Improved stress intensity factors for selected configurations in cracked plates

Accepted Manuscript Improved Stress Intensity Factors for Selected Configurations in Cracked Plates R. Evans, A. Clarke, R. Gravina, M. Heller, R. Stewart PII: DOI: Reference:

S0013-7944(14)00187-8 http://dx.doi.org/10.1016/j.engfracmech.2014.06.003 EFM 4310

To appear in:

Engineering Fracture Mechanics

Received Date: Revised Date: Accepted Date:

6 January 2014 22 May 2014 3 June 2014

Please cite this article as: Evans, R., Clarke, A., Gravina, R., Heller, M., Stewart, R., Improved Stress Intensity Factors for Selected Configurations in Cracked Plates, Engineering Fracture Mechanics (2014), doi: http:// dx.doi.org/10.1016/j.engfracmech.2014.06.003

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Improved Stress Intensity Factors for Selected Configurations in Cracked Plates Authors

R. Evans a1, A Clarkeb, R. Gravinab, M. Hellera, R. Stewartb

a

Aerospace Division, Defence Science and Technology Organisation, 506 Lorimer St Fishermans Bend, 3207, Australia b QinetiQ Australia, 210 Kings Way South Melbourne, 3205, Australia Abstract In this paper improved stress intensity geometry factors are determined for four key geometric configurations. They were developed using a p-version finite element method program. Twodimensional uniaxially loaded plates are investigated, with either: an edge crack, a crack approaching a hole, or a crack propagating from a hole after ligament failure. The threedimensional problem of a through crack in an integral stiffener approaching a junction, under uniaxial tension, is also considered. The resulting normalised stress intensity factor data are provided as compact equations or presented in tabular form. Keywords Stress intensity factor; Finite element analysis; Holes Nomenclature

a b c E e h hs K Kt L p R s tp ts uy w ỹ β δ ν σ

Length of edge crack or half-length of imbedded crack Distance between hole edge and crack centre Distance between the centre of the hole and the centre of the crack Young’s modulus Half-width of notch at plate edge Half-height of plate Stiffener height Stress intensity factor Stress concentration factor Nominal notch length Stiffener pitch Radius of hole or radius of notch end Interaction zone Plate thickness Stiffener thickness Uniform displacement in y-direction Width of plate Normalised position on crack front Beta factor, normalised stress intensity factor Displacement Poisson’s ratio Reference stress

1.

Introduction

Damage Tolerance Analysis (DTA) of aircraft structures is a primary tool in managing aircraft safety. One of the fundamental inputs to a DTA is the stress intensity factor (K), which is used to determine the crack growth life as well as the critical crack length. The general equation used to define the stress intensity is:

1

Corresponding author. Tel +61 3 9626 7919; fax +61 3 9626 7089 email address: [email protected]

K = σβ πa

(1)

where σ is a reference stress, a is the crack length and β is the beta factor. The beta factor is considered as the normalised stress intensity factor and accounts for geometry effects. The beta factor for common simple geometries is available from handbooks [1-5], whilst many DTA software codes also include such beta factor solutions (e.g. [6,7]). Rearranging Eq. (1), beta factor is given by:

β=

K

σ πa

(2)

It is known that in such handbooks (and the related underpinning journal articles), the accuracy and range of beta factor values available can be variable. These solutions are based on a mixture of analytical and numerical approaches. One reason for the inaccuracies is that certain solutions were developed decades ago where the capability of numerical methods, such as Finite Element Analysis (FEA), were significantly limited. Some solutions have however been revised over the years. Since publication of the historic work there has been continual advancement in FEA capability, especially for three dimensional (3D) problems. For example, p-element methods [8] can be applied to achieve more accurate beta factor solutions. An extensive application of such pelement modelling is given by Fawaz and Andersson [9] for plates with corner cracks at a hole. Based on a review of the literature and the authors’ experience in developing and applying beta solutions for airframe life assessment, four generic cases were identified as needing improvement. These cases are uniaxially loaded plates with either: (i) a through edge crack, (ii) a through crack approaching a hole, (iii) a through crack propagating from a hole after ligament failure, or (iv) a through crack in an integral stiffener approaching a junction. It is important to note that to achieve results for more complicated practical geometries, compounding of multiple generic handbook solutions is typically used. For example, prior work by the authors demonstrates the use of compounding to analyse C-130J-30 airframe configurations such as cracking in: skin panels stiffened by hat stringers, ‘L’ section spar caps, and panels with integral stringers, [10]. This work also includes some preliminary and limited results for generic cases (i), (ii) and (iv). The present paper focuses on the four cases listed above. The geometries of cases (i) to (iii) are 2D and have been investigated in the literature to various degrees [1-7,11-13]. Whereas case (iv) is three dimensional. Initially this paper extends the accuracy and range of values previously provided for cases (i) and (ii). Then new solutions are determined for cases (iii) and (iv), where to the authors’ knowledge, no published solutions are available. For these two cases, current practice usually involves the use of approximate geometries. To improve the usefulness of the new FE generated beta factors, a wide range of key parameters were considered. Compact equations are given for the cases of an edge crack, and a crack from a hole after ligament failure; the remaining two solutions obtained are presented in tabular form. The background for each of the four geometric cases is provided more fully, along with comparisons where possible with literature results. 2.

Methods and Assumptions

2.1

Finite Element Modelling

The StressCheck® commercial FE software package, Version 8.0.1 [14] was used for all 2D and 3D FEA. This code uses the variable order polynomial elements (p-element) approach, and can be used to determine the Mode I stress intensity for cracked components. Such p-version software can reduce the FEA discretisation error for a fixed mesh, by automatically increasing the p-order of the element shape function and displacement function. This is in comparison to h-version FE software, which requires mesh refinement to reduce the error. Typically, the maximum order polynomial in h-version FEA is two, while p-element FEA allows up to eighth order polynomials. 2

The use of p-elements allows a relatively coarse geometric mesh to be used, enabling quick mesh creation with a high level of accuracy. In the FEA undertaken here, the stress intensity factors are computed using the super-convergent contour integral method. Stress intensity factors are output as a standard result for linear elastic analysis where a crack boundary has been defined. In 3D analyses, they can be extracted at any position along the crack boundary. In this paper, a 2D plane stress model was used for all cases examined, except for the case of the integral stiffener model, where a 3D model was required. Beta factors were calculated from the extracted K, using Eq. (2). 2.2

Finite Element Analysis Convergence Checks

Each FE model was assessed for convergence as an indicator to determine the suitability of the modelling discretisation (mesh and p-level). The modelling discretisation was assessed by analysing at least one crack length for a range of polynomial levels. The error in energy-norm was used to indicate whether the solution had converged. For 3D FE analyses, the energy-norm error was less than 5%. Convergence of K was checked to assess the minimum p-level required for an acceptable result. For each crack length analysed, the K for the highest p-level was compared to the K for the previous p-level. For all crack lengths, this resulted in a difference of less than 2%. Based on this, a p-level of six was used for the analysis of the 3D integral stiffener. It was practical to run all the 2D FE models with a p-level of 8, which resulted in an error in energy-norm less than 1% and a convergence of the K values of better than 0.4%. 3.

