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Fatigue crack paths in light aircraft wing spars
International Journal of Fatigue 123 (2019) 96–104
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International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue
Fatigue crack paths in light aircraft wing spars a,⁎,⁎
Aleksandar Grbović Mihajlo D. Popovića a b
b
T a
a
, Gordana Kastratović , Aleksandar Sedmak , Igor Balać ,
University of Belgrade, Faculty of Mechanical Engineering, Serbia University of Belgrade, Faculty of Transport and Traffic Engineering, Serbia
A R T I C LE I N FO
A B S T R A C T
Keywords: Integral and differential structures Extended finite element method Crack paths Crack growth rate
Different cracks paths in three different wing spar designs are analysed (differential, integral and optimized integral) to explore how those cracks paths affect fatigue life estimation. First, numerical analysis was carried out and verified, using the experimental data for differential wing spar, followed by numerical analysis of both optimized integral wing spars. The optimized integral spar was obtained by analysing three different crosssections regarding fatigue life. Crack propagation simulation and fatigue life estimation were carried out by the extended finite element method, using Morfeo/Crack for Abaqus. Results provided better understanding and prediction of multiple cracks propagation in 3D structures.
1. Introduction The main load-carrying member of the aircraft wing is spar, [1]. It is positioned perpendicularly to the flight direction and is extended from the fuselage to the wing tip. Spar usually has I-beam shape, made of thin shear panel (web) and flanges (caps) at the top and bottom, assembled by rivets. Light airplanes mostly have wings with one spar, carrying almost all bending and shear load, [2]. Under service loading a fatigue crack may initiate from the most severe stress concentration on the spar bottom cap. It can grow unnoticeably, first in the spar cap and – after the cap failure – in the spar web, eventually leading to catastrophic failure if not detected and repaired, [2]. Since fatigue crack initiation and growth on riveted structures were thoroughly investigated over the past decades, it is difficult to get significant improvements concerning the extension of their fatigue life. On the other hand, integral structures are lighter and cheaper, easier to inspect and have less holes sensitive to high stress concentration. Since their manufacturing is more expensive, integral spars must have much longer fatigue life to justify higher initial costs. In recent decades, fracture mechanics parameters became essential for the prediction of crack initiation and propagation, i.e. for fatigue life estimation. The stress intensity factor (SIF) is one of the most important parameter which provides data on crack initiation and propagation. In complex geometries, such as wing spar, it is almost impossible to find an exact solution for SIFs; therefore, the numerical methods are needed for their estimation.
Over the years, many numerical techniques, such as finite element method (FEM), boundary element method (BEM), mesh-free methods and extended finite element method (XFEM), have been developed to simulate the fracture mechanics problems. Souiyah et al. [3], calculated SIFs for a crack emanating from circular hole in rectangular plate and double edge notched plate, using an adaptive mesh finite element technique. The same authors in [4] employed finite element method to predict the crack propagations directions and to calculate SIFs, and then validated the verification of the predicted SIF and crack trajectory, with the relevant numerical and analytical results. The FEM was one of the used methods in [5], where numerical investigation was carried out in order to investigate the influence of friction stir welding process parameters on fatigue crack growth in AA2024-T3 butt joints. Authors in [6] used 3D finite element analyses to study a coupled fracture mode generated by a nominal anti-plane (Mode III) loading, applied to linear elastic plates weakened by a straight through-the-thickness crack. The XFEM was developed in 1999 by Belytschko and Blackin [7], and Moës, Dolbow and Belytschko [8]. They proposed a new technique for modelling cracks in the finite element framework, where standard displacement‐based approximation was enriched near a crack by incorporating both discontinuous fields and the near tip asymptotic fields through a partition of unity method. They developed a methodology that was able to construct the enriched approximation from the interaction of the crack geometry with the mesh. This technique allowed the entire crack to be represented independently of the mesh, so that re-
⁎
Corresponding author. E-mail addresses: [email protected] (A. Grbović), [email protected] (G. Kastratović), [email protected] (A. Sedmak), [email protected] (I. Balać), [email protected] (M.D. Popović). https://doi.org/10.1016/j.ijfatigue.2019.02.013 Received 10 December 2018; Received in revised form 30 January 2019; Accepted 11 February 2019 Available online 12 February 2019 0142-1123/ © 2019 Published by Elsevier Ltd.
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Fig. 1. Differential spar used in experiment (dimensions in mm), [25].
foam materials under mixed mode loading, using an asymmetric semicircular bend and four-point mixed mode specimens was analysed, and numerical simulations of crack propagation were conducted by XFEM. Paper [22] presented the numerical simulation and validation of a fatigue propagation test of a semi-elliptical crack located at the side of the rectangular section of a beam subjected to four-point bending. For the numerical simulations, XFEM implemented in the Abaqus software has been used. The comparison between the experimental and numerical results showed very good correlation regarding crack shape and number of cycles to failure. In [23], XFEM capability available in ABAQUS was used to calculate the stress intensity factor in straight lugs of Aluminum 7075-T6. Crack growth and fatigue life of single through-thickness and single quarter elliptical corner cracks in attachment lug were estimated and then compared with the available experimental data for two different load ratios equal to 0.1 and 0.5. The SIF calculated from XFEM showed that the introduction of different loading boundary conditions significantly affected the estimated fatigue life. Finally, a comprehensive overview of XFEM simulations is given in [24]. In this paper, numerical study of possible crack paths in wing spar under load is presented. To investigate and improve fatigue life (and hence overall performance) of light aircraft UTVA 75, three different wing spar designs were examined: the existing differential spar, the integral spar with same dimensions as differential spar and redesigned integral spar, with the same mass as existing spar. Redesigned integral spar was obtained thru optimization of its cross-section, where three different cross-sections were also analysed. The optimization was carried out in order to improve fatigue life of the integral spar. The special attention was paid to the analysis of cracks growth and their paths along the spar in each case. All computations for crack propagation simulation and fatigue life estimation were carried out by XFEM, using Morfeo/Crack for Abaqus code.
