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Three-dimensional stress intensity factor calibration for a stiffened cracked plate
Engineering Fracture Mechanics 76 (2009) 2298–2308
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Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
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Three-dimensional stress intensity factor calibration for a stiffened cracked plate P.M.G.P. Moreira a,*, S.D. Pastrama b, P.M.S.T. de Castro c a
INEGI-Institute of Mechanical Engineering and Industrial Management, Porto, Portugal Department of Strength of Materials, University ‘‘Politehnica” of Bucharest, Romania c Department of Mechanical Engineering, University of Porto, Portugal b
a r t i c l e
i n f o
Article history: Received 11 September 2008 Received in revised form 3 June 2009 Accepted 1 July 2009 Available online 4 July 2009 Keywords: Stress intensity factor Stiffened panel J-Integral Plane stain behaviour Opening stress
a b s t r a c t Three-dimensional finite element analyses are used in this paper to calibrate the stress intensity factor in a cracked stiffened plate subjected to remote uniform traction. An accurate numerical determination of the stress field and stress intensity variation through the thickness of a central cracked plate was first carried out in order to evaluate three-dimensional effects. A stiffened cracked plate was then analysed, taking into account the results and the conclusions obtained in the previous study. Such a structure was chosen due to the growing interest for large integral metallic structures for aircraft applications, following the continuous need for low cost and the emergence of new technologies. The J-Integral technique was used to calculate the values of the stress intensity factor along the plate thickness. The plane strain behaviour near the crack front and the variation of the opening stress are discussed. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction An aircraft fuselage structure includes, among other parts, the external skin and longitudinal stiffeners (stringers and longerons) eg. [1]. Since the 1920s, investigations on the strength and behaviour of aluminium alloy stiffened sheet specimens used in aircraft construction were carried out [2,3]. Stiffened panels are light and highly resistant sheets reinforced by stringers designed to cope with a variety of loading conditions. Stiffeners improve the strength and stability of the structure and provide a mean of slowing down or arresting the growth of cracks in a panel. Most common stiffener cross-sections are bulb, flat bar or T- and L-sections, that can be bonded, extruded, connected by means of fasteners, machined or welded to form a panel. Riveted and bolted stiffeners tend to remain intact as the crack propagates beneath them, providing an alternative path for the panel load to pass. Also, riveted stiffeners continue to limit crack growth after the crack propagation over the stiffener, since a crack cannot propagate directly into the stiffener. The continuous need for low cost and the emergence of new technologies has brought interest for large integral metallic structures for aircraft applications. Evaluation programs for replacement of traditional fastening with these new emerging technologies have been carried out all over the aircraft sector, e.g. [4]. Studies show that in an integral stiffener (machined, extruded or welded) a crack propagates simultaneously in the stiffener and in the skin beneath the stiffener at approximately the same rate. In this case, the crack may propagate into the stiffener and break it [5]. Also, it was observed that the rate of crack growth is significantly reduced in the presence of stiffeners [6].
* Corresponding author. E-mail address: [email protected] (P.M.G.P. Moreira). 0013-7944/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2009.07.003
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Nomenclature 2D 3D COD DBEM FEM SIF SST b c Dpe h K K2D K3D Kaverage KI t d
rx ry rz r m
two-dimensional three-dimensional crack opening displacement dual boundary element method finite element method stress intensity factor singularity subtraction technique half plate width half crack length degree of plain strain half plate height stress intensity factor two-dimensional stress intensity factor three-dimensional stress intensity factor average value of stress intensity factor mode I stress intensity factor plate thickness plane strain zone size stress in the xx direction opening stress stress in the thickness direction remote stress Poisson’s ratio
Linear elastic fracture mechanics in conjunction with the Paris law [7] are widely used to analyze and predict crack growth and fracture behaviour of aircraft panels. During the past decades, several studies have been conducted to calculate stress intensity factors in cracked stiffened panels. Configurations involving cracks in infinite and semi-infinite plates with integral or discretely attached stiffeners have been studied by several authors, e.g. [8]. A compilation of results is presented in a parametric form in [9]. It was noticed that the stress intensity factor (SIF) decreases as the crack approaches a stiffener, indicating that the stiffener aided in restraining the crack or slowing down the propagation. Experiments on box girders with welded stiffeners to study crack growth aspects and remaining life prediction concluded that rigorous finite element analysis has to be performed to compute SIF for a structural component having a complex geometry [10]. In this case, it is important to take into account the effect of nearby boundaries by using numerical methods of structural analysis. Stress intensity factors for riveted stiffened cracked panels were calculated using the finite element method (FEM) in conjunction with strain energy release rate and the crack tip opening displacement [11,12]. The use of special crack tip enriched elements to solve problems involving edge cracks in stiffened panels was reported in [13]. The application of the complex variable method combined with compatible deformations to finite stiffened panels, by using boundary collocation method was extended in [14]. The purpose of this work is to obtain three-dimensional (3D) stress intensity factor solutions for a cracked stiffened plate using the FEM. In order to check the 3D solutions, results previously obtained by the authors using the compounding method were used [15]. In the compounding method, a complex cracked structure containing n boundaries is separated into a number of simpler ancillary configurations, each of these containing usually only one boundary that interacts with the crack, and has known SIF solutions from handbooks. The stress intensity factor for the complex geometry is obtained by compounding these solutions according to the general equation:
K ¼ K0 þ
n X ðK i K 0 Þ
ð1Þ
i¼1
where Ki is the stress intensity factor for the cracked structure having only the ith boundary and K0 is the stress intensity factor in the absence of all boundaries. This method was developed by Cartwright and Rooke [16]. Extensive descriptions of the method and different examples may can be found also in [17] and [18]. The numerical investigations of cracked structures feature both two-dimensional (2D) and 3D analyses. The 2D ones are simpler and, in most cases, have a reasonable degree of accuracy. Since the stress state near the crack tip is always 3D, this complex analysis of cracked structures has been used extensively in the last years, in order to produce more accurate numerical solutions. Three-dimensional analyses of thin cracked plates were presented in [19,20], where, in order to obtain stress distributions and stress intensity factor values, refined 3D finite element analyses were performed. Kwon and Sun presented refined 3D analyses where the stress field near the crack tip, the degree of plane strain and the crack tip singularity were investigated [21]. They suggested also a simple technique to determine 3D stress intensity factor at the mid-plane of a thin plate from a 2D analysis.
