Fatigue behaviour of slender corrugated webs subjected to plate breathing

Accepted Manuscript Fatigue behaviour of slender corrugated webs subjected to plate breathing Z.Y. Wang, T. Zhang, Y. Chen, F. Yuan, X.F. Zhou, Z.F. Liu PII: DOI: Reference:

S0142-1123(18)30590-5 https://doi.org/10.1016/j.ijfatigue.2018.12.028 JIJF 4942

To appear in:

International Journal of Fatigue

Received Date: Revised Date: Accepted Date:

27 September 2018 3 December 2018 27 December 2018

Please cite this article as: Wang, Z.Y., Zhang, T., Chen, Y., Yuan, F., Zhou, X.F., Liu, Z.F., Fatigue behaviour of slender corrugated webs subjected to plate breathing, International Journal of Fatigue (2018), doi: https://doi.org/ 10.1016/j.ijfatigue.2018.12.028

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Fatigue behaviour of slender corrugated webs subjected to plate breathing Z. Y. Wang1,2,3*, T. Zhang1, Y. Chen1, F. Yuan1, X. F. Zhou1, Z. F. Liu1 1

Key Laboratory of Deep Underground Science and Engineering (Ministry of Education), School of Architecture and Environment, Sichuan University, Chengdu, PR China 2

Key Laboratory of Performance Evolution and Control for Engineering Structures (Ministry of Education), Tongji University, Shanghai, PR China 3

*

State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian, PR China

Corresponding author at: School of Architecture and Environment, Sichuan University, Chengdu, PR China E-mail address: [email protected] (Zhi-Yu Wang)

Abstract This paper presents a study on the fatigue behaviour of slender corrugated webs subjected to plate breathing. It outlines a fatigue test of nine test girders with buckling in interactive buckling manner of the corrugated webs. Characteristic fatigue crack propagation manners and fatigue endurances of the corrugated webs are analyzed. Finite element models considering crack propagation details and out-of-plane deflection have been developed to investigate the stress intensity factor under shear and its combined modes. Proposed theoretical prediction allowing for the interacted shear and tearing fracture modes is confirmed with good degree of accuracy and its further improvement is also discussed.

Keywords: Corrugated web; shear buckling; web breathing; post-buckling; fatigue

1. Introduction Corrugated webs are well suited for bridge girders due to their adequate out-of-plane stiffness [1-2] and post buckling reserve of strength [3]; therefore, the weight of superstructures can be considerably decreased with the use of thin corrugated plates. The shear forces of such bridge girders are predominantly carried by corrugated webs as a result of their accordion effect which has been extensively concerned by contemporary researchers [4]. The shear buckling behaviour of the corrugated webs has been investigated in the previous studies, exclusively under static loading; however, its resultant failure mechanism may change under out-of-plane deflection of slender webs induced by repeated in-plane loading, which is known as “plate breathing”. Moreover, the fatigue cracking and premature failure may take place on webs under plate breathing scenarios. The concerns of plate breathing effect date back to the 1960s for stiffened slender plate girders subjected to combined bending and shear [5]. The web panel aspect ratio (i.e. width/depth=1.5) and the slenderness ratio (i.e. depth/thickness=190 and 380 for transversely and longitudinally stiffened respectively) were validated as design recommendations [6]. Later in the 1980s, the fatigue strengths of thin-walled plate subjected to out-of-plane 1

deflection and secondary bending stress were examined by finite element analysis. A representative report by Okura and Maeda [7-8] has shown that the shape of initial deflection of plate girder in shear has small effect on the increase of secondary bending stress. Also, the fatigue strength increases with the decrease of aspect ratio while it is not influenced by the initial deflection and becomes larger than corresponding buckling strength as the slenderness ratio exceeds about 200. Since the middle of 1990s, extensive research work have been conducted in the Europe towards the development of the Eurocode 3 Part 2 [9-10], including reports from the UK [11-20], the Czech Republic [21-27], Germany [28-31], Italy [32], Belgium [33] and Spain [34]. Firstly, amongst these were the identification of fatigue crack formation and its influencing factors. Based on the experimental observations from several pioneer researchers [8, 11, 13], typical crack formation on the flat webs of slender plate girders under combined bending and shear, as illustrated in Fig. 1. Fatigue cracks generally took place along the toe of the filled weld between the flat web and its adjoining boundary members. Their differences can be identified as: the crack types (2) and (3) are commonly the initiations near the tension flange due to membrane bending stresses below the neutral axis, while the crack types (1) and (4) are mostly induced by out-of-plane deflection. The crack type (4) was found by Roberts and Davies [10] to turn across the pseudo-diagonal tension field corner of the flat web and the combined tension and tearing modes are involved with its propagation. It was also expected to be pronounced when great stresses in the pseudo-diagonal tension field and the post-critical behaviour of the web became significant due to high load level as reported by Škaloud and Zörnerová [27]. Based on above reported experimental analysis, the limitation of the web slenderness and aspect ratio are further refined in Ref. [10, 26-27, 29, 32] and codified in the Eurocode 3 Part 2 [9] as: 2

2

 σ x,Ed,ser   1.1τ x,Ed,ser       1.1  k σ e   k τ σ e 

(1)

where, σx,Ed,ser and τx,Ed,ser are the stress membrane compressive and shear stresses for frequent load combination respectively. kσσe and kτσe are the linear elastic buckling stresses of an assumed simply supported panel respectively which can be referred to the Eurocode 3 Part 1-5 [35]. Additionally, the weld defects existed in the out-of-plane deflection and secondary bending stress zones may reduce the fatigue life of flat webs of slender plate girders, especially under large surface stress range [32]. On the other hand, the appearance of advanced computing technology has made it possible to simulate complex problem of web breathing. For example, Davies and Roberts [17] reported a finite element (FE) representation of the web panel with triangular elements and each boundary with bar element of proper axial, torsional and in-plane stiffness. The simulated principal surface, secondary bending and membrane stresses showed good correlation with experiments. Günther and Kuhlmann [31] performed numerical analysis of web breathing of longitudinally unstiffened and stiffened plate girders. Through a parametric study, slenderness limitations of the webs to exclude the web breathing effect have been derived for longitudinally unstiffened web panels. These results have been adopted by the Eurocode 3 Part 2 [9] as: hw 30  4 L  300 For highway bridge girders  tw 55  3.3L  250 For railway bridge girders

(2)

where, hw and tw are the depth and thickness of web respectively. L is the span length in meters but no less than 20 meters. Also, Crocetti [32] adopted FE model with eight node shell elements in modelling post-buckling behaviour of thin-walled flat webs of plate girders. It was shown that the membrane stresses at the boundary of 2

the web panel is a nonlinear function even at low load levels. Based on the experimental data, S-N relationships have also been established in relation with the Eurocode 3 Part 2 [9] for the simple fatigue analysis of web breathing of thin-walled plate girders [10, 13, 23, 29-32]. Alternatively, the fracture mechanics concepts were employed by Osman and Roberts [20] in the prediction of slender plate girder subjected to repeated shear loading. It was shown that their solution matched the experimental curves when the realistic crack size, aspect ratio and stress concentration at weld toe are specified. Also, a simple formula was developed by Skaloud and Roberts [23] in the determination of fatigue crack propagation life of welded plates subjected to breathing. In addition, as a further application for steel box girders as reported by Pateron et al. [36], the relationship between initial out-of-flatness, local stresses and fatigue life of flange plate were given and a simple prediction using S-N lines with consideration of anticipated variation of applied nominal stresses was recommended. With regard to the shear buckling of the corrugated webs, however, extensive research work has been performed in predominantly static loading manner, such as reports in Refs. [37-38, 40].

