Fatigue strength application of fracture mechanics to orthotropic steel decks

Abstract

Modern fatigue design in civil constructions is mainly limited to the use of S-N curves and the hypothesis of Palmgren-Miner, as described in design standards and Eurocode. While using the latter, the fatigue evaluation may be conservative, since the outdated S-N curves are compared to current construction technology and weld properties. This shortcoming has a direct influence on the current design of orthotropic steel decks. To increase the understanding of the fatigue behaviour, an improved analysing tool using linear elastic fracture mechanics and extended finite element model is proposed. As a result, thickness effects are evaluated for both the longitudinal stiffener and the deck plate. These calculations indicated that increasing the thickness of the deck plate and the longitudinal stiffener increases the fatigue life of the structure. However, the thickness should be limited to maintain the advantage of a light-weighted construction.

Keywords

crack propagation extended finite element model fatigue linear elastic fracture mechanics orthotropic steel deck

Introduction

Although fatigue in steel construction is the most important form of fracture because of its complexity, it is the least understood. In the past, fatigue problems have sometimes been overlooked during design. Fatigue as a failure mode first came to light in the mid-19th century with Wöhler’s research work on the premature failure of railroad axles at stresses below the elastic limit in Germany (Polák, 2003). His work is still used in the form of S-N curves provided in Eurocode NBN EN 1993-1-9 (CEN/TC 250, 2005). However, many of these curves are already outdated when considering present construction technology, weld preparation and weld finishing. In addition, classical fatigue assessment in civil structures uses the Palmgren-Miner hypothesis, which assumes that the fatigue crack propagation is due to a strictly linear damage accumulation phenomenon. This also implies that the load sequence has no effect at all. Questions have been raised in recent research about the validity of this classical fatigue philosophy, especially because of the combined effect of outdated S-N curves and the simplicity of the Palmgren-Miner hypothesis leading to a conservative fatigue design (Kolstein, 2007). This remains especially true for orthotropic steel decks (OSDs). These bridge decks consist of a complex welded network of longitudinal stiffeners, transverse stiffeners and a deck plate (Figure 1). As the orthotropic plate is a load-carrying element in both longitudinal and transversal directions, it contributes to the overall bridge strength, which results in an extremely lightweight and durable deck concept. Therefore, these decks are often used in long-span bridges. However, these bridge decks suffer from many sensitive crack locations, due to the use of a considerable number of welding details (De Jong, 2004; Pfeil et al., 2005; Ya et al., 2011). In addition, some examples of fatigue cracks in OSDs were observed in the last decade, especially in the Netherlands, where many of these bridge decks were constructed (De Jong, 2004; Maljaars et al., 2012). An example in Belgium is the Temse Bridge across the river Scheldt (Figures 2 to 4). Most of the observed fatigue cracks were caused by the increased traffic intensity and their growing axle loads.

Figure 1.

Cross-section and 3D view of an orthotropic steel deck.

Figure 2.

Movable part of the Temse Bridge across the river Scheldt in Belgium.

Figure 3.

OSD of the Temse Bridge.

Figure 4.

Longitudinal crack through the deck plate at the stiffener-to-deck plate connection.

A common location where fatigue cracks occur and are evaluated are the fatigue details located at the connection of the deck plate to a trapezoidal closed longitudinal rib. More precisely, the possible location for longitudinal cracks is at the span centre between two transversal stiffeners. This is also the crack location that will be discussed in particular (Figure 4). At the manufacturing level, welding details have to follow the available standards. Based on the Eurocode (CEN/TC 250, 2007) recommendations, a weld penetration of minimum 67%–75% should be achieved. In addition, the AASHTO Specification note describes a weld penetration of 80% (AASHTO, 2014). In new OSD designs, manufacturers already try to achieve full weld penetration as much as possible. Obviously, when welds are executed with care, the reference value Δσ_c for the fatigue strength has to increase.

According to Eurocode (CEN/TC 250, 2005), a reference value Δσ_c of 71 MPa (Figure 5) applies for the welding detail of the stiffener-to-deck plate connection (Figure 6). However, recent fatigue tests indicate that higher reference values up to 105 MPa should be used, depending on the crack path (Kolstein, 2007). This indicates that the fatigue resistance could be higher than expected. Unfortunately, designers rigidly following the Eurocode guidelines will have a tendency to use ever thicker deck plates and stiffeners, thus reducing the advantage of having a light-weighted construction. To investigate and quantify the real fatigue behaviour, a much more in-depth method can be used, based on linear elastic fracture mechanics (LEFM). With this method, detailed crack behaviour can be evaluated and the total fatigue life as well as the to be expected crack pattern can be estimated. This approach is well known in automobile, marine and aeronautic industries, although it is unfortunately not frequently used in civil engineering since it is a more complex and labour-intensive method. Nevertheless, LEFM is a more in-depth fatigue assessment tool that can overcome the shortcomings in current fatigue assumptions. In addition, the remaining safety of a bridge deck can be evaluated when fatigue cracks are detected. The latter is a huge improvement compared to the current fatigue calculations with the Palmgren-Miner hypothesis. When using this hypothesis for assessing the remaining safety of a bridge deck, it is necessary to have an accurate prediction of the load cycle history of the bridge deck. This is, however, not easy to determine.