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In order to understand the three-dimensional effects when a three-dimensional solution for the stress intensity factor is pursued, a preliminary study for accurate determination of the three-dimensional stress field for a plate with a through-thethickness central crack was carried out. The variation of the stress intensity factor along the thickness of the plate was also analysed and results were compared with values from the literature and also with results of 2D finite element analyses. The influence of the specimen geometry and mesh refinement on the SIF values and also the drop of the SIF values at the intersection of the crack front with the free surface of the plate are discussed. The plane strain behaviour near the crack front and the variation of the opening stress are also investigated. Calculations are made in order to verify the possible use of the formula proposed by Kwon and Sun [21] for the 3D stress intensity factor using 2D solutions, in the case of relatively thick plates. 2. Accurate finite element analyses of centre cracked plates The configuration of the centre cracked plate can be defined according to Fig. 1. The studied geometries are as follows: h/b = 0.875 and c/b = 0.5, for four values of the ratio t/c, namely 0.75; 1.5; 3 and 6. pffiffiffiffiffiffi According to Tada et al. [22], the non-dimensional stress intensity factor for this case is K=r pc ¼ 1:422. For this configuration, a two-dimensional finite element analysis using 8760 quadratic quadrilateral elements and a DBEM (Dual Boundary element method) analysis with 60 quadratic elements were performed. For the two-dimensional DBEM analysis, stress intensity factors obtained using the J-Integral technique [23,24] and the SST (Singularity Subtraction Technique) [25] are presented. Table 1 shows results of stress intensity factor obtained in the two-dimensional analysis using the DBEM and the FEM. To obtain the stress intensity factor for the three-dimensional analysis, a mesh with 67,200 20-nodes brick isoparametric elements (C3D20) was used on ABAQUS [26] finite element method code. Only half of the plate was modelled using 12 elements along the thickness. 2.1. 3D stress intensity factor for different plate thicknesses Keeping the plate width and height constant, four different ratios of t/c were studied: t/c = (0.75; 1.5; 3; 6). For a central cracked tension specimen with h/b = 0.875, c/b = 0.5 and t/c = 3 three-dimensional stress intensity factors solutions were found in the literature [27]. In this case, since the plate can be considered as thick, a plane strain state is expected to be found in the middle plane and a plane stress state is expected near the external surface. A comparison between the three-dimensional stress intensity factors obtained for each plate thickness with the twodimensional reference results is presented in Table 2. When z/t equals zero (mid-plane), for t/c = 3, the three-dimensional
Fig. 1. Plate with central crack.
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P.M.G.P. Moreira et al. / Engineering Fracture Mechanics 76 (2009) 2298–2308 Table 1 2D results, FEM and DBEM for centred cracked plates (dimensionless SIF rpKffiffiffiffi pc). Tada [23]
FEM (J-Integral)
1.4220
DBEM (J-Integral)
DBEM (SST)
Solution
Difference (%)
Solution
Difference (%)
Solution
Difference (%)
1.4223
0.02
1.4350
0.92
1.4001
1.54
Table 2 Comparison between three-dimensional stress intensity factors and two-dimensional reference results for centre cracked plates. Dimensionless SIF (shape factor) rpKffiffiffiffi pc
z/t
3D FEM
0 0.0625 0.125 0.1875 0.25 0.3125 0.375 0.4375 0.5
2D
t/c = 0.75
t/c = 1.5
t/c = 3
t/c = 6
Tada [23]
FEM
DBEM
1.500 1.500 1.500 1.499 1.498 1.492 1.483 1.456 1.220
1.476 1.477 1.479 1.482 1.484 1.487 1.487 1.474 1.290
1.425 1.427 1.433 1.443 1.458 1.477 1.500 1.519 1.391
1.400 1.401 1.404 1.410 1.422 1.443 1.479 1.512 1.556
1.422
1.422
1.435
stress intensity factor are similar to the plane strain reference values. When t < c, the three-dimensional stress intensity factors at the mid-plane are higher than the two-dimensional stress intensity factors. For thick or very thick plates the threeand two-dimensional stress intensity factors are comparable at the mid-plane. Similar trends are also reported in [21]. Kwon et al. [21] proposed a simple equation to obtain approximate three-dimensional stress intensity factors from twodimensional solutions:
K 3D ¼ K 2D
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 m2
ð2Þ
where m is Poisson’s ratio of the material. This equation was proposed taking into account that the stress state at the midplane of the plate approaches plane strain, as shown above. A comparison of the K3D estimated using Eq. (2) and the K3D calculated in the finite element analysis is presented in Table 3. In this study it was verified that this technique is accurate only for thin plates, t < 1.5c. For this comparison, the DBEM J-Integral solution was used as the reference value. 2.2. Characteristics of the three-dimensional stress fields The stress field near the crack front is three-dimensional. However, very close to the crack front, it is almost in a state of plane strain, while away from the crack front it approaches a state of plane stress. The results of the finite element analyses were used to characterize the stress field near the crack front, to determine the stress intensity factor variation along the thickness, and to prove the variation of the stress state in the vicinity of the crack. The three-dimensional region in the cracked plate can be characterized by the degree of plane strain, Dpe, a parameter which measures the variation of the constraint factor with respect to the thickness direction [19]:
Dpe ¼
rzz mðrxx þ ryy Þ
ð3Þ
The degree of plane strain near the crack front is shown in Fig. 2, for t/c = 0.75. Similar plots were obtained for the other two studied configurations. The plane strain condition seemed to be invalid near the plate free surface, which raises the question about the validity of using the plane strain condition along the crack front for calculating the stress intensity factor from near tip displacements.
Table 3 Difference between the estimated K3D and the FEM calculated K3D at the plate mid thickness for centre cracked plates (dimensionless SIF rpKffiffiffiffi pc). t/c = 0.75 Reference K2D (DBEM) K3D (Eq. (1)) Difference (%)
t/c = 1.5
t/c = 3
t/c = 6
1.4248 5.25
1.4014 7.01
1.4996 1.5004 0.06
1.4757 1.62
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Fig. 2. Degree of plane strain, for t/c = 0.75: (a) variation in the x direction and (b) variation in the z direction.
