Analytical prediction of Mode I stress intensity factors for cracked panels containing bonded stiffeners

Engineering Fracture Mechanics 97 (2013) 12–29

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Analytical prediction of Mode I stress intensity factors for cracked panels containing bonded stiffeners Calvin Rans a,⇑, Riccardo Rodi b, René Alderliesten b a b

Department of Mechanical and Aerospace Engineering, Carleton University, 1125 Colonel By Drive, Ottawa, Canada K1S 5B6 Faculty of Aerospace Engineering, Delft University of Technology, Kluyverweg 1, 2629 HS Delft, The Netherlands

a r t i c l e

i n f o

Article history: Received 23 May 2012 Received in revised form 31 October 2012 Accepted 1 November 2012 Available online 19 November 2012 Keywords: Fatigue Damage tolerance Crack growth Bonded stiffener Stress intensity factor

a b s t r a c t An analytical model for stress intensity factors in cracked skin panels containing bonded stiffening elements, based on the principles of superposition and displacement compatibility, is presented. The methodology permits assessment of the influence of stiffening elements ahead, behind, and over top of the skin panel crack tip, and allows the influence of stiffener interface disbond size to be studied. Validation of the model is made using experimental data from various literature sources. The influence of stiffener geometry and interface disbond size on damage tolerance is discussed using predictions from the presented model. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Aviation safety and the methodologies used to ensure integrity of aircraft structures over their lifetime have evolved over time. Currently, airworthiness regulations recommend the use of the damage tolerance philosophy for the design of aircraft primary structures [1,2]. Goranson defines damage tolerance as the ability of structure to sustain anticipated loads in the presence of fatigue, corrosion or accidental damage until such damage is detected, through inspections or malfunctions, and repaired [1]. Although this definition is generally agreed upon, numerous interpretations of damage tolerance exist, particularly related to differences in composite and metallic aircraft structures. In metallic structures, inspection intervals for repairs are set based upon a detection window defined as the service life required for a damage to grow from a detectable size (based on inspection capabilities) to a critical size (based on limit load carrying capability). Central to this is the concept of slowgrowth and the ability to predict damage growth behaviour. Conversely, in composite structures, a no-growth approach is typically adopted where damage growth under service conditions is not permitted, and inspection intervals are specified based on the statistical likelihood of damage-causing events. This paper focuses on the slow-growth interpretation most commonly adopted for metallic structures. The benefit of having a structure comprised of multiple elements, known as a built-up structure, on damage tolerance has long been understood. Built-up structures provide multiple load paths and the opportunity to isolate damages within particular structural elements. Early damage tolerance analyses for built-up mechanically fastened aircraft structures exploited this benefit, and robust prediction tools based on fracture mechanics, the principle of superposition, and displacement compatibility between structural elements at discrete fastener locations were developed [3–7].

⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (C. Rans). 0013-7944/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engfracmech.2012.11.001

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Nomenclature a A b E Fbridging Fbroken Fintact G Kbridging Kbroken Kintact Ktotal Papplied Pskin Pst,bridging Pst,broken Pst,intact R t

vbridging vbroken vintact W X Zi ; Zi b dad dst

r ryy m

half-crack length (mm) cross-sectional area (mm2) delamination length (mm) Young’s modulus (GPa) load transfer due to bridging stiffener (kN) load transfer due to broken stiffener (kN) load transfer due to intact stiffener (kN) shear modulus (GPa) p stress intensity factor for skin panel loaded by Fbridging (MPa mm) p stress intensity factor for skin panel loaded by Fbroken (MPa mm) stress intensity factor for skin panel, with intact stiffening elements ahead of the crack tip, subjected to farfield p loading (MPa mm) p resultant stress intensity factor in cracked stiffened panel (MPa mm) total load applied to stiffened panel (kN) far field load in the skin panel (kN) far field load in a bridging stiffener (kN) far field load in a broken stiffener (kN) far field load in an intact stiffener (kN) fatigue stress ratio thickness (mm) crack opening for skin panel loaded by Fbridging (mm) crack opening for skin panel loaded by Fbroken (mm) crack opening for a skin panel, with intact stiffening elements ahead of the crack tip, subjected to farfield loading (mm) width (mm) element centreline position (mm) Westergaard Stress Functions ratio of stiffened and unstiffened panel stress intensity factors adhesive shear deformation (mm) elongation of delaminated region of a stiffener (mm) stress (MPa) Westergaard stress (MPa) Poisson’s ratio

Subscripts ad referring element referring skin referring st referring

to to to to

the adhesive layer a generic structural element (skin or stiffener) the skin panel a stiffening element

Despite the damage tolerance benefit of built-up mechanically fastened structures, they are not ideal from a manufacturing and cost perspective. The high part count and manufacturing effort required for each fastener installation add cost and time to the manufacturing process. Integral structures overcome this drawback, but at the cost of damage tolerance. Another alternative is to maintain the built-up nature of the structure and replace the joining technique from mechanical fastening to adhesive bonding. This may reduce cost by reducing part count and machining steps; however, strict surface preparation and processing control requirements can offset this benefit. Although many obstacles for the widespread application of adhesive bonding for primary structural joints in aircraft still exist, the use of adhesive bonding for redundant structural elements such as stiffeners and damage tolerance features such as doublers and tear straps is increasing. Experimentally, the influence of bonded stiffeners, doublers, and tear straps on crack growth behaviour has been thoroughly studied [8–13]. Analytically, the superposition method of Poe [3] for riveted structures has been adapted to predict the residual strength of adhesively bonded stiffened panels [4,7]. These models simulate the adhesive interface as discrete adhesive patches, directly analogous to individual fasteners, requiring the adhesive interface of a bonded stiffener to be discretized into several elements. Furthermore, the treatment of the adhesive interface as discrete adhesive patches (or elements) makes the application of these models to cases of cracks beneath the bonded stiffeners difficult, where further discretization of the adhesive would be necessary. As a result, the current trend in analyzing bonded stiffened panels is to adopt finite element techniques, such as the study by Zhang et al. [14]. Finite element techniques do provide a means to analyze the problem, but they lack the simplicity and flexibility of simplified analytical models. The goal to develop simple and robust analytical models for built-up bonded structures is not unattainable. Alderliesten [15–21] successfully described the crack growth behaviour of the hybrid material family known as Fibre Metal Laminates

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(FMLs) using the same fracture mechanics, superposition, and displacement compatibility approach used by Poe [3], Swift [4], and Vlieger [6,7] for built-up structures. The breakthrough emerged by treating the laminated material not as a single material, but as an adhesively bonded built-up structure. The goal of this paper is to build upon the success of Poe, Swift, Vlieger and Alderliesten by applying the same approach to the analysis of skin panel containing adhesively bonded stiffening elements. A new analytical model for bonded stiffened panels is proposed and validated.