meshing was not necessary to model crack growth. In [9], an improvement of XFEM was presented, which enabled more efficient calculation of SIFs in 2D problems. The XFEM has also been used to calculate SIFs for problems involving multiple, interacting cracks, resulting from multiple site damage (MSD). In [10], it is shown how XFEM could be used for obtaining not only an accurate estimation of the fatigue life of the assembly, but also to predict the most probable directions in which cracks will develop through the structure. In [11,12], SIFs calculations based on implementation of XFEM in Abaqus, were conducted for a typical aero structural configuration with MSD. Analysed model was a unique 3D configuration with 22 cracks that propagate at the same time, whereas stress intensity factors were computed along the crack fronts for all 22 cracks. The fatigue life estimation has been conducted using XFEM, as well as for the simulating crack paths development in more complex models. Using extended finite element method, Sghayer et al. [13,14], simulated the fatigue behavior and crack propagation of the real laser beam welded skin-stringer panel with four stringers. Also in [14], fatigue life of simple flat plate, as well of the skin-stringer panel, were numerically simulated by using XFEM to investigate the effect of stringers. Rojjati-Talemi and Waha [15], proposed a modified fretting fatigue contact model in conjunction with extended finite element method, in order to monitor the effect of mixed mode on fretting fatigue crack propagation. Curà et al. [16] investigated the influence of centrifugal load on crack propagation path in thin rimmed and webbed gears. The investigation has been carried out by means of numerical models involving both 2D finite element and extended finite element models (XFEM). In [17], authors tried to determine the influence of weld residual stresses on crack growth rate and crack arrest, using Paris’ law and XFEM. Fatigue crack growth in friction stir welded T joints under three point bending were simulated numerically in [18], by using XFEM, also used in [19] to verify the SIFs solutions calculated by proposed approximation method, based on superposition. The aim of the study [20] was to predict fatigue crack growth life under variable amplitude loading through XFEM and to explicitly illustrate both fatigue crack growth life and crack propagation. The results were compared with NASGRO crack propagation software and experimental fatigue crack growth life test data on 7075-T6 aluminum alloy under various over load and over load-under load conditions which exhibited a good agreement. In [21] crack initiation and quasi-static propagation in polyurethane
2. Experimental and numerical analysis of the differential wing spar Numerical analysis was first carried out and verified using the experimental data for differential wing spar made of AA2024-T3, [25], with dimensions showed in Fig. 1. The load used in the experiment (minimum value +391.2 N, maximum value +2028 N, frequency 12.5 Hz), produced high tensile stress in caps; consequently, visible crack appeared on the left cap below the support after only 8542 cycles. The crack began to spread rapidly toward the spar web, then changed 97
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Fig. 2. Cracks on differential spar caps, view from above, [25].
Fig. 3. Damaged differential spar with strain gauges, [25].
its direction and continued along the cap at an angle of 90° with respect to the original direction. After 39,450 cycles, another crack was spotted on the right spar cap, initiated at the fastener hole, Fig. 2. Cracks then continued to grow but were not visible. After 58,520 cycles, the test was stopped, and the spar was examined, as shown in Figs. 2 and 3. There was no visible damage in the spar web. It should be noted that in 10 samples tested under different load histories crack always appeared in the same zone, at the edge of left upper cap zone, but at various positions within it, which is consistent with the stochastic nature of examined phenomenon. Therefore, numerical analysis of differential spar was conducted using Morfeo/Crack for Abaqus code, with the initial crack positioned at the edge of left upper cap zone. Crack was then propagated along the longest possible crack path toward the wall of spar in order to monitor and analyse changes in the stress intensity factors, [25]. After opening, the crack growth was simulated in steps of maximum 1 mm. The growth direction was not restricted, so the crack could grow in direction predicted by calculated kink angle (see Eq. (4)). After 19 steps, crack had the shape shown in Fig. 4a which clearly displays that the path on the horizontal part of the spar cap was winding, like the path in the experiment, Fig. 3. That was a solid proof that the boundary conditions and displacements taken from experiment were properly applied and that the numerical model was appropriate. However, crack path on the vertical spar wall (Fig. 4b) did not match completely the shape obtained in the experiment, because the real spar was under the influence of residual stresses induced during manufacturing, [10].
Fig. 4. (a) Crack on horizontal cap wall after 19th step of growth; (b) Crack on vertical cap wall after 45th step of expansion.
3. Numerical analysis of integral spar As the next step, the FE model of integral spar made of AA2024-T3, with exactly the same dimensions and mass as differential spar, was analysed. FE model consisted of 151,700 linear hexahedron elements of type C3D8R. The simultaneous growth of two penny shaped cracks located at left and right edge of spar cap, was simulated again in Morfeo/Crack for Abaqus. Initial cracks length was 1 mm. The path of the left crack was not straight, but curved (Fig. 5a), while the right crack did not propagate at all after the 8th step because the deformation of spar, caused by the 1st crack, literally “closed” it. After reaching the vertical spar wall, the left crack formed two fronts and continued to propagate simultaneously in the horizontal (1st front) and the vertical direction (2nd front), Fig. 5. 98
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4. Optimization of integral spar cross section based on fatigue life Next step was to optimize the cross-sectional shape of integral spar (numerically investigated in previous chapter) using fatigue crack growth life approach. 3D Software packages (CATIA & Abaqus) were used to design the integral spars’ structure and XFEM was again used to calculate the SIFs and the number of cycles of crack propagation. Several iterations were conducted for the design optimization of the integral spars. Performance of the beam was studied by comparison of the fatigue crack life for two different cross-sectional shapes of integral spar, but of the same area (i.e. the mass of the spar was kept constant). The first was I-beam spar investigated in the previous chapter, named here the main integral spar (case A). The second was I-beam integral spar with intermediate cap (case B). In addition, three sizes (different dimensions) of Ibeam integral spar with intermediate cap (B1, B2, B3) were analysed. The overall dimensions of the cases A and B are shown in Fig. 6, while dimensions for cases B1, B2, B3 are given in Table 1.
4.1. Evaluation of SIFs and crack trajectory for all cases After opening, the cracks propagated in steps of approximately 1 mm for all cases. Growth was again not restricted to a single plane; instead, the crack could grow in the direction defined by calculated kink angle. The shapes of these cracks can be seen in Figs. 7–10. After 25 steps of propagation the FE model of case B1 had shape shown in Fig. 7, indicating that paths of the left and right cracks on the horizontal wall were not straight; so, at the end of simulation they did not linkup. On the contrary, the left and right crack in the FE model of case B2 linked up at 32nd step of propagation (Fig. 8) and after that new crack propagated along the vertical cap until the end of numerical simulation at 39th step (Fig. 9). Left crack in the FE model of case B3 also propagated in horizontal cap along curved path until the end of simulation at 31st step. As in the case A, the right crack stopped after the 10th step when it reached the length of 4 mm, as shown in Fig. 10.
Fig. 5. (a) 1st crack tip after 35 growth steps; (b) 2nd crack tip after 35 growth steps.