The degree of plane strain is zero for a state of plane stress, and it is unity where plane strain occurs. The plots clearly show that plane strain behaviour occurs at and near the middle of the plate, where a degree of plane strain of almost 1.00 was obtained. As the distance from the crack front increases, the degree of plane strain tends to smaller values, showing three-dimensional and then plane stress behaviour. Based on the plots from Fig. 2, it was verified that the size d of the plane strain zone depends on the plate thickness. The size of the plane strain zone is defined as the value of x/t when Dpe becomes zero. As presented in Table 4, for t < c the plane strain behaviour is present on almost half of the thickness, as shown also in [19,21]. The ratio d/t of the size of the plane strain zone decreases as the thickness of the plate increases. A detailed stress analysis was carried out in the vicinity of the crack front, in order to emphasize the characteristics of the three-dimensional stress field. Due to the fact that, in Mode I, the crack propagation is determined by the stress perpendicular to the crack, namely ry, this stress component was dealt with in the analysis. The plots in Fig. 3 represent the variation of the opening stress ry and the stress rz in the thickness direction with the distance from the crack tip, at different z/t values, for the case t/c = 1.5. Similar trends were found for the other studied cases. A decrease of the stress ry values with the distance from the crack front, which is normal due to the strong stress concentration effect near the crack, is noticed. As the distance from the crack front increases, the values of the stress tend to the remote stress. It is clear that in the vicinity of the crack tip (for very small values of x) the first term is dominant while the others are negligible. As the distance from the crack tip increases, the other terms have an important influence and the oneterm stress field approach is not valid anymore. This tendency is illustrated in Fig. 3a, where one can notice a decreasing tendency up to a certain value of x/t. Then, the non-dimensional values of the stress begin to increase, due to the influence of the second and higher order terms. 2.3. Discussion As a contribution to the understanding of three-dimensional effects, detailed three-dimensional finite element analyses of cracked plates were conducted in order to characterize the near-tip stress field and to determine the variation of the stress intensity factor through the thickness of the plate. In the three-dimensional analysis, the stress intensity factor has different values through the thickness. Thus, a twodimensional stress intensity factor analysis is only an approximation of the exact solution since there is no difference on plane stress or plane strain stress intensity factor solutions for linear elastic fracture mechanics. The stress intensity factor varies over the thickness and drops to zero at the plate surface. A boundary layer exists near the plate free surface and its size is a function of thickness to crack length ratio. The best agreement with the reference values was obtained for the middle layer. The opening stress remains almost constant through the thickness except near the free surface where an important decrease can be noticed in the vicinity of the crack front. This behaviour was not obtained far from the crack front, where the opening stress is constant and tends to the applied remote stress. The plane strain behaviour near the middle of the plate was demonstrated by calculating the degree of plane strain. It was shown also that the degree of plane strain tends to smaller values when the distance from the crack front increases. Plane
Table 4 Size of the plane strain zone for centre cracked plates.
d/t
t/c = 0.75
t/c = 1.5
t/c = 3
0.46
0.4
0.26 (extrapolated value)
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Fig. 3. Stress distribution along the x direction near the crack front, for t/c = 1.5: (a) variation of ry and (b) variation of rz.
strain behaviour occurs at and near the middle of the plate, where a degree of plane strain near unity was obtained. As the distance from the crack front increases, the degree of plane strain tends to smaller values, showing thus three-dimensional and then plane stress behaviour. 3. Accurate numerical analyses of cracked stiffened panels 3.1. The studied structure The main purpose of this analysis is to calibrate the SIF for a double-stiffened cracked 700 mm long plate subjected to remote uniform traction. Both the cases of unbroken and broken stiffener are studied. The crack crosses the left stiffener, being symmetric with respect to it, as it can be seen in the cross view presented in Fig. 4. The left crack tip is identified as crack tip A and right crack tip as B. The calibration of the SIF was done by carrying out a parametric study of 7 crack lengths: 2c = 32.5; 65; 97.5; 130; 162.5; 195; and 227 mm. Owing to the symmetry, the analysis was carried out for half of the plate, using a mesh with 136,819 elements (20 nodes brick isoparametric elements). The plate thickness was modelled with seven layers of elements. The order of the layers of elements along the thickness is presented in Fig. 5a, while the used coordinate system and a detail of the mesh for 2c = 32.5 mm are shown in Fig. 5b and c. For the 3D finite element analyses, stress intensity factors calculated using the J-Integral technique are presented in two different ways. In a first study, results for layers of nodes lying at different depths along the thickness were calculated. In the second study, using the three results of SIF for each element, and according to Fig. 6, an average value of SIF is calculated:
K average ¼
K A þ 4K B þ K C 6
ð4Þ
3.2. SIF evaluation For the case of an unbroken stiffener, the non-dimensional stress intensity factors obtained through the thickness using the finite element method are listed in Tables 5 and 6, respectively, as a function of crack length. The values for the case of broken stiffener are shown in Tables 7 and 8. In all the above mentioned tables, the values previously calculated using the compounding technique [15] are also listed for comparison. The FEM average stress intensity factors obtained according to Eq. (4) are presented and the comparison is made using the values that are closer to the compounding ones (the columns that contain these values are marked in the tables with different shades of grey).
Fig. 4. Cross section through the stiffened plate with a crack crossing a stiffener.
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Fig. 5. Element layers through thickness order and coordinate axis origin: (a) order of element layers through thickness; (b) the coordinate system and (c) mesh detail for 2c = 32.5 mm.
Fig. 6. Nodes for calculating the average stress intensity factor for an element.