2. Prediction methodology The general approach adopted within this paper for modelling crack growth in a stiffened panel relies on linear elastic fracture mechanics (LEFM) and the principle of superposition. Mode I stress intensity factor solutions for different damage states in a bonded stiffened panel can be calculated using these methods. These stress intensity factor solutions can then be used as similitude parameters to infer expected crack growth behaviour from empirical coupon data. To aid in the LEFM description of the problem, the influence of a stiffener on crack growth within a panel can be classified by three stiffener conditions:  Intact stiffeners ahead of the crack tip. These stiffeners provide additional stiffness and load redistribution ahead of the crack tip, reducing the crack growth in the skin panel.  Broken stiffeners behind the crack tip. These stiffeners transmit load into the skin panel along the crack flanks, exacerbating crack growth in the skin panel.  Bridging stiffeners over the crack. These stiffeners bridge load over the crack, reducing crack growth in the skin panel in a manner analogous to that of a bonded patch repair. By breaking down a complex cracked stiffened panel into these components, stress intensity factor solutions for each of the more simple conditions stated above can be developed. The overall stress intensity factor for the cracked stiffened panel can then be determined by breaking the problem up into its simpler components, determining the stress intensity factor for each component, and summing the stress intensity factors of each component as illustrated in Fig. 1.

Fig. 1. Decomposition of stiffened panel into the three stiffener conditions.

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To avoid confusion, the stress intensity factor solutions for the various stiffener conditions will be developed in terms of loads rather than stresses. Pskin, Pst,intact, Pst,broken, and Pst,bridging represent the far field load in the skin, intact stiffener, broken stiffener, and bridging stiffener elements in the undamaged condition. Each of these loads is given by:

ðEAÞelement i¼1 ½ðEAÞelement i

Pelement ¼ Papplied Pn

ð1Þ

where EA is the structural element stiffness, n is the number of structural elements in the stiffened panel, and the subscript element refers to a particular skin or stiffener element. Conversely, Fintact, Fbroken, and Fbridging are unknown loads resulting from the redistribution of loads between the stiffeners and skin when the skin is cracked. This methodology and classification of stiffener conditions is analogous to the approach adopted by Poe [3] for panels with riveted stiffeners, and later modified by Swift and Vlieger [4,7] for panels with bonded stiffeners. Indeed, Fig. 1 bares a resemblance to Fig. A1 of Vlieger [7]; however, the treatment of the stiffener conditions is different. Vlieger discretizes the adhesive interfaces into elements, calculating individual values of adhesive element load redistribution from displacement compatibility between the sheet and stiffeners at each element location. These adhesive element loads, which Vlieger denotes as interaction forces or Fijk, are components of Fintact, Fbroken, and Fbridging illustrated in Fig. 1, depending on the applicable stiffener condition for the particular stiffener element denoted by ijk. The total stress intensity factor for the panel is then given by a summation of the solution for an unstiffened cracked sheet loaded with the far field stress, and an unstiffened sheet with a crack loaded along its edges by a stress distribution equal in magnitude but opposite in sign to the summation of all Fijk. The present approach provides alternative stress intensity factor solution methodologies which eliminate the need for discretization of individual stiffening elements, which will be discussed in the remaining subsections. 2.1. Appropriateness of LEFM for the analysis of stiffened panels Applying LEFM to calculate Mode I stress intensity factors for the purpose of predicting crack growth behaviour in stiffened panels raises two important questions: Is a linear elastic approach appropriate for a problem where plasticity may be present, and does Mode I loading accurately represent expected loading and fracture mechanisms? These questions highlight the important fact that many engineering models employ assumptions and/or simplifications which may limit their applicability. These limitations for the proposed model will briefly be discussed. With respect to the application of LEFM, it is well understood that linear elastic stress solutions for sharp cracks result in a singularity at the crack tip. This singularity cannot be accommodated in a real material, thus resulting in some degree of plasticity (in the case of a ductile material) at the crack tip and load redistribution. Despite this, the convenience of a LEFM description of the crack tip stress field over an elastic–plastic description makes it desirable from a simplified modelling perspective. This has led to the concept of the K-dominated zone [22], which describes a region around a crack tip where load redistribution due to small-scale yielding (SSY) is small enough that LEFM still provides a meaningful indication of the severity of stress at the crack tip. The Irwin plasticity model [23] has been used to derive a SSY criterion that is commonly applied as an engineering rule to define this limit [24]:

pffiffiffiffiffiffiffiffiffiffi KðaÞ 6 ry 0:4a

ð2Þ

In the context of stiffened panels, the consequence of large-scale yielding at the crack tip is not trivial. The stiffening elements provide multiple alternative load paths for the load redistribution resulting from the plasticity. Capturing this behaviour would require more complicated methods than presented in this paper. However, this restriction predominantly applies to cases of residual strength assessment, where quasi-static loads up to failure are considered. The validity of stress intensity factors for cases of crack propagation under fatigue loading is considered evident. Thus, the simplicity and ease of the presented model warrant its use as a design tool with the caveat of these potential consequences. The suitability of a Mode I description for fracture in a stiffened panel could also be questioned based on the range of applications stiffened panels are used for. In the context of aerospace structures, stiffened wing skins are predominately tension loaded, while fuselage skin panels are biaxial loaded. Furthermore, pillowing resulting from pressurization in fuselage shells can introduce local out-of-plane loading around stiffening elements. With respect to effects of biaxial loading, this is an interesting problem that is not is not well addressed even for unstiffened sheets. Crack growth properties are typically determined from uni-axial coupon testing in which the Paris relation constitutes recorded crack growth plotted against DK. The relation often used for DK in most literature relates to Westergaard’s stress field equations [29], which he initially developed for the conditions of bi-axial loading in pressure vessels. Although Irwin proposed correcting these equations for the case of uni-axial loading [25] most studies do not consider this. The penalty for applying bi-axial stress solutions to uniaxial load conditions is however limited if both the reference data (experiments used to determine the Paris relation) and the predicted cases (for example the current model) consistently neglect this aspect. In case the current model would be extended towards bi-axial loading, as it occurs in fuselage stiffened skin structures, one should consider the appropriateness of the relations for the reference experiments. Another question regarding the suitability of a Mode I description of fracture relates to the formation of shear lips and transition from tensile to shear dominated fracture mechanisms. Indeed, this transition can be expected in many stiffened skin applications, but its presence does not limit the applicability of the proposed approach. An investigation into the shear