4.2. Comparisons of SIFs For all cases, stress intensity factors Mode I, II, III and equivalent stress intensity factor K eq were calculated within Morfeo/Crack for Abaqus for each node on crack fronts. Here, the equivalent SIF is used as a reference for comparisons. For each step of propagation, the average values of K eq in all crack fronts’ nodes were calculated and shown in Table 2. SIF values are given in MPamm0.5. The SIFs were computed along the crack fronts using the interaction integral method, [26], defined as:
Fig. 6. Cross section shapes (Case A left, Case B right) used in optimization (dimensions in mm).
After the 35th step of propagation (counting from the crack opening) the simulation stopped because the 1st crack front left the area with refined mesh (Fig. 5a), and software couldn’t create new nodes for the 1st crack front with required accuracy. Position of the 2nd crack front after 35 steps of propagation is shown in Fig. 5b.
I=
− ∫V qi, j (σkl εklaux δij − σkjaux uki − σkj ukiaux ) dV
∫s δqn ds
(1)
where σij, εij, ui are the stress, strain and displacement, σijaux , εijaux , uiaux are the stress, strain and displacement of the auxiliary field, and qi is the crack-extension vector. The interaction integral is associated with the stress-intensity factors through:
Table 1 The overall dimensions for I-beam spar with intermediate cap. Case no.
a1 [mm]
b1 [mm]
a2 [mm]
a3 [mm]
b3 [mm]
a4 [mm]
a5 [mm]
b5 [mm]
H [mm]
Case B1 Case B2 Case B3
1.6 1.6 1.6
41 54.7 53.6
1.6 1.6 1.6
13.4 5 5
4.2 3 3
23.4 5 5
58.4 85.2 90.2
1 1 1
100 100 105
99
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Fig. 7. Cracks’ shape after the 25th step of propagation (case B1).
Fig. 8. Cracks’ shape after the 32nd step of propagation (case B2).
Fig. 9. Cracks’ shape after the 39th step of propagation (case B2).
I=
2 1 aux (KI KIaux + KII KIIaux ) + KIII KIII E∗ μ
ΔK eqv =
(2)
1 θ cos( )[ΔKI (1 + cosθ) − 3ΔKII sinθ] 2 2
where:
aux where KI . KII and KIII are mode I, II, and III SIFs, KIaux . KIIaux and KIII are E auxiliary mode I, II, and III SIFs, E ∗ = E for plane stress and E ∗ = 1 − ν2 for plane strain, E is Young’s modulus, ν is Poisson’s ratio and μ is shear modulus. Equivalent stress-intensity-factor range is calculated using
ΔKI = KImax − KImin
ΔKII = KIImax − KIImin 100
(3)
International Journal of Fatigue 123 (2019) 96–104
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Fig. 10. Cracks after the 31st step of propagation (case B3).
propagation. SIF values in cases B2 and B3 are much lower than in the case A. Also, SIF values are very similar when the cracks’ lengths are 10–21 mm. On the other hand, values of K eq in case B3 are somewhat lower than in case B2. Anyhow, case B2 was chosen as the optimum, because its dimensions correspond better with dimensions of existing differential spar, i.e. it could fit into the same place of the wing root (height 100 mm). Also, crack growth in this case is more realistic than in case B3, since the 2nd crack did not stop; it met 1st crack after 17 growth steps and they formed one front which moved along spar web (Fig. 9). After 37th step crack front reached additional cap and split into two fronts: 1st front continued to move along the web (Fig. 11), while the 2nd started to move along the cap. Simulation was interrupted after 40th step because crack front on the web reached the same point as the crack in integral spar with original design.
Table 2 Average K eq values for each step of cracks’ propagations. Step
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Case A
Case B1
Case B2
Case B3
a (mm)
Keq
a (mm)
Keq
a (mm)
Keq
a (mm)
Keq
1.00 2.12 3.16 4.16 5.17 6.17 7.17 8.16 9.16 10.15 11.14 12.14 13.11 14.08 15.06 16.02 17.00 17.96 18.93 19.89 20.86 21.80 22.72 23.66 24.30 24.85 27.47 27.85 28.64 29.37
91.7 151.4 193.8 228.3 260.5 291.4 316.9 352.1 373.4 414.7 449.1 473.7 521.6 554.5 582.3 637.8 660.4 718.6 731.8 775.8 779.3 792.3 774.8 705.8 537.6 531.4 531.1 587.8 688.6 706.4
0.82 1.70 2.72 3.52 4.40 5.33 6.26 7.25 8.21 9.05 9.86 10.62 11.51 12.44 13.19 14.13 14.89 15.47 16.26 17.06 17.84 18.54 19.39 19.90 20.40 20.46
93.3 146.5 185.4 215.2 242.7 271.0 300.8 318.8 334.4 342.4 363.0 383.8 411.7 439.8 444.4 469.1 460.7 418.9 511.8 543.9 547.9 535.5 448.9 451.8 507.5 506.0
0.78 1.68 2.27 2.82 3.54 4.35 5.28 6.33 7.27 8.29 9.18 10.17 11.20 12.23 13.24 14.10 14.99 15.83 16.66 17.45 18.22 18.95 19.71 20.56 21.46 22.40 23.36 24.33 25.31 26.26
99.4 139.0 155.6 173.4 189.2 203.3 212.2 223.2 230.4 243.0 256.6 271.2 288.2 297.9 312.7 330.1 332.9 347.3 349.3 358.2 367.0 372.8 385.8 395.1 414.4 429.2 441.6 458.3 470.3 436.6
0.79 1.51 2.09 2.79 3.65 4.50 5.53 6.43 7.20 7.99 8.89 9.88 10.71 11.66 12.63 13.59 14.59 15.57 16.53 17.52 18.48 19.46 20.45 21.41 22.38 23.36 24.32 25.31 26.13 26.60
91.2 112.6 105.0 82.8 95.1 105.2 102.3 115.8 132.6 147.0 191.0 217.3 236.9 251.5 268.4 281.9 297.2 310.5 319.4 336.1 341.5 349.9 360.6 353.8 362.8 362.4 363.9 349.7 274.8 273.0
5. Analysis of results Notwithstanding the different cracks’ paths obtained in 3D simulations compared to the real crack path obtained in experiment, it was decided to estimate the numbers of cycles that will grow initial crack until final cap failure and, then, to compare these numbers with the number of cycles observed in the experiment. For that purpose, modified Paris-Erdogan law was used with the stress ratio R = 0.15 identical to the ratio obtained in experiment. The graph of crack growth (in mm) as a function of number of cycles obtained in simulation with differential spar is shown in Fig. 12 (blue dots), along with experimental data (red1 dots). Graph shows that the initial crack 1 mm long extends to 2 mm after approximately 27,000 cycles and reaches a length of 3 mm after next 5850 cycles. It moves from horizontal to vertical spar cap wall after approximately 45,000 cycles from initiation, while the final crack length (45 mm, observed in the experiment) is achieved after additional 5750 cycles, Number of cycles required to initialize the crack in the spar cap (7944 cycles) was estimated by Ansys Workbench, [27]. So, the total spar fatigue life is 58,694 cycles, being very close to the number of cycles obtained in experiment (58,520). Although paths were not exactly the same, this result shows very good agreement between numerical and experimental results. The same procedure was used for crack growth rate estimation in the case of integral spar. The number of cycles obtained for the 1st crack on integral spar showed that the initial crack (1 mm long) will extend to 2 mm after approximately 198,000 cycles and to 3 mm after another 52,230 cycles. It will enter the area between the horizontal and
and direction of propagation (kink angle) is defined as:
θ = cos−1 (
3(KIImax )2 + (KImax ) (KImax )2 + 8(KIImax )2 (KImax )2 + 9(KIImax )2
(4)
Table 2 indicates that the K eq values in all cases increase until a certain step: for instance, 22nd step in case A – K eq = 792.3 MPamm0.5, 21st step in case B1 – K eq = 547.9 MPamm0.5, 29th step in case B2 – and 27th step in case B3 – K eq = 470.3 MPamm0.5 K eq = 363.9 MPamm0.5. After these steps, K eq starts to decline, which coincides with the cracks reaching the vertical wall of spar. The equivalent SIF values continue to decline until the cracks leave the area between horizontal and vertical wall (which is thicker than the other parts of spar), and then start to grow again until the last steps of
