Computational modeling of aircraft lugs failure under fatigue loading

Accepted Manuscript Computational modeling of aircraft lugs failure under fatigue loading Slobodanka Boljanović PII: DOI: Reference:

S0142-1123(18)30208-1 https://doi.org/10.1016/j.ijfatigue.2018.05.022 JIJF 4694

To appear in:

International Journal of Fatigue

Received Date: Revised Date: Accepted Date:

18 January 2018 5 May 2018 21 May 2018

Please cite this article as: Boljanović, S., Computational modeling of aircraft lugs failure under fatigue loading, International Journal of Fatigue (2018), doi: https://doi.org/10.1016/j.ijfatigue.2018.05.022

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Computational modeling of aircraft lugs failure under fatigue loading Slobodanka Boljanović 1

Mathematical Institute of the Serbian Academy of Sciences and Arts, Kneza Mihaila 36, Belgrade, Serbia, e-mail: [email protected]

* Corresponding author; phone: +381 63 805 60 85, fax: +381 11 35 11 282 Abstract In the present paper, a computational model of failure is developed to assess the fatigue behaviour of damaged aircraft lugs. In such a fracture mechanics-based analytical/numerical research work, the residual strength of lugs with either a throughthe-thickness crack or a quarter-elliptical corner crack is estimated, and then, through the experimental observations found in the literature, some applications are discussed. A stress field of pin-loaded linkage is numerically analyzed by using the finite element method. The failure resistance of lugs is quantified through the Huang-Moan crack growth concept in terms of fatigue life and crack path. The relevant crack driving forces are examined through a new analytical model, taking into account the effect of a lug head and the width-to-diameter ratio effect.

Keywords: Aircraft lug, Fatigue strength, FEM, Quarter-elliptical/through cracks, Crack path

1

Nomenclature a crack length in depth direction b

crack length in surface direction

C

material constant in fatigue crack growth law

da/dN

crack growth rate in depth direction

db/dN

crack growth rate in surface direction

D

diameter of the lug

E

elastic modulus

F0

correction factor related to the crack shape and lug width ratio

Flh

correction factor related to the lug-head height

gg



correction factors for angle location and crack configuration

H

height of the lug head

m

material constant in fatigue crack growth law

N

number of loading cycles to failure

R

load/stress ratio

K

stress intensity factor

S

applied stress

Su

ultimate strength

t

thickness of the lug

w

width of the lug

K

stress intensity factor range

P

applied force range

S

applied stress range



angle location



Poisson’s ratio

Subscripts A

depth position

B

surface position

f

failure

max

maximum value related to the applied load

0

initial

2

1. Introduction Aircraft structures contain several lug-type joints that act as safety connections between components and as a load transfer for relevant operations by means of a fork and a pin. Under service loading, the bearing capacity of such a mechanical assembly may be threatened because of appearance of fretting and corrosion in a stress-raised zone at the lug hole. Such a complex stress state can lead to the formation and propagation of fatigue cracks, and even to an abrupt failure. Therefore, the performance assessment of initial fatigue damages at a lug hole through adequate computational models of failure is crucial to guarantee the reliability and safety of the lug-linked components. The presence of a contact pressure at a stress-raised linkage considerably exacerbates the already complex nature of environment-load history interactions during service. According to the damage tolerance design strategies, such a dangerous phenomenon has to be examined by means of a fracture mechanics-based methodology, in which fatigue damages may theoretically be analyzed as the through-the-thickness crack or the surface (quarter-elliptical corner, semi-elliptical) crack problems [1-8]. The fracture strength of a lug with either a quarter-elliptical corner crack or a througththe-thickness crack has been modeled by Kathiresan et al. [9] by applying Forman’s and Walker’s crack growth concepts [10, 11], the Green’s function method [12] and the finite element method [13]. According to Narayana et al. [14] a modified crack closure integral model together with the finite element method have been used to estimate the fatigue behaviour of a through-the-thickness crack at a hole. Further, the aircraft lug with a quarter-elliptical crack has been analyzed by Kim et al. [15] through the Forman’s crack growth law [6] associated with the effective stress intensity factor, the weight function method [8] and the boundary element method [16]. The crack propagation in the case of the same lug configuration has been examined by Lanciotti et al. [17] employing the software programs NASGRO and AFGROW [18, 19], and the ABAQUS software package [20] (in which the finite element method is implemented) for residual life estimation and stress-intensity evaluation, respectively. The failure of lug with a through-the-thickness crack at a hole has theoretically been investigated by Antioni and Gaisne [21] using an analytical crack growth model. Then, Mikheevskiy et al. [22] have employed the UniGrow crack growth model proposed by Noroozi et al. [23] and the weight function [24] in order to assess the lug failure due to the quarter-elliptical crack growth. The pin-loaded linkage with a quarter-elliptical crack or through-the-thickness crack has been analyzed by Boljanović and Maksimović [25] through the two-parameter driving force model associated with the Newman’s relationships [26] and the finite element method incorporated in the MSC/NASTRAN software package [27]. Further, for the fatigue stability analysis of the quarter-elliptical crack at the lug hole Boljanović et al. [28] have developed software programs, in which the crack growth law proposed by Zhan et al. [29] and the relationships suggested by Raju and Newman [30] are used together with the J-integral method [31]. The variety of connections in aircraft structures is realized by the lugs, and the lug head contact areas can have a significant impact on the safety performance of lug-linked