2. Problem statement and research objectives Despite significant research of above mentioned thin-walled flat webs in plate breathing, however, their fatigue behaviours have been established based on all around welded web boundary which seems to be insufficient for the understanding of the fatigue behaviour of corrugated webs with different boundaries. Problems can arise owing to the facts that: 1. Only the shear forces are actually sustained by the corrugated webs which are different from combined bending and shear forces carried by the flat webs of conventional I-section plate girders. The primary fatigue cracks can be expected to be induced by shear forces and distributed vertically along the web depth as shown in Fig. 2(a) which are similar to the crack type (4) as plotted in Fig. 1 for flat webs of conventional I-section plate girders. 2. Apart from global buckling mode with tensile membrane stresses and anchorage in surrounding flanges and transverse stiffeners, the local buckling mode or its related interactive buckling mode of the corrugated webs may arise owing to the out-of-plane deflection of slender webs within each subpanel or interfering with adjacent multiple folds respectively. The pseudo-diagonal tension field is thus greatly anchored at the vertical intersection line between the longitudinal fold and the inclined fold. As a result, the buckled parts of longitudinal fold of the corrugated web are more prone to deflect about such an intersection line, as shown in Fig. 2(b), instead of the web side of the weld for flat webs in conventional I-section plate girders. 3. Buckling and out-of-flatness of compression flange have great effect on the out-of-plane deflection and the ultimate load of slender webs [27, 36]. In other words, the buckling strength of corrugated web girders is also influenced by flange transverse bending [40]. This is evidenced from a photograph as shown in Fig. 3 in which the slender flange plate of an exemplified corrugated web girder loses interacting instability due to transverse bending. As a result, the corrugated web girder is likely to be suffered from undesired failure of flange plate buckling prior to the occurrence of web breathing. On the other hand, for corrugated web girders under pure bending and under combined bending and shear, considerable concerns have previously been given by several researchers [3] and the authors [39] on the fatigue damage of web-to-flange welds. It is noted that in all these reports, the fatigue cracks are initiated from the weld 3

toe of the tension flange to corrugated web weld and the weld size becomes a decisive factor, while the corrugated webs remain intact without involvement of shear buckling. Consequently, the reported fatigue life results of these girders are to some extent limited to relative stocky webs with vulnerable web-to-flange welds since they do not take into consideration phenomenon of slender webs which undergo elastic buckling first and then yield in the post-buckling stage. Given there has still been a dearth of knowledge concerning the slender corrugated webs subjected to fatigue induced plate breathing to the author’s best knowledge, it is therefore the purpose of this study to further the understanding of this topic in the literature in terms of: 1. Verifying appropriate fracture analytical solution under pure shear condition for slender corrugated webs in shear. 2. Performing experimental test based fatigue failure analysis on the post-buckled slender corrugated webs in girders shaped like a dumbbell with double concrete filled RHS tubes (DTG). This configuration is adopted so as to isolate the corrugated web in shear buckling as illustrated in Fig. 4 [38] while the fatigue failure of welds and local flange buckling are noncritical. 3. Evaluating characteristic stress distribution and fatigue life of corrugated webs with related data of the flat webs of slender plate girders and comparing with well established design S-N curves. 4. Examining applicability of fracture mechanics approach in the prediction of crack propagation life of breathing corrugated webs with specified stress intensity factors under predominant shear mode or interacted mode.

3. Plate fracture characteristics under pure shear Since the corrugated webs in girders is predominantly subjected to shear force, it necessary to study the variation of stress intensity factors under pure shear state on the original crack plane. A well established linear elastic fracture mechanics procedure related to the stress field in the vicinity of a crack tip is briefly reviewed in this section. To characterize the fracture of a central crack 2a under pure shear state, the elastic stress field in Cartesian coordinates near the crack tip [41-42] as shown in Fig. 5 can be expressed by:

σx  

(3)

τ a θ θ 3θ  sin  2  cos cos  2 2 2 2r

σy 

τ a θ θ 3θ sin cos cos 2 2 2 2r

(4)

τ xy 

τ a θ θ 3θ  cos 1  sin sin  2 2 2 2r

(5)

where, a and r are the crack length and the distance from the crack tip respectively. θ is the angle in the polar coordinates based on the right-hand crack tip. The corresponding polar stress components can be rewritten in the form as:

σr 

(6)

τ a θ θ  sin 3cos 2  2  2 2  2r

4

σ   τ r 

(7)

1.5τ a θ sincos 2 2r

(8)

τ a θ cos 3cosθ  1 2 2 2r

The polar stress components near the crack tip can then be utilized for determining two stress intensity factors. The opening mode (KⅠ) which is characterized by local displacement of two fracture surfaces perpendicular to each other in opposite directions and the shear mode (KⅡ) which is characterized by two fracture surfaces sliding over each other in a direction perpendicular to the line of the crack tip. The calculation of KⅠ and KⅡ satisfy the boundary condition at the crack tip, namely:

 K (θ )  lim  τ

 2r 

KⅠ(θ )  lim σ  2r r 0

Ⅱ

r 0

r

(9)

By substituting Eq. (7) and Eq. (8) into Eq. (9) respectively, KⅠ and KⅡ in terms of θ can be obtained as:

3 θ KⅠ(θ )   τ asinθcos 2 2 1 θ KⅡ(θ )  τ a  3cosθ  1 cos 2 2

(10)

The maximum KⅠ and KⅡ can be obtained by integrations with respect to θ as:

KⅠ(θ ) 0 θ KⅡ(θ ) 0 θ

(11)

Substitute Eq. (9) into Eq. (11) yields:

3 θ θ 1 τ a  sinθsin  cosθcos   0 2 2 2 2 2  θ  θ sin 9  cos   2   0 2   2 

(12)

which reduces immediately to :

KⅠ,max

θ 70.42

KⅡ,max

θ 0

 1.155τ a

(13)

 τ a

Above calculated maximum KⅠ(θ) and KⅡ(θ) at θ=±70.42˚ and θ=0˚ respectively can be also evidenced from a comparison of normalized stress intensity factor with θ varied from -90˚ to 90˚, as shown in Fig. 6.

5

4. Description of fatigue test 4.1 Test specimen details As already mentioned, the fatigue failure of corrugated web in shear buckling is the focus of the study so that the test corrugated webs were designed with relatively high slenderness ratio while the flanges and its related web-to-flange welds were adequately designed and are noncritical. A typical a corrugated web girder is configured as a corrugated web welded to the upper flange and lower flange, as shown in Fig. 7. The width and overall thickness of the flanges are denoted as bf and tf respectively. For the rectangular hollow section (RHS), tf is the overall flange depth, tst,t, and thickness of concrete infill, dcon,t. The length of a single wave of the corrugated web consists of parts of the longitudinal fold width (bl) and the inclined fold width (bi) at an angle of θc which is also called as corrugation angle. The fold widths are designed allowing for the fact that the longitudinal fold and the inclined fold are reciprocally restrained to each other. The longitudinal fold with relatively larger bl becomes critical for local shear buckling as the inclined fold of relatively smaller bi has greater stiffness against buckling and vice versa. As a result, the longitudinal fold is likely to suffer greater shear buckling induced damage. Furthermore, the predicted shear buckling strength is expected to be underestimated using classical shear strength formulae for the case when bl and bi are not equal to each other. In this regard, an equal fold width, i.e. b=bl=bi, was accounted for in the test specimens to facilitate a contrast with general design consideration which has also been adopted by most researchers, such as reports in Ref. [37-38, 40]. The depth and the thickness of the corrugated webs are denoted as hw and tw respectively while the depth of corrugation is denoted as hr. The girders with double concrete filled RHS tubes (DTGs) were considered in the test; moreover, only one girder with flat steel plate flange (DFG) was fabricated for a comparison. The corrugated webs in the test girders were designed to represent much lower shear buckling strength than shear yield strength which can be, in other words, regarded as excluding overall yielding of the corrugated webs. Furthermore, it was accounted that the designed strengths for the local buckling mode or its related interactive buckling mode were much higher than those for global buckling of the corrugated webs, in which the latter is involved with much lower post-buckling strength and recommended to be precluded for design in engineering [37-38]. The geometric details of test girders are tabulated in Table 1 in which the design parameters for corrugated webs are listed as: panel aspect ratio (0.5lw/hw), panel slenderness ratio (hw/tw) and corrugation depth-to-thickness ratio (hr/tw). Out-of-flatness of each subpanel of the corrugated webs was carefully measured prior to tests. The corrugated webs were welded to the tubular or flat plate flanges with filled welds using CO 2 shielded semi-automatic gas metal arc welding. For the web juncture with the flanges, the fillet welds were designed with the leg length of 3.5 mm which is almost three times larger in geometry with respect of the thickness of the web so as to reduce the maximum stress over the sections of the welds susceptible to fatigue. Similar welding joining was performed for 8mm thick transverse stiffeners at the centre of span and reaction. For the purpose of comparison, reported data from the UK [12-17] and the Czech Republic [22-27] for breathing webs in DFGs are also included in Table 1 which are named as Cardiff tests (Ref. CDFGs) and Prague tests (Ref. PDFGs) respectively. Compared against reported test data, two groups of dimensions were chosen, i.e. hw/tw=323 for F-DTG-140-1.1 and hw/tw=479 for F-DTG-75-0.7 which are close and 20% higher to corresponding range of Cardiff tests respectively. Mechanical properties of the corrugated webs were determined from three coupon tests as listed in Table 2. The notations of σ, τ, E, F and N stand for uniaxial tensile stress, shear stress, elastic modulus, applied force and fatigue life respectively while the subscript ‘y’ indicates that the value correspond to yield. Likewise, the uniaxial tensile yield stress for steel tube and cube compressive strength for concrete infill 6

were obtained as 327 MPa and 56 MPa respectively.