1 For interpretation of color in Fig. 12, the reader is referred to the web version of this article.
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Fig. 11. Cracks paths in redesigned integral spar.
Fig. 12. Crack length vs. number of cycles for the numerical model of the differential spar.
vertical cap wall after approximately 341,250 cycles and, finally, it will reach the total length of 36 mm after 345,800 cycles (Fig. 13). It must be noted that final crack length in simulation (36 mm) was shorter than crack length in experiment (45 mm); nevertheless, estimated fatigue life of integral spar is obviously much longer. The number of cycles for integral spar is approximately 7 times greater than the value obtained for differential spar. Finally, Fig. 14 shows estimated fatigue life in the redesigned case B2, being much longer (over 2 million cycles) compared to other spars,
indicating significant increase in fatigue life. It should be mentioned that in the case of integral spar with original dimensions, the crack growth from 10 to 30 mm took only 14% of estimated fatigue life (8250/58,600), while in the case of redesigned integral spar the same crack growth took circa 70% of estimated life (Fig. 14). Taking this fact into account, it may be concluded that additional cap will not only affect the way cracks propagate through the spar, but it will also significantly increase spar fatigue life.
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Fig. 13. Crack length vs. number of cycles for integral spar.
Fig. 14. Crack length vs. number of cycles for redesigned integral spar.
6. Conclusions Numerical simulation of fatigue crack growth wing spars was performed, using extended finite element method, providing crack paths, stress intensity factor values and fatigue life for each crack path. Based on the results obtained for differential and integral designs simultaneously compared with experimental results, the following can be concluded:
•
• In the experiment, two cracks propagated in spar caps riveted to •
spar web until complete failure, while – at the same time – web remained undamaged during the test. In numerical analysis of the differential spar crack grew from horizontal cap wall to vertical wall; this path was somehow different
• 103
from that observed in experiment. In the integral spar of the same cross-sectional design and same mass, one crack stopped after few steps of propagation, whilst the other continued to grow and produced considerable damage in vertical wall. On the other hand, in the redesigned spar two cracks initiated at the left and right edge of the bottom cap merged after several steps and propagated along spar web, only to be separated again: one new front continued to move along the web until it reached newly added cap, while the other front continued to move along vertical wall. This “joining” and “splitting” resulted in significant increase in estimated number of cycles. Finally, it is highly recommended to replace existing differential spar with integral spar of the same mass, but optimized cross-section shape.
International Journal of Fatigue 123 (2019) 96–104
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Appendix A. Supplementary material
[13] Sghayer A, Grbović A, Sedmak A, Dinulović M, Doncheva E, Petrovski B. Fatigue life analysis of the integral skin-stringer panel using XFEM. Struct Integrity and Life 2017;17:7–10. [14] Sghayer A, Grbović A, Sedmak A, Dinulović M, Grozdanovic I, Sedmak S, et al. Experimental and numerical analysis of fatigue crack growth in integral skinstringer panels. Technical Gazete 2018;25(3):785–91. [15] Hojjati-Talemi R, Waha MA. XFEM for fretting fatigue: straight VS mixed mode crack propagation. In: Proceedings of the 4th international conference on crack paths (CP 2012), Gaeta, Italy; 2012. p. 351–8. [16] Curà F, Mura A, Rosso C. Effect of centrifugal load on crack path in thin-rimmed and webbed gears. Proceedings of the 5th International Conference on Crack Paths (CP 2015), Ferrara, Italy. 2015. p. 512–20. [17] Taheri S, Julan E, Tran V-X. Fatigue crack growth and arrest under high-cycle thermal loading using XFEM in presence of weld residual stresses. Proceedings of the 5th International Conference on Crack Paths (CP 2015), Ferrara, Italy. 2015. p. 908–17. [18] Kraedegh A, Li W, Sedmak A, Grbović A, Trišović N, Mitrović R, et al. Simulation of fatigue crack growth in A2024–T351 T welded joint. Structural Integrity and Life 2017;17(1):3–7. [19] Kastratović G, Grbović A, Vidanović N. Approximate method for stress intensity factors determination in case of multiple site damage. Appl Math Model 2015;39(19):6050–9. [20] Dirik H, Yalçinkaya T. Fatigue crack growth under variable amplitude loading through XFEM. Proc Struct Integrity 2016;2:3073–80. [21] Marsavina L, Constantinescu DM, Linul E, Stuparu FA, Apostol DA. Experimental and numerical crack paths in PUR foams. Eng Fract Mech 2016;167:68–83. [22] Bergara A, Dorado JI, Martin-Meizoso A, Martínez-Esnaola JM. Fatigue crack propagation in complex stress fields: experiments and numerical simulations using the extended finite element method (XFEM). Int J Fatigue 2017;103:112–21. [23] Naderi M, Iyyer N. Fatigue life prediction of cracked attachment lugs using XFEM. Int J Fatigue 2015;77:186–93. [24] Sedmak A. Computational fracture mechanics – an overview from early efforts to recent developments. Fatigue Fract Eng Mater Struct. 2018;41:2438–74. [25] Petrašinović D, Rašuo B, Petrašinović N. Extended finite element method (XFEM) applied to aircraft duralumin spar fatigue life estimation. Technical Gazette 2012;19(3):557–62. [26] Kastratović G, Vidanović N, Grbović A, Rašuo B. Approximate determination of stress intensity factor for multiple surface cracks. FME Trans 2018;46:39–45. [27] ANSYS Workbench, Release 18.0, ANSYS, Inc.