Edge Crack

The problem considered here is a finite width plate with an edge crack, as defined in Fig. 1(a). The geometric parameters are: crack length a, plate width w, and half plate height h. Two remote loading conditions were considered: uniform stress σ, and uniform displacement uy; at y = ±h. The local detail of a typical FE mesh is shown in Fig. 1(b). Half the plate was modelled and symmetry boundary conditions were utilised along the mid-plane, as well as a nodal x constraint to prevent rigid body motion. The mesh around the crack tip was designed as a boundary layer to ensure accurate stress intensity values. The material properties for aluminium alloy 2024-T3, with a Young’s modulus of E = 1.06x107 psi and Poisson’s ratio of ν= 0.33, were used. 3.1

Uniform Stress

Beta factor results for an edge crack in a finite width sheet with a uniform applied stress and bending unrestrained are available in numerous handbooks ([1-7]). The impact of the plate aspect ratio (h/w) is negligible when h/w ≥ 1.0 and many handbooks only show solutions in this range. However, solutions for h/w < 1.0 are often only shown for a few cases, such as h/w = 0.5 in [3]. Hence, the solutions investigated here cover a greater range of plate aspect ratios, compared to those available in the literature. The results are given in Fig. 2 for h/w values from 0.2 to 1.0, as a function of the ratio of crack length to plate width (a/w), where the beta factor was determined for a reference stress equal to the applied stress, σ. The limiting beta curve was reached when h/w = 1, which is consistent with the literature ([1-3]) and hence, there was no need to provide values for h/w > 1. All results were fitted with a sixth order polynomial equation, given by: 2

β = A0 + A1

3

4

5

a ⎛a⎞ ⎛a⎞ ⎛a⎞ ⎛a⎞ ⎛a⎞ + A2 ⎜ ⎟ + A3 ⎜ ⎟ + A4 ⎜ ⎟ + A5 ⎜ ⎟ + A6 ⎜ ⎟ w w w w w ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝w ⎠

6

(3)

where the parameters A0-6 are presented in Table 1. The curve fitting resulted in an error of less than 1% for all h/w ratios from 0.2 to 1.0.

3

Comparison to existing handbook solutions is shown in Fig. 3, where the lines represent the handbook solution [3]. The obtained beta factors are within 1% of the handbook solution for h/w = 1.0, and typically less than 2.0% for h/w = 0.5 (for 0 < a/w < 0.7). For higher a/w values the difference is a max of 5.5%. The difference for h/w = 1.0 is consistent with the error stated for the handbook solution [3]. For the case of h/w = 0.5, the handbook solution [3] is taken from Bowie and Neal 1965. Here the quoted accuracy is claimed to be 2%, noting that no details are given. Overall the new FE results are considered more accurate than the existing solutions. 3.2

Uniform Displacement

Handbook beta factors for an edge crack in a finite width sheet with a uniform applied displacement are less common than for the uniform applied stress case. Fett and Bahr [12] developed beta factor solutions for the uniform displacement case for h/w ratios from 0.25 to 1.0, whilst Newman, Wu, Venneri and Li [13] provide a solution for h/w = 2.0. Thus, in this paper, beta solutions are determined for the extended range of h/w ratios of 0.20 to 100. The beta factor was calculated using a reference stress based on the applied displacement, defined as:

σ =

uy h

E

(4)

This is consistent with the definition of reference stress used by Fett and Bahr [12]. In the early stages of the present work, the average stress along the applied displacement edge was considered for use as the reference stress (the average was calculated by integration of the stress along the edge). This approach is consistent with that of Newman et al. [13]. However, with this definition, the reference stress changes with crack length. Hence, this method is not as convenient to use in practice as Eq. (4). The uniform displacement results, utilising Eq. (4), are given in Fig. 4 for h/w ratios of 0.2 to 100. One of the general trends of Fig. 4 is that the value of beta factor increases with increasing h/w for a given a/w ratio, i.e. as the plate becomes more compliant. In Fig. 4(a), where h/w ≤ 1.0, beta factor decreases with increasing a/w, for a given h/w ratio. This is due to the deformation in the plate being significantly restrained. However in Fig. 4(b), where h/w ≥ 2.0, initially the beta factor is approximately constant for all cases, then as a/w increases, the beta factor is higher with increasing h/w. The limiting case is approximately h/w = 100. Thus for h/w = 100, the uniform displacement solution approaches the uniform stress solution (refer to Fig. 3). All results were fitted with sixth order polynomial curves, as given by Eq. (3), with the parameters A0-6 provided in Table 2. The curve-fitting error was less than 3.3% for h/w ratios from 0.2 to 0.3, and less than 1% for h/w ratios from 0.5 to 100. The obtained uniform displacement FE beta factors are compared to the literature data of [12,13] in Fig.5. In Ref. [12] the authors found that K was influenced by ν. Thus the data plotted in Fig.5 for Ref [12] has been linearly interpolated (from Tables 2-4 of [12]) for ν= 0.33. The FE results are within 6.2% of Fett and Bahr [12] for h/w = 0.25, 4.6% for h/w = 0.5, and less than 1.3% for h/w = 1.0. The stated accuracy of values in [12] is not given. For h/w = 2.0, comparison of the obtained FE beta factors with Newman et al. [13], show that they are within 1% (assuming ν= 0.33 for high strength aluminium). Note that for h/w = 2.0, the FE beta factors in this figure are normalised using the average stress (rather than the reference stress of Eq. (4)) to be consistent with Newman et al. [13]. 4.

Crack Approaching a Circular Hole

The geometry and notation for a crack approaching a hole are shown in Fig. 6(a). The geometric parameters are: hole radius R, crack half-length a, distance between the crack centre and the edge of the hole b, distance between the centre of the hole and the centre of the crack c, interaction zone s, half plate height h, and plate width w. The plate was subject to a remote uniform stress σ in 4

the y direction at y = ±h. Beta factors were determined at the crack tip growing away from and approaching the hole; points A and B respectively in Fig. 6(a). The crack (or flaw) approaching a hole in an infinite plate is a common handbook solution ([1-3]), with most of them based on the work of Isida [11]. A more recent investigation into this configuration was undertaken by Harter and Taluk [15], using p-element FEA. The latter results are generally consistent with Isida [11], however they provide a greater range of results (0.0625 ≤ R/c ≤ 0.90). The current investigation extends this range further, to 0.01 ≤ R/c ≤ 0.90, with a higher density of results for normalised crack lengths (a/b) between 0.750 and 0.995. This normalised crack length range is investigated in more detail, as this is where larger changes in beta factor occur. An example of the local detail of a typical FE mesh is shown in Fig. 6(b). Half the plate was modelled and symmetry boundary conditions were utilised along the mid-plane, as well as a nodal x constraint to prevent rigid body motion. A sufficiently large plate was selected such that finite width effects on K were negligible. Thus a value of w/s = 20 was used to approximate the infinite plate solution in all FEA. A square plate was modelled and hence h/w = 0.5. The material properties for aluminium alloy 7075-T6, with E = 1.05x107 psi and ν= 0.30, were used. Beta factors were generated for a range of R/c values between 0.01 and 0.90, for a/b values of 0.050 to 0.995. Full tabular results are provided in Table 3 and Table 4 for both crack tips (A and B) for completeness, however crack tip B is typically the location of interest. The tabulated results can be linearly interpolated for specific R/c values. Fig. 7 shows some typical beta factor results obtained at the A and B crack tips, Fig. 7(a) and 7(b) respectively. Included in this figure are solutions from Isida [11] and also Harter and Taluk [15]. For all comparable cases, the difference between the beta factors in Tables 3 and 4 and those from Isida [11] is 1% or less. Also, the equivalent difference to the Harter and Taluk [15] values is generally less than 1%. However, when R/c = 0.5 and 0.7, differences are up to 5%. The biggest difference for both crack tips occurs when R/c = 0.7, where the Harter and Taluk [15] values are greater. Interestingly, the difference for the Harter and Taluk [15] values reduce to 1.6% when R/c = 0.90 (not shown in Fig. 7), where the opposite occurs and the Harter and Taluk [15] values are lower than those in Table 4. Hence, there is no obvious pattern in the differences observed in beta factor between this study and those from Harter and Taluk [15]. Harter and Taluk [15] state that their beta factors are valid up to a/b values of 0.98. However, they only provide values when a/b equals 0.90 and 1.0, where a/b = 1.0 beta factor values are an approximation for use in a look-up table. In reality as a/b approaches 1.0, the beta factor at crack tip B tends to infinity. In the crack range 0.90 ≤ a/b ≤ 1.0, the Harter and Taluk [15] values are lower than those in Tables 3 and 4. The differences are probably a result of mesh density between the crack tip and the hole. As a/b approaches 1.0, the crack tip almost touches the hole and hence the number of elements between the crack tip and hole is small. Thus, small differences in meshing would result in noticeable differences in beta value as a/b approaches 1.0. Harter and Taluk [15] indicate that a crack approaching a hole with an R/c ratio lower than 0.0625 can be effectively modelled as a centre crack in a plate, with the effect of the hole being ignored. The results in Table 3 and Fig. 7 support this conclusion. The results presented in Table 3 are however more accurate for these low R/c ratios due to the presence of the hole. 5.