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lip transition by Schijve [26] revealed that the transition results in a gradual change in the slope of the crack growth rate curve when plotted against the Mode I stress intensity factor (Paris Relation) and is related primarily to the stress intensity factor range at the crack tip. In this case, the use of the Mode I stress intensity factor may appear to disagree with the change in fracture mode associated with the shear lip transition. Practically speaking, however, it is still a valid similitude parameter for comparing crack growth in a stiffened panel to an experimentally determined Paris relation which implicitly includes the shear lip transition. 2.2. K due to intact stiffeners ahead of the crack tip The overall stiffness of a structure, or structural stiffness, is related to both material stiffness (Young’s modulus) and geometric stiffness of the structure. In an unstiffened skin panel, the presence of a crack reduces geometric stiffness by reducing the net sectional area of the skin panel. As no other structural elements are present in an unstiffened skin panel, the reduction in stiffness simply results in an increased structural deformation and the cracked skin panel continues to carry the entire applied load. Thus the load carried by the sheet remains constant with crack length and the stress intensity factor can be defined easily using analytical solutions for flat unstiffened plates [27]. In the case of a stiffened skin panel, the presence of a crack also reduces the geometric stiffness of the skin. This reduction in geometric stiffness in the skin, however, is compensated partially by the presence of intact stiffening elements ahead of the crack tip. Displacement compatibility between the intact stiffeners and cracked skin results in load transfer between the two structural elements in response to the reduced geometrical stiffness of the skin (Fintact in Fig. 1). The effect of Fintact, which will vary as a function of crack length, is to reduce the far field load carried by the cracked skin. If this value can be calculated, and the far field load carried by the skin in the cracked condition can be determined, then the stress intensity factor can be computed using the same analytical solutions for an unstiffened skin panel. A stiffened panel containing bonded stiffeners provides a convenient compatibility condition: isostrain between the stiffener and skin panel. This compatibility condition has been applied by Barsoum and Ravi Chandran in their analyses of stress intensity factors in layered and functionally graded materials [28,29]. The same approach can be applied to a skin panel with bonded stiffeners ahead of the crack tip. Consider a symmetric cracked skin panel with a bonded stringer ahead of the crack tip, as illustrated in Fig. 2. Assume that the stress state in the skin panel ahead of the crack tip can be expresses according to the Westergaard stress distribution [30], given as:

rskin for a 6 x 6 W ryy;skin ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  ða=xÞ2

ð3Þ

where x is the distance from the centre of the crack along the crack plane and rskin is the far field stress carried by the skin panel in the cracked condition:

rskin ¼

Pskin  F intact W  tskin

ð4Þ

In Eq. (4) rskin is an unknown quantity since the value of Fintact is unknown. Given the isostrain condition between the stiffener and skin panel, it follows that for the stiffener:

Fig. 2. Westergaard stress distribution for an intact stiffener ahead of the crack tip.

C. Rans et al. / Engineering Fracture Mechanics 97 (2013) 12–29

ryy;st ¼

    W st W st 6 x 6 X st þ for X st  2 2

Est rskin  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Eskin 1  ða=xÞ2

17

ð5Þ

The total load carried by the stiffened panel in Fig. 2 is equal to the sum of the far field loads in the skin and intact stringer. The integral of the stress distributions along the panel net section in Eqs. (3) and (5) over the net sectional area of the stiffened panels needs to be in equilibrium with the far field applied loads.

Pskin þ Pst;intact ¼

Z

W

ryy;skin  tskin  dx þ

Z

a

ryy;st 



W st

Ast W st

  dx

ð6Þ

where tskin is the thickness of the skin panel and Ast/Wst is an effective stiffener thickness, approximating the stiffener geometry as a bonded strap with uniform thickness across the bonded pad width of the actual stiffener. Rearranging for rskin:

rskin ¼ R W a

Pskin þ Pst;intact R  Est  Ast 

t skin pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi dx þ 2 1ða=xÞ

W st

Eskin

W st

ð7Þ

1 pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi dx 2 1ða=xÞ

According to the definition of stress intensity factor, the value of K in the cracked sheet with intact stiffeners ahead of the crack tip can be written in terms of the far field stress from the Westergaard stress distribution:

K intact ¼ rskin

pffiffiffiffiffiffi pa ¼ R W a

pffiffiffiffiffiffi ðP skin þ Pst;intact Þ pa    R t skin 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi dx þ W EEst WAst pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi dx 2 2 st st skin

1ða=xÞ

ð8Þ

1ða=xÞ

Eq. (8) is the stress intensity factor solution for a symmetric cracked skin panel with one stiffener ahead of the crack tip. This solution can be extended to n number of intact stiffeners ahead of the crack tip as follows:



K intact ¼

RW a

pffiffiffiffiffiffi P Pskin þ ni¼1 ðPst;intact Þi pa

n X R  Est  Ast  t skin 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi2 dx þ p ffiffiffiffiffiffiffiffiffiffiffiffi ffi dx W st E W st 2 1ða=xÞ

i¼1

skin

1ða:xÞ

ð9Þ i

The solution in Eq. (9) provides the stress intensity factor for the skin panel and intact stiffeners ahead of the crack tip. The application of the isostrain condition along the crack plane permits determination of this quantity without the need to discretize the adhesive interface between the intact stiffeners and skin panel. In the following sections, the treatment of the remaining two stiffener conditions, broken and bridging, will be presented. 2.3. K due to a broken stiffener When a bonded stiffener fails, its far field load is no longer transmitted over the crack by the stiffener, rather it is transmitted into the cracked skin. This load transfer is analogous to that of bonded stiffener run-out or half of a bonded single lap joint. The stress distribution over the bonded interface is highly non-linear and can be examined in detail using shear lag theory [31–33]. The distribution follows the so called bathtub shape where the stresses and stress gradients are high at the ends of the adhesive interface (disbond front) and the stresses are low remote from the ends of the adhesive interface. The concentration of stress at the ends of the adhesive interface result in almost all load transfer occurring within a few millimeters of these boundaries [34]. As a result, it is a reasonable to approximate the load transfer along the bonded interface of a broken stringer as a concentrated load at the end of the adhesive interface. This approximation eliminates the need to discretize the stiffener as was implemented by Vlieger. Using the above approximation, the load transfer into a cracked skin panel due to symmetric broken stringers is reduced to a cracked panel with symmetric point loads applied a distance b from the crack flanks and at the centre line of the broken stringers, as illustrated in Fig. 3. The parameter b represents the distance from the crack flank to the adhesive interface (or

Fig. 3. Four-point loaded crack (adapted from [27]).