Figure 5.

Fatigue strength curves for stiffener-to-deck plate connections according to EN 1993-1-9:2005 (CEN/TC 250, 2005) and recent fatigue tests (Kolstein, 2007).

Figure 6.

Welding procedure of a stiffener-to-deck plate connection according to EN 1993-2:2009 (CEN/TC 250, 2007).

Finally, the evaluation of the crack pattern can be very useful for welding details similar to the stiffener-to-deck plate connection because of the crack often starting at the weld root (Figure 7). The crack can thus grow through the deck plate (Figure 7: path (a)) or through the weld throat (Figure 7: path (d)). In both cases, the crack will grow without being visible to any visual inspection tool because it is at the inside of a closed longitudinal stiffener. When the crack becomes visible at the pavement surface or at the weld surface, sufficient damage to impose necessary reparation is already present (Van Bogaert and De Backer, 2008). Although weld toe cracks are not often found, they could initiate at the weld toe located at the deck plate (Figure 7: path (b)) or at the weld toe located at the stiffener (Figure 7: path (c)). For these types of cracks, no problems occur regarding visual inspection possibilities.

Figure 7.

Possible crack paths in a stiffener-to-deck plate connection: (a) through deck plate crack initiated at weld root, (b) through deck plate crack initiated at weld toe, (c) through stiffener crack initiated at weld toe and (d) through weld bead crack initiated at weld root.

LEFM implementation in finite element models

General

To evaluate the fatigue life of steel bridges, stress cycles due to traffic are calculated with detailed finite element (FE) models. Nevertheless, when LEFM is introduced, standard FE models cause many practical and computational problems, especially if automatic crack propagation is required. Therefore, two-level numerical models can offer a solution (Kiss and Dunai, 2002). The ‘first level model’ or ‘full-scale model’ is the uncracked bridge deck. It is used to locate the areas of possible crack initiation. For this research, the area of interest is the stiffener-to-deck plate connection, at the span centre between two crossbeams. The ‘second-level model’ or ‘small-scale model’ is a more detailed model of a small part of the bridge. This second model significantly reduces the complexity of the model and reduces the number of degrees of freedom. With such a model, it is easier to determine all the necessary LEFM parameters for the fatigue assessment. However, modelling and evaluating numerous crack front sizes and shapes can still be a limitation. In such cases, extended finite element models (XFEM) can offer a solution. When using XFEM, it is possible to evaluate automated crack propagation, with all its LEFM calculated parameters, without adapting the initial model and corresponding mesh for every crack propagation step. Nonetheless, the downside of this is the heavy computational effort. In order to have realistic and accurate results, the mesh in the vicinity of the crack tip has to be sufficiently small to realistically capture the stress peak. In addition, the larger the crack tip, the more the degrees of freedom are generated. Due to the latter, XFEM simulations are often limited to simulating a small crack front when considering automatic crack propagation methods. This is, however, not a problem, as the XFEM calculations can give a clear visualization of the initial crack propagation up to stable crack growth.

Combination of an FE model and XFEM

For the ‘full-scale model’, an FE model was made based on the dimensions of the Temse Bridge in Belgium (Figure 8). This bridge deck, a movable truss across the river Scheldt, is part of an important transport route for heavy lorries to the Port of Antwerp. The Temse Bridge was built in 1955, the movable part having initially an aluminium deck plate. After many repairs, the movable part was rebuilt in 1994 using an OSD. The Temse Bridge has a span of 53.90 m and a width of 7.00 m. The deck plate of the bridge is only 12-mm thick, and the closed trapezoidal stiffeners are 8-mm thick. The stiffeners are 350-mm high, 300-mm wide at the top and 90-mm wide at the lower soffit. The distance between the longitudinal stiffeners equals 300 mm. An overview of all these dimensions is given in Figure 9. In addition, no asphalt layer is installed on this bridge deck, but only a thin epoxy layer. For this reason, no load spreading is taken into account when introducing the wheel loads on the model.

Figure 8.

Part of the full-scale FE model of the Temse Bridge seen from below.

Figure 9.

Dimensions of the OSD of the Temse Bridge.

In 2004, a fatigue crack with a length of 600 mm was detected in this bridge deck at a stiffener-to-deck plate connection at the span centre between two crossbeams. The main reason for the initiation of this crack is the occurrence of multiple welded joints intersecting each other (Van Bogaert and De Backer, 2008). In addition, these welds were not chamfered and even the prescribed degree of lack of penetration can be questioned here. In this particular case, the crack propagated through the deck plate starting from the weld root.