Crack from a Hole after Ligament Failure

The geometry and notation for a crack from a hole after ligament failure is defined in Fig. 8(a). This geometry represents the continuing damage crack growth scenario. The geometric parameters are: hole radius R, crack length a, nominal notch length L, plate width w, and plate half-height h. The plate was subject to a remote uniform stress, σ at y = ±h. To adapt handbook solutions such as those in [1,2] requires the assumption that growth after ligament failure is equivalent to growth from either a ‘U’ or elliptical shaped notch. The relevant 5

geometry is shown in Fig. 8(b), where e is the half-width of the notch at the edge of the plate. Other notch solutions, such as those in Ref. [13], are for semi-circular edge notches. Due to their geometry they have limited application for the present problem. This investigation modelled the actual geometry thereby providing comparison to the assumption of a notch. Both finite width and semi-infinite width beta factor solutions were developed. The obtained beta factors are provided as an equation for the finite width plate configuration, whilst tabulated values are provided for the semi-infinite width plate solution. The local detail for a typical FE mesh is shown in Fig. 9. Half the plate was modelled and symmetry boundary conditions were utilised along the mid-plane, as well as a nodal x constraint to prevent rigid body motion. Based on the results from Section 3.1 for the edge-crack solution, h/w was set to 1.0. The material properties for aluminium alloy 2024-T3, with E = 1.06x107 psi and ν= 0.33, were used. 5.1

Finite Width Solutions

Beta factors were calculated as a function of R/w, a/R and (a+L)/w. Sample results are shown in Fig. 10. To improve the usefulness of the results, curve fits were applied as defined by:

β = A0 + A1 (1 − e

− A2 (

a ) R )

+ A3 (1 − e

− A4 (

a ) R )

(5)

with an error of less than 0.4%. Parameters A0-4 for Eq. (5) are provided in Table 5 for (a+L)/w values between 0.025 and 0.70, and R/w values of 0.001, 0.005, 0.05 and 0.125. The beta factors resulting from Eq. (5) are based on Eq. (2), where a (in Eq. (2)) is the equivalent edge-crack length (i.e. a+L in Fig. 8(a)), rather than the crack length from the edge of the hole. In order to use Eq. (5), bi-linear interpolation is used for the R/w and (a+L)/w ratios. Since variation in beta factor with R/w is almost linear, interpolation between data points for R/w values should be performed first. The error from interpolation generally increases as (a+L)/w increases. 5.2

Semi-infinite Width Solutions

The method adopted for determining semi-infinite width beta factors was to divide the FEA finite width solutions by a finite width correction factor. This correction factor is the finite width edgecrack solution (from Section 3.1) divided by the semi-infinite edge-crack solution (i.e. 1.12). To increase the usefulness of the semi-infinite width beta factors and for consistency with other published work, the beta factors were developed for constant R/L values. The first step in generating semi-infinite width solutions was to use Eq. (5) (with linear interpolation) to compute finite width solutions as a function of R/L and a/(a+L). Then the finite width correction factors were applied. Finally, the beta factors were converted for a crack length a, defined in Fig. 8(a). For each semi-infinite width geometry (a, R and L) up to four estimates of beta factor are available, corresponding to R/w values of 0.001, 0.005, 0.05 and 0.125. Inspection of the semiinfinite width beta factors showed a variation of less than 2% for the same geometry for derivations based on R/w values of 0.001, 0.005 and 0.05. The variation was up to 10% when beta factors based on R/w = 0.125 were considered. Results are presented in Table 6, where the beta factor given is from the lowest applicable value of R/w. Beta factors for R/L values between 0.025 and 0.25 are provided. These R/L values represent a range of distances from the plate edge to the hole centre of 1.5 to 19.5 times the hole diameter, which covers most applications. Typical results for a semi-infinite width plate are shown in Fig. 11. The results show that as the crack length increases, the beta factor converges to a value of 1.12 for all R/L values. This is as expected because the effect of the hole decreases with crack length. The diminishing effect of the hole is further illustrated in Fig. 12, where the solid lines represent the semi-infinite edge-crack solution with the crack length defined by a in Fig. 8(a). This plot shows that as the crack size increases relative to the size of the hole, the value of the beta factor approaches that of the edgecrack solution. The difference between the two solutions is negligible when a/R exceeds 1.2. 6

The approach described above to extract semi-infinite width solutions works well except for relatively high values of R/w. Therefore in these cases, the standard edge-crack finite width corrections may not be suitable. 5.3

Comparison with Handbook Solutions

Comparison of the semi-infinite width beta factors (Section 5.2) was made to handbook solutions to determine the impact of the edge-notch assumptions underpinning the handbook solutions. The results are shown in Fig. 13 for the case of R/L = 0.05. Beta factors obtained from Table 6 are compared to handbook solutions from Murakami [1] and Tada et al. [2]. Both the Murakami and Tada solutions are for a semi-infinite width plate with a crack emanating from an elliptical notch, as shown in Fig. 8(b). Comparison is also made with an estimated solution, which is obtained using the method described in Ref. [16]. Initially for a = 0, K is estimated by multiplying the local stress concentration (Kt) of the uncracked geometry with the free surface correction of 1.12. Here Kt was obtained from Ref. [17] (semi-infinite case). Then for longer cracks, the edge-crack solution is plotted. Hence to represent the K transition behaviour, a tangent line is drawn from the K result for a = 0 to the curve for the edge-crack solution. The largest differences between values from Table 6 and Murakami [1], Tada et al. [2] and the tangent Kt method [16,17] are 5, 16 and 7% respectively. Interestingly the greatest differences do not occur for very small crack lengths. They occur between an a/(a+L) of 0.012 and 0.022 (small crack lengths). The differences in beta factor results at small crack lengths are expected to be due to the notch assumption in prior work or the numerical techniques used to obtain stress intensity and hence beta factor values. For example, Murakami [1] uses the body-force method, whilst the solutions in Tada et al. [2] are obtained from stress relaxation and superposition. When the crack length is very small, all solutions are close as they converge to Kt multiplied by the free surface correction of 1.12. For example, at these crack lengths the Tada et al. [2] solution is within 1.0%. Also overall, as the crack length increases, there is good agreement between the FEA and handbook approaches. For example, excluding the Tada et al. [2] solution, the other handbook solutions ([1,16,17]) are within 0.6% of the current FEA values (Table 6) when a/(a+L) > 0.05. In summary, the results presented here provide a greater level of fidelity in beta factors, whilst being in a format easier to extract than current handbook solutions. 6.

Crack in an Integral Stiffener Approaching a Junction

The crack in an integral stiffener approaching a junction is defined in Fig. 14(a) for a uniform stress, σ in the x direction. The geometric parameters are: crack length a, plate width w, stiffener height hs, stiffener thickness ts, plate thickness tp and stiffener pitch p. A full 3D analysis was undertaken, where the overall idealisation is shown in Fig. 14(b). Published work regarding integral stiffeners has focussed on cracking in the plate either between stiffeners or across a stiffener [3,18]. The geometric case investigated here however, represents a through crack in an integral stiffener approaching a junction. The typical cracking scenarios represented by this case are a crack starting at the stiffener edge or more likely a crack from a fastener hole in the stiffener after ligament failure. To the authors’ knowledge, there are no published solutions for this case. Without development of a specific model the crack retarding effect of the plate junction would usually be approximated or ignored. The approximation involves compounding the beta factors for a crack near a junction of two sheets ([11]) and a semi-infinite edge-crack beta factor of 1.12 and is often used in practice. Fig. 15 shows the adapted Isida [11] model considered. To account for the stiffening effect of the skin, the stiffness ratio of the two plates needs to be considered carefully and could be interpreted in various ways. Prior to the current investigation, the accuracy of this approximation was unclear. As shown in Fig. 14(b), the FE models were loaded at one end via a uniformly distributed stress. The plate displacement was restrained in the x and z directions at the unloaded end, in the z 7