4.2 Test set-up and instrumentation As shown in Fig. 8, all test girders were simply supported with smooth flange surface at loading and reaction locations. To avoid lateral torsional buckling phenomena, the test girders were laterally braced accordingly. A central concentrated load was applied at the top surface of the upper flange above a double transverse stiffener using a servo-controlled hydraulic testing machine of 200kN capacity. The resultant quasi-static shear force for left or right panel can thus be obtained as a half applied load. The test girders were loaded under sinusoidally varying loads with the range between Fmin and Fmax (as listed in Table 2) at a frequency of 6 Hz. The fatigue test was proceed until structural failure which is represented by quickly increased midspan deflection but the applied load of Fmax is unable to be sustained by the test girder. Deflections of the test girders were recorded using four linear variable differential transformers (LVDTs) labelled as DTM, in which average measuring value of DTM1 and DTM2 is used for midspan deflection while the measuring value of DTM3 and DTM4 is used for correcting the support deformation, as shown in Fig. 8. Out-of-plane deflection was primarily recorded on the second longitudinal fold of the corrugated web away from the support using a specially fabricated potentiometer bar as referred to Ref. [32]. A digital camera was used in monitoring the initiation and propagation of the cracks during the fatigue loading progress. Strains were measured as the test specimen was loaded monotonically at the interval of every 0.2 million cycles. The strain distributions are mainly concerned at the first and second longitudinal folds of the corrugated web away from the support which are expected to undergo considerable out-of-plane deflection. Three element rosettes were placed with their centres at the middle and corner of the pseudo out-of-plane deflection pattern which is defined as the acute angle of 60˚ between the pseudo-diagonal and horizontal axis as observed from monotonic loading trial tests, as shown in Fig. 9. From theoretical point of view, the local shear buckling stress of the corrugated webs, τL,el, can be generally expressed in line with the classical buckling theory of a rectangular plate under identical shear loads acting on surrounding sides [37-38]. τcr for the corrugated webs can be written as:  cr 

kL 2 Es  tw    12(1   2 )  b 

2

(14)

where, kL is the buckling coefficient which can be applied to the boundary condition of all sides simply supported as:  b  kL  5.34  4    hw 

2

(15)

Based on Eq. (14), τcr for refer breathing flat webs in DFGs from Cardiff tests (Ref. CDFGs) [12-17] and Prague tests (Ref. PDFGs) [22-27] can also be calculated when b is replaced by 0.5lw and kL is chosen for all sides clamped case which can be given by:  0.5lw  kL  8.98  5.6    hw 

2

(16)

Accordingly, τcr for test corrugated webs and refer flat webs are listed in Table 2. The applied shear load levels can be accounted as the ratios of the nominal membrane shear stress range, Δτavr (a half applied load range, 0.5ΔF, divided by the cross-section area of the web) to the calculated values of τcr. 7

5. Fatigue test results 5.1 Failure phenomenon and formation of fatigue cracks As expected, the problem of flange buckling induced by interacting instability for aforementioned test No. F-DFG as shown in Fig. 3 was eliminated for all test girders with double concrete filled RHS tubes (DTGs) which possess substantial flange stiffness. Moreover, no fatigue cracks in the web-to-flange welds of test girders were identified through non-destructive crack inspection and monitoring after the completion of experiments. Rather, only the corrugated webs exhibited obvious fatigue failures, i.e. the local parts close to the support were buckled first and then propagated to the rest parts away from the support. In other words, all DTGs in the fatigue tests demonstrated significant plate breathing along with significant shear buckles occurring and recurring along the diagonal tension field of the subpanel of the corrugated webs close to the support. Typical fatigue cracks observed in the tests can be schematically plotted in Fig. 10 from a summary of crack distributions recorded for three exemplified test specimens F-DTG-140-1.1 as shown in Fig. 11 and F-DTG-78-0.7 as shown in Fig. 12. In general, observed fatigue cracks formed along the boundaries of the pseudo-diagonal tension field induced by out-of-plane deflection of the subpanel of the corrugated webs. As shown in Fig. 10, the measured elliptical out-of-plane deflection contour shaped by orthogonal distributed depth (Sse) and width (Sle) whose angle to the horizontal labelled as θodc is approximated to 61.4˚ as listed in Table 3. Correlated fatigue cracks with out-of-plane deflection are observed in isolated and combined manners. The former, as shown in Fig. 10(a), consists of three isolated crack types as:

·

Type ⅰ crack is initiated in the vicinity of the upper or lower corner of the out-of-plane deflection contour in juncture with the vertical intersection line between the longitudinal fold and the inclined fold of the corrugated web. Its propagation under predominant tension action, i.e. fracture mode Ⅰ, within the length of aⅰ is about the vertical by approximately θⅰ=69.3˚ which is close to the case of θ=70.42˚ for maximum KⅠ(θ) as obtained from Eq. (13).

·

Type ⅱ crack is initiated along the vertical intersection line between the longitudinal fold and the inclined fold of the corrugated web. Its propagation vertically within the length of aⅱ due to predominantly fracture mode Ⅱ shear condition for maximum KⅡ(θ) which is in line with the case of θ=0˚ as deduced from Eq. (13).

·

Type ⅲ crack is initiated at the central part of the elliptical out-of-plane deflection contour due to its resultant local bending stresses. Successively, it is propagated along the depth (Sse) and width (Sle) of the elliptical out-of-plane deflection contour whose direction can be referred to as θ=θodc.

The latter, as shown in Fig. 10(b), consists of two combined crack types as:

·

·

Type C1 crack is a combination of the cracks in the type ⅰ and the type ⅱ since the direction of crack propagation rotates from the vertical intersection line between the longitudinal fold and the inclined fold of the corrugated web to an inclined direction. The external form of this type of crack is similar to the crack type (4) as plotted in Fig. 1; however, its appearance is expected to be owing to the significant concentration of shear stresses at the crack tip and the shear stress is transferred due to crack opening instead of the corner of welded web for crack type (4). Type C2 crack is a combination of the cracks when the bending stresses become significant along orthogonal axes of elliptical out-of-plane deflection contour. The simultaneous occurrence of these cracks in a central zone of elliptical out-of-plane deflection contour. The web in this zone is flexible and thus sensitive to plate breathing.

As shown in Fig. 11 and Fig. 12, the subpanels of the corrugated web closer to the support exhibit more 8

significantly fatigue failure modes due to plate breathing. It seems that the longitudinal folds are subjected to more significant propagation of fatigue cracks than the inclined folds of the corrugated web since the former parallel to the longitudinal stress of test girder are likely to undergo more deflection than the latter. Moreover, it can be seen that aforementioned crack types corresponding to failed corrugated webs can be primarily due to the fractures in the opening mode (Ⅰ) and the shear mode (Ⅱ). The crack propagation behaviour of the cracks in fracture modes Ⅰ and Ⅱ with respect to the number of loading cycles to failure life is shown in Fig. 13. The results indicate that the fatigue crack propagation for the fracture due to the shear mode (Ⅱ) is more significant in contrast to that for the opening mode (Ⅰ). Moreover, as compared with the test specimens F-DTG-140-1.1, the test specimens F-DTG-78-0.7 with relatively compact and narrow fold widths exhibit more gradual crack propagation.