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delamination length). The analytical solution for the loading case in Fig. 3 is provided by Tada et al. [27] in terms of a differential equation of Westergaard Stress Functions and has been simplified to the form given below:

2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a

a þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ a2 ðxc ibÞ2 Pst;broken 6 pffiffiffiffiffiffi 4 ¼ tskin pa  b ð1 þ mÞ aðxc þibÞi 3  2 2

3

a2 ðxc þibÞ2

K broken

ða2 ðxc þibÞ

Þ2

7 5

aðxc ibÞi

ða2 ðx

ð10Þ

3 2 c ibÞ Þ2

where i is the imaginary unit which is employed in [27] as a means of distinguishing the xy-coordinates of the point of load application. The load introduction by a broken stiffener is more accurately represented by a uniform stress applied along the width of the broken stringer, particularly for wider stiffening elements. This case can be determined by dividing the above expression by the stiffener width and integrating it as a function of x over the width of the stiffener, as given below:

K broken ¼

Z W st

2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a ffi

a ffiþ þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 ðxibÞ2 Pst;broken 6 pffiffiffiffiffiffi 4 W st t skin pa  b ð1 þ mÞ aðxþibÞi 3  2 2

3

a2 ðxþibÞ2

ða2 ðxþibÞ Þ2

aðxibÞi

7 5  dx

ð11Þ

3 ða2 ðxibÞ2 Þ2

In reality, the loading across a broken stringer can be more complex and both of the above solutions represent a simplification of the problem. The appropriate usage of either solution will be dependent on the nature of the result desired. For the point load approximation in Eq. (10), the stress intensity factor will be over-estimated in the immediate vicinity of the stiffener. For predictions of overall fatigue crack growth life, this over-estimation will result in a conservative result that may be acceptable for the designer. For examining the potential for static failure, this over-estimation may not be suitable. Applying the uniform stress solution in Eq. (11) may appear to solve this issue, however, there are instances where this is also an oversimplification. Specific stiffener geometries, such as bonded stiffeners with concentrated vertical shear webs, may have a load transfer distribution closer to the point applied load approximation. The influence of these approximations is also highly influenced by the proximity of the crack tip to the applied loading (i.e.: disbond length and crack length beyond the broken stiffener). As with all engineering approximations, care must be taken in choosing the most suitable approximation for the case at hand. In the predictions presented later within this paper, the stress intensity factor solution given in Eq. (10) was applied. 2.4. K due to bridging stiffener When a crack in the skin panel propagates underneath a stiffener, the stiffness of the intact stiffener limits the opening of the fatigue crack by means of bridging load over the crack. This process is analogous to the bridging function of the fibre layer in Fibre Metal Laminates (FMLs), such as Glare, and has been extensively described in the literature [9,15–20,35–45]. The same compatibility approach used by Alderliesten [15,16] for FMLs can be employed to solve for the bridging load, and its resultant stress intensity factor, for a bridging stiffener on a cracked skin panel. Consider the generic bridging stiffener case illustrated in Fig. 4 where the stiffener of width Wst contains an arbitrary disbond b(x). At every point along the width of the bridging stiffener, compatibility between stiffener and sheet deformation over the delaminated length must hold. Deformation of the cracked sheet in this region is dominated by the crack opening displacement, as the strain immediately above and below the crack flanks is negligible. This displacement in the sheet needs

Fig. 4. Bridging stiffener with an arbitrary stiffener-skin disbond.

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to be compatible with the overall displacement resulting from cumulative strain in the stiffener over its delaminated length and shear deformation of the adhesive at the disbond boundary. 2.4.1. Crack opening displacement The total crack opening displacement in the sheet can be broken down into the individual crack opening displacements for each of the loading cases illustrated in Fig. 1. For the intact stringer load case the far field load in the skin panel and intact stringers ahead of the crack tip are accounted for. In determining the stress intensity factor solution for this loading case an isostrain condition was assumed ahead of the crack tip. This leads to the definition of an effective far field skin stress accounting for load redistribution by the intact stringers as given in Eq. (7). The crack opening displacement for this loading case can be calculated using the LEFM solution for crack opening profile for a far field stress of rskin [27]:

v intact ðxÞ ¼

2rskin pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2  x2 Eskin

ð12Þ

For the broken stringer load case the far field load in broken stringers are accounted for. This loading case was approximated as either a point load introduction along the centreline of the stringer or a uniform stress applied along the width of the broken stringer. Again, the crack opening displacements for these two loading cases can be determined from their LEFM crack opening profile solutions. The solution for the crack opening profile for the four-point loaded crack illustrated in Fig. 3 is given in Tada et al. [27] in differential form in terms of the Westergaard Stress Functions for that load case. The differential equation was expanded by Wilson et al. [45] resulting in:

v broken ðxÞ ¼

h  i 2 1  1þm m  ImZ I  b  ReZ I 2G

ð13Þ

where ZI and Z I are the Westergaard Stress Functions, in terms of the far field broken stiffener load Pst,broken, which are written out in full in Appendix A. If multiple broken stiffeners exist in the panel, individual contributions from each broken stiffener can be calculated using Eq. (13) and summed to determine the total contribution from all broken stiffeners. For the bridging stiffener load case the far field and bridging load in the bridging stiffener are accounted for. This loading case is analogous to that of a broken stiffener, with the difference arising due to the direction of the load. Therefore, the crack opening displacement due to a bridging stiffener should also be given by Eq. (13), accounting for the sign difference in the loading. In doing so, however, a difficulty arises in the point load approximation of the stiffener force. This approximation results in overly large displacements near the location of the point load, which is precisely where the displacements of interest occur. As a result, it is necessary to approximate the load introduction of the bridging stiffener as a distributed load. Assuming the distribution of this load to be uniform along the width, the crack opening due to the bridging stiffener can be determined by:

