 7

394 25 L [36]

566 25 L [36]

−12.41|−12.24 0.115| 0.171 N [28]

−25.92|−17.32 0.279| 0.320 N [28]

8.16|5.10 0 D [28]

2.88 0 D [28]

170|76 34|15 L [28]

See Table 7

127 32 G

a

D = deterministic, N = normal distribution, L = lognormal distribution, G = Gumbel.

Fig. 13. Fatigue reference strength related to the hot-spot stress at 2 · 106 cycles and a 95% survival probability resulting from the model.

Fatigue inspections can be carried out from the top side of the deck plate using TOFD, but only after removal of the road pavement. Consequently, the replacement interval of the road pavement dictates the interval of accurate fatigue inspections. To allow for an assessment whether this inspection interval is sufficient, simulations are carried out with the model in which the remaining fatigue life after TOFD inspection is determined. The scenario considered is that all cracks detected with a depth greater than 3 mm are repaired so that the remaining life is associated with cracks starting from a measured depth smaller than or equal to 3 mm. Depending on the probability of detection (PoD) and the sizing accuracy, however, the actual crack depth may be different. The significant variation of the PoD for TOFD in literature – [38–40] – demonstrates that the PoD depends on the geometry of the detail. Ample experience with this type of crack indicates that small crack sizes can be detected with high probability (personal communication with a bridge assets owner and an inspector). In this assessment, the Weibull distribution according to [38] is used for the PoD:

a −0.7 mm 0.65⎞ ⎞ Fad = 1−exp ⎛−⎛ d ⎝ ⎝ 0.63 mm ⎠ ⎠ ⎜

⎟

(11)

where ad is the crack depth that can be detected and Fad is its cumulative distribution function. The distribution parameters imply that a crack with depth a = 2 mm is detected with 80% PoD. The sizing error εa is assumed to be normal distributed, unbiased, and having a standard deviation of 1 mm [38], i.e. εa ∊ N (0 mm, 1 mm). The same FORM assessment as before is carried out, but now in each run, the number of cycles Ni is counted from the crack size onwards that satisfies both conditions: a ≥ ad and a + εa > 3 mm, i.e. for each run:

Ni = Nt −Nd Nd = min[N ∨ (a ⩾ ad ) ∧ (a + εa ⩾ 3 mm)]

(12)

where Nt is the number of cycles between initiation and the final crack size and Nd is the number of cycles between initiation and the smallest crack that satisfies both inspection conditions. The FAT95% class is derived from Ni and provided in Fig. 13 (right-hand set of columns). The figure shows a small reduction in FAT class for the TOFD inspection case as compared to the as-welded case, which is again attributed to the fast initial crack growth of this type of crack (Fig. 10). 6. Conclusions Fatigue cracks can grow into the deck plate starting from the root of the weld with the stringer in orthotropic deck structures. The very high fatigue resistance reported in tests for this type of crack is confirmed with the fracture mechanics model in this study. An initial fast crack growth rate in depth direction is followed by a much smaller crack growth rate, the transition takes place at a crack depth of approximately 75% of the plate thickness. The crack growth rate is almost constant in length direction. Local plastic failure of the remaining ligament above the crack at certain crack size is again followed by an almost constant growth rate in the length direction. The non-progressive rate is caused by a reduced stress as the crack tip grows away from the crossbeam. It causes a 13

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significant remaining life once the crack is surface breaking. Adding a stiff road pavement – such as a concrete layer – significantly reduces the calculated hot-spot stress. It is also responsible for a reduction in the fatigue resistance class if a surface breaking crack is considered or for a slight increase in resistance if a larger (250 mm) crack length is allowed. This difference is caused by the road pavement transferring a larger portion of the load as the crack causes the deck plate to weaken. The overall effect of the stiff road pavement is a significant increase in fatigue life as compared to a bare steel deck plate. Because the deck plate acts as the top flange of the crossbeam, in-plane deformations of the bended crossbeam cause an increase of the deck plate stress (i.e. load effect), but the fatigue resistance (i.e. strength) remains unaffected. A TOFD inspection and repair criterion is considered, where all cracks detected and sized with a depth larger than 3 mm are repaired. Using this criterion, the fatigue resistance only moderately reduces. The practical relevancy of this result is that it is expected that such an inspection and repair procedure can be planned if the road pavement has to be replaced. Acknowledgements The Dutch bridge asset owner Rijkswaterstaat sponsored this research. The authors acknowledge Rijkswaterstaat and engineering office Arup for reviewing the research. References [1] De Jong FBP. Overview fatigue phenomenon in orthotropic bridge decks in The Netherlands. In: Orthotropic bridge conference, Sacramento, California, USA – August 25–27; 2004. p. 489–512. [2] Yamada K, Ya S, Xiao Z-G. Fatigue assessment of orthotropic steel deck – case study: trough-to-deck detail. 8th Japan-Korea seminar on steel bridges, 2005, Japan. [3] Mehue P. Cracks in steel orthotropic decks. 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