5.2 Load range and fatigue endurance To compare the fatigue endurance of test details, a calculation is performed as the number of cycles (N) against stress ranges (Δσ) to plot a strength curve. The S-N curves codified by the Eurocode 3 [43] for different 9 detail categories ranging from EC3-45 to EC3-160 are shown in Fig. 14. The general form of them follows:

N  C ( ) m

(17)

where, m is the slope of the S-N relation and C is the material constant. Two slope ranges are divided by ΔσC and ΔσD corresponding to fatigue strengths at 2 million cycles and 5 million cycles respectively. Each detail category is labelled in relation to its corresponding fatigue strength at 2 million cycles, e.g., EC3-112 represents a detail category for which ΔσC=112MPa can be attained at 2 million cycles. They are also shown to decrease in two stages with slopes (m) equal to 3 and 5 respectively as the number of cycles (N) increases until the cut off limit. Similar to Eq. (17), the S-N relations in the Eurocode 3 [43] are given by:

N     2 106   C  3

(18)

3

N     5 106   D  5

(19)

5

The recorded number of cycles of failure corresponding to the stress range data resulted from fatigue tests are plotted in Fig. 15~Fig. 18 and listed in Table 2. Also plotted is the refer data for breathing webs in DFGs from Cardiff tests (Ref. CDFGs) [12-17] and Prague tests (Ref. PDFGs) [22-27]. The load ranges in the plot are chosen in terms of measured principal stress range, Δσp, and nominal membrane shear stress range, Δτavr, of the corrugated web. Δτavr for all test specimens are kept as below 80% of τy, i.e. 162.58MPa, which indicates that only the elastic buckling takes places as commented in Ref. [37]. For comparison, the S-N data with two standard deviations below the mean curve, knowing as the lower 95.4% confidence, is also plotted in Fig. 15. As far as the principal surface stress range is concerned, it can be seen that the measured data of Δσp at certain N for current test is slightly lower than those for breathing webs in DFGs as reported from Ref. CDFGs. A slightly lower detail category curve of EC3-110 can be suggested for breathing corrugated webs as compared against counterpart curve of EC3-125 for breathing flat webs with welded boundaries [12-17], as shown in Fig. 16. It appears that the local shear buckling stress is more pronounced within several subpanels of the corrugated web as compared against global buckling for referred flat webs of larger widths anchored in surrounding welded flanges and transverse stiffeners. As far as the nominal membrane shear stress range is concerned, a part of test data along with those from Ref. CDFGs and Ref. PDFGs falls below codified detail category curve of EC3-80 for shear stress ranges. This indicates that plate breathing related out-of-plane deflection and its secondary bending stress also have significant effect on the fatigue behaviour of the corrugated webs. Unlike referred data from Ref. CDFGs and Ref. PDFGs, however, all test data are above codified detail category curve of EC3-45 which is higher in contrast to 9

EC3-36 for unclassified details in Eurocode 3, as shown in Fig. 18. Additionally, it seems that nonconservative nominal membrane shear strengths of current tests can be obtained from the mean curves of Cardiff tests (Ref. CDFGs) [12-17] in contrast to the mean curves of Prague tests (Ref. PDFGs) [22-27]. Based on above discussion, therefore, it is recommended the breathing plate of corrugated webs be analyzed with measured principal stress range by referring the detail category curve of EC3-112 which can be defined as:

Log ( N )  12.45  3Log ( p )

(N  5 106 )

Log ( N )  16.28  5Log ( p )

(N  5 106 )

(20)

while these with nominal membrane shear stress range by referring the detail category curve of EC3-45 which can be defined as:

Log ( N )  11.26  3Log ( avr )

(N  5 106 )

Log ( N )  14.30  5Log ( avr )

(N  5 106 )

(21)

6. FE analyses of shear fracture model It has been known that the principal stresses on breathing plates are difficult to be predicted theoretically, requiring the use of numerical techniques such as the finite element (FE) methods [12]. Given detailed FE analyses of buckling behaviour of corrugated webs has been reported in author’s previous study as referred to Ref. [38], the aim of FE analyses herewith is thus to gain a better understanding of fracture in shear mode (KⅡ) which has been show experimentally in the Section 5 as a predominant factor for corrugated webs in plate breathing. The FE analyses were performed using the ANSYS Academic Research, v.12.1 software package [44] in the determination of the stress intensity factor (SIF) of shear mode (K ). Multi-linear elastic-plastic stress strain curve is adopted for web plate constitutive model which is based on corresponding average results of five identical coupon tensile tests. The defined parameters of material curve are listed as: yield stress and strain of 352MPa and 0.17% respectively, strain at the end of yielding plateau of 1.62%, ultimate stress and strain of 464MPa and 9.78% respectively. 4-node shell elements which have six degrees-of-freedom per node were used in the finite element model. Three layers of integration were defined for elements with capacity to allow for out-of-plane deflection. As shown in Fig. 19(a), the developed finite element model has upper and lower plates with the length of 0.5lw and the widths of blcosθc and bl representing the lengths for projected inclined fold and longitudinal fold respectively. Since the shear stress, τ, takes place as upper and lower plates have a trend of sliding in the opposite direction, the edge DF is simply supported while the uniformly distributed pressure of q0=100Mpa is applied on edge AC in horizontal direction. The material properties are considered as Young’s modulus, E, and Poisson’s ratio, υ, equal to 206GPa and 0.3 respectively. For a notched model studied herein, a preset crack in depth, a, was inserted between two plates. Mesh refinement was performed for the crack tip with mesh-size control-concentrated keypoint and quadratic singular elements. A typical resultant shear stress contour of the FE model for the case of a/(0.5lw)=0.2 is presented in Fig. 19(b). As expected, the crack tip is significantly stressed with certain plasticity under pure shear loading. K are calculated numerically using J-integral method from path integration around crack tip. As the crack length ratio, a/(0.5lw), is varied between 0 and 0.5, the values of K obtained from FE modelling is shown to correlate well with these from theoretical prediction from Eq. (13), as shown in Fig. 20. Based on validated FE models, the fracture behaviour of breathing plates of corrugated webs is subsequently simulated. Ⅱ

Ⅱ

Ⅱ

10

The measured out-of-plane deflection of longitudinal fold due to plate breathing was initially applied using predefined spline surface with its affiliated points which approximate geometric details of experimental measurements. Afterwards, aforementioned uniformly distributed pressure of q0 was applied onto the model. Typical equivalent stress distribution of notched models is illustrated in Fig. 21. Under the loading situation of initial out-of-plane deflection of longitudinal fold due to plate breathing, certain plasticity is formed at the crack tip and the distribution of shear stress in the vicinity is also notably increased, as shown in Fig. 22. At the subsequent stage of shear crack propagation, a wider area around the crack tip is stressed and also extended to the upper plate with higher magnitude than former case. This observation can be a result of a interacted action of tearing due to local out-of-plane deflection which corresponds to the stress intensity factor (SIF) of tearing fracture mode (K ) [45]. To allow for such an interacted fracture mode, the effective stress intensity factor, Keff, recommended by BS7910 [46] is introduced and expressed as: Ⅲ

Keff  KⅠ2  KⅡ2 

KⅢ2 1 ν

(22)

Since the value of KⅠ for opening mode is relatively small, Keff is considered to be mostly related to K and K . Fig. 23 shows the effect of the normalized crack length a/(0.5lw) on the normalized stress intensity factors related to K and Keff. It is observed that as the normalized crack length is varied between 0 and 0.5, the normalized K remains constant while the normalized Keff is moderately increased with the decrease of the thickness of the plate from 0.75mm to 4mm. This can be expected as thinner plates have greater flexibility and thus more prone to out-of-plane deflection induced by plate breathing. Based on regression analyses of FE modelling results, the normalized effective stress intensity factor, Keff, can be expressed relating to the normalized crack length and normalized fold width to plate thicknesses, i.e. a/tw and bl/tw, respectively as: Ⅱ