Supplementary data to this article can be found online at https:// doi.org/10.1016/j.ijfatigue.2019.02.013. References [1] Maheswaran N, Venkatesan S, Sampath Kumar M, Velmurugan G. Study of weight optimization on spar beam for the wing of an aircraft. Int J Softw Hardw Res Eng 2015;3(3):72–9. [2] Dutt KM, Kumar DBN. Damage tolerance evaluation of the front spar in a transport aircraft wing. Int J Innovat Res Sci, Eng Technol 2013;2(9):5048–55. [3] Souiyah M, Alshoaibi A, Muchtar A, et al. Stress intensity factor evaluation for crack emanating from circular-hole using finite element method. Int Rev Mech Eng 2008;2. [4] Souiyah M, Muchtar A, Alshoaibi A, et al. Finite element analysis of the crack propagation for solid materials. Am J Appl Sci 2009;6(7):1396–402. [5] Carlone P, Citarella R, Lepore M, Palazzo GS. A FEM-DBEM investigation of the influence of process parameters on crack growth in aluminium friction stir welded butt joints. Int J Mater Form 2015;8(4):591–9. [6] Pook LP, Campagnolo A, Berto F, Lazzarin P. Coupled fracture mode of a cracked plate under anti-plane loading. Eng Fract Mech 2015;134:391–403. [7] Belytschko T, Black T. Elastic crack growth in finite elements with minimal remeshing. Int J Numer Mech Eng 1999;45:601–20. [8] Moës N, Dolbow J, Belytschko T. A finite element method for crack growth without remeshing. Int J Numer Mech Eng 1999;46:131–50. [9] Jovičić G, Živković M, Sedmak A, Jovičić N, Milovanović D. Improvement of algorithm for numerical crack modeling. Arch Civil Mech Eng 2010;10:19–35. [10] Grbovic A, Rašuo B. Use of modern numerical methods for fatigue life predictions Recent Trends in Fatigue Design. New York: Nova Science Publishers; 2015 Chapter 2. [11] Aldarwish M, Grbović A, Kastratović G, Sedmak A, Lazić M. Stress intensity factors evaluation at tips of multi-site cracks in unstiffened 2024–T3 aluminum panel using XFEM. Technical Gazete 2018. in press. [12] Aldarwish M, Grbović A, Kastratović G, Sedmak A, Vidanović N. Numerical assessment of stress intensity factors at tips of multi-site cracks in unstiffened panel. Struct Integrity Life 2017;17:11–4.
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International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue
Fatigue crack paths in light aircraft wing spars a,⁎,⁎
Aleksandar Grbović Mihajlo D. Popovića a b
b
T a
a
, Gordana Kastratović , Aleksandar Sedmak , Igor Balać ,
University of Belgrade, Faculty of Mechanical Engineering, Serbia University of Belgrade, Faculty of Transport and Traffic Engineering, Serbia
A R T I C LE I N FO
A B S T R A C T
Keywords: Integral and differential structures Extended finite element method Crack paths Crack growth rate
Different cracks paths in three different wing spar designs are analysed (differential, integral and optimized integral) to explore how those cracks paths affect fatigue life estimation. First, numerical analysis was carried out and verified, using the experimental data for differential wing spar, followed by numerical analysis of both optimized integral wing spars. The optimized integral spar was obtained by analysing three different crosssections regarding fatigue life. Crack propagation simulation and fatigue life estimation were carried out by the extended finite element method, using Morfeo/Crack for Abaqus. Results provided better understanding and prediction of multiple cracks propagation in 3D structures.
1. Introduction The main load-carrying member of the aircraft wing is spar, [1]. It is positioned perpendicularly to the flight direction and is extended from the fuselage to the wing tip. Spar usually has I-beam shape, made of thin shear panel (web) and flanges (caps) at the top and bottom, assembled by rivets. Light airplanes mostly have wings with one spar, carrying almost all bending and shear load, [2]. Under service loading a fatigue crack may initiate from the most severe stress concentration on the spar bottom cap. It can grow unnoticeably, first in the spar cap and – after the cap failure – in the spar web, eventually leading to catastrophic failure if not detected and repaired, [2]. Since fatigue crack initiation and growth on riveted structures were thoroughly investigated over the past decades, it is difficult to get significant improvements concerning the extension of their fatigue life. On the other hand, integral structures are lighter and cheaper, easier to inspect and have less holes sensitive to high stress concentration. Since their manufacturing is more expensive, integral spars must have much longer fatigue life to justify higher initial costs. In recent decades, fracture mechanics parameters became essential for the prediction of crack initiation and propagation, i.e. for fatigue life estimation. The stress intensity factor (SIF) is one of the most important parameter which provides data on crack initiation and propagation. In complex geometries, such as wing spar, it is almost impossible to find an exact solution for SIFs; therefore, the numerical methods are needed for their estimation.
Over the years, many numerical techniques, such as finite element method (FEM), boundary element method (BEM), mesh-free methods and extended finite element method (XFEM), have been developed to simulate the fracture mechanics problems. Souiyah et al. [3], calculated SIFs for a crack emanating from circular hole in rectangular plate and double edge notched plate, using an adaptive mesh finite element technique. The same authors in [4] employed finite element method to predict the crack propagations directions and to calculate SIFs, and then validated the verification of the predicted SIF and crack trajectory, with the relevant numerical and analytical results. The FEM was one of the used methods in [5], where numerical investigation was carried out in order to investigate the influence of friction stir welding process parameters on fatigue crack growth in AA2024-T3 butt joints. Authors in [6] used 3D finite element analyses to study a coupled fracture mode generated by a nominal anti-plane (Mode III) loading, applied to linear elastic plates weakened by a straight through-the-thickness crack. The XFEM was developed in 1999 by Belytschko and Blackin [7], and Moës, Dolbow and Belytschko [8]. They proposed a new technique for modelling cracks in the finite element framework, where standard displacement‐based approximation was enriched near a crack by incorporating both discontinuous fields and the near tip asymptotic fields through a partition of unity method. They developed a methodology that was able to construct the enriched approximation from the interaction of the crack geometry with the mesh. This technique allowed the entire crack to be represented independently of the mesh, so that re-
⁎
Corresponding author. E-mail addresses: [email protected] (A. Grbović), [email protected] (G. Kastratović), [email protected] (A. Sedmak), [email protected] (I. Balać), [email protected] (M.D. Popović). https://doi.org/10.1016/j.ijfatigue.2019.02.013 Received 10 December 2018; Received in revised form 30 January 2019; Accepted 11 February 2019 Available online 12 February 2019 0142-1123/ © 2019 Published by Elsevier Ltd.