The FE model is developed using the FE software LMS Samcef Solver Suite (Siemens, 2014). This model consists of a combination of shell elements for the deck and beams for the truss. The advantage of using a full FE model is having boundary conditions as close as possible representing the real structure. All the wheel loads are on the right-hand side of the detail being considered due to the majority of trucks driving at the centre of the traffic lane. The wheel loads are located between longitudinal stiffeners, just next to the weld location. Fatigue load model 4 of Eurocode is used for the traffic loads (CEN/TC 250, 2004). The Temse Bridge has a known traffic intensity of 1.4 E + 6 lorries each year for which the lorry percentage of medium distance is used. This latter choice corresponds with the traffic measurements by the authorities (Agenschap Wegen en Verkeer, 2004).

In a next step, this model is linked to a ‘small-scale model’ which is a much more detailed XFEM. This XFEM is a small piece of a stiffener-to-deck plate connection, made out of volume elements (Figure 10). The XFEM used is developed by Cenaero (2015) and is incorporated in the FE software Samcef (Siemens, 2014). To link both models together, the displacements from the larger FE model are introduced as boundary conditions on the edges of the XFEM. As mentioned before, thicker plates are often chosen by designers when trying to increase the fatigue life of the OSD. To validate whether this is justified, the dimensions of the Temse Bridge are also modified using different stiffener web and deck plate thicknesses. This allows for comparing the different possible design choices.

Figure 10.

Small-scale XFEM of the stiffener-to-deck plate detail.

Evaluation of LEFM parameters within the XFEM

Fatigue crack growth is mainly defined by three different phases: crack initiation, crack propagation and crack failure. In general, the fatigue crack initiation period is much longer than the fatigue crack propagation period. It can sometimes be up to 90% of the total fatigue life. However, when using the LEFM method, only the crack propagation stage at macro-scale can be evaluated. A group of multi-scale crack models have been developed in the last decade by Sih and Tang (Tang, 2014; Tang and Peng, 2015; Tang and Wei, 2015) which take the continuous process of crack initiation at micro-scale and crack propagation at macro-scale into account.

When considering welded connections, the crack initiation phase can be relatively small. Sometimes, it can even be eliminated due to welding defects, partially rewelded tack welds, stress concentrations and residual stresses being present. Lack of penetration, lack of fusion, slag inclusions and porosity are some of the possible welding defects. Due to these welding defects and especially the residual stresses, it is commonly assumed that in welded connections the crack initiation phase can be neglected (Acevedo and Nussbaumer, 2010; Zerbst et al., 2012). For this reason, only the crack propagation stage is evaluated using the LEFM method. Therefore, the total fatigue life always refers to only the crack propagation stage for this research.

Another very important factor in LEFM calculations is the initial crack length. The choice of this length could have a large influence on the total fatigue life of the structure. Depending on the welding detail, construction technologies used, lifetime of the structure and so on, an initial crack between 0.1 and 1 mm is often chosen (Polák, 2003; Zerbst et al., 2012). However, in this case, the longitudinal ribs are welded on one side without knowing the exact penetration depth. When looking, for example, at some weld macros of recently executed stiffener-to-deck plate welds, relevant lack of penetration can be noticed (Figure 11). This lack of penetration is a perfect initial crack with a length of approximately 1.5 mm. However, it is possible that a bit further in the welding detail almost full penetration or excessive full penetration could be available (De Backer, 2006; De Backer et al., 2013). Therefore, it is assumed in the XFEM that the welding detail is shaped perfectly, with full weld penetration. Afterwards, a defect is added inside the material as crack initiator. This initial crack is limited in size. This implies that a semi-elliptical shape is chosen with a half-length a of 1.5 mm along the minor axis and a half-length c of 3 mm along the major axis (Figure 12). More in particular, the minor axis is oriented perpendicular to the stiffener wall and parallel to the deck plate. Consequently, the major axis is oriented according to the welding direction.

Figure 11.

Weld macros of a stiffener-to-deck plate connection executed by a Belgian steel constructor in 2009 (deck plate = 12 mm, stiffener = 6 mm). Left: unwelded root gap of 1.33 mm; right: unwelded root gap of 1.88 mm.

Figure 12.

XFEM crack simulation until failure including the initial semi-elliptical crack length for a longitudinal stiffener of 8 mm and a deck plate of 12 mm. Blue line: longitudinal crack growth direction; orange line: transversal crack growth direction.