Ⅲ

Ⅱ

Keff

 πa

Ⅱ



  a tw

4e

b  0.05 l   tw 

(23)

 g

7. Fatigue crack propagation life model In the fatigue crack propagation life assessment, it is often assumed based on the linear elastic fracture mechanics approach that the crack propagates relating the crack growth rate, da/dN, and the stress intensity factor range, ΔK. The crack growth is insignificant when the value of ΔK is below a threshold stress intensity factor range, ΔK0. Therefore, it is often sufficient to assume that the fatigue crack growth rate follows the equation below [45]:

da  C (K )m dN

(24)

where, N is the calculated cyclic life. C and m are material constants which are referred from BS7910 [46] as 3.25×10-12 m/[cycle*(MPa*m0.5)3] and 2.88 respectively. The fatigue life can be obtained by integrating the following equation:

N

1 af 1 da C ai K m

(25)

where, ai and af are initial crack length and final crack length respectively. Substituting Eq. (13) into Eq. (25), the 11

fatigue life related to the fracture induced by the shear mode (Ⅱ) can be calculated as:

N

1 C  m

1

af

  a  ai

0.5 m

da

(26)

where, Δτ is the shear stress range. For the fatigue life prediction, ai=9mm and ai=2.5mm are used for test specimens F-DTG-140-1.1 and F-DTG-78-0.7 respectively as the smallest crack size measured at the vertical intersection line between the longitudinal fold and the inclined fold. The predicted fatigue crack propagation lives are plotted along with test data, as shown in Fig. 24. In contrast between above mentioned prediction and test data, it can be observed that the predicted crack length, a, and it propagation lives are obviously greater in contrast to test results under a given crack length, as shown in Fig. 24. This overestimation can be due to the lack of allowance of the interacted tearing fracture mode (K ). In this regard, normalized effective stress intensity factor, Keff, and its related χg are introduced in correcting above predictions and then Eq. (26) can be rewritten as: Ⅲ

N

1 C  m



1

af

ai



m g

 a 

0.5 m

da

(27)

where, χg is proposed correction factor considered in the preceding section. Accordingly, the predictions can be recast as shown in Fig. 25. Although a few local underestimations exist, the general corrected prediction reaches a closer agreement with test results. It can be thought that the corrected prediction allowing for interacted tearing fracture mode would demonstrate the applicability of the effective stress intensity factor to practical situation of corrugated web in plate breathing. Moreover, concerning the propagation of interacted cracks prior to fatigue failure, the difference between above prediction and test observation can be discussed as follows. It was observed that type ⅰ crack within the length of aⅰ was initiated quickly as the Type ⅱ crack has a certain propagation under the action of tearing fracture. Significant propagation rate of this kind of crack would be expected if the ratio of the plate width to thickness is high. In this regard, further research work is still needed to expand the understanding of related influential parameters for such an interacted fracture mode.

8. Summary and conclusions In this study, the fatigue behaviour of slender corrugated webs subjected to plate breathing has been investigated by using fatigue experimental tests and comparison with related test data of thin-walled flat webs. Distinct failure mechanism and fatigue endurance of breathing corrugated webs are analyzed. Moreover, numerical analysis has been performed to understand the fracture mode of the plate in breathing. As a consequence, the effect of interacted tearing due to local out-of-plane deflection on the plate breathing has been revealed and introduced to the equation for fatigue life prediction. The applicability of this equation to the presented test results has also been discussed. Based on above research findings, the following conclusions can be drawn:

·

The fatigue test results show that three types of fatigue cracks on the breathing corrugated webs are due to out-of-plane deflection and shear loading. Apart from a few crack growths along the orthogonal axes of elliptical out-of-plane deflection contour, most cracks are found along or moderately rotated from the vertical intersection line between the longitudinal fold and the inclined fold. The direction of crack propagation with respect to intersection line can be correlated following the theory of linear elastic fracture 12

mechanics for opening and shear fracture modes.

·

·

The comparison of experimental fatigue results to Eurocode 3 fatigue assessment procedure reveals that the detail category 112 appears to be suitable for fatigue evaluation in terms of principal stress range. Such an identified detail category is slightly lower than the detail category 125 for referred breathing flat webs with wider range of anchorage in surrounding welded flanges and transverse stiffeners. The former can be due to more pronounced surface stressing as a result of local buckling on the narrow subpanels of the corrugated web. It is shown that the developed finite element model can provide a good agreement with theoretical stress intensity factor of shear mode (KⅡ) as the corrugated web is simplified as a plate in pure shear loading. This loading case, however, to some extent disagrees with that under out-of-plane deflection which gives rise to certain tearing effect on the increase of the crack propagation. Instead, an interacted fracture mode for plate breathing is justified in the finite element analyses with the introduction of effective stress intensity factor relating to the normalized crack length and normalized plate width to thickness.

·

Using the linear elastic fracture mechanics approach, the fatigue life predictions for test specimens are found to be overestimated without appropriate allowance of interacted tearing fracture. Accordingly, a correction factor for such an effect is thus considered in the proposed approach. As a result, the corrected prediction is shown to be in a good agreement with test fatigue crack propagation lives.

Finally, it is worth mentioning that the particular focus in this study has been set on very slender corrugated webs under plate breathing rather than the web-to-flange welds susceptible to fatigue crack as well documented in the literature. In fact, the latter may become non-negligible for shallow depth parts of variable depth corrugated web girders or the weld imperfection locations. A detailed allowance of the web-to-flange welds would result in an enormous increase of complexity of the fatigue behavioural model of the corrugated webs subjected to plate breathing. Although the details of the welds are disregarded in the model studied herein, an extension of current investigation is still required to couple their influences for possible engineering applications in further research works.

Acknowledgements The authors are most grateful to the financial supports provided by the Open Fund of Key Laboratory of Performance Evolution and Control for Engineering Structures (Tongji University), Ministry of Education (Grant No. 2018KF-5); and the Open Fund of State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology (Grant No. LP1822).

References 1. Easley JT. Buckling formulas for corrugated metal shear diaphragms. Journal of Structural Engineering, SECF, 1975, ST7: 1403–17. 2. Kövesdi B, Dunai L. Fatigue life of girders with trapezoidally corrugated webs: An experimental study. International Journal of Fatigue, 2014, 64: 22-32. 3. El-Metwally AS, Loov RE. Corrugated steel webs for prestressed concrete girders, Materials and Structures, 2003, 36: 127-134. 4. Huang L, Hikosaka H, Komine K. Simulation of accordion effect in corrugated steel web. Computer and structures, 2004, 82: 2061-2069. 5. Mueller JA, Yen B. Girder web boundary stresses and fatigue. Welding Research Council Bulletin, 13