direction at the loaded end, and in the y direction along the sides. For this investigation the skin surface of the plate was restrained against movement in the z direction (bending restrained). The material properties for aluminium alloy 7075-T6, with E = 1.05x107 psi and ν= 0.30, were used. A sensitivity study was conducted to determine the required width and length of the FE model to eliminate finite width and length effects. Plate widths of three and four times the height of the stiffener were analysed with all other geometric parameters kept constant. Little difference was found between the results of the two models. Therefore, four times the height of the stiffener was chosen as the plate width for all FE analyses. A similar sensitivity study was conducted for plate lengths of five and ten times the height of the stiffener with little difference to the result. Therefore, ten times the height of the stiffener was chosen as the plate length. In all FE models only a single stiffener was modelled. In these 3D analyses, K can be extracted at any position along the crack front. Here, K was extracted at through-thickness positions (y) along the idealised straight-line crack front. This location is shown in Fig. 14(a), where the normalised position is given by ỹ = y/ts. The corresponding beta factor was calculated from Eq. (2). The average beta value was determined for the range of beta values along the crack front from ỹ = 0.06 to 0.94. Fig. 16 shows an example of the variation of beta factor over the crack front compared to the average, where typically, the variation is less than 3%. Beta factors for a crack in an integral stiffener approaching a skin plate were developed in terms of the stiffener height hs, stiffener thickness ts, and skin plate thickness tp. Beta factors were determined for ratios of a/hs for hs/tp values of 5, 10 and 20, and ts/tp values of 0.5, 1.0, 1.5 and 2.0. The results are presented in Table 7. In order to use these FE results, bi-linear interpolation is used for the ts/tp ratio and the hs/tp ratio of the geometry. The ts/tp interpolation should be performed first as it is the most influential parameter. Separate to this generic configuration analysis (results in Table 7), an independent detailed 3D parametric model of an integrally stiffened skin panel was created and analysed. The panel modelled was five stringer pitches in width, with adjacent stringers and stringer fillets modelled. A small study of four aircraft stiffener skin configurations, which had stiffener pitch to stiffener thickness ratios (p/ts) of 11.9 to 18.1, and stiffener to plate stiffness ratios (hsts/ptp) of 0.38 to 0.55, was performed. The interpolated beta solutions (utilising Table 7 results) were within 1.5% of the independent detailed parametric FE solution for the comparative section and crack geometries. The results in Table 7 do not account for the impact on beta factor from adjacent stiffeners. Hence the small 1.5% difference would be a result of both interpolation and any affects from adjacent stiffeners. The result for one geometry case is shown in Fig. 17, for p/ts of 17.1 and a stiffener to plate stiffness ratio of 0.38. Sample results from Table 7 are plotted in Fig. 18, where they are compared to the idealised handbook solution by Isida [11] for a crack near a junction of two plates compounded by 1.12 to account for the free edge. For application of the Isida [11] solution to this geometry it was assumed that the section thickness after the junction was equal to twice the plate thickness. This assumption ensures the cross sectional area remains the same. As seen in Figs. 14 and 15, the actual stiffener geometry is different to that of the Isida model. The results of Figs. 17 and 18 show that the beta factor decreases for both the handbook method and the FE analyses as the crack front approaches the junction with the plate (i.e. as a/hs increases). Note that when ts/tp = 2.0, the Isida approximation becomes a uniform plate with no stiffness change. It can be seen that there are significant differences (of up to 20%) between the new FE beta factor and those based on Isida [11]. The Isida solution [11] over-estimates the beta factor when compared to the new FE values. Variation of the FE results to the handbook solution is not unexpected since the handbook solution geometry does not match the actual crack geometry well. 7.

Conclusion

In this work accurate normalised stress intensity factors have been developed for four common structural geometries using 2D and 3D p-element FE methods. 8

Normalised stress intensity factors for an edge crack under uniform stress with bending unrestrained or under uniform displacement were developed and represented by equations. The new solutions cover a greater range of plate size and at a higher fidelity than solutions currently available in the literature. Where comparisons could be made, the new solutions were typically within 2.0% of the handbook solution. Normalised stress intensity factors for a crack approaching a hole in a large plate were provided in tabular form. The parameters considered extend the range found in the literature. A higher density of solutions was also provided for cracks in close vicinity to the hole. Where comparisons could be made, the new solutions were principally within 1% of available solutions but had a greater difference (of up to 5%) in normalised stress intensity factors for hole-size to crack-distance ratios of 0.5 and 0.7. The normalised stress intensity factors for a crack propagating from a hole after ligament failure in a finite width plate were obtained and presented as an equation. Because the hole and ligament were modelled, these analyses are much more representative of the real problem compared to the literature. The results were also used to estimate those for a semi-infinite width plate; which can be easier to apply for certain practical geometries. These were provided in tabular form. Tabulated normalised stress intensity factors were provided for a through crack in a stiffener approaching a junction. A handbook comparison of a crack near a junction of two plates indicated that in approximating this geometry the solution would be over-estimated by up to 20%. Considering the assumptions needed to use the handbook solution as well as the error, it was concluded that there is no suitable handbook solution available for this geometry. Overall it is considered that these four new solutions are beneficial since they expand the range of solutions currently openly available. They can be used as either stand alone or with compounding to analyse more complex geometries.

Acknowledgement The authors gratefully acknowledge the contributions made by Ron Wescott in providing support and technical advice. References [1] [2] [3] [4]

[5] [6] [7] [8] [9]

Murakami Y (editor-in-chief). Stress Intensity Factors Handbook. Vol. 1. Committee on Fracture Mechanics. Oxford: Pergamon Press; 1987. Tada H, Paris PC, Irwin GR. The Stress Analysis of Cracks Handbook. Missouri: Del Research Corporation; 1973. Rooke DP, Cartwright DJ. Compendium of Stress Intensity Factors. Middlesex: The Hillingdon Press; 1976. Gallagher JP, Giessler FJ, Berens, AP (University of Dayton Research Institute, Dayton Ohio) Engle, Jr RM. USAF damage tolerant design handbook: guidelines for the analysis and design of damage tolerant aircraft structures. Air Force Wright Aeronautical Laboratories Flight Dynamics Laboratory, Wright-Patterson AFB, Ohio. May 1984. Report No.: AFWALTR-82-3073. Contract No.: F33615-80-C-3229. Air Force Research Laboratory. Handbook for Damage Tolerance Design. 11 December 2002. Air Force Research Laboratory. AFGROW, Version 4.0012.15. 10 July 2008. NASA Johnson Space Center. NASGRO Fracture Analysis Software version 3.0. March 1995. Babuska I, Szabo B. On the Rates of Convergence of the Finite Element Method. Int J Numer Meth Eng 1982;18:323-41. Fawaz SA, Andersson B. Accurate Stress Intensity Factor Solutions for Corner Cracks at a Hole. Eng Fract Mech 2004;71:1235-54. 9

[10] Evans R, Gravina R, Heller H, Clarke A, Rock C, Burchill M. Computational Approaches for the Development of Improved Beta Factor Solutions for C-130J-30 DTA Locations. [Internet]. Aircraft Airworthiness & Sustainment Conference; 2010 May 10-13; Austin, USA [cited 2014 Mar 24]. Available from: http://www.meetingdata.utcdayton.com/agenda/airworthiness/2010/proceedings/techpapers/ TP3253.pdf [11] Isida M. On the Determination of Stress Intensity Factors for Some Common Structural Problems. Eng Fracture Mech 1970;2:61-79. [12] Fett T, Bahr H-A. Mode I Stress Intensity Factors and Weight Functions for Short Plates Under Different Boundary Conditions. Eng Fracture Mech 1999;62:593-606. [13] Newman Jr JC, Wu XR, Venneri SL, Li CG. Small Crack Effects in High Strength Aluminum Alloys. NASA Reference Publication 1309; May 1994. [14] Engineering Software Research and Development Inc. StressCheck 8.0.1. November 2008. [15] Harter JA (AFRL/VASM), Taluk D (Eagle Aeronautics, Inc., Newport News, VA). Life Analysis Development and Verification. Air Force Research Laboratory Air Vehicles Directorate, Wright-Patterson AFB, Ohio. Oct 2004. Report No.: AFRL-VA-WP-TR-2004-3112. Delivery Order 00012: Damage Tolerance Application of Multiple Through Cracks in Plates With and Without Holes. [16] Broek D. The Practical Use of Fracture Mechanics. Dordrecht: Kluwer Academic Publishers; 1989: 260-6. [17] Pilkey WD. Peterson’s Stress Concentration Factors. 2nd ed. New Jersey: John Wiley & Sons, Inc.; 1997. [18] Adeel M. Study on Damage Tolerant Behaviour of Integrally Stiffened Panel and Conventional Stiffened Panel. World Academy of Science: Eng and Tech 2008;21:315-9.