It should be mentioned that some values of the stress intensity factor are negative, due to the negative ry stress values that exist in the layer of elements opposite to the stiffener surface as presented in Fig. 7. 3.3. Characteristics of the three-dimensional stress fields for a broken stiffener In this section, the plate with a broken stiffener was studied since it is the case most likely to occur in real integral structures. The results were used to characterize the stress intensity factor variation along the thickness, and to emphasize the variation of the stress state in the vicinity of the crack. Again, the degree of plane strain was used to measure the variation of the constraint factor with respect to the thickness direction [19]. The origin of the coordinate system of axes was defined at the crack tip node on the surface containing the stiffener, (Fig. 5b). The degree of plane strain near the crack front is shown in Fig. 8, in the case 2c = 162.5, similar curves being obtained for all the studied crack lengths and both crack tips. As for the centre cracked plates, in the case of stiffened plates, the plane strain conditions seemed to be invalid near the plate free surface.
Table 5 Nondimensional SIFs for crack tip A obtained with FEM and the compounding technique in the case of unbroken stiffener.
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Table 6 Nondimensional SIFs for crack tip B obtained with FEM and the compounding technique in the case of unbroken stiffener.
Table 7 Nondimensional SIFs for crack tip A obtained with FEM and the compounding technique in the case of broken stiffener.
Table 8 Nondimensional SIFs for crack tip B obtained with FEM and the compounding technique in the case of broken stiffener.
For both crack tips and all crack lengths, the plots show that a state close to plane strain occurs between z/t = 0.36 and z/t = 0.64, where the degree of plane strain has its higher values. Nevertheless, it should be pointed out that for the smallest crack length the degree of plane strain reaches its peak before the middle of the plate thickness. As the distance from the crack front increases, the degree of plane strain tends to smaller values, showing three-dimensional and then plane stress behaviour. It was observed that the size of the plane strain zone, d, is somehow dependent on the crack length. In order to emphasize the characteristics of the three-dimensional stress field, a detailed stress analysis in the vicinity of the crack front was carried out. The non-dimensional stresses in the y and z direction were obtained for all the studied cases. In Fig. 9, the plots are shown for crack tip A, in the case 2c = 162.5. A decrease of the ry values with the distance from the crack front, due to the strong stress concentration effect near the crack, is noticed. As the distance from the crack front increases, the values of the stress for different plate depths show that a stiffened plate has a high three-dimensional stress distribution. This is less pronounced at crack tip B, for higher crack lengths, where the remaining intact stiffener carries the load. For all crack lengths, near the plate mid thickness, the stress in the z direction presents significant values only until a value of x/t of approximately 0.4 is reached. The stress in the z direction presents always higher values at crack tip A.
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Fig. 7. Variation of the opening stress ry in the stiffener side (top layer) and in the opposite side (back layer) for c = 32.5 mm.
Fig. 8. Degree of plane strain in the x direction for 2c = 162.5: (a) crack tip A and (b) crack tip B.
Fig. 9. Stress distribution near the crack front for 2c = 162.5 (crack tip A): (a) variation of ry and (b) variation of rz.
3.4. Discussion Refined three-dimensional finite element analyses were performed in order to calibrate the stress intensity factors for a symmetrically stiffened plate subjected to uniform tensile stress. Cracks that could develop in such structures were assessed
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by considering the case of a crack that crosses one of the stiffeners in two possible scenarios: (i) the stiffener crossed by the crack is unbroken and (ii) the stiffener is unbroken. Several important conclusions can be drawn from this work: The three-dimensional finite element analyses show that the stress intensity factor varies through the thickness even in this case of a very thin plate. For the unbroken stiffener, the results that are in best agreement with the ones obtained with the compounding technique are those from the layer opposite to the stiffener. For the broken stiffener, the results that are in best agreement with the ones obtained with the compounding technique are those from the middle elements layer. The obtained results justify the non-straight crack front of advancing fatigue cracks in fatigue cracks in stiffened panels subjected to cyclic loading [28]. Since such structures are often used in the aeronautical industry, where the fatigue phenomenon is always present, the obtained results can be used in a subsequent fatigue analysis based on crack growth. 4. Conclusions As a contribution to the understanding of three-dimensional effects, three-dimensional stress intensity factor solutions for a plate with a central crack and for a cracked stiffened plate were obtained using the finite element method. In order to characterize the near-tip stress field and to determine the variation of the stress intensity factor through the thickness of the plate, very refined finite element analyses were conducted. When possible, the obtained results were compared with those in the literature and with the values obtained following a two-dimensional finite element analysis and dual boundary element analysis. The stress intensity factors for a symmetrically stiffened plate subjected to uniform tensile stress were assessed by considering the case of a crack that crosses one of the stiffeners in two possible scenarios: (i) the stiffener crossed by the crack is unbroken and (ii) the stiffener is unbroken. Conclusions could be drawn from this numerical study, as follows: At the plate free surface, where the singularity r1/2 does not occur, there is a boundary layer whose size is a function of thickness/crack length ratio. For the centre cracked plate, the opening stress ry remains almost constant through the thickness except near the free surface where an important decrease can be noticed in the vicinity of the crack front. For the same structure, the plane strain behaviour near the middle of the plate was demonstrated by calculating the degree of plane strain. It was shown also that the degree of plane strain tends to smaller values when the distance from the crack front increases. In a three-dimensional analysis, stress intensity factor has different values through the thickness. Thus, a two-dimensional stress intensity factor analysis is only an approximation of the exact solution. For the stiffened cracked panel, a non-uniform stress intensity factor distribution through the thickness was obtained which justifies the non-straight crack front of advancing fatigue cracks in stiffened panels subjected to cyclic loading.
References [1] Murphy A et al. The computational post-buckling analysis of fuselage stiffened panels loaded in shear. Thin Wall Struct 2005;43(9):1455–74. [2] Wagner H. Structures of thin sheet metal, their design and construction. Technical note 490, National Advisory Committee for Aeronautics; 1928. [3] Holt M. Tests of aluminium alloy stiffened sheet specimens cut from an airplane wing. Technical note 883, National Advisory Committee for Aeronautics; 1943. [4] Hoffman E et al., Compression buckling behavior of large-scale friction stir welded and riveted 2090-T83 A1–Li alloy skin-stiffener panels. NASA/TM2002-211770; 2002. [5] Mahmoud H, Dexter R. Propagation rate of large cracks in stiffened panels under tension loading. Mar Struct 2005;18(3):265–88. [6] Mellings S et al. Durability prediction using automatic crack growth simulation in stiffened panel structures; 2002.