1968(127): 1-22. 6. Patterson PJ, Corrado JA, Huang JS, Yen BT. Fatigue and static tests of two welded plategirders. Welding Research Council Bulletin 1970(155): 1-18. 7. Okura I, Maeda Y. Analysis of deformation-induced fatigue of thin-walled plate girder in shear. Proceeding of JSCE, Structural Engineering/Earthquake Engineering, 1985, 2(2): 131-138 8. Maeda Y, Okura I. Influence of initial deflection of plate girder webs on fatigue crack initiation. Engineering Structures, 1983, 5: 58-66 9. CEN, BS EN 1993-2: 2006-Eurocode 3: Part 2: Steel bridges, Brussels: CEN, 2006. 10. Roberts TM, Davies AW. Fatigue induced by plate breathing. Journal of Constructional Steel Research, 2002, 58: 1495-1508. 11. Davies AW, Roberts TM, Evans HR, Bennett JH. Fatigue of slender web plates subjected to combined membrane and secondary bending stresses. Journal of Constructional Steel Research, 1994, 30: 85-101 12. Roberts TM, Davies AW, Skaloud M, Zornerova M. Fatigue resistant design of slender webs subjected to plate breathing. III Plate and box girders in Stability of Steel Structures, Budpest, Hungary, 1995. 13. Roberts TM, Davies AW, Bennett JH. Fatigue shear strength of slender web plate. Journal of Structural Engineering, ASCE, 1995, 121(10): 1396-1401. 14. Roberts TM, Osman MH, Skaloud M, Zornerova M. Fatigue crack propagation and residual shear strength of slender web panels. In: International Colloquium on Stability of Steel Structures, Technical University of Budapest, Hungary; 1995. vol. 1, p. 293–300 15. Roberts TM. Analysis of geometric fatigue stresses in slender web plates. Journal of Constructional Steel Research, 1996, 37(1): 33-45. 16. Roberts TM, Osman MH, Skaloud M, Zornerova. Residual shear strength of fatigue cracked slender web panels. Thin-Walled Structures, 1996, 24: 157-172. 17. Davies AW, Roberts TM. Numerical studies of fatigue induced by breathing of slender web plates. Thin-Walled Structures, 1996, 25(4): 319-333. 18. Costa J, Roberts TM. Numerical studies of fatigue induced by breathing of slender web plates. American Society of Mechanical Engineers, Petroleum Division (Publication) PD, 1996, 73(1): 79-86 19. Davies AW, Roberts TM. Finite element studies of fatigue induced by breathing of slender web plates. American Society of Mechanical Engineers, Petroleum Division (Publication) PD, 1996, 73(1): 63-70 20. Osman MH, Roberts TM. Prediction of the fatigue life of slender web plates using fracture mechanics concepts. Thin-Walled Structures, 1999, 35: 81-100. 21. Kutmanova-Karnikova I, Skaloud M, Janus K. 'Breathing' of thin webs under variable repeated patch loading. Stability of Steel Structures, 1991: 477. 22. Skaloud M. Interaction of breathing with the cumulation of damage in the webs of steel plate girders. Thin-Walled Structures, 1994, 20: 83-95. 23. Skaloud M, Roberts TM. Fatigue crack initiation and propagation in slender webs breathing under repeated loading. Journal of Constructional Steel Research, 1998, 46(1-3): 417-419. 24. Skaloud M. Fatigue behaviour of the slender webs of steel girders breathing under alternated loading. Stahlbau, 1999, 68(1): 3-8. 25. Maquoi R, Skaloud M. Stability of plates and plated structures-General report. Journal of Constructional Steel Research, 2000, 35: 45-68. 26. Skaloud M, Zörnerova M, Kuhlmann U, Spiegelhalder U. Prague and Stuttgart experimental research on web breathing. Journal of Structural Engineering (Madras), 2000, 27(1): 33-40. 27. Škaloud M, Zörnerová M. The fatigue behaviour of the breathing webs of steel bridge girders. Journal of Civil Engineering and Management, 2005, 11(4): 323-336. 28. Ndogmo J. Fatigue behaviour of thin web from plate-girder bridges-Web breathing(in German). Stahlbau, 2001, 70(5): 352-356. 29. Kuhlmann U, Spiegelhalder U, Günther HP. Fatigue assessment of steel plated girders susceptible to web breathing (in German). Stahlbau, 2002, 71(5): 358-366. 30. Kuhlmann U, Spiegelhalder U. Experimental studies on web breathing. Stahlbau, 2002, 71(4): 233-243. 31. Günther HP, Kuhlmann U. Numerical studies on web breathing of unstiffened and stiffened plate 14

girders. Journal of Constructional Steel Research, 2004, 60(3-5): 549-559. 32. Crocetti R. Web breathing of full-scale slender I-girders subjected to combined action of bending and shear. Journal of Constructional Steel Research, 2003, 59(3): 271-290. 33. Duchene Y, Maquoi R. Fatigue resistance to web breathing of slender plate girders subjected to shear. Journal of Constructional Steel Research, 1998, 46(1-3): 416. 34. Costa J, Mirambell E. Fatigue induced by breathing on slender webs. Journal of Constructional Steel Research, 1998, 46, 1-3: 415 35. CEN, BS EN 1993-1-5:2006 - Eurocode 3: Design of steel structures - Part 1-5: General rules - Plated structural elements, 2006. 36. Pateron D, Yen BT, Fisher JW. Fatigue induced by repeated out of plane deflection of welded plates, 2005, 5: 199-210. 37. Sause R, Braxtan TN. Shear strength of trapezoidal corrugated steel webs, Journal of Constructional Steel Research, 2011, 67: 223–236. 38. Wang ZY, Li X, Gai W, Jiang R, Wang QY, Zhao Q, Dong J, Zhang T. Shear response of trapezoidal profiled webs in girders with concrete-filled RHS flanges, Engineering Structures, 2018(174): 212-228. 39. Wang ZY, Tan L, Wang QY. Fatigue strength evaluation of welded structural details in corrugated steel web girders. International Journal of Steel Structures. 2013, 13(4): 707-721 40. Lindner J, Huang B. Beulwerte für trapezförmig profilierte bleche unter schubbeanspruchung. Stahlbau, 1995, 64(12): 370–374. 41. Parker AP. The mechanics of fracture and fatigue. 1981, London, E. & F. N. Spon Ltd. 42. Barsom JM, Rolfe ST. Fracture and fatigue control in structures: applications of fracture mechanics, 1999, Philadelphia, American Society for Testing and Materials. 43. prEN 1993-1-9: 2005. Eurocode 3: Design of Steel Structures, Part 1.9: Fatigue, European Standard, CEN, Brussels, 2005 44. ANSYS@Academic Research, Release 12.0, Help System, Structural Analysis Guide, ANSYS, Inc., 2011. 45. Dowling NE. Mechanical behaviour of materials. Engineering methods for deformation, fracture, and fatigue, New Jersey: Pearson Education, Inc., 2007. 46. BS 7910 2005: Guide to methods for assessing the acceptability of flaws in metallic structures, 2005

15

Flange plate

A Crack (1)

Crack (4)

Crack (2) Crack (3)

A

w

w

Fatigue crack

Flat web

0.5

Stiffener

0.5

w

(a) Tensile membrane bending induced cracks

Stiffener w

(b) Plate breathing induced cracks

(c) Detail A

Fig. 1 Illustration of web breathing in slender plate girders with stiffened flat webs

Stress state

Corrugated web

Potential crack Out-of-plane deflection

A

Inclined fold

w

Longitudinal fold

(a) Desired crack locations on corrugated web

Fatigue crack

(b) Detail A

Fig. 2 Illustration of desired web breathing in girders with corrugated webs

Normalized out-of-plane deflection

V

(Test No. F-DFG with slender flange plate)

Fig. 3 Undesired involvement of buckled flange plate for a corrugated web girder

16

V

Fig. 4 Typical first eigenmode shape of corrugated web in predominant out-of-plane deflection

1.5

Normalized stress intensity factor

y xy

e lan kp

x

c ra

r

Crack tip o

r

c ry itra rb

A

r

Original crack plane

1.5

Peak+: (-70.42˚,1.155) Peak+: (0˚,1)

1

1

0.5

0.5

0

0

-90 -0.5 -1

-60

-30

0

    K KⅡ/  τ a  K /  τ a 

60

90 -0.5

KⅠ / τ a KⅠ / τ a KⅠ Ⅱ

-1.5

30

Ⅱ

-1

Peak-: (70.42˚,1.155)

Fig. 5 Elastic stress field and its related components in the vicinity of crack tip

Fig. 6 Comparison of normalized stress intensity factor varied with varying θ

Upper flange Longitudinal fold Inclined fold

Corrugated web w

Lower flange

Test girder with flat plate flanges (DFG)

st,t f

-1.5

θ (degree)

0.5

f

con,t

f

st,t f f

Tubular flange

Plate flange i w c

i

r

Single wave Corrugation

Test girder with tubular flanges (DTG) Fig. 7 Geometric notation of corrugation and flange in test girder

17

50 450

Load pin

350

Actuator Load cell

Lateral brace

100

Test specimen DTM3

DTM4 DTM1 DTM2

Support pin

60

9 82 9 50

60

48

113 140 113 140 20 140 113 140 113 47 47 1221

Strong floor

Fig. 8 Illustration of test girder with support and loading conditions

Upper flange Fold: Inclined

Longitudinal i

Fig. 9 Arrangement of three element rosettes along with pseudo out-of-plane deflection pattern (unit: mm)