10

σ = const. or uy = const.

y

a

h

x h

w

σ = const. or uy = const.

(a)

(b)

Fig. 1. Edge crack in a finite width plate: (a) geometry and notation, (b) typical local detail of p-element FE model (h/w = 1.0, a/w = 0.3). 12 h/w = 0.2 h/w = 0.25 h/w = 0.3 h/w = 0.4 h/w = 0.5 h/w = 0.75 h/w = 1.0 Curve fits (Eq. (3))

10

β

8 6 4 2 0 0.0

0.2

0.4 a/w

0.6

0.8

Fig. 2. Beta factor for an edge crack in a finite width plate: uniform stress case. 10 h/w = 0.5, bending unrestrained h/w ≥ 1.0, bending unrestrained

β

8

h/w = 0.5, Rooke & Cartwright [3] 6

h/w = 1.0, Rooke & Cartwright [3]

4 2 0 0.0

0.2

0.4 a/w

0.6

0.8

Fig. 3. Comparison of beta factors for an edge crack in a finite width plate: uniform stress case. 11

1.2 1.0

β

0.8 0.6

h/w = 0.2 h/w = 0.25 h/w = 0.3 h/w = 0.4 h/w = 0.5 h/w = 0.75 h/w = 1.0 Curve fits (Eq. (3))

0.4 0.2 0.0 0.0

0.2

0.4 a/w

0.6

0.8

0.6

0.8

(a) 8 h/w = 2 h/w = 4 h/w = 5 h/w = 10 h/w = 12 h/w = 80 h/w = 100 Curve fits (Eq. (3))

β

6

4

2

0 0.0

0.2

0.4 a/w

(b) Fig. 4. Beta factor for an edge crack in a finite width plate for uniform displacement case: (a) h/w = 0.2 to 1.0, (b) h/w = 2 to 100.

12

3 h/w = 0.25, Eq. (4) stress h/w = 0.5, Eq. (4) stress h/w = 1.0, Eq. (4) stress h/w = 0.25, Fett & Bahr [12] h/w = 0.5, Fett & Bahr [12] h/w = 1.0, Fett & Bahr [12]

β

2

h/w = 2.0, average stress h/w = 2.0, Newman et al. [13]

1

0 0.0

0.2

0.4 a/w

0.6

0.8

Fig. 5. Comparison of beta factors for an edge crack in a finite width plate with different reference stresses: uniform displacement case.

y

σ

h A

x

B

R

a b c h

s/2

s/2 w σ

(a)

(b)

Fig. 6. Crack approaching a circular hole: (a) geometry and notation, (b) typical local detail of p-element FE model (R/c = 0.06, a/b = 0.925).

13

2

β

Tip A 1.5

1 R/c = 0.01 R/c = 0.1 R/c = 0.1, Isida [11] R/c = 0.1, Harter et al. [15] R/c = 0.5

0.5

R/c = 0.5, Isida [11] R/c = 0.5, Harter et al. [15] R/c = 0.7 R/c = 0.7, Isida [11] R/c = 0.7, Harter et al. [15]

0 0.0

0.2

0.4

0.6

0.8

1.0

a/b

(a) 4

R/c = 0.01 R/c = 0.1 R/c = 0.1, Isida [11] R/c = 0.1, Harter et al. [15] R/c = 0.5

β

3

R/c = 0.5, Isida [11] R/c = 0.5, Harter et al. [15] R/c = 0.7 R/c = 0.7, Isida [11] R/c = 0.7, Harter et al. [15]

2

1

Tip B 0 0.0

0.2

0.4

0.6

0.8

1.0

a/b

(b) Fig. 7. Comparison of typical beta factors with published solutions for a crack approaching a circular hole: (a) tip A - furthest from hole, (b) tip B - closest to hole.

14

y

σ

σ

L

h

h L

e

x

2R a

h

R a

h

w

w

σ

σ

(a)

(b)

Fig. 8. Geometry and notation for a crack from a hole with ligament failure in a finite width plate: (a) nominal case, (b) simplified representation.

Fig. 9. Typical local detail of p-element FE model for a crack from a hole with ligament failure (a/R = 0.2, (a+L)/w = 0.3, R/w = 0.05). 8

6

β

R/w = 0.001, (a+L)/w = 0.025 R/w = 0.05, (a+L)/w = 0.3

4

R/w = 0.125, (a+L)/w = 0.7 Curve fits (Eq. (5))

2

0 0.0

0.2

0.4

0.6 a/R

0.8

1.0

1.2

Fig. 10. Sample beta factor results for a crack from a hole after ligament failure in a finite width plate as a function of the equivalent edge-crack length a+L.

15

8

β

6

R/L = 0.025 R/L = 0.1 R/L = 0.25

4

2 1.12 0 0.0

0.2

0.4

0.6

0.8

1.0

a/(a+L)

Fig. 11. Typical beta factors based on crack length a for crack from a hole after ligament failure in a semi-infinite width plate. 20 R/L = 0.025 R/L = 0.1 R/L = 0.25 Edge-crack solution

β

15

10

5

0 0

1

2

3 a/R

4

5

6

Fig. 12. Beta factor results for crack from a hole with ligament failure in a semi-infinite width plate as a function of crack length a compared to the edge-crack solution.

16

12

Present study

10

Murakami [1], edge notch Tada et al [2], edge notch

8 β

Tangent Kt Method, edge crack 6 4 2 0 0.0

0.1

0.2 a/(a+L)

0.3

0.4

Fig. 13. Comparison of beta factor results (Table 6) for a crack from a hole after ligament failure in a semi-infinite width plate where R/L = 0.05.

17

Crack

a ỹ = 0.0

ỹ = 1.0

hs

ts

tp

w

(a) z δx=δz=0, restrained

x

δy=0, restrained along both sides

y

Crack

δz=0, restrained lower surface Remote stress δz=0, restrained

(b) Fig. 14. Crack in a stiffener approaching a junction: (a) cross-sectional geometry and notation, (b) overall arrangement.

hs a ts

2tp

Fig. 15. Isida [11] geometry for a crack near a junction of two plates.

18

1.2 Average

β

1.0

0.8

0.6 0.0

0.2

0.4

ỹ

0.6

0.8

1.0

Fig. 16. Typical beta factor distribution along the crack front compared to the average: ts/tp = 0.50, h/ts = 5, and a/hs = 0.10. 1.2

β

1.0

0.8

0.6

Interpolated Table 7 Aircraft location

0.4 0.0

0.2

0.4

0.6

0.8

1.0

a/h s

Fig. 17. Comparison of the crack in a stiffener approaching a junction solution (Table 7) to an aircraft location with five stiffeners (where ts/tp = 0.62, p/ts = 17.1 and stiffener pitch to plate stiffness ratio = 0.38).

19

1.2

β

1.0

0.8 ts/tp ==2.0 ts/tp 2.0 ts/tp ==0.5 ts/tp 0.5 ts/tp ==2.0 ts/tp 2.0(Handbook) (Hdbk) ts/tp 0.5(Handbook) (Hdbk) ts/tp ==0.5

0.6

0.4 0.0

0.2

0.4

0.6

0.8

1.0

a/h s

Fig. 18. Comparison of the beta factors for a crack in a stiffener approaching a junction: h/tp = 5.0. Handbook solution is factored Isida [11].