When the initial crack is implemented in the XFEM, the Paris crack propagation law is used to simulate the crack growth according to the path that uses the least energy to crack

\frac{da}{dN} = C \cdot {(Δ K_{I}^{eff})}^{m}

(1)

The parameters C and m are material properties. For structural steel, C equals 3 × 10⁻¹³ (for da/dN in mm/cycle and ΔK in N/mm^3/2) and m equals 3 (–) (Maljaars et al., 2012). For the assessment of existing (old) steel bridges, C = 4 × 10⁻¹³ and m = 3 could be used (Kühn et al., 2008). The stress intensity factor (SIF) $Δ K_{I}^{eff}$ in this equation is a function of the three main SIF modes ΔK_I, ΔK_II, ΔK_III, the bifurcation angle θp and the used material (Siemens, 2014)

Δ K_{I}^{eff} = \sqrt{K_{I}^{2} + K_{II}^{2} + \frac{(1 + ν)}{(1 - ν^{2})} K_{III}^{2}}

(2)

The parameter ν in this equation is Poisson’s ratio. In addition, the SIF values are generally expressed as a function of the applied load or stress range Δσ, the crack size a and the geometry-dependent parameter f(a)

Δ K_{I}^{eff} = f (a) \cdot Δ σ \cdot \sqrt{π \cdot a}

(3)

Knowing the calculated SIF values from equation (2) and the applied stress Δσ from the full-scale model, the geometry-dependent parameter f(a) can be evaluated using equation (2) for every crack propagation step.

The applied stress variation Δσ from this equation is derived from the full-scale model according to Eurocode (CEN/TC 250, 2005). Therefore, the designer can use these equations in the same manner as using S-N curves. The main difference being that besides the bending stresses, the normal stresses are also taken into account. The location of the resulting stress peaks should closely match the expected ones and the examined initial crack location. In addition, the stress peaks should be derived from the deck plate or the longitudinal stiffener, depending on which element the initial crack is located. In this particular situation, the derived stress from the bending moments and the normal forces is the one at the weld root on the stiffener web. Furthermore, the initial stress state is of major importance when using LEFM. Therefore, the dead load should be added to the model. This opposes the use of S-N curves, the applied stress cycle being the difference between the loaded and unloaded situation of the bridge deck, thus eliminating the initial stress state due to dead load. In reality, the initial stress state can influence the stress trajectories inside the material, thus influencing the crack propagation angle in LEFM simulations.

The geometrical parameter f(a) in equation (3) depends on the crack length a, as well as on the overall dimensions of the bridge deck. Therefore, once this parameter is evaluated for a particular welding detail using a particular configuration of longitudinal stiffeners and transverse web spacing, this can be used for several other bridges with comparable characteristics. In addition, with this geometrical parameter, both the total fatigue life of new bridges and the remaining fatigue life of existing bridges could be estimated, if the initial crack length is given. The total fatigue life can be calculated with the following equation

N_{f} = \int_{a_{i}}^{a_{f}} \frac{da}{C \cdot Δ K_{I}^{{eff}^{m}}} = \int_{a_{i}}^{a_{f}} \frac{da}{C \cdot {(f (a) \cdot Δ σ \cdot \sqrt{π \cdot a})}^{m}}

(4)

In this equation, a_i is the initial crack length and a_f is the final crack length at failure. Finally, the variation of the crack length in time can be derived from equation (4)

a = \sum_{i} C \cdot {(f (a_{i}) \cdot Δ σ_{i} \cdot {\sqrt{π \cdot a}}_{i})}^{m} \cdot dN

(5)

It is assumed that the fatigue life has been reached when the crack has grown through half the thickness of the deck plate or through the web of the stiffener. It is believed that for these types of welding details, already sufficient integrity of the welding detail has been lost at that point (Bignonnet et al., 1990, 1991; Darcis et al., 2004). Figure 12 illustrates a crack propagation through the stiffener web until the crack front reaches the half thickness line t/2. Due to the irregular shape of the crack, it is therefore possible that the final crack length is longer than the actual half thickness value.

Results

Sensitivity check

Although XFEMs are very practical considering the automatic mesh refinement at every crack propagation step, the chosen mesh size can have a large influence on the accuracy of the calculated SIF values. Therefore, it is necessary to perform a sensitivity check on the XFEM to find an acceptable equilibrium between the mesh size, the accuracy of the SIF values and the computational effort. The XFE software tool Samcef (Siemens, 2014) is used for simulating the crack propagation and the LEFM parameters. The advantage of this software tool is the built-in automatic refinement option as a function of the chosen refinement radius, the crack advancement da and the crack length. With this option enabled, the mesh size in the vicinity of the crack front should always be sufficiently small to obtain accurate results. As a test, two different overall mesh sizes were used for generating the volume elements in the XFEM: 1 and 2 mm. Figure 13 illustrates the geometrical parameter f(a) as a function of the crack length a for both mesh sizes. As can be noted, the initial mesh size does not really affect the results, indicating that the automatic refinement tool is reliable. Taking into account the available computational power, a mesh size of 1 mm is used for further calculations.

Figure 13.