Upper flange Fold: Inclined

Inclined Longitudinal

Longitudinal i

Crack i Crack C1

Crack iii

Inclined Longitudinal Propagation stage (2) Propagation stage (1) Initiation

le

le

Crack C2 Crack ii

Crack ii

odc se

47

Transverse stiffener of end support

odc se

Lower flange

Transverse stiffener of end support

Lower flange

(a) Isolated cracks

(b) Combined cracks

Fig. 10 Typical observed fatigue cracks for corrugated webs

18

Upper flange Fold: Inclined

Longitudinal

Inclined Longitudinal Crack C1

Crack C1 Crack C2

Crack ii Crack C1

Transverse stiffener of end support Lower flange

(a) F-DTG-140-1.1-1 Upper flange Fold: Inclined

Longitudinal

Inclined Longitudinal

Crack C1

Transverse stiffener of end support

Crack ii Lower flange

(b) F-DTG-140-1.1-2 Upper flange Fold: Inclined

Longitudinal

Inclined Longitudinal Crack C1

Crack C2

Crack ii

Transverse stiffener of end support

Lower flange

(c) F-DTG-140-1.1-3 Fig. 11 Typical fatigue crack distribution and out-of-plane deflection contour for F-DTG-140-1.1

19

Upper flange Fold: Longitudinal Longitudinal Longitudinal Longitudinal Inclined Inclined Inclined Crack C2

Crack C1

Crack C1 Crack C2 Crack ii

Crack C1 Transverse stiffener of end support

Crack ii

Crack ii Lower flange

(a) F-DTG-78-0.7-1 Upper flange Fold: Longitudinal Longitudinal Longitudinal Longitudinal Inclined Inclined Inclined Crack C1 Crack C1 Crack ii

Crack C1

Crack C1 Crack C2 Transverse stiffener of end support

Crack ii Lower flange

(b) F-DTG-78-0.7-2 Upper flange Fold: Longitudinal Longitudinal Longitudinal Longitudinal Inclined Inclined Inclined Crack C2 Crack C2 Crack C1

Crack C1

Crack C1 Crack C2 Crack ii

Transverse stiffener of end support

Lower flange

(c) F-DTG-78-0.7-3 Fig. 12 Typical fatigue crack distribution and out-of-plane deflection contour for F-DTG-78-0.7

20

80

80

Test specimen

Fracture mode Ⅰ

Ⅱ

60

40

a (mm)

F-DTG-140-1.1-1 F-DTG-140-1.1-2 F-DTG-140-1.1-3

20

Ⅰ

Ⅱ

F-DTG-78-0.7-1 F-DTG-78-0.7-2 F-DTG-78-0.7-3

40

20

0

0 0

0.2

0.4

0.6

0.8

1

1.2

0

0.2

0.4

0.6

N/Nu

N/Nu

(a) F-DTG-140-1.1

(b) F-DTG-78-0.7

EC3-160 EC3-100 Fig. 13 Propagation of typical cracksEC3-125 for fracture modes Ⅰ and Ⅱ

EC3-90 EC3-63

EC3-80 EC3-56

EC3-71 EC3-45

10 3 1000 Stress range, ΔσR (MPa)

a (mm)

60

Fracture mode

Test specimen

Δσc ΔσD

EC3 -160 EC3 -125

100 10 2

1 m=3

EC3 -100 EC3 -9 0 EC3 -8 0 EC3 -71 EC3 -6 3 EC3 -5 6 EC3 -45

ΔσL

m=5 10 10 105

106 107 108 Endurance, number of cycles, N

109

Fig. 14 Fatigue curves and category classes suggested by Eurocode 3

21

0.8

1

1.2

1000 10 3

Stress range, ΔσP (MPa)

Stress range, ΔσP (MPa)

1000 10 3

2s

10 1002 2s F-DTG-78-0.7 F-DTG-140-1.1

100 10 2

F-DTG-78-0.7 F-DTG-140-1.1

↗ No fatigue cracks

Cardiff tests (Ref. CDFGs)

Cardiff tests (Ref. CDFGs)

10 10 100000 10 5

1000000 10 6

10000000 10 7

10 4 10000

Number of cycles, N

10 7 10000000

10 3 1000

F-DTG-140-1.1 F-DFG

2 10 100

F-DTG-78-0.7

F-DTG-140-1.1

Cardiff tests (Ref. CDFGs)

F-DFG

Prague tests (Ref. PDFGs)

2

10 100

↗ No fatigue cracks

↗ No fatigue cracks 10 10 10 4 10000

10 6 1000000

Fig. 16 ΔσP based fatigue experimental results and codified detail categories for DTGs and CTGs

Stress range, Δτavr (MPa)

Stress range, Δτavr (MPa)

F-DTG-78-0.7 Cardiff tests (Ref. CDFGs) Prague tests (Ref. PDFGs)

10 5 100000

Number of cycles, N

Fig. 15 ΔσP based best fit and lower bound S-N fatigue curves for DTGs and CTGs 1000 10 3

↗ No fatigue cracks

10 10

10 10

10 5 100000

10 6 1000000

10 4 10000

107 10000000

10 5 100000

10 6 1000000

10 7 10000000

Number of cycles, N

Number of cycles, N

Fig. 17 Δτavr based best fit and lower bound S-N fatigue curves for DTGs, CTGs and PTGs

Fig. 18 Δτavr based fatigue experimental results and codified detail categories for DTGs, CTGs and PTGs

22

B

A

0

blcosθc τ

o C

a a

D

τ 0.5lw-a

bl

E

F

(a) Element discretization

(b) Typical shear stress contour

Fig. 19 Illustration of finite element models in fracture mode Ⅱ (a/(0.5lw)=0.2)

25

KⅡ (MPa·m0.5)

20

15

10 Theoretical solution

5

FE modelling 0

0

0.1

0.2

0.3

0.4

0.5

a / (0.5lw)

Fig. 20 Comparison of KⅡ obtained from theoretical solution and FE modelling

23

(a) Equivalent stress contour induced by measured out-of-plane deflection

(b) Equivalent stress contour induced by subsequent shear crack propagation

Fig. 21 Transition of equivalent stress around the crack tip (a/(0.5lw)=0.4)

Normalized stress intensity factor

2

a g     tw 

1.5

 b  605.7-4.7   t w 

tw (mm) 0.75 1.5 2 4

1

kii KⅡ /  πa eff Keff /  πa 0.5

0

0.1

0.2

0.3

0.4

0.5

a / (0.5lw)

Fig. 22 Typical shear stress contour of crack plane Fig. 23 Comparison of normalized stress intensity factor of KⅡ and Keff varied with a/(0.5lw) in combined shear and tearing mode

24

60

45

45

a (mm)

a (mm)

60

30

Prediction Test with KⅡ data 67MPa F-DTG-140-1.1-1 72MPa F-DTG-140-1.1-2 80MPa F-DTG-140-1.1-3

15

0

105 100000

106 1000000

30

15

Prediction Test with KⅡ data

F-DTG-78-0.7-1 93MPa F-DTG-78-0.7-2 99MPa F-DTG-78-0.7-3 107MPa

0

105 100000

107 10000000

106 1000000

Number of cycles, N

Number of cycles, N

(a) F-DTG-140-1.1

(b) F-DTG-78-0.7

107 10000000

60

60

45

45

a (mm)

a (mm)

Fig. 24 Comparison of test results and prediction of crack propagation life with KⅡ

30

Corrected Test rediction data 15

67MPa F-DTG-140-1.1-1 72MPa F-DTG-140-1.1-2 F-DTG-140-1.1-3 80MPa

0

30

15

Corrected Test prediction data 93MPa F-DTG-78-0.7-1 99MPa F-DTG-78-0.7-2 107MPa F-DTG-78-0.7-3

0

105 100000

106 1000000

107 10000000

105 100000

106 1000000

Number of cycles, N

Number of cycles, N

(a) F-DTG-140-1.1

(b) F-DTG-78-0.7

Fig. 25 Comparison of test results and corrected prediction of crack propagation life with Keff

25

107 10000000

Table 1 Geometric details and design parameters of test specimens Web (mm)

Flange (mm)