20

Table 1: Parameters for Eq. (3): beta factor for an edge crack in a finite width plate for uniform stress case. h/w 0.2 0.25 0.3 0.4 0.5 0.75 1.0

A0 1.12 1.12 1.12 1.12 1.12 1.12 1.12

A1 0.013 -0.484 -0.547 -0.584 -0.565 -0.554 -0.561

A2 75.164 60.836 47.477 32.764 25.732 20.828 20.554

A3 -320.597 -268.129 -206.483 -143.479 -119.409 -106.784 -107.683

A4 788.947 687.687 535.725 383.676 340.826 331.633 338.127

A5 -972.706 -894.201 -717.137 -526.878 -479.586 -477.604 -487.554

A6 470.646 461.498 388.885 300.956 278.815 277.965 282.730

Table 2: Parameters for Eq. (3): beta factor for an edge crack in a finite width plate for uniform displacement case. h/w 0.2 0.25 0.3 0.4 0.5 0.75 1 2 4 5 10 12 80 100

A0 1.12 1.12 1.12 1.12 1.12 1.12 1.12 1.12 1.12 1.12 1.12 1.12 1.12 1.12

A1 -2.044 -0.764 -0.130 0.252 0.238 0.094 0.057 0.063 0.056 0.031 0.037 0.055 0.140 0.073

A2 -23.105 -30.166 -30.094 -22.767 -15.145 -5.036 -1.303 2.768 5.221 6.179 6.824 6.445 5.404 6.922

A3 159.246 172.504 154.948 98.163 53.458 6.556 -4.293 -6.269 -11.449 -15.631 -13.652 -9.312 -0.543 -10.906

A4 -397.987 -400.815 -339.741 -189.151 -85.322 5.454 15.537 1.118 23.424 37.523 28.647 13.325 -3.171 30.760

A5 445.959 429.850 349.969 176.162 66.604 -16.051 -17.107 2.913 -36.320 -53.778 -24.450 1.921 11.713 -39.492

A6 -187.621 -175.406 -138.563 -64.354 -20.641 8.584 6.629 -0.902 19.298 25.825 1.207 -14.121 3.928 33.876

21

Table 3: Beta factor for a crack approaching a hole under uniform stress for R/c values of 0.01 to 0.10. R/c a/b

0.01 Tip A

Tip B

0.02 Tip A

Tip B

0.03 Tip A

Tip B

0.04 Tip A

Tip B

0.05 Tip A

Tip B

0.06 Tip A

Tip B

0.08 Tip A

Tip B

0.10 Tip A

Tip B

0.050 0.100 0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500 0.550 0.600 0.650 0.700 0.750 0.760 0.770 0.780 0.790 0.800 0.810 0.820 0.830 0.840 0.850 0.860 0.870 0.880 0.890 0.900 0.910 0.920 0.925 0.930 0.935 0.940 0.945 0.950 0.955 0.960 0.965 0.970 0.975 0.980 0.985 0.990 0.995

1.000 1.000 1.000 1.000 1.001 1.001 1.001 1.001 1.001 1.001 1.002 1.002 1.002 1.002 1.003 1.003 1.003 1.003 1.003 1.003 1.003 1.003 1.003 1.003 1.003 1.003 1.003 1.003 1.003 1.003 1.003 1.003 1.003 1.004 1.003 1.004 1.004 1.004 1.004 1.004 1.004 1.004 1.004 1.004 1.004 1.005 1.005

1.000 1.000 1.000 1.000 1.001 1.001 1.001 1.001 1.001 1.002 1.002 1.002 1.002 1.003 1.003 1.003 1.003 1.003 1.004 1.004 1.004 1.004 1.004 1.004 1.005 1.005 1.005 1.005 1.006 1.007 1.007 1.009 1.009 1.010 1.011 1.012 1.014 1.016 1.018 1.022 1.026 1.033 1.043 1.059 1.086 1.143 1.296

1.000 1.000 1.000 1.000 1.001 1.001 1.001 1.001 1.002 1.002 1.002 1.002 1.002 1.003 1.003 1.003 1.003 1.003 1.003 1.003 1.003 1.003 1.003 1.003 1.003 1.003 1.003 1.004 1.004 1.004 1.004 1.004 1.004 1.004 1.004 1.004 1.004 1.004 1.005 1.005 1.005 1.005 1.005 1.006 1.006 1.007 1.008

1.000 1.000 1.000 1.000 1.001 1.000 1.001 1.001 1.002 1.002 1.002 1.003 1.003 1.004 1.004 1.005 1.005 1.005 1.006 1.006 1.006 1.007 1.007 1.008 1.009 1.009 1.010 1.012 1.013 1.015 1.018 1.021 1.023 1.025 1.028 1.032 1.036 1.042 1.049 1.058 1.069 1.086 1.108 1.143 1.198 1.299 1.529

1.000 1.000 1.001 1.000 1.001 1.001 1.001 1.002 1.001 1.002 1.002 1.002 1.003 1.003 1.003 1.003 1.003 1.003 1.003 1.004 1.004 1.003 1.004 1.004 1.004 1.004 1.004 1.004 1.004 1.004 1.005 1.005 1.005 1.005 1.005 1.005 1.006 1.006 1.006 1.006 1.006 1.007 1.007 1.008 1.009 1.010 1.011

1.000 1.000 1.001 1.001 1.001 1.001 1.001 1.002 1.002 1.003 1.003 1.004 1.004 1.005 1.007 1.007 1.008 1.009 1.009 1.010 1.010 1.011 1.012 1.013 1.015 1.016 1.018 1.020 1.023 1.027 1.031 1.038 1.038 1.044 1.050 1.057 1.065 1.075 1.086 1.101 1.120 1.145 1.180 1.229 1.304 1.433 1.707

1.001 1.001 1.000 1.001 1.001 1.001 1.002 1.002 1.002 1.002 1.002 1.003 1.003 1.003 1.003 1.004 1.004 1.004 1.004 1.004 1.004 1.004 1.005 1.005 1.005 1.005 1.005 1.005 1.005 1.005 1.006 1.006 1.006 1.006 1.007 1.007 1.007 1.007 1.008 1.008 1.008 1.009 1.009 1.011 1.011 1.013 1.015

1.001 1.001 1.000 1.001 1.002 1.002 1.002 1.003 1.003 1.003 1.004 1.005 1.006 1.008 1.010 1.011 1.012 1.012 1.013 1.014 1.016 1.017 1.019 1.020 1.022 1.025 1.028 1.032 1.036 1.041 1.048 1.057 1.061 1.068 1.076 1.085 1.096 1.109 1.125 1.145 1.170 1.202 1.246 1.306 1.396 1.544 1.848

1.001 1.001 1.001 1.001 1.001 1.001 1.002 1.002 1.002 1.003 1.003 1.003 1.003 1.004 1.004 1.004 1.004 1.005 1.005 1.005 1.005 1.005 1.005 1.005 1.006 1.006 1.006 1.006 1.007 1.007 1.007 1.008 1.008 1.008 1.008 1.009 1.009 1.009 1.010 1.010 1.011 1.012 1.012 1.013 1.015 1.016 1.018

1.001 1.001 1.001 1.002 1.002 1.002 1.003 1.003 1.004 1.005 1.006 1.007 1.008 1.011 1.014 1.015 1.016 1.017 1.019 1.020 1.022 1.024 1.026 1.028 1.031 1.035 1.039 1.044 1.050 1.057 1.067 1.079 1.084 1.093 1.104 1.115 1.129 1.146 1.166 1.190 1.220 1.259 1.310 1.380 1.481 1.643 1.973

1.002 1.002 1.002 1.002 1.002 1.002 1.003 1.003 1.003 1.003 1.003 1.004 1.004 1.004 1.005 1.005 1.005 1.005 1.006 1.006 1.006 1.006 1.006 1.006 1.007 1.007 1.007 1.008 1.008 1.008 1.009 1.009 1.010 1.010 1.010 1.010 1.011 1.011 1.012 1.013 1.013 1.014 1.015 1.016 1.017 1.019 1.023