Sensitivity check of the XFEM simulation for an overall mesh size of 1 and 2 mm. S: stiffener (number: thickness of the stiffener); D: deck plate (number: thickness of the deck plate); 1 mm/2 mm: chosen overall mesh size.

LEFM parameters

As previously mentioned, the evaluation of the geometry-dependent parameter f(a) is of major importance quantifying the fatigue behaviour of the detail being studied. As illustrated in Figure 12, the XFEM calculations result in a detailed visualization of the crack propagation behaviour through the stiffener web. For every time step and curvilinear location along every crack front, the SIF values are determined. Using equation (3), the geometry-dependent parameter f(a) is calculated. With the used XFEM techniques, the crack propagation could be simulated up to a half-length of approximately 24 mm in the longitudinal direction, which is actually the welding direction. To investigate larger crack fronts, larger initial crack fronts, complying with the final crack growth direction of the automatic XFEM simulation, are evaluated with a half-length of, respectively, 50, 100, 150 and 200 mm. It is assumed for these larger crack sizes that the crack has already fully penetrated the plate thickness. Therefore, the crack front inside the material is reduced and less degrees of freedom are necessary. In addition, the larger crack fronts are evaluated without automatic crack propagation. Therefore, only the SIF values are calculated at the additional initial inserted crack front. However, the automatic refinement is still used to increase the accuracy of the results.

In the transversal direction, cracks could be evaluated up to approximately half the thickness of the longitudinal stiffener (Figure 12), which is the assumed point of failure. Because not all f(a) curves are simulated with XFEM techniques up to a complete half thickness crack, the SIF values for these curves are evaluated at more points until complete crack penetration. This is done in a similar manner as the additional initial inserted longitudinal crack fronts. The reason why crack fronts up to fracture are used relies on the used best-fit calculation. To have an accurate best-fit curve and therefore an accurate simulation of the crack propagation, sufficient points are needed beyond the neutral axis.

To assess the influence of plate thicknesses of the deck plate and the longitudinal stiffeners, different plate thicknesses are used in the calculations: 6, 7 and 8 mm for the longitudinal stiffener and 12, 14 and 16 mm for the deck plate. Figure 14 illustrates all the calculated transverse f(a) values for these thicknesses. The results for the longitudinal direction are illustrated in Figure 15.

Figure 14.

Geometry-dependent parameters f(a) for different deck plate and stiffener thicknesses in transversal direction. Graphs (a), (c) and (e): f(a) curves grouped per stiffener thickness. Graphs (b), (d) and (f): f(a) curves grouped per deck plate thickness. S: stiffener (number: thickness of the stiffener); D: deck plate (number: thickness of the deck plate).

Figure 15.

Geometry-dependent parameters f(a) for different deck plate and stiffener thicknesses in longitudinal direction. Graphs (a), (b) and (c): f(a) curves grouped per deck plate thickness.

For the evaluation of the fatigue life and the corresponding crack length, a best-fit curve has been fitted to these data points. Using curve-fitting tools, the highest precision could be achieved with the following equation

f (a) = \frac{p_{1} \cdot a^{5} + p_{2} \cdot a^{4} + p_{3} \cdot a^{3} + p_{4} \cdot a^{2} + p_{5} \cdot a + p_{6}}{a^{5} + q_{1} \cdot a^{4} + q_{2} \cdot a^{3} + q_{3} \cdot a^{2} + q_{4} \cdot a + q_{5}}

(6)

The p and q values in this equation are constants that are determined by curve fitting. In addition to the plotted best-fit curves in Figures 14 and 15, Table 1 gives the corresponding R² values. A clear trend is visible and the best-fit curves show a high reliability.

Table 1.

R² values of the best-fit curves for both the transversal and longitudinal directions.

Stiffener-to-deck plate detail	S6_D12	S6_D14	S6_D16	S7_D12	S7_D14	S7_D16	S8_D12	S8_D14	S8_D16
R ² (transversal)	0.9994	0.9978	0.9998	0.9947	0.9969	0.9994	0.9999	0.9991	0.9993
R ² (longitudinal)	0.9861	0.9943	0.9965	0.9895	0.9985	0.997	0.9921	0.9986	0.9941

S: stiffener (number: thickness of the stiffener); D: deck plate (number: thickness of the deck plate).

In the f(a) curves for the transversal direction, it is clear that the deck plate thickness has no major influence on the profile of the curve. The curves are also almost identical except for a stiffener thickness of 6 mm. It has been noted that although higher values of f(a) result in faster crack propagation, these higher values only apply to thicker deck plates. In these deck plates, the stresses at the weld root are much smaller and will therefore result in lower crack propagation than for thinner deck plates. When evaluating the curves for equal deck plate thickness, it is clear that the longitudinal stiffener thickness does not really affect the shape of the curve until the neutral axis. Cracks within thinner stiffeners seem to propagate faster than thicker stiffeners. In addition, only for deck plates of 16 mm, the curves do not coincide. The same conclusions can be drawn from the f(a) curves in the longitudinal direction. The only difference is that all the curves are practically the same until a certain crack length. When the crack increases, larger changes are visible. This is especially true for different stiffener thicknesses with a deck plate thickness of 16 mm.