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components. Therefore, in the present paper, a computational modeling tool is proposed for the fatigue analysis of pin-loaded lug with either a through-the-thickness crack or a quarter-elliptical corner crack, in which the interaction between the lug geometry and the operating conditions is examined. Such a new fracture mechanics-based analytical/numerical methodology takes into account the stress-intensity evaluation, the fatigue life assessment, and the crack path evolution, by involving the effect of lug head, the width-to-diameter effect and the stress ratio effect. The Huang-Moan crack growth concept is employed to estimate the residual life to failure for both lug configurations examined, whereas the crack path is evaluated for the quarter-elliptical corner crack at a lug hole. The non-linear stress state field is analyzed by using the singular finite elements. Finally, some failure assessments related to fatigue damages at a lug hole are presented and verified through appropriate experimental observations available in the literature. Previous research studies reported by the present author and colleagues [8, 25, 28] have suggested relevant fracture mechanics-based computational methodologies for analyzing the fatigue strength of the lug with through-the-thickness crack and/or quarter-elliptical corner crack, in which the effect of lug head and the width-to-diameter ratio effect were not taken into account. The computational model of failure herein presented, where such effects are examined, provides new contribution to the knowledge of the mechanical performance of lug-pin joints. Also the important benefit of the fracture mechanics-based analytical model developed is that it can simultaneously assess the failure behaviour of above damaged linkages without expensive and time-consuming experimental observations related to the quarter-elliptical crack configuration.

2. The Huang-Moan crack growth concept for evaluating the fatigue life Under service loading, due to the high notch effect of the contact linkage zone, the lugpin joints are decidedly susceptible to fatigue damages. Therefore, it is of a great importance to assess the stability of such crack-like flaws, through fracture mechanicsbased computational models. In the present study, a quarter-elliptical corner crack configuration (Fig. 1, case 1) is analyzed by taking into account the crack growth rates related to two critical directions (A and B crack positions) and applying the crack growth concept [32], as follows: da m (1a)  C A M K A  A dN db m (1b)  CB M K B  B dN where a and b denote depth and surface crack length, respectively, KA andKB are the stress intensity factor range related to the relevant points A and B at the crack front, whereas CA, mA and CB, mB represent material parameters (experimentally obtained) for crack growth in the depth and surface direction, respectively, M is a material parameter, and N denotes the number of loading cycles.

4

It is worth noting that a complex interaction between applied loads and environment, which may seriously endanger a bearing-capacity of mechanical components during service operations, is theoretically tackled here through the fracture mechanics parameter M, defined in terms of the stress ratio R, as follows: 1 R  1    M  1 R   2   1.05 1.4 R  0.6 R 

 5  R 0 0  R  0.5

(2)

0.5  R 1

where 1 and  as material parameters are employed to describe the fatigue crack growth behaviour in the case of negative and positive stress ratios through one and two cyclic loading domains [32], respectively. In order to quantify the residual strength performance of damaged lug against fatigue, the number of loading cycles to failure is herein evaluated. As a matter of fact, the relationships for crack growth rate are integrated from initial crack lengths (a0 , b0) to final crack lengths (af , bf) in depth and surface direction (Fig. 1, case 1), respectively, i.e.: af

da mA a 0 C A M K A 

N

(3a)

bf

db mB b0 C B M K B 

N

`

(3b)

From a damage tolerance point of view, one of the primary causes of lug failures is the propagation of a through-the-thickness crack. Hence, such fatigue damage located at a lug hole (Fig. 1, case 2) is also analyzed through the same crack growth concept by employing Eq. (3b). The durability of the safety-critical linkage is assessed through relevant software program, developed by the present author, in which the fatigue behaviour of the throughthe-thickness crack as well as quarter-elliptical corner crack is taken into consideration. In such a computational modeling tool, due to the integral form of Equations (3a) and (3b), the number of loading cycles to failure is evaluated by employing the numerical integration method based on the Simpson’s algorithm. Further, according to the fracture mechanics condition, it is assumed that the failure occurs when the value of stress intensity factor reaches the fracture toughness, for computing the final crack length under cyclic loading.