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[22] Tada H, Paris P, Irwin G. The stress analysis of cracks handbook. 3rd ed. New York, London: ASME Press Professional Engineering Publishing; 2000. XX, 677. [23] Eshelby J. The continuum theory of lattice defects. Solid State Phys – Adv Res Appl 1956;3:79–144. [24] Rice J. Mathematical analysis in the mechanics of fracture. In: Fracture, an advanced treatise, New York; 1968. p. 191–308. [25] de Matos P et al. Dual boundary element analysis of cracked plates: post-processing implementation of the singularity subtraction technique. Comput Struct 2004;82(17–19):1443–9. [26] Hibbitt Karlson, Sorensen. ABAQUS, fracture mechanics (course notes); 1991. [27] Raju I, Newman J. Three dimensional finite-element analysis of finite-thickness fracture specimens. TN-D-8414, NASA Center for AeroSpace Information (CASI); 1977. [28] Moreira P. Lightweight stiffened panels: mechanical characterization of emerging fabrication technologies. PhD thesis, Faculdade de Engenharia da Universidade do Porto, Universidade do Porto, Porto; 2008. p. 481.
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Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
Note
Three-dimensional stress intensity factor calibration for a stiffened cracked plate P.M.G.P. Moreira a,*, S.D. Pastrama b, P.M.S.T. de Castro c a
INEGI-Institute of Mechanical Engineering and Industrial Management, Porto, Portugal Department of Strength of Materials, University ‘‘Politehnica” of Bucharest, Romania c Department of Mechanical Engineering, University of Porto, Portugal b
a r t i c l e
i n f o
Article history: Received 11 September 2008 Received in revised form 3 June 2009 Accepted 1 July 2009 Available online 4 July 2009 Keywords: Stress intensity factor Stiffened panel J-Integral Plane stain behaviour Opening stress
a b s t r a c t Three-dimensional finite element analyses are used in this paper to calibrate the stress intensity factor in a cracked stiffened plate subjected to remote uniform traction. An accurate numerical determination of the stress field and stress intensity variation through the thickness of a central cracked plate was first carried out in order to evaluate three-dimensional effects. A stiffened cracked plate was then analysed, taking into account the results and the conclusions obtained in the previous study. Such a structure was chosen due to the growing interest for large integral metallic structures for aircraft applications, following the continuous need for low cost and the emergence of new technologies. The J-Integral technique was used to calculate the values of the stress intensity factor along the plate thickness. The plane strain behaviour near the crack front and the variation of the opening stress are discussed. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction An aircraft fuselage structure includes, among other parts, the external skin and longitudinal stiffeners (stringers and longerons) eg. [1]. Since the 1920s, investigations on the strength and behaviour of aluminium alloy stiffened sheet specimens used in aircraft construction were carried out [2,3]. Stiffened panels are light and highly resistant sheets reinforced by stringers designed to cope with a variety of loading conditions. Stiffeners improve the strength and stability of the structure and provide a mean of slowing down or arresting the growth of cracks in a panel. Most common stiffener cross-sections are bulb, flat bar or T- and L-sections, that can be bonded, extruded, connected by means of fasteners, machined or welded to form a panel. Riveted and bolted stiffeners tend to remain intact as the crack propagates beneath them, providing an alternative path for the panel load to pass. Also, riveted stiffeners continue to limit crack growth after the crack propagation over the stiffener, since a crack cannot propagate directly into the stiffener. The continuous need for low cost and the emergence of new technologies has brought interest for large integral metallic structures for aircraft applications. Evaluation programs for replacement of traditional fastening with these new emerging technologies have been carried out all over the aircraft sector, e.g. [4]. Studies show that in an integral stiffener (machined, extruded or welded) a crack propagates simultaneously in the stiffener and in the skin beneath the stiffener at approximately the same rate. In this case, the crack may propagate into the stiffener and break it [5]. Also, it was observed that the rate of crack growth is significantly reduced in the presence of stiffeners [6].
* Corresponding author. E-mail address: [email protected] (P.M.G.P. Moreira). 0013-7944/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2009.07.003
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Nomenclature 2D 3D COD DBEM FEM SIF SST b c Dpe h K K2D K3D Kaverage KI t d
rx ry rz r m
two-dimensional three-dimensional crack opening displacement dual boundary element method finite element method stress intensity factor singularity subtraction technique half plate width half crack length degree of plain strain half plate height stress intensity factor two-dimensional stress intensity factor three-dimensional stress intensity factor average value of stress intensity factor mode I stress intensity factor plate thickness plane strain zone size stress in the xx direction opening stress stress in the thickness direction remote stress Poisson’s ratio
Linear elastic fracture mechanics in conjunction with the Paris law [7] are widely used to analyze and predict crack growth and fracture behaviour of aircraft panels. During the past decades, several studies have been conducted to calculate stress intensity factors in cracked stiffened panels. Configurations involving cracks in infinite and semi-infinite plates with integral or discretely attached stiffeners have been studied by several authors, e.g. [8]. A compilation of results is presented in a parametric form in [9]. It was noticed that the stress intensity factor (SIF) decreases as the crack approaches a stiffener, indicating that the stiffener aided in restraining the crack or slowing down the propagation. Experiments on box girders with welded stiffeners to study crack growth aspects and remaining life prediction concluded that rigorous finite element analysis has to be performed to compute SIF for a structural component having a complex geometry [10]. In this case, it is important to take into account the effect of nearby boundaries by using numerical methods of structural analysis. Stress intensity factors for riveted stiffened cracked panels were calculated using the finite element method (FEM) in conjunction with strain energy release rate and the crack tip opening displacement [11,12]. The use of special crack tip enriched elements to solve problems involving edge cracks in stiffened panels was reported in [13]. The application of the complex variable method combined with compatible deformations to finite stiffened panels, by using boundary collocation method was extended in [14]. The purpose of this work is to obtain three-dimensional (3D) stress intensity factor solutions for a cracked stiffened plate using the FEM. In order to check the 3D solutions, results previously obtained by the authors using the compounding method were used [15]. In the compounding method, a complex cracked structure containing n boundaries is separated into a number of simpler ancillary configurations, each of these containing usually only one boundary that interacts with the crack, and has known SIF solutions from handbooks. The stress intensity factor for the complex geometry is obtained by compounding these solutions according to the general equation:
K ¼ K0 þ
n X ðK i K 0 Þ
ð1Þ
i¼1
where Ki is the stress intensity factor for the cracked structure having only the ith boundary and K0 is the stress intensity factor in the absence of all boundaries. This method was developed by Cartwright and Rooke [16]. Extensive descriptions of the method and different examples may can be found also in [17] and [18]. The numerical investigations of cracked structures feature both two-dimensional (2D) and 3D analyses. The 2D ones are simpler and, in most cases, have a reasonable degree of accuracy. Since the stress state near the crack tip is always 3D, this complex analysis of cracked structures has been used extensively in the last years, in order to produce more accurate numerical solutions. Three-dimensional analyses of thin cracked plates were presented in [19,20], where, in order to obtain stress distributions and stress intensity factor values, refined 3D finite element analyses were performed. Kwon and Sun presented refined 3D analyses where the stress field near the crack tip, the degree of plane strain and the crack tip singularity were investigated [21]. They suggested also a simple technique to determine 3D stress intensity factor at the mid-plane of a thin plate from a 2D analysis.