1.002 1.002 1.002 1.002 1.002 1.003 1.004 1.005 1.005 1.007 1.008 1.009 1.011 1.014 1.019 1.020 1.022 1.023 1.025 1.027 1.029 1.031 1.035 1.038 1.041 1.046 1.052 1.058 1.065 1.075 1.087 1.102 1.110 1.120 1.132 1.146 1.163 1.183 1.206 1.234 1.269 1.313 1.370 1.447 1.557 1.732 2.084

1.003 1.003 1.003 1.003 1.003 1.004 1.004 1.004 1.004 1.004 1.005 1.006 1.006 1.007 1.008 1.007 1.007 1.007 1.007 1.008 1.008 1.009 1.009 1.009 1.009 1.010 1.010 1.011 1.011 1.012 1.012 1.013 1.014 1.014 1.014 1.015 1.015 1.016 1.017 1.018 1.019 1.020 1.021 1.023 1.024 1.027 1.031

1.002 1.004 1.004 1.004 1.005 1.006 1.007 1.007 1.009 1.010 1.012 1.015 1.018 1.023 1.031 1.032 1.034 1.037 1.039 1.042 1.046 1.050 1.054 1.059 1.065 1.072 1.080 1.089 1.100 1.114 1.130 1.151 1.163 1.177 1.193 1.211 1.232 1.257 1.286 1.321 1.364 1.416 1.483 1.571 1.695 1.890 2.280

1.005 1.005 1.005 1.005 1.005 1.005 1.005 1.005 1.006 1.006 1.006 1.007 1.008 1.009 1.009 1.010 1.010 1.010 1.010 1.011 1.011 1.011 1.012 1.012 1.013 1.013 1.014 1.014 1.015 1.016 1.017 1.017 1.018 1.019 1.019 1.020 1.020 1.021 1.022 1.023 1.024 1.026 1.027 1.029 1.032 1.035 1.039

1.005 1.006 1.006 1.007 1.008 1.009 1.009 1.011 1.012 1.015 1.018 1.021 1.026 1.033 1.044 1.046 1.049 1.052 1.057 1.061 1.065 1.070 1.076 1.084 1.091 1.100 1.111 1.123 1.137 1.155 1.176 1.202 1.216 1.233 1.253 1.275 1.301 1.330 1.364 1.405 1.453 1.512 1.586 1.684 1.820 2.031 2.455

22

Table 4: Beta factor for a crack approaching a hole under uniform stress for R/c values of 0.15 to 0.90. R/c a/b

0.15 Tip A

Tip B

0.20 Tip A

Tip B

0.30 Tip A

Tip B

0.40 Tip A

Tip B

0.050 0.100 0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500 0.550 0.600 0.650 0.700 0.750 0.760 0.770 0.780 0.790 0.800 0.810 0.820 0.830 0.840 0.850 0.860 0.870 0.880 0.890 0.900 0.910 0.920 0.925 0.930 0.935 0.940 0.945 0.950 0.955 0.960 0.965 0.970 0.975 0.980 0.985 0.990 0.995

1.012 1.011 1.011 1.011 1.011 1.011 1.011 1.012 1.012 1.013 1.014 1.014 1.015 1.017 1.019 1.018 1.019 1.019 1.019 1.020 1.021 1.021 1.022 1.023 1.023 1.024 1.025 1.026 1.027 1.029 1.030 1.031 1.033 1.033 1.034 1.035 1.036 1.037 1.039 1.040 1.042 1.044 1.047 1.049 1.053 1.056 1.062

1.011 1.013 1.015 1.016 1.017 1.019 1.021 1.024 1.027 1.031 1.037 1.044 1.053 1.067 1.087 1.091 1.097 1.102 1.109 1.116 1.124 1.134 1.144 1.155 1.168 1.182 1.199 1.218 1.241 1.266 1.297 1.334 1.353 1.376 1.403 1.432 1.466 1.503 1.547 1.597 1.656 1.727 1.816 1.931 2.090 2.337 2.838

1.022 1.021 1.021 1.020 1.020 1.020 1.020 1.021 1.021 1.022 1.023 1.024 1.025 1.027 1.030 1.030 1.031 1.031 1.033 1.033 1.034 1.035 1.036 1.037 1.038 1.039 1.041 1.042 1.043 1.045 1.047 1.049 1.050 1.052 1.053 1.054 1.056 1.057 1.060 1.062 1.063 1.066 1.069 1.072 1.076 1.081 1.088

1.024 1.025 1.027 1.029 1.031 1.034 1.038 1.042 1.048 1.055 1.063 1.075 1.090 1.111 1.140 1.148 1.156 1.165 1.174 1.185 1.196 1.210 1.223 1.239 1.257 1.276 1.298 1.324 1.352 1.385 1.423 1.469 1.491 1.520 1.551 1.586 1.625 1.668 1.718 1.775 1.843 1.924 2.025 2.154 2.332 2.613 3.184

1.056 1.054 1.053 1.052 1.051 1.051 1.050 1.051 1.051 1.052 1.054 1.056 1.058 1.062 1.066 1.067 1.068 1.069 1.070 1.071 1.073 1.074 1.076 1.078 1.079 1.081 1.083 1.086 1.088 1.091 1.094 1.097 1.101 1.103 1.105 1.107 1.110 1.112 1.115 1.118 1.121 1.124 1.128 1.133 1.139 1.145 1.155

1.059 1.064 1.068 1.072 1.078 1.084 1.092 1.101 1.113 1.127 1.144 1.166 1.194 1.232 1.282 1.294 1.307 1.321 1.336 1.353 1.370 1.390 1.412 1.434 1.459 1.487 1.518 1.551 1.590 1.634 1.684 1.740 1.774 1.811 1.851 1.894 1.942 1.995 2.057 2.127 2.208 2.307 2.427 2.584 2.803 3.148 3.862

1.116 1.112 1.109 1.107 1.105 1.104 1.103 1.103 1.104 1.105 1.107 1.109 1.113 1.118 1.124 1.125 1.126 1.128 1.130 1.132 1.133 1.135 1.138 1.140 1.142 1.145 1.148 1.151 1.154 1.158 1.161 1.166 1.169 1.171 1.174 1.176 1.179 1.182 1.186 1.190 1.194 1.198 1.203 1.209 1.216 1.225 1.237

1.124 1.131 1.138 1.147 1.156 1.168 1.181 1.197 1.216 1.238 1.266 1.300 1.341 1.395 1.464 1.480 1.498 1.516 1.536 1.558 1.580 1.605 1.631 1.659 1.691 1.725 1.763 1.804 1.850 1.901 1.959 2.028 2.066 2.107 2.153 2.203 2.257 2.320 2.390 2.471 2.565 2.680 2.822 3.008 3.267 3.679 4.537

0.50 Tip A

Tip B

1.214 1.208 1.203 1.198 1.195 1.193 1.191 1.190 1.190 1.191 1.193 1.196 1.200 1.206 1.214 1.215 1.217 1.219 1.221 1.224 1.226 1.228 1.231 1.234 1.237 1.240 1.244 1.248 1.252 1.256 1.261 1.266 1.269 1.272 1.275 1.278 1.282 1.286 1.290 1.294 1.299 1.305 1.311 1.318 1.326 1.337

1.228 1.240 1.251 1.264 1.279 1.296 1.316 1.339 1.366 1.398 1.435 1.480 1.534 1.602 1.688 1.707 1.728 1.751 1.774 1.800 1.827 1.856 1.887 1.920 1.956 1.995 2.039 2.086 2.139 2.198 2.265 2.344 2.386 2.433 2.485 2.541 2.606 2.676 2.759 2.852 2.962 3.094 3.261 3.478 3.786 4.273

0.60 Tip A

Tip B

1.368 1.358 1.351 1.344 1.339 1.334 1.331 1.329 1.328 1.328 1.330 1.333 1.338 1.345 1.354 1.355 1.357 1.360 1.362 1.365 1.368 1.371 1.374 1.377 1.381 1.385 1.389 1.393 1.398 1.403 1.409 1.415 1.419 1.422 1.426 1.430 1.434 1.439 1.444 1.449 1.455 1.462 1.469 1.478