PLEASE insert Figure 1 here.

5

3. Driving forces for fatigue damages at the lug hole Design of optimized and safe structures subjected to cyclic loading requires the failure performance analysis of relevant stress-raised components through fracture mechanicsbased analytical and/or numerical approaches [8, 25, 28, 33-39]. In this context, for the pin-loaded lug with a through-the-thickness crack (Fig. 1, case 2), the driving force behaviour is analytically here examined by taking into account the stress intensity factor, defined as follows: (4) K Bt  Fcor S  b where KBt is the stress intensity factor range in the case of the through-crack growth, b and Fcor are crack length and a corrective function, respectively, and S denotes applied gross/net stress range. Since the lug-pin joints are mainly used for bearing loads on structural supports, lifting heavy loads and dragging the lug, according to the damage-tolerance and operational requirements, the boundary conditions related to the lug performance are taken into account through the following corrective function:  w   w 2b   b D  (5) Fcor   1 F0  ,  Flh  ,   D   D D   l1 H  Note that, for the through-the-thickness crack configuration, the influence of crack shape, crack size and width-to-diameter ratio is analyzed by means of the correction factor F0. Then, it should be mentioned that the same lug failure correction factor has theoretically been examined by Hsu [40] taking into account also the width-to-diameter ratio effect. Based on such results, suitable polynomial expressions for different width-to-diameter ratios are here developed for straight lug, as follows: w  1.5 D 4

3

2

 2b   2b   2b   2b  F0 131.7790   111.0885   48.7716   12.1144   4.2025 D D D D w  2.08 D 5

4

3

2

5

4

3

2

(6a)

 2b   2b   2b   2b   2b  F0 14.4047   25.2199   10.0245   6.4725   6.1469   2.6556 (6b) D D D D         D

w  2.25 D  2b   2b   2b   2b   2b  F0  8.5216   18.2997   9.7174   4.1067   5.3056   2.3958 (6c) D D D D D w  2.40 D 6

5

4

3

2

 2b   2b   2b   2b   2b   2b  F0  2.6776   8.9931   14.5661   15.9789   12.5919   6.2259   2.3083 D D D D D D

6

(6d)

w  3.00 D 6

5

4

3

2

 2b   2b   2b   2b   2b   2b  F0  0.5935   3.1764   7.1832   9.2794   7.8423   4.3899   1.8244 D D D D D D

(6e)

where w and D are width and diameter of the lug, respectively. Further, the lug-hole location (with respect to the top of the lug head) may have a certain impact on the load-bearing capacity of the lug-linked components during service and such effect is analyzed through the correction factor Flh [41], defined by: b D f   10 3 b D l w D H with l1  (7) Flh  ,   1   b 2  l1 H  10 3 l1 D D D f    0.72  0.52    0.23   H H H

2

(8)

where H is the height of a lug head. Moreover, under cyclic loading the contact zone at the lug head represents also the position where crack-like flaws, such as a quarter-elliptical corner crack (Fig.1, case 1), may often appear. Therefore, the failure caused by such a fatigue damage is theoretically investigated here employing the following relationships related to the stress intensity factor [42]:

K  K B



b 2 1 0.11  cos   a



1/ 4

  a 2      cos 2   sin 2    b    



g1 g 2 g1B g 2 B

(9)

and 2  a  2 g1 1  0.1 0.35    1 sin     t   2   a    g1B 1  0.1 0.35     t   

(10b)



 1.4252  1.5783  2.1564 1 with   (11a) 2 2b 1  0.13 1 cos     D 2 3 4 1  0.358B  1.425B  1.578B  2.156B  with   1 (11b)  B 2b 1  0.132B  1 D

g 2 

g2B

1  0.358

(10a)







7

where a and b are the crack length in the depth and surface direction, respectively, K  denotes the stress intensity factor range defined in terms of an angle location (i.e. expressed as K=90=KA for a critical point examined in the crack depth direction), t represents the thickness of the pin-loaded lug,  is a parameter depending on the type of applied loading (for tensile  = 0.85) [42] and   denotes the correction factor in which the effects of crack length-to-radius ratio and angle location are involved. Further, it should be noted that the stress intensity factor range related to the crack growth in the surface direction KB is calculated by employing Eq. (4) examined in the case of through-crack configuration. Additionally, in all fatigue-linkage calculations the applied loading conditions are taken into consideration by means of the relevant net stress range. In the present study, the aircraft lug failure is also theoretically examined through the finite element analysis by empoying the MSC/NASTRAN software [27]. Then, the propagation of the above fatigue cracks, is analyzed in terms of the stress distribution and sress intensity factor, applying quarter-point singular finite elements [43]. Some stressintensity calculations using both analytical and numerical models are presented in next Section.