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Fig. 1. Differential spar used in experiment (dimensions in mm), [25].
foam materials under mixed mode loading, using an asymmetric semicircular bend and four-point mixed mode specimens was analysed, and numerical simulations of crack propagation were conducted by XFEM. Paper [22] presented the numerical simulation and validation of a fatigue propagation test of a semi-elliptical crack located at the side of the rectangular section of a beam subjected to four-point bending. For the numerical simulations, XFEM implemented in the Abaqus software has been used. The comparison between the experimental and numerical results showed very good correlation regarding crack shape and number of cycles to failure. In [23], XFEM capability available in ABAQUS was used to calculate the stress intensity factor in straight lugs of Aluminum 7075-T6. Crack growth and fatigue life of single through-thickness and single quarter elliptical corner cracks in attachment lug were estimated and then compared with the available experimental data for two different load ratios equal to 0.1 and 0.5. The SIF calculated from XFEM showed that the introduction of different loading boundary conditions significantly affected the estimated fatigue life. Finally, a comprehensive overview of XFEM simulations is given in [24]. In this paper, numerical study of possible crack paths in wing spar under load is presented. To investigate and improve fatigue life (and hence overall performance) of light aircraft UTVA 75, three different wing spar designs were examined: the existing differential spar, the integral spar with same dimensions as differential spar and redesigned integral spar, with the same mass as existing spar. Redesigned integral spar was obtained thru optimization of its cross-section, where three different cross-sections were also analysed. The optimization was carried out in order to improve fatigue life of the integral spar. The special attention was paid to the analysis of cracks growth and their paths along the spar in each case. All computations for crack propagation simulation and fatigue life estimation were carried out by XFEM, using Morfeo/Crack for Abaqus code.
meshing was not necessary to model crack growth. In [9], an improvement of XFEM was presented, which enabled more efficient calculation of SIFs in 2D problems. The XFEM has also been used to calculate SIFs for problems involving multiple, interacting cracks, resulting from multiple site damage (MSD). In [10], it is shown how XFEM could be used for obtaining not only an accurate estimation of the fatigue life of the assembly, but also to predict the most probable directions in which cracks will develop through the structure. In [11,12], SIFs calculations based on implementation of XFEM in Abaqus, were conducted for a typical aero structural configuration with MSD. Analysed model was a unique 3D configuration with 22 cracks that propagate at the same time, whereas stress intensity factors were computed along the crack fronts for all 22 cracks. The fatigue life estimation has been conducted using XFEM, as well as for the simulating crack paths development in more complex models. Using extended finite element method, Sghayer et al. [13,14], simulated the fatigue behavior and crack propagation of the real laser beam welded skin-stringer panel with four stringers. Also in [14], fatigue life of simple flat plate, as well of the skin-stringer panel, were numerically simulated by using XFEM to investigate the effect of stringers. Rojjati-Talemi and Waha [15], proposed a modified fretting fatigue contact model in conjunction with extended finite element method, in order to monitor the effect of mixed mode on fretting fatigue crack propagation. Curà et al. [16] investigated the influence of centrifugal load on crack propagation path in thin rimmed and webbed gears. The investigation has been carried out by means of numerical models involving both 2D finite element and extended finite element models (XFEM). In [17], authors tried to determine the influence of weld residual stresses on crack growth rate and crack arrest, using Paris’ law and XFEM. Fatigue crack growth in friction stir welded T joints under three point bending were simulated numerically in [18], by using XFEM, also used in [19] to verify the SIFs solutions calculated by proposed approximation method, based on superposition. The aim of the study [20] was to predict fatigue crack growth life under variable amplitude loading through XFEM and to explicitly illustrate both fatigue crack growth life and crack propagation. The results were compared with NASGRO crack propagation software and experimental fatigue crack growth life test data on 7075-T6 aluminum alloy under various over load and over load-under load conditions which exhibited a good agreement. In [21] crack initiation and quasi-static propagation in polyurethane
2. Experimental and numerical analysis of the differential wing spar Numerical analysis was first carried out and verified using the experimental data for differential wing spar made of AA2024-T3, [25], with dimensions showed in Fig. 1. The load used in the experiment (minimum value +391.2 N, maximum value +2028 N, frequency 12.5 Hz), produced high tensile stress in caps; consequently, visible crack appeared on the left cap below the support after only 8542 cycles. The crack began to spread rapidly toward the spar web, then changed 97
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Fig. 2. Cracks on differential spar caps, view from above, [25].
Fig. 3. Damaged differential spar with strain gauges, [25].
its direction and continued along the cap at an angle of 90° with respect to the original direction. After 39,450 cycles, another crack was spotted on the right spar cap, initiated at the fastener hole, Fig. 2. Cracks then continued to grow but were not visible. After 58,520 cycles, the test was stopped, and the spar was examined, as shown in Figs. 2 and 3. There was no visible damage in the spar web. It should be noted that in 10 samples tested under different load histories crack always appeared in the same zone, at the edge of left upper cap zone, but at various positions within it, which is consistent with the stochastic nature of examined phenomenon. Therefore, numerical analysis of differential spar was conducted using Morfeo/Crack for Abaqus code, with the initial crack positioned at the edge of left upper cap zone. Crack was then propagated along the longest possible crack path toward the wall of spar in order to monitor and analyse changes in the stress intensity factors, [25]. After opening, the crack growth was simulated in steps of maximum 1 mm. The growth direction was not restricted, so the crack could grow in direction predicted by calculated kink angle (see Eq. (4)). After 19 steps, crack had the shape shown in Fig. 4a which clearly displays that the path on the horizontal part of the spar cap was winding, like the path in the experiment, Fig. 3. That was a solid proof that the boundary conditions and displacements taken from experiment were properly applied and that the numerical model was appropriate. However, crack path on the vertical spar wall (Fig. 4b) did not match completely the shape obtained in the experiment, because the real spar was under the influence of residual stresses induced during manufacturing, [10].
Fig. 4. (a) Crack on horizontal cap wall after 19th step of growth; (b) Crack on vertical cap wall after 45th step of expansion.