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In order to understand the three-dimensional effects when a three-dimensional solution for the stress intensity factor is pursued, a preliminary study for accurate determination of the three-dimensional stress field for a plate with a through-thethickness central crack was carried out. The variation of the stress intensity factor along the thickness of the plate was also analysed and results were compared with values from the literature and also with results of 2D finite element analyses. The influence of the specimen geometry and mesh refinement on the SIF values and also the drop of the SIF values at the intersection of the crack front with the free surface of the plate are discussed. The plane strain behaviour near the crack front and the variation of the opening stress are also investigated. Calculations are made in order to verify the possible use of the formula proposed by Kwon and Sun [21] for the 3D stress intensity factor using 2D solutions, in the case of relatively thick plates. 2. Accurate finite element analyses of centre cracked plates The configuration of the centre cracked plate can be defined according to Fig. 1. The studied geometries are as follows: h/b = 0.875 and c/b = 0.5, for four values of the ratio t/c, namely 0.75; 1.5; 3 and 6. pffiffiffiffiffiffi According to Tada et al. [22], the non-dimensional stress intensity factor for this case is K=r pc ¼ 1:422. For this configuration, a two-dimensional finite element analysis using 8760 quadratic quadrilateral elements and a DBEM (Dual Boundary element method) analysis with 60 quadratic elements were performed. For the two-dimensional DBEM analysis, stress intensity factors obtained using the J-Integral technique [23,24] and the SST (Singularity Subtraction Technique) [25] are presented. Table 1 shows results of stress intensity factor obtained in the two-dimensional analysis using the DBEM and the FEM. To obtain the stress intensity factor for the three-dimensional analysis, a mesh with 67,200 20-nodes brick isoparametric elements (C3D20) was used on ABAQUS [26] finite element method code. Only half of the plate was modelled using 12 elements along the thickness. 2.1. 3D stress intensity factor for different plate thicknesses Keeping the plate width and height constant, four different ratios of t/c were studied: t/c = (0.75; 1.5; 3; 6). For a central cracked tension specimen with h/b = 0.875, c/b = 0.5 and t/c = 3 three-dimensional stress intensity factors solutions were found in the literature [27]. In this case, since the plate can be considered as thick, a plane strain state is expected to be found in the middle plane and a plane stress state is expected near the external surface. A comparison between the three-dimensional stress intensity factors obtained for each plate thickness with the twodimensional reference results is presented in Table 2. When z/t equals zero (mid-plane), for t/c = 3, the three-dimensional
Fig. 1. Plate with central crack.
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FEM (J-Integral)
1.4220
DBEM (J-Integral)
DBEM (SST)
Solution
Difference (%)
Solution
Difference (%)
Solution
Difference (%)
1.4223
0.02
1.4350
0.92
1.4001
1.54
Table 2 Comparison between three-dimensional stress intensity factors and two-dimensional reference results for centre cracked plates. Dimensionless SIF (shape factor) rpKffiffiffiffi pc
z/t
3D FEM
0 0.0625 0.125 0.1875 0.25 0.3125 0.375 0.4375 0.5
2D
t/c = 0.75
t/c = 1.5
t/c = 3
t/c = 6
Tada [23]
FEM
DBEM
1.500 1.500 1.500 1.499 1.498 1.492 1.483 1.456 1.220
1.476 1.477 1.479 1.482 1.484 1.487 1.487 1.474 1.290
1.425 1.427 1.433 1.443 1.458 1.477 1.500 1.519 1.391
1.400 1.401 1.404 1.410 1.422 1.443 1.479 1.512 1.556
1.422
1.422
1.435
stress intensity factor are similar to the plane strain reference values. When t < c, the three-dimensional stress intensity factors at the mid-plane are higher than the two-dimensional stress intensity factors. For thick or very thick plates the threeand two-dimensional stress intensity factors are comparable at the mid-plane. Similar trends are also reported in [21]. Kwon et al. [21] proposed a simple equation to obtain approximate three-dimensional stress intensity factors from twodimensional solutions:
K 3D ¼ K 2D
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 m2
ð2Þ
where m is Poisson’s ratio of the material. This equation was proposed taking into account that the stress state at the midplane of the plate approaches plane strain, as shown above. A comparison of the K3D estimated using Eq. (2) and the K3D calculated in the finite element analysis is presented in Table 3. In this study it was verified that this technique is accurate only for thin plates, t < 1.5c. For this comparison, the DBEM J-Integral solution was used as the reference value. 2.2. Characteristics of the three-dimensional stress fields The stress field near the crack front is three-dimensional. However, very close to the crack front, it is almost in a state of plane strain, while away from the crack front it approaches a state of plane stress. The results of the finite element analyses were used to characterize the stress field near the crack front, to determine the stress intensity factor variation along the thickness, and to prove the variation of the stress state in the vicinity of the crack. The three-dimensional region in the cracked plate can be characterized by the degree of plane strain, Dpe, a parameter which measures the variation of the constraint factor with respect to the thickness direction [19]:
Dpe ¼
rzz mðrxx þ ryy Þ
ð3Þ
The degree of plane strain near the crack front is shown in Fig. 2, for t/c = 0.75. Similar plots were obtained for the other two studied configurations. The plane strain condition seemed to be invalid near the plate free surface, which raises the question about the validity of using the plane strain condition along the crack front for calculating the stress intensity factor from near tip displacements.