4. Verification of the fatigue damage assessments The fatigue stability of pin-loaded lug is here analyzed through the stress intensity factor and residual life to failure, in order to estimate the capability of the developed computational modeling tool. Such a theoretical investigation, in which the effect of a lug-head height and the stress ratio effect are taken into account, examines the failure behaviour of a through-the-thickness crack and a quarter-elliptical corner crack. Further, in the fatigue performance assessment of the lug with quarter-elliptical corner crack, the crack path is evaluated.

4.1 Failure behaviour of the through-the-thickness crack at a lug hole Fatigue performance analysis, presented in this Section, tackles the case of an aircraft lug with a through-the-thickness crack (Fig. 1, case 2) in terms of stress intensity factor and residual life. The failure strength of lug (w = 83.3 mm, D = 40 mm, t = 15 mm), whose initial through-crack length is equal to b0 = 2.5 mm, is theoretically investigated by taking into account the effect of a lug-head height. Such calculations are performed under cyclic loading with constant amplitude (Pmax = 63720 N, R = 0.1) in the case of two different lug-head heights (H = 57.1 mm and 44.4 mm) [41], assuming that the lug is made of 7075 T351 aluminium alloy ( = 0.7, CB= 4.37 10-11, mB = 3.29 with da/dN in m/cycles, KB in MPam0.5). Furthermore, for the lug (D = 38.1 mm, t = 12.7 mm, b0 = 0.635 mm) [9], made of 7075 T651 aluminium alloy ( = 0.7, CB = 8 10-11, mB = 3.4 with da/dN in m/cycles, KB in MPam0.5), the influence of stress ratio on the residual strength is also analyzed. External

8

loading conditions and relevant lug geometry parameters (related to the width and the lug-head height) are shown in Table 1. As is discussed in Sections above, the fatigue behaviour of through-the-thickness crack located at a lug hole is theoretically examined here through stress intensity factor and residual life by employing Eqs. (4)-(8) and Eqs. (2) and (3b), respectively. The estimated life to failure, as a function of crack length, is plotted in Fig. 2a and b for lug with head height equal to H = 57.1 and 44.4 mm, respectively. Then, the results related to the lugs (ABPLC85, ABPLC89 and ABPLC84, ABPLC91) subjected to stress ratio R = 0.1 and 0.5, respectively, and those for the lugs (ABPLC62, ABPLC63 and ABPLC49, ABPLC79) characterized by two values of width w (57.15 mm and 85.72 mm) are shown in Fig.3a and b and Fig.4a and b, respectively. In the same Figures, experimental data found in the literature [9, 41] are also reported to verify the efficiency of the fracture mechanics-based analytical model here examined. Such comparisons show that the height of a lug head and the stress ratio affect the residual strength of damaged lug under fatigue loading. Further, according to the damagetolerance requirements, the conservative evaluations obtained indicate that the developed computational modeling tool can be employed for a reliable failure analysis of the lug with through-the-thickness crack. PLEASE insert Figure 2 here. PLEASE insert Figure 3 here. PLEASE insert Figure 4 here. Additionally, the same fatigue-damaged linkage (Fig. 1, case 1) is also analyzed by means of fracture mechanics-based numerical models. In this context, the crack propagation is investigated under cyclic loading (Pmax = 56225 N, R = 0) for the pinloaded lug (w = 83.3 mm, D = 40 mm, H = 57.1 mm, t = 15 mm, b0 = 6.66 mm), made of 7075 T651 aluminium alloy (E = 72 GPa,  = 0.33). The through-crack configuration is here theoretically examined using two-dimensional finite elements, which are implemented in MSC/NASTRAN software [27]. In such an analysis, the four-node shell finite elements were used for meshing the linkage model (with the geometry mentioned above). Also, the eight quarter-point six-node singular finite elements, i.e. super elements [43] were employed to describe a non-linear stress state field in the vicinity of the crack tip at the lug hole. Further, by assuming the linearelastic behaviour of material, the stress distributions and stress intensity factors are evaluated through the finite element meshes modeled for appropriate crack length increments. The stress distributions for crack length equal to b = 6.66 mm and 14.16 mm are shown in Fig.5a and b, respectively. PLEASE insert Table 1 here. PLEASE insert Figure 5 here. PLEASE insert Table 2 here.

9

Further, in the failure mode analysis here presented, the stress intensity factor is calculated by employing Eqs (4)-(8). Such evaluations related to the lug with throughthe-thickness crack are compared with those determined by using finite elements, as is shown in Table 2. It can be seen that analytical solutions correlate well with those deduced by means of the fracture mechanics-based numerical model.