v bridging ðxÞ ¼ 

Z W st

h  i m 1 2 1  1þm  ImZ I  b  ReZ I  dx W st 2G

ð14Þ

where the Westergaard Stress Functions become functions of the far field bridging stiffener load Pst,bidging and unknown stiffener bridging load Fbridging as detailed in Appendix A. The negative sign in the above equation is included to account for the sign difference in Fbridging. The total crack opening displacement due to all three loading cases can thus be expressed as a summation of the results in Eqs. (12)–(14). 2.4.2. Bridging stiffener deformation Determination of the deformation of the stiffener is more straight forward. The extension of the stiffener over the delaminated region can be calculated assuming a uniform strain field within this region. Elongation thus becomes the stiffener strain multiplied by the length, b, of the delaminated region. Thus, in terms of the far field bridging stiffener load and unknown bridging load:

dst ¼

b Pst;bridging þ F bridging  Est W st t st

ð15Þ

In addition to the elongation of the stiffener, shear deformation of the adhesive interface plays a role in the compatibility of deformation between the skin and stiffener as described in detail for FMLs in [15,17,45]. This deformation is a result of transmission of Fbridging from the cracked skin to the bridging stiffener through the adhesive layer and can be expressed as [45]:

dad

F bridging tad ¼ W st Gad

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   Gad 1 1 þ tad t skin Eskin t st Est

ð16Þ

where tad and Gad are the thickness and shear modulus of the adhesive layer connecting the skin and bridging stiffener.

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2.4.3. Compatibility The unknown bridging load in the stiffener can be determined by enforcing compatibility of the deformation in the cracked skin and bridging stiffener along the disbond interface between these two structural elements. Compatibility can be expressed in terms of the above derived displacements:

v intact þ v broken þ v bridging ¼ ðdst þ dad Þst;bridging

ð17Þ

In this equation, the deformation of both the cracked skin panel and bridging stiffener are dependent on the unknown bridging load. In order to solve for this unknown load, a location at which compatibility is enforced needs to be selected. As a result of the uniform stress assumption within the bridging stiffener, the deformation of the stiffener is constant along its width. Conversely, the crack opening displacements in the skin are all functions of x. Thus the uniform stress assumption over-constrains the system, and compatibility cannot be enforced along the entire width of the bridging stiffener. This can be solved by releasing the uniform bridging stress distribution and solving for compatibility along the entire disbond interface as has been done for FML crack growth [15,16]. Alternatively, the uniform stress assumption can be retained, a constant disbond length across the stiffener can be assumed, and compatibility can be enforced at a single point along the width of the bridging stiffener. The later approach has been selected here for simplicity. For consistency, the centreline of the bridging stiffener is always used for evaluating the compatibility condition. The compatibility condition in Eq. (16) is analogous to the compatibility conditions given in Eq. A1 by Vlieger [7]; however, its implementation is different. Vlieger discretizes each stiffener into multiple elements and enforces displacement compatibility between the skin and stiffener at the centre of each element. In the present analysis, this discretization is avoided and displacement compatibility is enforced along the disbond interface between the skin and stiffener. 2.4.4. Stress intensity factor In solving for the bridging load using compatibility as described above, a uniform bridging stress assumption was made. The LEFM solution for stress intensity factors for such a loading case was presented previously for the case of a broken stringer with a uniform stress distribution in Eq. (11). Taken into account the difference in load direction for a bridging stiffener, this stress intensity solution can be rewritten as:

K bridging ¼

Z W st

2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a a ffi þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiþ 2 2 2 2 F bridging 6 a ðxþibÞ a ðxibÞ pffiffiffiffiffiffi 4 W st t skin pa  b ð1 þ mÞ aðxþibÞi 3  2

ða2 ðxþibÞ2 Þ2

3 aðxibÞi

7 5  dx

ð18Þ

3 ða2 ðxibÞ2 Þ2

2.5. Partially bridging and cracked stiffener cases In addition to the intact, broken, and bridging stiffener cases described above, there are two additional possible stiffener conditions: the partially bridging stiffener and the cracked stiffener. Both of these stiffener conditions can be accounted for by a combination of the above cases. A partially bridging stiffener occurs when a crack grows underneath, but not beyond, a stiffener. In this instance, a portion of the stiffener bridges load behind the crack tip while the remainder of the stiffener acts as an intact stiffener ahead of the crack tip. This stiffener condition can be accounted for by dividing it into two at the location of the crack tip. In this way, the portion of the stiffener behind the crack tip can be treated as described in Section 2.4, while the portion ahead of the crack tip can be treated as described in Section 2.2. A cracked stiffener occurs when a crack initiates within a stiffener and links up with the crack in the underlying skin. The cracked width of the stiffener cannot transmit load across the crack tip. This can be treated in one of two ways. First, the cracked width of the stiffener can be treated as a broken stiffener as outlined in Section 2.3 while the intact portion is treated as an intact stiffener as outlined in Section 2.2. The singularity of the point load stress intensity factor solution for a broken stiffener, however, results in overly high predictions of crack growth. The alternative approach is to assume the intact portion of the cracked stiffener carries the entire far field stiffener load, and account for its contribution to the total stress intensity factor as outlined in Section 2.2. The later approach is utilized in this paper. It should be noted that it is possible for separate cracks to be present in the stiffener and skin. They do not necessarily have to link up and grow together. This possibility is not considered here, and the integrity of a stiffener in the remainder of the paper will be characterized as being fully intact, fully broken, or cracked, where the cracked stiffener case is precisely as described above. 2.6. Stiffener failure and disbond growth In the previous sections, the stress intensity factor solutions for the different stiffener conditions were formulated. These solutions are dependent on the panel geometry and loading; however, they also require knowledge about three damage states. Firstly, the crack length in the skin panel is needed. This is precisely the damage to which the stress intensity factor pertains and their interdependence is intuitive. Secondly, the damage state in each stiffener is needed. The contribution of each stiffener to the total stress intensity factor in the cracked skin panel is dependent on whether the stiffener is intact or