Table 3 Difference between the estimated K3D and the FEM calculated K3D at the plate mid thickness for centre cracked plates (dimensionless SIF rpKffiffiffiffi pc). t/c = 0.75 Reference K2D (DBEM) K3D (Eq. (1)) Difference (%)
t/c = 1.5
t/c = 3
t/c = 6
1.4248 5.25
1.4014 7.01
1.4996 1.5004 0.06
1.4757 1.62
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Fig. 2. Degree of plane strain, for t/c = 0.75: (a) variation in the x direction and (b) variation in the z direction.
The degree of plane strain is zero for a state of plane stress, and it is unity where plane strain occurs. The plots clearly show that plane strain behaviour occurs at and near the middle of the plate, where a degree of plane strain of almost 1.00 was obtained. As the distance from the crack front increases, the degree of plane strain tends to smaller values, showing three-dimensional and then plane stress behaviour. Based on the plots from Fig. 2, it was verified that the size d of the plane strain zone depends on the plate thickness. The size of the plane strain zone is defined as the value of x/t when Dpe becomes zero. As presented in Table 4, for t < c the plane strain behaviour is present on almost half of the thickness, as shown also in [19,21]. The ratio d/t of the size of the plane strain zone decreases as the thickness of the plate increases. A detailed stress analysis was carried out in the vicinity of the crack front, in order to emphasize the characteristics of the three-dimensional stress field. Due to the fact that, in Mode I, the crack propagation is determined by the stress perpendicular to the crack, namely ry, this stress component was dealt with in the analysis. The plots in Fig. 3 represent the variation of the opening stress ry and the stress rz in the thickness direction with the distance from the crack tip, at different z/t values, for the case t/c = 1.5. Similar trends were found for the other studied cases. A decrease of the stress ry values with the distance from the crack front, which is normal due to the strong stress concentration effect near the crack, is noticed. As the distance from the crack front increases, the values of the stress tend to the remote stress. It is clear that in the vicinity of the crack tip (for very small values of x) the first term is dominant while the others are negligible. As the distance from the crack tip increases, the other terms have an important influence and the oneterm stress field approach is not valid anymore. This tendency is illustrated in Fig. 3a, where one can notice a decreasing tendency up to a certain value of x/t. Then, the non-dimensional values of the stress begin to increase, due to the influence of the second and higher order terms. 2.3. Discussion As a contribution to the understanding of three-dimensional effects, detailed three-dimensional finite element analyses of cracked plates were conducted in order to characterize the near-tip stress field and to determine the variation of the stress intensity factor through the thickness of the plate. In the three-dimensional analysis, the stress intensity factor has different values through the thickness. Thus, a twodimensional stress intensity factor analysis is only an approximation of the exact solution since there is no difference on plane stress or plane strain stress intensity factor solutions for linear elastic fracture mechanics. The stress intensity factor varies over the thickness and drops to zero at the plate surface. A boundary layer exists near the plate free surface and its size is a function of thickness to crack length ratio. The best agreement with the reference values was obtained for the middle layer. The opening stress remains almost constant through the thickness except near the free surface where an important decrease can be noticed in the vicinity of the crack front. This behaviour was not obtained far from the crack front, where the opening stress is constant and tends to the applied remote stress. The plane strain behaviour near the middle of the plate was demonstrated by calculating the degree of plane strain. It was shown also that the degree of plane strain tends to smaller values when the distance from the crack front increases. Plane
Table 4 Size of the plane strain zone for centre cracked plates.
d/t
t/c = 0.75
t/c = 1.5
t/c = 3
0.46
0.4
0.26 (extrapolated value)
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Fig. 3. Stress distribution along the x direction near the crack front, for t/c = 1.5: (a) variation of ry and (b) variation of rz.
strain behaviour occurs at and near the middle of the plate, where a degree of plane strain near unity was obtained. As the distance from the crack front increases, the degree of plane strain tends to smaller values, showing thus three-dimensional and then plane stress behaviour. 3. Accurate numerical analyses of cracked stiffened panels 3.1. The studied structure The main purpose of this analysis is to calibrate the SIF for a double-stiffened cracked 700 mm long plate subjected to remote uniform traction. Both the cases of unbroken and broken stiffener are studied. The crack crosses the left stiffener, being symmetric with respect to it, as it can be seen in the cross view presented in Fig. 4. The left crack tip is identified as crack tip A and right crack tip as B. The calibration of the SIF was done by carrying out a parametric study of 7 crack lengths: 2c = 32.5; 65; 97.5; 130; 162.5; 195; and 227 mm. Owing to the symmetry, the analysis was carried out for half of the plate, using a mesh with 136,819 elements (20 nodes brick isoparametric elements). The plate thickness was modelled with seven layers of elements. The order of the layers of elements along the thickness is presented in Fig. 5a, while the used coordinate system and a detail of the mesh for 2c = 32.5 mm are shown in Fig. 5b and c. For the 3D finite element analyses, stress intensity factors calculated using the J-Integral technique are presented in two different ways. In a first study, results for layers of nodes lying at different depths along the thickness were calculated. In the second study, using the three results of SIF for each element, and according to Fig. 6, an average value of SIF is calculated:
K average ¼
K A þ 4K B þ K C 6
ð4Þ
3.2. SIF evaluation For the case of an unbroken stiffener, the non-dimensional stress intensity factors obtained through the thickness using the finite element method are listed in Tables 5 and 6, respectively, as a function of crack length. The values for the case of broken stiffener are shown in Tables 7 and 8. In all the above mentioned tables, the values previously calculated using the compounding technique [15] are also listed for comparison. The FEM average stress intensity factors obtained according to Eq. (4) are presented and the comparison is made using the values that are closer to the compounding ones (the columns that contain these values are marked in the tables with different shades of grey).