4.2 Fatigue assessment of the quarter-elliptical corner crack from a pin-loaded hole Now failure performance of the aircraft lug with a quarter-elliptical corner crack (Fig. 1, case 1) is analyzed through the residual life assessment. Such a damaged lug (D = 38.1 mm, t = 12.7 mm) [9], made of 7075 T651 aluminium alloy, is subjected to cyclic loading, characterized by two different stress ratios (R = 0.1 and 0.5). Relevant parameters related to the lug geometry and external loading are reported in Table 1. In the fatigue calculations, the initial crack lengths are equal to a0 = b0 = 0.635 mm in the depth and surface direction, respectively, and the following material and fatigue parameters are assumed: Su = 575 MPa; E = 72 GPa,  = 0.7, CA= 7.6 10-11, CB= 8 10-11, mA = mB = 3.4. The durability of the pin-loaded lug with fatigue damage, approximated as surface cracklike flaw, is here assessed for two critical directions on the crack front by means of Eqs. (2)-(8) and Eqs. (9)-(11), as is discussed in above Sections. In such an analysis, Eq. (6a) and (6c) are applied for the lug width equal to w = 57.1 mm and 85.7 mm, respectively. The computed life to failure, as a function of crack length, is shown in Figs.6a and 7a, and Figs.6b and 7b for the lug with a quarter-elliptical corner crack (subjected to stress ratio R = 0.1 and 0.5 mm) in depth and surface direction, respectively. Further, the same fatigue calculations related to the lug, characterized by width w = 85.7 mm (R = 0.1), are presented in Figs 8a and b for depth and surface crack directions, respectively. Since Kathiresan and Brussat [9] have experimentally investigated the same damaged lug configurations, such observations are used to estimate the reliability of the computational modeling tool developed in the present study. By examining Figs. 6 to 8, it can be inferred that the theoretical results, as conservative evaluations, are in a quite satisfactory agreement with relevant experimental results [9] for depth and surface crack directions.

PLEASE insert Figure 6 here. PLEASE insert Figure 7 here. PLEASE insert Figure 8 here. Furthermore, the quarter-elliptical corner crack (Fig. 1, case 1) located at a lug hole is here numerically analyzed by means of the MSC/NASTRAN software [27]. The stress state field is evaluated by employing the three-dimensional finite elements in the case of a lug (w = 83.3 mm, D = 40 mm, H = 57.1 mm, t = 15 mm, a = b = 7.5 mm), subjected to cyclic loading (Smax = 56225 MPa, R = 0).

10

Under cyclic loading, for appropriate crack length increments the relevant finite element meshes and stress distributions were modeled by means of eight 3-D singular finite elements, i.e. super elements (which are employed to simulate the singularity of stress around the crack tip) [43]. Note that, the final crack length is determined using the fracture mechanics condition (see Section 2). Finite element mesh and stress distribution, for the lug with fatigue damage (whose crack lengths are equal to a = b = 7.5 mm), are shown in Fig. 9a and b, respectively. PLEASE insert Figure 9 here. PLEASE insert Table 3 here. Then, the failure of lug with the quarter-elliptical corner crack is herein examined through the stress intensity factor by employing the fracture mechanics-based analytical model discussed in Section 3. The results determined by means of Eqs (4)-(11) and the numerical analysis are shown in Table 3. Such a comparison indicates that analytical calculations are in accordance with those deduced by means of finite element method.

4.3 Failure analysis of the lug with quarter-elliptical corner crack Finally, the lug performance against fatigue is examined by means of the crack path evaluation. The detailed geometry of a lug specimen (w = 60 mm, H = 40 mm, t = 9.6 mm), subjected to cyclic loading (Pmax=40320 N, R = 0.33), is shown in Fig.10. The initial quarter-elliptical corner crack and the lug-hole diameter are characterized by the following sizes: a0 = b0 = 0.25 mm and D = 25 mm, respectively. In the failure simulations, it is assumed that the lug is made of 7075 T6 aluminium alloy, and the following mechanical and fatigue parameters are adopted: Su = 568 MPa; S0.2 = 506.5 MPa,  = 0.7, CA = 7.6 10-11, CB = 8 10-11, mA = mB = 3.4, respectively. Further, to validate the fatigue assessments related to the crack path, the experimental observations reported by Friedrich and Schijve [44] are employed. PLEASE insert Figure 10 here. In the damage tolerance design analysis presented here, the crack path is modeled taking into account the effect of width-to-diameter ratio and the lug-head height effect through the appropriate correction factors F0 and Flh, respectively, as is discussed in the above Sections. Further, the propagation of a quarter-elliptical corner crack located at a lug hole is evaluated by means of the crack growth rate and the stress intensity factor, using Eqs. (2) and (3) and Eqs. (4)-(11), respectively. It is worth to mention that the relevant curve with respect to the width-to-diameter ratio (w/D = 2.40) is designed by taking into account the Hsu’s crack growth results related to the straight lug [40]. The crack growth paths evaluated here are compared with those experimentally tested [44] for six different crack lengths in depth direction, as is shown in Figs. 11a-f. Note that, in the Cartesian reference system, vertical axis and horizontal axis match with the lug hole and the front face of the lug, respectively. The results presented in Fig. 11