Finally, the difference between the three wheel loads (category A, B and C) according to load model 4 of Eurocode (CEN/TC 250, 2004) is evaluated to investigate whether f(a) values are influenced by the used wheel prints. From these calculations, it was clear that the wheel print does not influence the shape of the curve. Only a small shift was visible, this being due to larger wheel prints tending to spread the loads. These results are taken into account for further fatigue life calculations.

Fatigue life

When combining the best-fit curves for f(a) with equation (5), the crack length as a function of time can be plotted. Figure 16 illustrates the fatigue life up to a service life of 210 years. The curves in this figure are limited to the maximum crack length that could be simulated with the XFEM or to the half thickness crack length in case the XFEM calculation needed to be extended with the larger crack fronts. In addition, the abscissa in this figure is expressed in years, which is based on the known traffic intensity of the Temse Bridge of 1.4 E + 06 lorries each year. It can be concluded that the fatigue life seriously increases while increasing the deck plate thickness. This is mainly because a thicker deck plate reduces the weld root stresses. In addition, thicker deck plates are much stiffer and are capable of spreading the wheel loads over multiple longitudinal stiffeners. In Table 2, the fatigue life in years is summarized for different configurations. As mentioned before, this corresponds with a fatigue crack reaching half of the stiffener thickness. For the lower deck plates, a rather low fatigue life can be found. For the configuration of the Temse Bridge (S8_12) a service life of 7.41 years can be found. This is remarkably close to the real service life where a large fatigue crack was found after 10 years. This indicates that XFEMs are capable of predicting the fatigue life very accurately. At lower deck plate thicknesses, it can also be concluded that the lower the stiffener thickness, the larger the fatigue life becomes. The opposite is true when the deck plate increases. In that situation, a larger stiffener thickness increases the fatigue life. Therefore, the thicker the deck plate or the stiffener, the longer the fatigue life of the stiffener-to-deck plate detail. But this increase in thickness is of course not preferable in OSDs. To maintain the advantage of a lightweight structure, an optimum should be found between the required amount of steel and the increase in fatigue life. In this specific case, adding a thin asphalt layer would also reduce the weld root stresses and increase the total fatigue life of the detail.

Figure 16.

Crack propagation in time for different deck plate and stiffener thicknesses in transversal direction. Dotted reference lines S6, S7 and S8 indicate the point of failure for stiffener thicknesses of 6, 7 and 8 mm, respectively.

Table 2.

Fatigue life and corresponding transversal and longitudinal crack lengths for different OSD geometries.

Stiffener-to-deck plate detail	S6_D12	S6_D14	S6_D16	S7_D12	S7_D14	S7_D16	S8_D12	S8_D14	S8_D16
Fatigue life (years)	9.29	24.85	60.19	7.11	33.98	96.09	7.41	40.13	210.08
Transversal crack length a at failure (mm)	3.50	3.50	3.50	3.90	3.90	3.90	4.32	4.32	4.32
Longitudinal half-length c at failure (mm)	10.08	8.85	10.11	11.38	12.83	14.43	13.22	16.77	19.55

S: stiffener (number: thickness of the stiffener); D: deck plate (number: thickness of the deck plate).

Finally, the longitudinal crack propagation is illustrated in Figure 17. Identical to the remarks on Figure 16, the curves within this figure are limited to the same point of service life as Figure 16. First, the crack propagates much faster longitudinally for thin deck plates. In addition, the difference in crack propagation is noticeable for a deck plate of 12 mm. The thicker the deck plate, the lower the difference in crack propagation when different stiffener thickness is evaluated. When looking at the curves with the same stiffener thickness, the influence of the deck plate thickness is very clear. Therefore, when looking at the longitudinal crack propagation, only the deck plate plays an important role. Thicker deck plates increase the fatigue life of the detail. Identical conclusions are valid for the transversal crack propagation. In Table 2, the longitudinal half-length c of the semi-elliptical crack is given at the end of the fatigue life. As can be seen, the longitudinal crack lengths are not large yet. But Figure 17 illustrates that the defect at this point is already sufficient large for increased crack propagation speed. Larger through thickness cracks are expected within a short time.

Figure 17.

Crack propagation in time for different deck plate and stiffener thicknesses in longitudinal direction.