3. Numerical analysis of integral spar As the next step, the FE model of integral spar made of AA2024-T3, with exactly the same dimensions and mass as differential spar, was analysed. FE model consisted of 151,700 linear hexahedron elements of type C3D8R. The simultaneous growth of two penny shaped cracks located at left and right edge of spar cap, was simulated again in Morfeo/Crack for Abaqus. Initial cracks length was 1 mm. The path of the left crack was not straight, but curved (Fig. 5a), while the right crack did not propagate at all after the 8th step because the deformation of spar, caused by the 1st crack, literally “closed” it. After reaching the vertical spar wall, the left crack formed two fronts and continued to propagate simultaneously in the horizontal (1st front) and the vertical direction (2nd front), Fig. 5. 98
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4. Optimization of integral spar cross section based on fatigue life Next step was to optimize the cross-sectional shape of integral spar (numerically investigated in previous chapter) using fatigue crack growth life approach. 3D Software packages (CATIA & Abaqus) were used to design the integral spars’ structure and XFEM was again used to calculate the SIFs and the number of cycles of crack propagation. Several iterations were conducted for the design optimization of the integral spars. Performance of the beam was studied by comparison of the fatigue crack life for two different cross-sectional shapes of integral spar, but of the same area (i.e. the mass of the spar was kept constant). The first was I-beam spar investigated in the previous chapter, named here the main integral spar (case A). The second was I-beam integral spar with intermediate cap (case B). In addition, three sizes (different dimensions) of Ibeam integral spar with intermediate cap (B1, B2, B3) were analysed. The overall dimensions of the cases A and B are shown in Fig. 6, while dimensions for cases B1, B2, B3 are given in Table 1.
4.1. Evaluation of SIFs and crack trajectory for all cases After opening, the cracks propagated in steps of approximately 1 mm for all cases. Growth was again not restricted to a single plane; instead, the crack could grow in the direction defined by calculated kink angle. The shapes of these cracks can be seen in Figs. 7–10. After 25 steps of propagation the FE model of case B1 had shape shown in Fig. 7, indicating that paths of the left and right cracks on the horizontal wall were not straight; so, at the end of simulation they did not linkup. On the contrary, the left and right crack in the FE model of case B2 linked up at 32nd step of propagation (Fig. 8) and after that new crack propagated along the vertical cap until the end of numerical simulation at 39th step (Fig. 9). Left crack in the FE model of case B3 also propagated in horizontal cap along curved path until the end of simulation at 31st step. As in the case A, the right crack stopped after the 10th step when it reached the length of 4 mm, as shown in Fig. 10.
Fig. 5. (a) 1st crack tip after 35 growth steps; (b) 2nd crack tip after 35 growth steps.
4.2. Comparisons of SIFs For all cases, stress intensity factors Mode I, II, III and equivalent stress intensity factor K eq were calculated within Morfeo/Crack for Abaqus for each node on crack fronts. Here, the equivalent SIF is used as a reference for comparisons. For each step of propagation, the average values of K eq in all crack fronts’ nodes were calculated and shown in Table 2. SIF values are given in MPamm0.5. The SIFs were computed along the crack fronts using the interaction integral method, [26], defined as:
Fig. 6. Cross section shapes (Case A left, Case B right) used in optimization (dimensions in mm).
After the 35th step of propagation (counting from the crack opening) the simulation stopped because the 1st crack front left the area with refined mesh (Fig. 5a), and software couldn’t create new nodes for the 1st crack front with required accuracy. Position of the 2nd crack front after 35 steps of propagation is shown in Fig. 5b.
I=
− ∫V qi, j (σkl εklaux δij − σkjaux uki − σkj ukiaux ) dV
∫s δqn ds
(1)
where σij, εij, ui are the stress, strain and displacement, σijaux , εijaux , uiaux are the stress, strain and displacement of the auxiliary field, and qi is the crack-extension vector. The interaction integral is associated with the stress-intensity factors through:
Table 1 The overall dimensions for I-beam spar with intermediate cap. Case no.
a1 [mm]
b1 [mm]
a2 [mm]
a3 [mm]
b3 [mm]
a4 [mm]
a5 [mm]
b5 [mm]
H [mm]
Case B1 Case B2 Case B3
1.6 1.6 1.6
41 54.7 53.6
1.6 1.6 1.6
13.4 5 5
4.2 3 3
23.4 5 5
58.4 85.2 90.2
1 1 1
100 100 105
99
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Fig. 7. Cracks’ shape after the 25th step of propagation (case B1).
Fig. 8. Cracks’ shape after the 32nd step of propagation (case B2).
Fig. 9. Cracks’ shape after the 39th step of propagation (case B2).
I=
2 1 aux (KI KIaux + KII KIIaux ) + KIII KIII E∗ μ
ΔK eqv =
(2)
1 θ cos( )[ΔKI (1 + cosθ) − 3ΔKII sinθ] 2 2
where:
aux where KI . KII and KIII are mode I, II, and III SIFs, KIaux . KIIaux and KIII are E auxiliary mode I, II, and III SIFs, E ∗ = E for plane stress and E ∗ = 1 − ν2 for plane strain, E is Young’s modulus, ν is Poisson’s ratio and μ is shear modulus. Equivalent stress-intensity-factor range is calculated using
ΔKI = KImax − KImin
ΔKII = KIImax − KIImin 100
(3)
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Fig. 10. Cracks after the 31st step of propagation (case B3).
propagation. SIF values in cases B2 and B3 are much lower than in the case A. Also, SIF values are very similar when the cracks’ lengths are 10–21 mm. On the other hand, values of K eq in case B3 are somewhat lower than in case B2. Anyhow, case B2 was chosen as the optimum, because its dimensions correspond better with dimensions of existing differential spar, i.e. it could fit into the same place of the wing root (height 100 mm). Also, crack growth in this case is more realistic than in case B3, since the 2nd crack did not stop; it met 1st crack after 17 growth steps and they formed one front which moved along spar web (Fig. 9). After 37th step crack front reached additional cap and split into two fronts: 1st front continued to move along the web (Fig. 11), while the 2nd started to move along the cap. Simulation was interrupted after 40th step because crack front on the web reached the same point as the crack in integral spar with original design.