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indicate that the computational model of failure is able to generate the crack growth path in accordance with experimental observations related to the quarter-elliptical corner crack located at a lug hole. PLEASE insert Figure 11 here. 6. Conclusions In the present paper, the performance of a damaged lug-pin joint under cyclic loading has been quantitatively assessed through new computational modeling tool. The fatigue damages at a lug hole, represented by a through-the-thickness crack growth or a quarterelliptical crack growth, have been analyzed by means of the stress-intensity factor and the residual life to failure, using suitable fracture mechanics-based analytical and numerical approaches. Thus, the failure modes have been investigated through the Huang-Moan crack growth concept. Further, the stress-intensity analysis for a through-the-thickness crack has been performed by employing the Hsu’s crack growth results for the straight lug and the relationships suggested by Geier. Such relationships associated with those introduced by Newman and Raju have been employed to evaluate the life to failure and crack path in the case of quarter-elliptical corner crack. The non-linear stress fields of damaged lug configurations have been analyzed by using quarter-point singular finite elements. It is worth to point out that the implementation of the fracture mechanics-based analytical methodology here developed enables fast and simple fatigue analysis for a quarterelliptical corner crack, since the effects of lug-head height and the width-to-diameter ratio are taken into consideration in the same way as in the case of the through-the-thickness crack.

Acknowledgement The author gratefully acknowledges the support provided by the Mathematical Institute of the Serbian Academy of Sciences and Arts and the Ministry of Science and Technological Development of Serbia through the Project No. OI174001. References [1] Wang GS. Weight functions and stress intensity factors for the single crack roundended straight lug. Int J Fracture 1992;56:233-55. [2] Carpinteri A, Brighenti R, Spagnoli A. Fatigue growth simulation of part-through flaws in thick-walled pipes under rotary bending. Int J Fatigue, 2000;22(1):1-9. [3] Carpinteri A, Brighenti R, Vantadori S. Notched shells with surface cracks under complex loading. Int J Mech Sciences 2006;48(6):638-49. [4] Brown MA, Evans JL, Fatigue life variability due to variations in interference fit of steel bushings in 7075-T651 aluminum lugs. Int J Fatigue 2012;44:177-87. [5] Navarro C, Vázquez J, Dominguez J. 3D vs. 2D fatigue crack initiation and propagation in notched plates. Int J Fatigue 2014;58:40-6.

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[6] Yuan X, Yue ZF, Wen SF, Li L, Feng T. Numerical and experimental investigation of the cold expansion process with split sleeve in titanium alloy TC4. Int J Fatigue 2015;77:78-85. [7] Toribio J, Matos JC, González B. Aspect ratio evolution associated with surface cracks in sheets subjected to fatigue. Int J Fatigue 2016;92:588-95. [8] Boljanović S, Maksimović S. Fatigue failure analysis of pin-loaded lugs. Frattura ed Integrita Strutturale 2016;35:162-70. [9] Kathiresan K, Brussat TR. Advanced life analysis methods. AFWAL-TR-84-3080, Air Force Wright Aeronautical Laboratories, Ohio; 1984. [10] Forman RG, Kearney VE, Engle RM. Numerical analysis of crack propagation in cyclic-loaded structures. ASME J Basic Eng 1967;89:459-464. [11] Walker K. The effect of stress ratio during crack propagation and fatigue for 2024T3 and 7075-T6 aluminum, ASTM STP 462, American Society for Testing and Materials, PA, 1970, p.1-14. [12] Rice JR. Some remarks on elastic crack-tip stress fields. Int J Solids Struct 1972;8:751-758. [13] Tracey DM. 3-D elastic singularity element for evaluation of K along an arbitrary crack front. Int J Fracture 1973;9(3):340-343. [14] Narayana KB, Dayanada TS, Dattaguru B, Ramamurthy TS, Vijayakumar K. Cracks emanating from pin-loaded lugs. Eng Fract Mech 1994;47:29-38. [15] Kim JH, Lee SB, Hong SG. Fatigue crack growth behavior of Al7050-T7451 attachment lugs under flight spectrum variation. Theor Appl Fract Mech 2003;40(2):13544. [16] Computational Mechanics BEASY Ltd., BEASY Ver. 5.0, 1996. [17] Lanciotti A, Nigro F, Polese C. Fatigue crack propagation in the wing to fuselage connection of the new trainer aircraft M346. Fatigue Fract Eng Mater Struct 2006;29(12):1000-9. [18] NASGRO 4.02 Fracture mechanics and fatigue crack growth analysis software NASA Johnson Space Center and Southwest Research Institute, USA, 2002. [19] AFGROW Reference Manual, Wright-Patterson Air Force Base, AFRL/VASM, USA, 2003. [20] ABAQUS Version 6.4, Copyrigths c Hibbitt, Karlsson & Sorensen, Italy, 2003. [21] Antoni N, Gaisne F. Analytical modelling of static stress analysis of pin-loaded lugs with bush fitting. Appl Math Model 2011;35(1):1-21. [22] Mikheevskiy S, Glinka G, Algera D. Analysis of fatigue crack growth in an attachment lug based on the weight function technique and the UniGrow fatigue crack growth model. Int J Fatigue 2012;42:88-94. [23] Noroozi AH, Glinka G, Lambert S. A study of the stress ratio effects on fatigue crack growth using the unified two-parameters fatigue crack growth driving force. Int J Fatigue 2007;29:1616-33. [24] Shen G, Glinka G. Weight functions for surface semi-elliptical crack in a finite thickness plate. Theor Appl Fract Mech 1991;15(3):247-55. [25] Boljanović S, Maksimović S. Fatigue crack growth modeling of attachment lugs. Int J Fatigue 2014;58(1):66-74. [26] Newman JC Jr. Fracture analysis of surface- and through-cracked sheets and plates. Eng Fract Mech 1973;5(3):667-89.