Discussion

Crack path

All previous fatigue cracks propagated through the weld bead as illustrated in Figure 12. This is different from the observed crack in the Temse Bridge, where the crack propagated through the deck plate. However, fatigue cracks in the longitudinal weld between deck plate and stiffener wall have been observed in many other existing bridges (De Jong, 2004). The reason for this bifurcation problem is not clear. Due to the flexibility of the longitudinal stiffener during loading, bending moments will arise in the stiffener. Therefore, stress concentrations are present, both at the weld root and at the weld toe. In case the crack propagates through the weld, the crack path follows the orientation of the present principal stress, which in this case is mainly the path between the weld root and weld toe on the longitudinal stiffener. A possible explanation why the crack often grows into the deck plate could be the absence of residual stresses in the XFEM. As previously mentioned, the initial stress state can influence the stress trajectories inside the material. However, for orthotropic bridge decks, it is not clear how large these stresses are and if they are in tension or in compression. According to Connor et al. (2012), residual stresses are probably tensile yield stresses, as is generally accepted for fillet welds. However, recent studies on residual stresses in OSD details indicate yield stresses in compression (Nagy et al., 2015). Since no general agreement exists on the matter of residual stresses in the considered detail, they are not yet integrated in these models.

Initial crack length

Based on weld macros of the stiffener-to-deck plate detail (Figure 11), it has been decided to use an initial semi-elliptical shaped crack with a half-length of 1.5 mm along the minor axis and a half-length of 3 mm along the major axis. When the manufacturer could manage a higher degree of weld penetration during welding, this initial crack length may be reduced. Obviously, when the initial crack length is reduced, a longer fatigue life should be expected. To investigate this phenomenon, three different initial crack lengths are considered: 0.5, 1.0 and 1.5 mm. These are all half-lengths along the minor axis. The half-length along the major axis is always the double of the half-length along the minor axis. Figures 18 and 19 illustrate the different geometrical parameters f(a) and the crack propagation during service life, respectively. The smaller the initial crack, the higher the f(a) at the early stage of crack propagation. This is mainly due to the fact that the crack front for calculations with smaller initial cracks is closer to the surface of the inner wall of the stiffener. Closer to the surface, the stresses and therefore the SIF values are higher. When the crack length increases, the curves tend to coincide. As for the curve S8_D12_1.5 mm(a), a parallel shift is noticeable. This is due to the fact that with an initial crack length of 1.5 mm, the crack path is slightly shifted upwards within the weld geometry compared to the other initial crack lengths. Therefore, the crack front is propagating through slightly more stressed material.

Figure 18.

Geometry-dependent parameters f(a) for different initial crack sizes. S: stiffener (number: thickness of the stiffener); D: deck plate (number: thickness of the deck plate); # mm: size of the initial crack.

Figure 19.

Crack propagation in time for different initial crack sizes. Dotted reference lines indicate point of failure for the corresponding initial crack size. S: stiffener (number: thickness of the stiffener); D: deck plate (number: thickness of the deck plate); # mm: size of the initial crack.

When looking at the crack propagation in time in Figure 19 and Table 3, the fatigue life of the considered detail obviously increases with a decreasing initial crack length. The effect of this smaller initial crack is not linear. At first the fatigue life more than doubles for an initial crack length of 1.0 mm, but for an initial crack length of 0.5 mm, the increase in fatigue life is only 50%. Still, these results indicate that a smaller initial crack or a better weld penetration can seriously increase the total fatigue life of the detail.

Table 3.

Fatigue life and corresponding transversal crack lengths for different initial crack lengths.

Stiffener-to-deck plate detail	S8_D12_0.5 mm	S8_D12_1.0 mm	S8_D12_1.5 mm(a)	S8_D12_1.5 mm(b)
Fatigue life (years)	23.12	16.17	7.41	12.15
Transversal crack length a at failure (mm)	4.25	4.29	4.32	4.32

S: stiffener (number: thickness of the stiffener); D: deck plate (number: thickness of the deck plate).

In all previous calculations, it was assumed that the welding detail was perfectly welded with 100% full weld penetration. In addition, only a small defect inside the material was added at the weld root. To investigate if a larger weld defect could influence the fatigue life of that detail, XFEM was used. In this model, it was assumed that the entire weld was not fully penetrated through the stiffener wall (Figure 20). More precisely, the lack of penetration was set to 1.4 mm from the weld root. In addition, an initial crack of 0.1 mm was added at the weld root at the connection with the deck plate. This combination results in an initial crack of 1.5 mm similar to previous calculations. The results are also illustrated in the curve S8_D12_1.5 mm(b) in Figures 18 and 19. When the geometrical parameter f(a) of S8_D12_1.5 mm(b) is compared to the original curve of S8_D12_1.5 mm(a), it can be noted that the adapted model is less critical at the early stage of the crack propagation. After reaching a certain crack length, the curves are coinciding. When translating this to the crack propagation in time, the curve is not following the other curves anymore. At the early stage, there is almost no crack propagation. After 5 years, the crack suddenly increases much faster than all previous curves. Therefore, unstable crack growth is expected and a full-thickness crack will soon occur. Nevertheless, when looking at the fatigue life, it is almost doubled compared to the perfect weld with a local weld defect of 1.5 mm. This indicates that a not fully penetrated weld is not necessarily worse.