Table 2 Average K eq values for each step of cracks’ propagations. Step
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Case A
Case B1
Case B2
Case B3
a (mm)
Keq
a (mm)
Keq
a (mm)
Keq
a (mm)
Keq
1.00 2.12 3.16 4.16 5.17 6.17 7.17 8.16 9.16 10.15 11.14 12.14 13.11 14.08 15.06 16.02 17.00 17.96 18.93 19.89 20.86 21.80 22.72 23.66 24.30 24.85 27.47 27.85 28.64 29.37
91.7 151.4 193.8 228.3 260.5 291.4 316.9 352.1 373.4 414.7 449.1 473.7 521.6 554.5 582.3 637.8 660.4 718.6 731.8 775.8 779.3 792.3 774.8 705.8 537.6 531.4 531.1 587.8 688.6 706.4
0.82 1.70 2.72 3.52 4.40 5.33 6.26 7.25 8.21 9.05 9.86 10.62 11.51 12.44 13.19 14.13 14.89 15.47 16.26 17.06 17.84 18.54 19.39 19.90 20.40 20.46
93.3 146.5 185.4 215.2 242.7 271.0 300.8 318.8 334.4 342.4 363.0 383.8 411.7 439.8 444.4 469.1 460.7 418.9 511.8 543.9 547.9 535.5 448.9 451.8 507.5 506.0
0.78 1.68 2.27 2.82 3.54 4.35 5.28 6.33 7.27 8.29 9.18 10.17 11.20 12.23 13.24 14.10 14.99 15.83 16.66 17.45 18.22 18.95 19.71 20.56 21.46 22.40 23.36 24.33 25.31 26.26
99.4 139.0 155.6 173.4 189.2 203.3 212.2 223.2 230.4 243.0 256.6 271.2 288.2 297.9 312.7 330.1 332.9 347.3 349.3 358.2 367.0 372.8 385.8 395.1 414.4 429.2 441.6 458.3 470.3 436.6
0.79 1.51 2.09 2.79 3.65 4.50 5.53 6.43 7.20 7.99 8.89 9.88 10.71 11.66 12.63 13.59 14.59 15.57 16.53 17.52 18.48 19.46 20.45 21.41 22.38 23.36 24.32 25.31 26.13 26.60
91.2 112.6 105.0 82.8 95.1 105.2 102.3 115.8 132.6 147.0 191.0 217.3 236.9 251.5 268.4 281.9 297.2 310.5 319.4 336.1 341.5 349.9 360.6 353.8 362.8 362.4 363.9 349.7 274.8 273.0
5. Analysis of results Notwithstanding the different cracks’ paths obtained in 3D simulations compared to the real crack path obtained in experiment, it was decided to estimate the numbers of cycles that will grow initial crack until final cap failure and, then, to compare these numbers with the number of cycles observed in the experiment. For that purpose, modified Paris-Erdogan law was used with the stress ratio R = 0.15 identical to the ratio obtained in experiment. The graph of crack growth (in mm) as a function of number of cycles obtained in simulation with differential spar is shown in Fig. 12 (blue dots), along with experimental data (red1 dots). Graph shows that the initial crack 1 mm long extends to 2 mm after approximately 27,000 cycles and reaches a length of 3 mm after next 5850 cycles. It moves from horizontal to vertical spar cap wall after approximately 45,000 cycles from initiation, while the final crack length (45 mm, observed in the experiment) is achieved after additional 5750 cycles, Number of cycles required to initialize the crack in the spar cap (7944 cycles) was estimated by Ansys Workbench, [27]. So, the total spar fatigue life is 58,694 cycles, being very close to the number of cycles obtained in experiment (58,520). Although paths were not exactly the same, this result shows very good agreement between numerical and experimental results. The same procedure was used for crack growth rate estimation in the case of integral spar. The number of cycles obtained for the 1st crack on integral spar showed that the initial crack (1 mm long) will extend to 2 mm after approximately 198,000 cycles and to 3 mm after another 52,230 cycles. It will enter the area between the horizontal and
and direction of propagation (kink angle) is defined as:
θ = cos−1 (
3(KIImax )2 + (KImax ) (KImax )2 + 8(KIImax )2 (KImax )2 + 9(KIImax )2
(4)
Table 2 indicates that the K eq values in all cases increase until a certain step: for instance, 22nd step in case A – K eq = 792.3 MPamm0.5, 21st step in case B1 – K eq = 547.9 MPamm0.5, 29th step in case B2 – and 27th step in case B3 – K eq = 470.3 MPamm0.5 K eq = 363.9 MPamm0.5. After these steps, K eq starts to decline, which coincides with the cracks reaching the vertical wall of spar. The equivalent SIF values continue to decline until the cracks leave the area between horizontal and vertical wall (which is thicker than the other parts of spar), and then start to grow again until the last steps of
1 For interpretation of color in Fig. 12, the reader is referred to the web version of this article.
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Fig. 11. Cracks paths in redesigned integral spar.
Fig. 12. Crack length vs. number of cycles for the numerical model of the differential spar.
vertical cap wall after approximately 341,250 cycles and, finally, it will reach the total length of 36 mm after 345,800 cycles (Fig. 13). It must be noted that final crack length in simulation (36 mm) was shorter than crack length in experiment (45 mm); nevertheless, estimated fatigue life of integral spar is obviously much longer. The number of cycles for integral spar is approximately 7 times greater than the value obtained for differential spar. Finally, Fig. 14 shows estimated fatigue life in the redesigned case B2, being much longer (over 2 million cycles) compared to other spars,
indicating significant increase in fatigue life. It should be mentioned that in the case of integral spar with original dimensions, the crack growth from 10 to 30 mm took only 14% of estimated fatigue life (8250/58,600), while in the case of redesigned integral spar the same crack growth took circa 70% of estimated life (Fig. 14). Taking this fact into account, it may be concluded that additional cap will not only affect the way cracks propagate through the spar, but it will also significantly increase spar fatigue life.
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Fig. 13. Crack length vs. number of cycles for integral spar.
Fig. 14. Crack length vs. number of cycles for redesigned integral spar.
6. Conclusions Numerical simulation of fatigue crack growth wing spars was performed, using extended finite element method, providing crack paths, stress intensity factor values and fatigue life for each crack path. Based on the results obtained for differential and integral designs simultaneously compared with experimental results, the following can be concluded:
•
• In the experiment, two cracks propagated in spar caps riveted to •
spar web until complete failure, while – at the same time – web remained undamaged during the test. In numerical analysis of the differential spar crack grew from horizontal cap wall to vertical wall; this path was somehow different
• 103
from that observed in experiment. In the integral spar of the same cross-sectional design and same mass, one crack stopped after few steps of propagation, whilst the other continued to grow and produced considerable damage in vertical wall. On the other hand, in the redesigned spar two cracks initiated at the left and right edge of the bottom cap merged after several steps and propagated along spar web, only to be separated again: one new front continued to move along the web until it reached newly added cap, while the other front continued to move along vertical wall. This “joining” and “splitting” resulted in significant increase in estimated number of cycles. Finally, it is highly recommended to replace existing differential spar with integral spar of the same mass, but optimized cross-section shape.
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Appendix A. Supplementary material
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