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[27] MSC/NASTRAN software code, Theoretical Manuals, the MacNeal-Schwendler Corporation, Los Angeles, USA, 1994. [28] Boljanović S, Maksimović S, Carpinteri A, Jovanović B. Computational fatigue analysis of the pin-loaded lug with quarter-elliptical corner crack. Int J Appl Mech 2017;9(4):1-17(1750058). [29] Zhan W, Lu N, Zhang C. A new approximate model for the R-ratio effect on fatigue crack growth rate. Eng Fract Mech 2014;119:85-96. [30] Raju IS, Newman JC Jr. Stress intensity factor for two symmetric corner cracks. In Fracture Mechanics ASTM STR 667, ed. Smith CW, pp.411-30, 1979. [31] Rice JR. A path independent integral and the approximate analysis of strain concentration by notches and cracks. J Appl Mech 1968;35:379-86. [32] Huang X, Moan T. Improved modeling of the effect of R-ratio on crcak growth rate. Int J Fatigue 2007;29(4):591-602. [33] Carpinteri A. Shape change of surface cracks in round bars under cyclic axial loading. Int J Fatigue 1993;15(1):21-6. [34] Carpinteri A. Propagation of surface cracks under cyclic loading, In: Carpinteri A., editor. Handbook of Fatigue Crack Propagation in Metallic Structures. Amsterdam, Elsevier Sciece B V, 1994. [35] Carpinteri A, Brighenti R, Vantadori S. Notched double-curvature shells with cracks under pulsating internal pressure. Int J of Press Vessel Pip 2009;86(7):443-53. [36] Carpinteri A, Brighenti R, Vantadori S. Influence of the cold-drawing process on fatigue crack growth of a V-notched round bar. Int J Fatigue 2010;32(7):1136-45. [37] Vantadori S, Carpinteri A, Scorza D. Simplified analysis of fracture behaviour of a Francis hydraulic turbine runner blade. Fatigue Fract Eng Mater Struct 2013;36(7):67988. [38] Maksimović S, Posavljak S, Maksimović K, Nikolić V, Djurković V. Total fatigue life estimation of notched structural components using low-cycle fatigue properties. J Strain 2011;47(Suppl.2):341-9. [39] Belinha J, Azevedo JMC, Dinis LMJS, Natal Jorge RM. The natural neighbour radial point interpolation method extended to the crack growth simulation. Int J Appl. Mech 2016;8(1):1-32, 1650006. [40] Hsu TM. Analysis of cracks at attachment lugs. J Aircraft 1981;18(9):755-60. [41] Geier W. Strength behaviour of fatigue cracked lugs. Royal Aircraft Establishment, LT 20057, 1980. [42] Newman JC Jr, Raju IS. Stress-intensity factor equations for cracks in threedimensional finite bodies subjected to tension and bending loads. NASA Technical Memorandum 85793, April, 1984. [43] Barsoum RS. Triangular quarter-point elements as elastic and perfectly-plastic crack tip elements. Int J Numer Meth Engng 1977;11(1):85-98. [44] Friedrich S, Schijve J. Fatigue crack growth of corner cracks in lug specimens. Delft University of Technology Report LR-375, 1983.