Figure 20.

Adapted XFEM for investigating the effect of lack of penetration (dimensions in mm).

Crack propagation due to compression

Using the XFEM, transversal crack lengths up to approximately half the thickness of the longitudinal stiffener could be simulated during automatic crack propagation. After this point, the crack propagation seems to stop because the crack front is at the neutral axis of the stiffener web. After this neutral axis, compression forces are present indicating that at that stage, the fatigue crack is due to compression–compression loading (R > 1). With the classical LEFM, compressive stresses are assumed to have no effect on the character of failure and the fatigue crack growth rate (Shabanov, 2005). When the crack front is under compression, it is assumed that the crack will be fully closed and Δσ_min should equal zero. This implies that negative (compressive) K_I values are ignored in equation (2). Therefore, only K_II and K_III values influence the SIF values beyond the neutral axis, resulting in rather limited crack propagation. It is the opinion of the authors that this is not realistic because it is a well-known fact that fatigue cracks can propagate under compression and that the crack growth can even accelerate (Shabanov, 2005; Vasudevan and Sadananda, 2001). For a proper suggestion of the amount of compressive K_I values that should be allowed in equation (2), more practical research is necessary for these fatigue details. This restriction in the LEFM calculations does not have any influence on the results of the present fatigue calculations when assuming that the fatigue resistance has been reached when the crack has grown through half the thickness of the deck plate.

Random versus fixed load sequences

Finally, the effect of random loads is compared to fixed load sequences. With the current fatigue design methods, the Palmgren-Miner method is used. This method is not very accurate since the load history and the load sequences do not have any effect on the fatigue resistance. With LEFM, however, it is possible to take the load sequence into account. In Figure 21, a comparison is made between a random sequence and two fixed sequences. Sequence 1-2-3-4-5 corresponds to the set of equivalent lorries described in Eurocode using the same order. This agrees with using all of the passages of the smaller lorries first and ending with all of the heavy lorries. Sequence 5-4-3-2-1 is the opposite. The random sequence used is an arbitrary choice taking into account the distribution of the lorries described in Eurocode. It is not based on detailed statistic considerations, since Figure 21 is only used as an indication of the difference between a random and a fixed load sequence.

Figure 21.

Random sequence of applied loads versus fixed sequences.

It is clear that the load sequence has an influence on the crack propagation. When first using smaller loads and ending with heavy loads, the crack propagation is much lower at the beginning when compared with first using heavy loads and ending with smaller loads. For sequence 1-2-3-4-5, the fatigue life increases up to 12.67 years, while for sequence 5-4-3-2-1, the fatigue life decreases up to 6.09 years. Remarkably, if the crack propagation is evaluated at a later stage, the curves of the random sequence and the fixed sequences meet. Therefore, in the long term, it does not seem to matter if a random or a fixed sequence is used, provided the choice does not result in early failure. This means that up to a crack length of half of the plate thickness, the load sequence can have an important role. Therefore, the load sequence can have an effect on the total fatigue life of the structure. Later during service time, the memory effect of the traffic history of the bridge deck seems to fade away.

Conclusion

Using LEFM as a fatigue assessment tool has many advantages compared to the current fatigue designs with Eurocode. It is a more in-depth method resulting in the evaluation of both the crack growth direction and the crack propagation. The advantages of this method can be combined with the implementation of the initial stress state due to dead load and residual stresses. By doing this, a more realistic crack path can be evaluated. For this research, no residual stresses were added due to the lack of information on their sign and magnitude. For all different stiffener-to-deck plate geometries studied in this article, all cracks propagated through the longitudinal weld.

When comparing the crack propagation for different geometries, it can be concluded that an increased deck plate thickness and an increased stiffener thickness improves the fatigue life of the stiffener-to-deck plate detail. This increase should, however, be limited to maintain the advantage of a light-weighted construction.

The initial crack length can have profound influence on the fatigue life of the structure. From the results, it is clear that an improved weld quality with full penetration can seriously increase the total fatigue life of the stiffener-to-deck plate detail. This conclusion applies to perfectly executed welds with small defects. When a much larger longitudinal defect is added into the XFEM, but with the same transversal defect, it was noticeable that the crack propagation at early stage is much lower. After reaching a certain crack length, the crack propagates much faster and unstable crack growth is expected. Nevertheless, it can be concluded that a non-fully penetrated weld is not necessarily a worse situation.

Finally, using the Palmgren-Miner method in current design methods does not allow for accounting for the actual load sequences. When using LEFM, such an analysis can be performed. The effect of using fixed load sequences can have a major influence on the crack propagation. This was also visible in the results up to the criterion of fatigue failure. When first all the heavy trucks pass the bridge deck, obviously the fatigue life decreased compared to a random sequence of the used trucks.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship and/or publication of this article.

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