1.391 1.405 1.421 1.438 1.459 1.482 1.508 1.538 1.572 1.612 1.659 1.713 1.779 1.858 1.958 1.981 2.005 2.031 2.058 2.087 2.118 2.150 2.186 2.224 2.265 2.310 2.359 2.413 2.474 2.541 2.618 2.706 2.756 2.810 2.870 2.934 3.008 3.092 3.184 3.294 3.423 3.579 3.775 4.030

0.70 Tip A

Tip B

1.598 1.586 1.576 1.566 1.559 1.553 1.548 1.545 1.543 1.543 1.545 1.548 1.553 1.561 1.571 1.574 1.576 1.579 1.582 1.585 1.588 1.592 1.596 1.600 1.604 1.609 1.613 1.619 1.625 1.631 1.638 1.645 1.649 1.654 1.658 1.663 1.668 1.674 1.680 1.687 1.694 1.702

1.627 1.644 1.664 1.686 1.711 1.739 1.770 1.806 1.846 1.893 1.947 2.009 2.084 2.174 2.287 2.313 2.341 2.370 2.401 2.433 2.469 2.506 2.546 2.589 2.636 2.688 2.744 2.806 2.875 2.953 3.043 3.142 3.202 3.267 3.334 3.412 3.499 3.596 3.706 3.836 3.987 4.173

0.90 Tip A

Tip B

2.392 2.383 2.377 2.373 2.371 2.372 2.373 2.377 2.383 2.391 2.402 2.416 2.432 2.451 2.476 2.480 2.486 2.491 2.498 2.504 2.511 2.519 2.526 2.534

2.413 2.432 2.452 2.474 2.500 2.530 2.565 2.606 2.654 2.711 2.777 2.857 2.952 3.072 3.224 3.258 3.296 3.335 3.378 3.423 3.472 3.524 3.580 3.642

23

Table 5: Parameters for Eq. (5): beta factor for a crack from a hole after ligament failure in a finite width plate as a function of the equivalent edge-crack length a+L. R/w 0.001

0.005

0.05

0.125

(a+L)/w 0.025 0.05 0.1 0.2 0.3 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.025 0.05 0.1 0.2 0.3 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.2 0.3 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.45 0.5 0.55 0.6 0.65 0.7

A0 0.269 0.271 0.282 0.320 0.388 0.491 0.565 0.657 0.772 0.943 1.159 1.475 0.273 0.277 0.289 0.327 0.394 0.496 0.585 0.641 0.774 0.945 1.158 1.480 0.340 0.414 0.526 0.605 0.702 0.837 1.014 1.264 1.625 0.643 0.751 0.902 1.096 1.382 1.808

A1 0.423 0.434 0.454 0.518 0.629 0.799 0.921 1.081 1.271 1.558 1.914 2.425 0.387 0.419 0.452 0.521 0.633 0.803 0.945 1.057 1.262 1.542 1.890 2.428 0.468 0.612 0.801 0.934 1.094 1.303 1.578 1.981 2.539 0.915 1.078 1.307 1.588 1.996 2.547

A2 14.70 14.29 14.05 14.12 14.05 14.06 13.97 13.89 14.01 13.61 13.71 13.77 16.97 15.34 14.49 14.29 14.20 14.20 13.41 14.88 14.29 13.92 14.09 13.93 17.42 16.01 15.44 15.19 15.19 15.01 14.95 14.81 14.89 17.28 17.25 17.00 17.16 17.17 17.51

A3 0.432 0.431 0.448 0.521 0.632 0.807 0.922 1.067 1.273 1.501 1.870 2.401 0.467 0.444 0.447 0.515 0.627 0.801 0.882 1.107 1.281 1.516 1.895 2.394 0.562 0.635 0.782 0.881 1.022 1.193 1.425 1.722 2.167 0.867 0.992 1.132 1.343 1.600 1.994

A4 4.203 4.023 3.904 3.861 3.819 3.793 3.765 3.716 3.737 3.656 3.669 3.673 5.233 4.596 4.208 4.036 3.942 3.885 3.751 3.900 3.828 3.751 3.754 3.701 5.464 4.857 4.553 4.423 4.342 4.275 4.259 4.183 4.203 5.386 5.296 5.151 5.179 5.178 5.300

24

Table 6: Beta factors for a crack from a hole after ligament failure, semi-infinite width plate solution. R/L a/(a+L)

0.025

0.04

0.05

0.065

0.1

0.15

0.2

0.25

0.0005

18.355

16.262

15.506

14.869

14.083

13.551

13.253

13.232

0.0007

17.318

15.015

14.144

13.406

12.491

11.874

11.530

11.453

0.001

16.508

14.042

13.061

12.220

11.164

10.448

10.049

9.912

0.002

15.267

12.828

11.731

10.750

9.452

8.538

8.021

7.765

0.003

14.404

12.263

11.202

10.217

8.844

7.833

7.249

6.925

0.005

12.933

11.392

10.511

9.644

8.313

7.247

6.598

6.200

0.01

10.433

9.678

9.154

8.621

7.642

6.703

6.063

5.617

0.025

7.087

6.950

6.760

6.602

6.229

5.758

5.357

5.011

0.05

5.039

5.038

4.975

4.961

4.869

4.691

4.507

4.304

0.075

4.114

4.120

4.077

4.090

4.080

4.013

3.921

3.789

0.1

3.562

3.568

3.530

3.546

3.558

3.539

3.494

3.404

0.15

2.907

2.913

2.880

2.893

2.911

2.916

2.909

2.863

0.2

2.516

2.523

2.492

2.503

2.519

2.528

2.529

2.525

0.3

2.052

2.059

2.031

2.039

2.053

2.063

2.066

2.066

0.4

1.774

1.782

1.759

1.784

1.774

1.784

1.788

1.789

0.5

1.582

1.592

1.577

1.596

1.582

1.592

1.597

1.599

0.6

1.439

1.449

1.444

1.456

1.439

1.449

1.454

1.458

0.7

1.327

1.337

1.339

1.345

1.329

1.334

1.341

1.345

0.8

1.243

1.242

1.247

1.252

1.250

1.261

1.246

1.258

0.9

1.173

1.176

1.172

1.169

1.171

1.182

1.179

1.175

0.92

1.158

1.167

1.163

1.158

1.163

1.166

1.169

1.158

0.93

1.156

1.157

1.159

1.155

1.150

1.157

1.159

1.150

0.94

1.150

1.147

1.153

1.151

1.144

1.148

1.149

1.144

0.95

-

1.140

1.142

1.148

1.141

1.139

1.140

1.141

0.96

-

1.138

1.134

1.137

1.139

1.132

1.138

1.139

0.97

-

-

1.132

1.129

1.135

1.131

1.126

1.135

0.98

-

-

-

-

1.122

1.130

1.127

1.122

25

Table 7: Beta factor for a crack in an integral stiffener approaching a junction. ts/tp 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0

0.10 hs/tp = 5 1.064 1.087 1.095 1.098 hs/tp = 10 1.066 1.089 1.100 1.106 hs/tp = 20 1.057 1.091 1.098 1.104

0.30

0.50

a/hs 0.70

1.011 1.043 1.058 1.069

0.941 0.985 1.011 1.029

0.861 0.923 0.963 0.994

0.810 0.886 0.937 0.978

0.738 0.835 0.905 0.960

0.662* 0.784* 0.873* 0.944*

1.007 1.039 1.055 1.067

0.934 0.978 1.004 1.025

0.850 0.913 0.955 0.987

0.796 0.874 0.929 0.973

0.719 0.818 0.892 0.954

0.606 0.740 0.842 0.927

1.001 1.038 1.047 1.067

0.928 0.974 1.001 1.024

0.841 0.904 0.951 0.986

0.790 0.866 0.924 0.971

0.707 0.807 0.886 0.950

0.565 0.704 0.810 0.915

0.80

0.90

0.98

* These values were extrapolated from the results for a/h ratios of 0.90 and 0.96.

26

Highlights • • • • •

Improved geometry (or beta) factors were developed using 2D and 3D p-version FEA. Four key generic geometric configurations were considered. A broader range of parameters than currently available were analysed. Also, a higher density of solutions were calculated. Results are plotted and provided as compact equations or tabular data.

27