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TABLE CAPTIONS

Table 1 Geometrical and loading parameters for the aircraft lug with fatigue damages

Table 2 Stress intensity factors determined by employing analytical and numerical approaches, in the case of through-the-thickness crack at a lug hole

Table 3 Stress intensity factors determined for a quarter-elliptical corner crack by means of analytical and numerical approaches

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FIGURE CAPTIONS Fig. 1. Geometry of the damaged lug: (a) through-the-thickness crack and (b) quarterelliptical corner crack. Fig. 2. Residual life evaluation of the lug with through-the-thickness crack: (a) H = 57.1 mm (experiment – Lug 6) and (b) H = 44.4 mm (experiment – Lug 2). Experimental results reported in Ref. [41] and the present results (calculated curves). Fig. 3. Residual life evaluation of the lug with through-the-thickness crack: (a) w = 114.3 mm, R = 0.1 (experiment 1 – ABPLC85, experiment 2 – ABPLC89) and (b) w = 114.3 mm, R = 0.5 (experiment 1 – ABPLC84, experiment 2 – ABPLC91). Experimental results reported in Ref. [9] and the present results (calculated curves). Fig. 4. Residual life evaluation of the lug with through-the-thickness crack: (a) w = 57.1 mm, R = 0.1 (experiment 1 – ABPLC62, experiment 2 – ABPLC63) and (b) w = 85.7 mm, R = 0.5 (experiment 1 – ABPLC49, experiment 2 – ABPLC79). Experimental results reported in Ref. [9] and the present results (calculated curves). Fig. 5. Stress distribution numerically determined for the lug with through-the-thickness crack (a) b = 6.66 mm and (b) b = 14.16 mm. Fig. 6. Residual life evaluation of the lug with quarter-elliptical corner crack (w = 85.7 mm, R = 0.1): (a) a against N; (b) b against N (experiment 1 – ABPLC36, experiment 2 – ABPLC30). Experimental results reported in Ref. [9] and the present results (calculated curves). Fig. 7. Residual life evaluation of the lug with quarter-elliptical corner crack (w = 57.15 mm, R = 0.5): (a) a against N; (b) b against N (experiment 1 – ABPLC18, experiment 2 – ABPLC22). Experimental results reported in Ref. [9] and the present results (calculated curves). Fig. 8. Residual life evaluation of the lug with quarter-elliptical corner crack (w = 57.15 mm, R = 0.1): (a) a against N; (b) b against N (experiment 1 – ABPLC21, experiment 2 – ABPLC17). Experimental results reported in Ref. [9] and the present results (calculated curves). Fig. 9. Finite element analysis of the lug with a quarter-elliptical corner crack (a = b = 7.5 mm): (a) finite element mesh; (b) stress distribution. Fig. 10. Detailed geometry of the lug specimen with a quarter-elliptical corner crack [44]. Sizes in millimeters. Fig. 11. Crack path evolution for quarter-elliptical corner crack at a lug hole: (a) a = 2.07 mm, (b) a = 3.16 mm, (c) a = 3.64 mm, (d) a = 4.14 mm, (e) a = 4.95 mm, (f) a = 6.06 mm. Experimental results reported in Ref. [44] (Lug 2, initial crack a0 = b0 = 0.25 mm) and the present results (calculated curves).

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Table 1 Experiments [9] Through-the-thick. crack: ABPLC85,ABPLC89 ABPLC84, ABPLC91 ABPLC62, ABPLC63 ABPLC49, ABPLC79 Quarter-elliptical crack: ABPLC30, ABPLC36 ABPLC18, ABPLC22 ABPLC17, ABPLC21

w (mm) 114.3 114.3 57.15 85.72 85.80 57.15 57.15

Table 2 Step

b 10-3 (m)

1 2 3 4 5

6.66 7.50 9.16 14.16 18.30

Kmax (MPam1/2) Numerical 21.92 22.41 23.44 26.83 37.46

Analytical 19.04 19.57 20.78 25.51 36.95

Table 3

 () 0 15 30 45 60 75 90

Kmax (MPam1/2) Numerical 16.76 15.78 15.53 16.03 17.27 20.21 23.94

Analytical 19.57 18.31 17.79 18.06 19.19 21.32 24.73

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H (mm) 57.15 57.15 28.57 42.86 42.90 28.57 28.57

R 0.1 0.5 0.1 0.5 0.1 0.5 0.1

Pmax (N) 60070.52 60070.52 30035.26 112632.23 45029.82 29993.21 29993.21

HIGHLIGHTS    

The strength of aircraft lugs with fatigue cracks is estimated by new computational methodologies. The crack driving forces are analyzed using the fracture mechanics-based models. The life and crack path are assessed through the Huang-Moan crack growth concept. The failure assessments agree with relevant experimental observations.

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