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broken and on the location of the stiffener relative to the crack tip in the skin (in front or behind). This is reflected in the various stress intensity factor solutions for the different stiffener conditions. Finally, the size of the interface disbond between each stiffener and the skin panel is needed. This damage parameter constrains the compatibility condition for bridging stiffeners and determines the location of load introduction due to load transfer between stiffeners and skin. The state of all three damages is needed in order to determine the total stress intensity factor for the cracked skin. The method presented in this paper is intended for analyzing and predicting the growth of a crack within the skin of a stiffened panel. Prediction within the stiffener of crack initiation, growth, and final stiffener failure as well as disbond initiation and growth are beyond the scope of this paper. Indeed, it is possible to utilize S–N curves to predict crack initiation in the stiffener elements [46] and strain energy release rate methods to predict disbond growth [21], however these topics will not be covered here. It will be assumed (and implemented) for the remainder of the paper that stiffener failure occurs at known times, defined by the length of the crack in the skin panel, and that disbonds initiate when the stiffener is in the bridging condition. The length of these disbonds will be assumed to be constant along the width of the stiffener and will not grow under fatigue loading. The consequence of this simplification is that the stiffener conditions and disbond sizes are treated as known geometric parameters, and the stress intensity factor solution for a stiffened panel can be reduced to a function of applied stress and crack length. This solution can be normalized by the stress intensity factor solution for a unstiffened plate, removing the dependency of the solution on applied load.

b¼

K stiffened K unstiffened

ð19Þ

3. Model verification Verification of the described prediction methodology was carried out using a number of in-house and literature based experimental data sets. For the purposes of this paper, three verification cases will be presented. The first case is based on an experimental study carried out by Schijve [10] on narrow aluminum panels containing symmetric rectangular bonded stiffeners near the edges of the panel. The second case is based on wide panel fatigue test results published in [8] for a 7stringer stiffened panel containing bonded titanium tear straps between each of the bonded stringers. The third case comes from the same study as the second case, and uses the same panel geometry where the titanium tear straps have been replaced with bonded aluminum tear straps. Each of these test cases are briefly described below and compared to predictions using the above described methodology. The input parameters for each of the three cases are summarized in Table 1.

3.1. Crack growth in a panel containing symmetric edge stiffeners A study conducted by Schijve [10] examined the effect of integral and bonded stiffeners of various material types on the crack growth performance of flat 2024-T3 aluminum sheet material. Results from test series 2 from this study were selected for verification. The panel from this test series consisted of a 160 mm wide 2024-T3 aluminum sheet of thickness 1 mm. A pair of 25 mm wide bonded rectangular plate stiffeners was centred on the panel with a pitch of 105 mm. Each stiffener

Table 1 Summary of model input variables. Element

Variable

Case 1

Case 2

Case 3

Skin

E (Gpa) G (Gpa) t (mm) W (mm) m ()

73 28 1 80 0.3

73 28 1.4 600 0.3

73 28 1.4 600 0.3

Adhesive

G (Gpa) t (mm)

5.5 0.2

5.5 0.2

5.5 0.2

Stringer

E (Gpa) W (mm) A (mm2) b (mm) Pitch (mm)

73 25 100 5 80

73 25 126 5 185

73 25 126 5 185

Tear strap

E (Gpa) W (mm) A (mm2) b (mm) Pitch (mm)

– – – – –

114 25 20 5 185

73 35 28 5 185

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Fig. 5. Model predictions of test series 2 results from Schijve [10].

consisted of two 2 mm thick straps bonded on opposing faces of the panel to create a symmetric stiffener. The panel contained an initial central crack (2ao = 3 mm) and was fatigue tested with a maximum stress of 118 MPa and an R-ratio of 0.33. Results from this study, including predictions made using the methodology described in this paper, are given in Fig. 5. The progression of failure in the test panel was the growth of the skin fatigue crack up to then underneath the bonded stiffener. After approximately 5 mm of crack growth beneath the stiffener, crack initiation in the stiffener was detected. Due to the proximity of the stiffener to the free edge of the panel, static failure of the panel quickly followed crack initiation in the stiffener. The study provided details regarding the number of cycles required for crack nucleation within the stiffener, allowing the cracked stiffener case detailed in Section 2.5 to be accurately included in the predictions. No evidence of disbonding was reported in the study, suggesting that disbond lengths were sufficiently small that they were not observed or did not warrant reporting. Working on this assumption, several predictions for disbond lengths ranging from 1 mm to 5 mm were used in the comparison. Predictions made with a disbond size of 2 mm agree well with the results from the study, as shown in Fig. 5. The small size of the disbond necessary to produce accurate predictions also agrees with the absence of any observations of large scale disbonding in the study. The model predicts all phases of the crack propagation accurately up to the stiffener cracking. Although the simulated crack stiffener case does predict a significant increase in the crack propagation rate, it levels off at a level far below that observed in the study. This deviation occurs due to the contribution of static fracture to the final moments of crack propagation within the test panel. The predictive model does not contain a static failure criterion (particularly for the stiffener), thus crack propagation continues at a reduced rate until complete stiffener failure is triggered in the model. 3.2. Crack growth in a 7-stringer stiffened panel Meneghin et al. carried out an experimental investigation of crack growth in stiffened panels containing bonded stringers and inter-stringer tear-straps. Two panels from this study have been selected for comparison. Each panel consisted of 1.4 mm thick 2024-T3 aluminum skin with bonded 7075-T73511 ‘‘J’’ – profile stringers. A bonded tear strap was located between each stringer pair. Results for the first panel containing Ti–6Al–4V tear straps are given in Fig. 6 and for the second panel containing 2024-T3 aluminum tear straps in Fig. 7. Each panel contained an initial central crack (2ao = 50 mm) and was fatigue tested with a maximum stress of 100 MPa and an R-ratio of 0.1. It should be noted that results in [8] were published in the form of crack growth rates, not b factors. These results have been converted to b factors in Figs. 6 and 7 by the following relation:

ðda=dNÞstiffened 1 C DK stiffened 10nlog pffiffiffiffiffiffi b¼ ¼ DK unstiffened r pað1  RÞ

ð20Þ

Here, the numerator is determined by assuming Paris Law behaviour in the skin material and back calculating the stress intensity factor related to the crack growth rate observed in the stiffened panel. The denominator is the stress intensity factor range solution for an infinite sheet. The material constants C and n were determined to be 2.05  1014 (mm/cycle)/ p (MPa mm)n and 3.74 respectively using data from [8,47]. The stiffened panel containing titanium tear straps provides a useful benchmark for evaluating the bridging stiffener solution in the prediction model. The titanium tear strap, due to its lower fatigue sensitivity, remains intact after the crack in the skin panel has grown underneath and beyond the tear strap itself. While the crack is underneath the intact stiffening

C. Rans et al. / Engineering Fracture Mechanics 97 (2013) 12–29

23

Fig. 6. Model prediction of results for 7-stringer panel with Ti tear straps from Meneghin et al. [8].