Fig. 4. Cross section through the stiffened plate with a crack crossing a stiffener.
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Fig. 5. Element layers through thickness order and coordinate axis origin: (a) order of element layers through thickness; (b) the coordinate system and (c) mesh detail for 2c = 32.5 mm.
Fig. 6. Nodes for calculating the average stress intensity factor for an element.
It should be mentioned that some values of the stress intensity factor are negative, due to the negative ry stress values that exist in the layer of elements opposite to the stiffener surface as presented in Fig. 7. 3.3. Characteristics of the three-dimensional stress fields for a broken stiffener In this section, the plate with a broken stiffener was studied since it is the case most likely to occur in real integral structures. The results were used to characterize the stress intensity factor variation along the thickness, and to emphasize the variation of the stress state in the vicinity of the crack. Again, the degree of plane strain was used to measure the variation of the constraint factor with respect to the thickness direction [19]. The origin of the coordinate system of axes was defined at the crack tip node on the surface containing the stiffener, (Fig. 5b). The degree of plane strain near the crack front is shown in Fig. 8, in the case 2c = 162.5, similar curves being obtained for all the studied crack lengths and both crack tips. As for the centre cracked plates, in the case of stiffened plates, the plane strain conditions seemed to be invalid near the plate free surface.
Table 5 Nondimensional SIFs for crack tip A obtained with FEM and the compounding technique in the case of unbroken stiffener.
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Table 6 Nondimensional SIFs for crack tip B obtained with FEM and the compounding technique in the case of unbroken stiffener.
Table 7 Nondimensional SIFs for crack tip A obtained with FEM and the compounding technique in the case of broken stiffener.
Table 8 Nondimensional SIFs for crack tip B obtained with FEM and the compounding technique in the case of broken stiffener.
For both crack tips and all crack lengths, the plots show that a state close to plane strain occurs between z/t = 0.36 and z/t = 0.64, where the degree of plane strain has its higher values. Nevertheless, it should be pointed out that for the smallest crack length the degree of plane strain reaches its peak before the middle of the plate thickness. As the distance from the crack front increases, the degree of plane strain tends to smaller values, showing three-dimensional and then plane stress behaviour. It was observed that the size of the plane strain zone, d, is somehow dependent on the crack length. In order to emphasize the characteristics of the three-dimensional stress field, a detailed stress analysis in the vicinity of the crack front was carried out. The non-dimensional stresses in the y and z direction were obtained for all the studied cases. In Fig. 9, the plots are shown for crack tip A, in the case 2c = 162.5. A decrease of the ry values with the distance from the crack front, due to the strong stress concentration effect near the crack, is noticed. As the distance from the crack front increases, the values of the stress for different plate depths show that a stiffened plate has a high three-dimensional stress distribution. This is less pronounced at crack tip B, for higher crack lengths, where the remaining intact stiffener carries the load. For all crack lengths, near the plate mid thickness, the stress in the z direction presents significant values only until a value of x/t of approximately 0.4 is reached. The stress in the z direction presents always higher values at crack tip A.
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Fig. 7. Variation of the opening stress ry in the stiffener side (top layer) and in the opposite side (back layer) for c = 32.5 mm.
Fig. 8. Degree of plane strain in the x direction for 2c = 162.5: (a) crack tip A and (b) crack tip B.
Fig. 9. Stress distribution near the crack front for 2c = 162.5 (crack tip A): (a) variation of ry and (b) variation of rz.
3.4. Discussion Refined three-dimensional finite element analyses were performed in order to calibrate the stress intensity factors for a symmetrically stiffened plate subjected to uniform tensile stress. Cracks that could develop in such structures were assessed
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by considering the case of a crack that crosses one of the stiffeners in two possible scenarios: (i) the stiffener crossed by the crack is unbroken and (ii) the stiffener is unbroken. Several important conclusions can be drawn from this work: The three-dimensional finite element analyses show that the stress intensity factor varies through the thickness even in this case of a very thin plate. For the unbroken stiffener, the results that are in best agreement with the ones obtained with the compounding technique are those from the layer opposite to the stiffener. For the broken stiffener, the results that are in best agreement with the ones obtained with the compounding technique are those from the middle elements layer. The obtained results justify the non-straight crack front of advancing fatigue cracks in fatigue cracks in stiffened panels subjected to cyclic loading [28]. Since such structures are often used in the aeronautical industry, where the fatigue phenomenon is always present, the obtained results can be used in a subsequent fatigue analysis based on crack growth. 4. Conclusions As a contribution to the understanding of three-dimensional effects, three-dimensional stress intensity factor solutions for a plate with a central crack and for a cracked stiffened plate were obtained using the finite element method. In order to characterize the near-tip stress field and to determine the variation of the stress intensity factor through the thickness of the plate, very refined finite element analyses were conducted. When possible, the obtained results were compared with those in the literature and with the values obtained following a two-dimensional finite element analysis and dual boundary element analysis. The stress intensity factors for a symmetrically stiffened plate subjected to uniform tensile stress were assessed by considering the case of a crack that crosses one of the stiffeners in two possible scenarios: (i) the stiffener crossed by the crack is unbroken and (ii) the stiffener is unbroken. Conclusions could be drawn from this numerical study, as follows: At the plate free surface, where the singularity r1/2 does not occur, there is a boundary layer whose size is a function of thickness/crack length ratio. For the centre cracked plate, the opening stress ry remains almost constant through the thickness except near the free surface where an important decrease can be noticed in the vicinity of the crack front. For the same structure, the plane strain behaviour near the middle of the plate was demonstrated by calculating the degree of plane strain. It was shown also that the degree of plane strain tends to smaller values when the distance from the crack front increases. In a three-dimensional analysis, stress intensity factor has different values through the thickness. Thus, a two-dimensional stress intensity factor analysis is only an approximation of the exact solution. For the stiffened cracked panel, a non-uniform stress intensity factor distribution through the thickness was obtained which justifies the non-straight crack front of advancing fatigue cracks in stiffened panels subjected to cyclic loading.
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