Design parameter

Test No. *

0.5lw

hw

tw

b

F-DTG-78-0.7-1

601

350

0.73

78.0

F-DTG-78-0.7-2

600

350

0.73

F-DTG-78-0.7-3

600

350

F-DTG-78-0.7-4

600

F-DTG-140-1.1-1

θc

*

*

tf

tst,t

bf

0.5lw/hw

hw/b

37.5

50

44.2

100

1.72

78.0

37.5

50

44.1

100

0.73

78.0

37.5

50

44.1

350

0.73

77.9

37.5

50

602

349

1.08

140.1

37.5

F-DTG-140-1.1-2

600

350

1.08

140.1

F-DTG-140-1.1-3

600

349

1.08

F-DTG-140-1.1-4

600

349

F-DTG-140-1.1-5

601

F-DFG

*

*

hw/tw

hr/tw

4.49

479.45

65.02

1.71

4.49

479.45

65. 02

100

1.71

4.49

479.45

65. 02

44.1

100

1.71

4.49

479.45

64.93

50

44.2

100

1.72

2.49

323.15

78.88

37.5

50

44.1

100

1.71

2.50

324.07

78.88

141.1

37.5

50

44.1

100

1.72

2.48

323.15

79.44

1.08

141.1

37.5

50

44.1

100

1.72

2.48

323.15

79.44

350

1.08

140.1

37.5

50

44.1

100

1.71

2.50

324.07

78.88

700

422

3.70

84.0

30.7

4

-

84

1.66

5.02

114.05

11.59

Ref. CDFG-A

800

800

2.91

-

-

12

-

250

1.00

-

274.91

-

Ref. CDFG-B

1200

800

3.05

-

-

12

-

250

1.50

-

262.30

-

Ref. CDFG-C

1200

600

3.07

-

-

12

-

250

2.00

-

195.44

-

Ref. CDFG-D

400

800

2.06

-

-

12

-

250

0.50

-

388.35

-

Ref. CDFG-E(1)

600

800

2.00

-

-

10

-

250

0.75

-

400.00

-

Ref. CDFG-E(2)

600

800

3.00

-

-

10

-

250

0.75

-

266.67

-

Ref. CDFG-E(3)

1200

800

3.00

-

-

10

-

250

1.50

-

266.67

-

Ref. PDFG

500

500

1.94

-

-

10

-

50

1.00

-

257.73

-

Note: * is only valid for girders with corrugated webs. CDFG and PDFG stand for reported test data from laboratories in Cardiff [12-17] and Prague [22-27] respectively.

Table 2 Mechanical properties, applied load condition and test measured results Test No.

Web material (MPa)

Applied load 26

Test measurement

Buckling stress

σy

τy

E

Fmin-Fmax (kN)

ΔF (kN) Δτavr(MPa)

Δσp(MPa)

N

τcr(MPa)

Δτavr/τcr

F-DTG-78-0.7-1

352

203

206000

5-58

53

103.7

172

1050852

90.23

1.15

F-DTG-78-0.7-2

352

203

206000

5-61

56

109.6

194

720267

90.23

1.21

F-DTG-78-0.7-3

352

203

206000

5-65

60

117.4

199

547891

90.23

1.30

F-DTG-78-0.7-4

352

203

206000

5-45

40

78.28

94

5041463

90.23

0.87

F-DTG-140-1.1-1

352

203

206000

5-61

56

74.07

164

1036695

66.19

1.12

F-DTG-140-1.1-2

352

203

206000

10-70

60

79.37

175

781525

66.19

1.20

F-DTG-140-1.1-3

352

203

206000

5-71

66

87.30

182

537338

66.19

1.32

F-DTG-140-1.1-4

352

203

206000

5-56

51

67.46

104

2567089

66.19

1.02

F-DTG-140-1.1-5

352

203

206000

10-64

54

71.43

143

1644989

66.19

1.08

F-DFG

420

242

206000

11-110

99

31.70

403

420003

160.6

0.39

Ref. A-2

258

149

211000

5-193

188

80.76

279

353460

36.58

2.21

Ref. A-3

258

149

211000

5-165

160

68.73

251

905450

36.58

1.88

Ref. A-4

258

149

211000

5-110

105

45.10

208

1600000

36.58

1.23

Ref. B-1

337

195

208000

5-81

76

31.15

169

6022190

31.61

0.99

Ref. B-2

337

195

208000

5-112

107

43.85

206

1401620

31.61

1.39

Ref. B-3

337

195

208000

5-116

111

45.49

254

540740

31.61

1.44

Ref. B-4

337

195

208000

5-221

216

88.52

284

670190

31.61

2.80

Ref. C-1

351

203

208000

5-146

141

76.55

230

1339030

72.37

1.06

Ref. C-2

351

203

208000

5-150

145

78.72

295

487460

72.37

1.09

Ref. C-3

351

203

208000

5-149

144

78.18

190

1490230

72.37

1.08

Ref. D-1

267

154

208000

5-125

120

72.82

285

177000

18.33

3.97

Ref. D-3

267

154

208000

5-110

105

63.71

263

445000

18.33

3.48

Ref. E(1)-1

197

114

208000

-

-

-

205

1371000

17.28

-

Ref. E(1)-2

197

114

208000

-

-

-

275

175000

17.28

-

27

Ref. E(2)-3

271

156

208000

-

-

-

215

1352000

38.88

-

Ref. E(2)-4

271

156

208000

-

-

-

280

1141000

38.88

-

Ref. E(3)-6

271

156

208000

-

-

-

210

2758000

38.88

-

Ref. PTG-5

275

159

210000

5-75

70

72.16

-

92000

41.62

1.73

Ref. PTG-9

275

159

210000

5-70

65

67.01

-

112000

41.62

1.61

Ref. PTG-15

275

159

210000

5-50

45

46.39

-

439000

41.62

1.11

Ref. PTG-16

275

159

210000

5-60

55

56.70

-

201000

41.62

1.36

Ref. PTG-17

275

159

210000

5-60

55

56.70

-

143000

41.62

1.36

Ref. PTG-18

275

159

210000

5-50

45

46.39

-

453000

41.62

1.11

Ref. PTG-19

275

159

210000

5-50

45

46.39

-

797000

41.62

1.11

Ref. PTG-20

275

159

210000

5-40

35

36.08

-

510000

41.62

0.87

Ref. PTG-21

275

159

210000

5-70

65

67.01

-

304000

41.62

1.61

Ref. PTG-22

275

159

210000

5-60

55

56.70

-

297000

41.62

1.36

Ref. PTG-23

275

159

210000

5-70

65

67.01

-

404000

41.62

1.61

Ref. PTG-25

275

159

210000

5-30

25

25.77

-

6823000

41.62

0.62

Ref. PTG-26

275

159

210000

5-65

60

61.86

-

112000

41.62

1.49

Ref. PTG-28

275

159

210000

5-50

45

46.39

-

614000

41.62

1.11

Ref. PTG-29

275

159

210000

5-70

65

67.01

-

127000

41.62

1.61

Ref. PTG-30

275

159

210000

5-80

75

77.32

-

128000

41.62

1.86

28

Table 3 Lists of measured θodc and θⅰ Web (mm)

Test measurement (degree)

Specimen index 0.5lw

hw

tw

b

θc

θodc

θⅰ

F-DTG-78-0.7-1

601

350

0.73

78.0

37.5

60.9

63.9

F-DTG-78-0.7-2

600

350

0.73

78.0

37.5

59.1

67.1

F-DTG-78-0.7-3

600

350

0.73

78.0

37.5

62.7

70.2

F-DTG-78-0.7-4

600

350

0.73

77.9

37.5

57.2

67.8

F-DTG-140-1.1-1

602

349

1.08

140.1

37.5

63.2

72.1

F-DTG-140-1.1-2

600

350

1.08

140.1

37.5

64.4

68.1

F-DTG-140-1.1-3

600

349

1.08

141.1

37.5

65.1

69.3

F-DTG-140-1.1-4

600

349

1.08

141.1

37.5

60.5

72.2

F-DTG-140-1.1-5

601

350

1.08

140.1

37.5

63.3

72.9

Average

61.8

69.3

St. Deviation

2.44

2.74

29

Highlights Fatigue crack modes of corrugated webs in plate breathing are evaluated. Fatigue design formulae are proposed for breathing corrugated webs. FE models allowing for combined shear and tearing are developed for plate buckling. Proposed fatigue life prediction of breathing corrugated webs is shown to be desired.

30