Fig. 7. Model prediction of results for 7-stringer panel with Al tear straps from Meneghin et al. [8].

element, a trend in reducing crack growth rate (or decreasing b) with increasing crack length is observed in Fig. 6. This results from a larger portion of the stiffener bridging load in the wake of the crack as the crack continues to grow underneath the stiffener. Once the crack grows beyond the stiffening element, an overall reduction in crack growth rate remains (b < 1); however, this benefit decreases as the crack tip grows away from the bridging element. The trend and magnitude of this described behaviour was captured by the prediction model as illustrated in Fig. 6. This prediction was obtained for an assumed constant disbond length of 5 mm, as no information about the disbond shape or size was provided in the study. Results from the second panel containing aluminum tear straps provide a useful benchmark for the model solutions related to a broken or cracked stiffener. The aluminum tear strap fails earlier than the titanium tear straps due to its relatively lower fatigue strength. As the stiffness ratio and loading in both panels was kept constant in the study, the primary cause of differences in the behaviour observed between Figs. 6 and 7 is the earlier failure of the aluminum tear strap. Unfortunately, Meneghin et al. [8] did not report explicit details regarding the failure of the aluminum tear strap. As a result, assumptions regarding tear strap failure needed to be made for the predictions. Two failure scenarios were considered. For prediction 1 in Fig. 7, the tear strap was assumed to remain fully intact until the crack in the skin reached the far side of the tear strap, at which point the tear strap was assumed to completely fail. Thus prediction 1 is identical to

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the prediction for the titanium tear strap up until the simulated tear strap failure. At this point, the stress intensity factor contribution of the bridging stiffener is replaced by that of the case for a broken stiffener, resulting in a sudden and large increase in b. For prediction 2 in Fig. 7, damage initiation in the tear strap was assumed to occur when at a skin crack length of 85 mm, corresponding to an observed sudden increase in crack growth rate in the experimental results. At this crack length, the tear strap is assumed to contain a crack which follows the crack in the underlying skin as described in Section 2.5. For both predictions, a constant disbond length of 5 mm was assumed. It is evident from Fig. 7 that prediction 2, simulating cracking of the tear strap at a = 85 mm, proved to contain the better assumptions regarding stiffener failure for capturing the experimental results. Indeed, capturing the failure of stiffening elements, and the associated switch from a beneficial state of bridging to a detrimental state of opening a fatigue crack in the skin, is essential to accurately model the fatigue behaviour of stiffened panels. Although a robust methodology for including stiffener failure has not yet been incorporated into the presented analysis methodology, the presented stress intensity factor solutions seem capable of capturing the complex interaction between a cracked skin panel and bonded stiffening elements. 4. Results The developed prediction methodology provides a simple alternative to finite element modelling techniques for predicting the driving force for fatigue crack growth in bonded stiffened panels. Additionally, while utilizing the same basic approach of superposition and displacement compatibility, the present methodology provides improvements in simplicity and performance compared to previous models by Poe and Swift [3,4]. Most notably, the present methodology eliminates the need for stiffener discretization through use of the isostrain condition described in Section 2.2, and includes the stiffening/bridging effects of each stiffener across its physical width rather than concentrated along the stiffener centreline as implemented by Poe and Swift. The results of this second improvement will be examined in the following section. 4.1. Influence of stiffener area Increasing the stiffness, EA, of a stiffener will directly increase its influence on crack growth in an underlying skin panel. In the case of a riveted stiffener, this effect is concentrated along the centreline of the fasteners where load transfer between the stiffener and skin occurs. Considering the case of a simple rectangular stiffener cross-section, variation in EA resulting from changes in material stiffness, E, or cross-sectional area, A, should produce similar results for a riveted stiffener as long as it remains connected to the skin panel through a single row of fasteners. The same is not true for a bonded stiffener, where load transfer occurs along the entire bondline width; thus variations in EA resulting from changes in stiffener width will have an appreciable difference to those cause by changes in stiffener thickness or material stiffness.

Fig. 8. Influence of stiffener area on Stress Intensity Factor for a panel containing two bonded titanium stiffening elements with (a) constant width and (b) constant thickness.

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25

To illustrate this effect, predictions for a skin panel with pair of symmetric stiffeners of varying EA are presented. For simplicity of presentation, a similar configuration as used in validation case 2 has been used, utilizing the same skin and adhesive properties as given in Table 1. The stiffener pair uses the same pitch, material stiffness, and disbond length as the titanium tear straps given in Table 1, while the stiffener area is altered by varying either thickness or width. The influence of varying stiffener area on b is shown in Fig. 8 while the influence on total stiffener load, given as a multiple of the far field stiffener load, is shown in Fig. 9. The results in Fig. 8 strongly indicate a more favorable improvement in crack growth performance resulting from a change in stiffener area by increase in stiffener width (subplot (b)) compared to an increase in stiffener thickness (subplot (a)). The wider stiffener increases the range of crack lengths over which the crack is bridged by the stiffener. Thus, crack bridging occurs earlier in the crack propagation life of the panel and is sustained for a longer portion of the crack propagation life. The effects of the earlier bridging on the stiffener can be observed in Fig. 9 where increases in stiffener load occur earlier than in narrow stiffeners. It should be noted that the stiffener load concentration factors in Fig. 9 are based on overall load, and not stress:

Ls ¼

Pst þ F intact þ F bridging P st

ð21Þ

Where Pst is the far field stiffener load, and Fintact and Fbridging are redistribution of loads into the stiffener as illustrated in Fig. 1. The trend of lower Ls magnitudes for stiffeners with larger areas (and consequently larger stiffness, EA) is a result of the load distribution in the panel in the undamaged state. For larger stiffener EA values, the stiffeners will carry a larger percentage of the load relative to the skin (larger Pst), reducing the amount of load transfer from the skin to the stiffener in the presence of a crack (smaller Fintact and Fbridging). The plotted results must be interpreted understanding the assumptions and limitations in the present analysis. The first assumption made is that stiffener failure does not occur. Failure of a stiffener will exacerbate crack growth as it will introduce load into the cracked panel rather than bridge it. For the cases simulated, when the stiffener area is small, large load concentration factors are observed which may lead to static failure of the stiffener. For stiffeners with large widths, increases in stiffener load occur earlier as smaller crack lengths, increasing the potential for stiffener fatigue failure. Stiffener failure criterion may be introduced into the present analysis methodology; however, this lies beyond the scope of the current paper which focuses on the stress intensity factor solutions and their applicability to bonded stiffened panels. The second assumption made is that the disbond has a constant length of 5 mm and does not grow. The effects of this assumption will be expanded on in the next section. 4.2. Influence of disbond length The effectiveness of a stiffener in bridging load is directly related to the size of the interface disbond between the stiffener and cracked skin panel. The relevance of disbond shape, size, and growth on load bridging is thoroughly discussed in the

Fig. 9. Influence of stiffener area on Stiffener Load Concentration Factor for a panel containing two bonded titanium stiffening elements with (a) constant width and (b) constant thickness.

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C. Rans et al. / Engineering Fracture Mechanics 97 (2013) 12–29

context of Fibre Metal Laminates by Alderliesten et al. [15,16,21]. In order to facilitate the discussion of the influence of disbonds in the context of the presented methodology, predictions for the same panel configuration given in the previous section and a stiffener area of 20 mm2 have been made for various fixed disbond lengths. The resulting influence of disbond length on b is plotted in Fig. 10, while the influence on total stiffener load is given in Fig. 11. The results in Fig. 10 indicate that a larger interface disbond reduces the benefit of the bridging stiffener. The length of the disbond dictates the length of the stiffener that must conform to the crack opening displacement. As a result, a longer disbond implies a lower required bridging strain in the stiffener. This is illustrated by the reduction in stiffener load with disbond length shown in Fig. 11. This behaviour is important to consider as it places a limit on the improvement to crack growth behaviour with increasing stiffener width shown in Fig. 8b. Increasing the width of a bridging stiffener will indeed increase the range of crack lengths where the benefit of bridging is experienced. This in turn will also increase the range of crack lengths over which the bonded interface is fatigued by the bridging load, increasing the likelihood for disbond propagation

Fig. 10. Influence of stiffener disbond length on Stress Intensity Factor for a panel containing two bonded titanium stiffening elements.

Fig. 11. Influence of stiffener disbond length on Stiffener Load Concentration Factor for a panel containing two bonded titanium stiffening elements.

C. Rans et al. / Engineering Fracture Mechanics 97 (2013) 12–29

27

and reduction in the beneficial bridging load. The precise interplay between disbond growth and its influence on crack growth will be dependent on the fatigue resistance of the materials and adhesives used in the stiffened panels, in addition to the geometries and stiffness considered here, which is beyond the scope of this paper. The family of curves in Figs. 10 and 11, however, provide a clear illustration of the sensitivity of the bridging mechanism to disbond size/growth in the context of assessing the damage tolerance of bonded stiffened panels. 5. Conclusions A set of stress intensity factor solutions related to potential states of bonded stiffening elements in a typical fuselage skin panel have been presented. Implementation of the stress intensity factor solutions for a panel configuration requires the determination of the load distribution due to crack bridging of the stiffeners, which is solved by enforcing displacement compatibility between the cracked skin panel and bonded stiffener. Although this approach is similar to that adopted by Poe [3], Swift [4], and Vlieger [6,7], it provides several improvements including removing the need to discretize individual stiffeners into various elements, and accounting for the bridging effect of the stiffener across its entire width rather than concentrated at the stiffener centreline. The accuracy of the presented stress intensity factor solutions was validated by comparing to experimental studies of stiffened panel crack growth available in the literature. The versatility of the stress intensity factor solutions for studying the influence of stiffener geometry and of disbond state has been demonstrated. The solutions can account for various stiffener damage states and can handle different disbond sizes, allowing their influence on panel damage tolerance to be quickly assessed. The bridging load determined within the presented stress intensity factor solutions can be used as an input for prediction of disbond growth and stiffener failure criterion; however, modelling of these specific phenomena are beyond the scope of this paper. Acknowledgments The work presented in this paper was sponsored by the Netherlands Organization for Scientific Research (NWO) and Technical Science Foundation (STW) under the Innovational Research Incentives Scheme Veni Grant. Appendix A The Westergaard Stress Functions for the four-point loaded crack illustrated in Fig. 3 are given by:

0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 20qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 2 a2  z20 a2  z20 P 4@ a  z0 i  z B 0 A  ay0 @ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ i  2z0 ZI ¼ þ 2 2 2 2 p z  z0 z  z0 ðz2  z20 Þ2 ðz2  z2 Þ a2  z2 0

0

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 13 a2  z20 i  z0 1 A5 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i  2z0 þ 2 2 2  ðz  z0 Þ ðz2  z20 Þ a2  z20 1  ða=zÞ2 " !# P y0 y0 ðz  x0 Þ2  y20 ðz þ x0 Þ2  y20 þ  þ a y þ 0 p ðz  x0 Þ2 þ y20 ðz þ x0 Þ2 þ y20 ððz  x0 Þ2 þ y20 Þ2 ððz þ x0 Þ2 þ y20 Þ2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 6 1 z2  a2 z2  a2 1 Z I ¼ 4tan qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ tan qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 2 2 a  z0 a2  z20 0 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    

z2  a2 z2  a2 C i  z0 z z B i  z0 1 z 1 z ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q q þ  i  tan h  a y þ  i  tan h  ay0 @ 2 A 0 z0 z0 z  z20 a2  z2 z2  z20 a2  z2 z20  z2 z20  z2 0

ð22Þ

ð23Þ

0

where P is the load per unit thickness, and z; z0 ; andz0 are complex numbers representing the xy-coordinates of the location of evaluation of the stress functions and the locations of the applied point loads (see Fig. 3):

z¼xþiy z0 ¼ x0 þ i  y0 ¼ xc þ i  b z0 ¼ x0  i  y0 ¼ xc  i  b

ð24Þ

In applying Eqs. (22)–(24) for the case of a broken stiffener in Eq. (13), the following substitutions are needed:

z¼xþib P st;broken P¼ t skin

ð25Þ

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Conversely, in applying Eqs. (22)–(24) for the case of a bridging stiffener in Eq. (14), the following substitutions are needed:

z¼xþib Pst;bridging þ F bridging P¼ t skin

ð26Þ

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