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invoking technical community comment on the results and conclusions of the research.

Reliability of Damage Tolerant Composite Structures Using

Fasteners as Disbond Arrest Mechanism

Chi Ho E. Cheung1 and Kuen Y. Lin

2

University of Washington, Seattle, WA, 98195-2400

A FEA model for analyzing effectiveness of fastener as crack arrest mechanism has been

established. The effect of the fastener is modeled using fastener flexibility approach

consisting of linear springs. It is shown that the fastener provide significant crack

retardation capability in Mode I, but has limited resistance for Mode II propagation. The

development of analytical solution for the crack arrest problem is important to the design of

bonded/co-cured/co-bonded composite aircraft structures.

A procedure for assessing the reliability of the fastener arrest mechanism has been

demonstrated. A probabilistic approach is used because traditional damage tolerance

methods for metallic structures are not generally applicable to composite structures.

Probabilistic method can provide a more quantitative evaluation of reliability and safety of a

structure.

Analytical method for analyzing the crack arrest problem is being developed. The

method consists of two parts: 1) crack-tip stress analysis; and, 2) nonlinear split-beam

analysis with fastener. The analytical method makes possible accurate and efficient solution

of crack arrest problems and is useful for structural design and optimization.

I. Introduction

HE use of composites in aircraft has enabled the use of bonded (or co-cured, co-bonded) structures, the main

advantages of which are reduction of part counts and weight. The critical damage mode in this type of structure

is disbond due to impact damage. Complete disbonding of components (e.g. skin-stringer) can cause failure at

the structural level. Therefore, any bonded structures must demonstrate fail-safety by providing adequate disbond

arrest capability to ensure safety.

In addition, due to the probabilistic nature of impact damage occurrence, traditional damage tolerance

methodology for metals does not apply and thus is unable to provide meaningful reliability estimate for the

structure. Damage tolerance methodology for metallic structures is based on the assumption that an initial defect or

crack pre-exists somewhere in a structures and this defect propagates stably at a known rate under fatigue loading.

Proper inspection and maintenance schemes are established to detect and repair the damaged structure before the

crack grows to a critical size. However, impact damage and disbonds in composite structures are often caused by

discrete events randomly, rendering traditional deterministic damage tolerance approach ineffective. To overcome

this difficulty, a probabilistic approach to damage tolerance of composite structures has been developed by Lin and

Styuart, et al [1-3] by taking into account the life-cycle discrete events of damage occurrence, peak load occurrence,

as well as inspection and repair. The probabilistic method has been demonstrated to quantitatively assess the

structural reliability and maintenance scheduling of an aircraft composite structure.

This paper will apply the probabilistic approach to study damage tolerance of a practical problem in composites:

disbond arrestment in a fuselage skin-stringer structure by fasteners. The effectiveness of fasteners as disbond arrest

mechanism, a relatively understudied area, will be first analyzed using deterministic means. The reliability of the

structural component, fuselage skin-stringer assembly, will then be evaluated using the developed probabilistic

method. The procedure needed to evaluate the reliability of a damage tolerant composite structure will be

demonstrated.

1 Pre-Doctoral Research Assistant, Department of Aeronautics and Astronautics.

2 Professor, Department of Aeronautics and Astronautics.

T

II. Fastener Effectiveness as Disbond Arrest Mechanism

Bonded structures (co-cured, co-bonded or bonded) are common in composite structures, especially thin

structures, due to numerous advantages including weight, part count and assembly costs. The bond alone, which is

the primary load path, seldom possesses necessary geometric or mechanical arrest capability. This can be a difficult

problem when designing the structure to be damage tolerant. In aircraft structures, it is common to use fasteners on

geometrically complex locations (e.g. fuselage skin-frame shear tie). These fasteners are co-located with the skin-

stringer bond, and thus also perform as disbond arrest mechanism, without the added cost and complexity

alternatives such as z-pin and z-stitching. Disbond in Mode I is well understood [4] and typically less problematic

because any mechanism arrest feature would be so effective that the laminates will fail in other modes first, e.g.

bending, pull through. However, Mode II disbond failure with arrest mechanism is less understood. It is therefore

important to understand the effectiveness of these fasteners in arresting disbond to maximize their benefits and

ensure safety of the structure.

The load case, shown in Figure 1, represents the typical condition in which the fastener would perform as a

crack arrest mechanism. For example, the lower plate represents the fuselage skin while the upper plate represents a

stringer leg (Figure 2). A crack (disbond) exists at the edge of the skin-stringer bond. The over-hanging portion of

the stringer leg is free from any load, while the skin is loaded with general axial tension (N) and moment (M) loads.

As the crack advances to the fastener location, the fastener would retard or arrest the growth of the crack. A proper

design should comprise of a fastener capable of retarding or arresting the crack up to limit load or until other failure

modes occur, such as bending, bearing, fastener pull-through, etc. Therefore, failure would be defined as the

continual advancement of the crack below the critical loads of the other failure modes. The critical loads of the other

failure modes and how they interact with the effectiveness of the fastener is beyond the scope of this paper.

Figure 1. Typical Load Case of Fastener as Disbond Arrest Mechanism

Figure 2. Schematic of Damaged Fuselage Skin-Stringer with Fastener

A. Structural Properties The structure used in this paper comprises of a skin and stringer of identical laminate. A 16-ply (45/02/-

45/02/902)s (50% 0 degree plies) laminate is used. Ply thickness is t = 0.1905mm (0.0075in); laminate thickness is

3.048mm (0.12in). AS4/3501-6 material properties used are: E1 = 127.5GPa, E2 = 11.3GPa, G12 = 6.0GPa, ν = 0.3,

Xt = 2282MPa, Xc = 1440MPa, Yt = 57MPa, Yc = 228MPa, S = 71MPa (E1 = 18.5×106psi, E2 = 1.64×10

6 psi, G12

= 0.871×106 psi, ν = 0.3, Xt = 331×10

3psi, Xc = 208.9×10

3psi, Yt = 8.3×10

3psi, Yc = 33.1×10

3psi, S = 10.3×10

3psi)

[6,7]. Fracture properties used are: GIC = 0.2627N/mm, GIIC = 1.226N/mm (GIC = 1.5 lb/in, GIIC = 7 lb/in) and

fracture criterion BK law (1) with mixed-mode parameter η = 1.75 [7]. Titanium Ti-Al6-V4 fastener with d =

6.35mm (0.25in) and E = 114GPa (16.5×106 psi) is used.

( ) IIequivC IC IIC IC

I II

GG G G G

G G

η

= + − +

(1)

The effect of fastener is modeled using fastener flexibility approach (H. Huth [5]). The equation for compliance

of the fastener in un-bonded bolted joints in the sliding direction (Mode II) was obtained from empirical data as

equation (2). The parameters used are: ti = laminate thickness, d = fastener diameter, n = number of fasteners (n = 1),

E1/2 = laminate stiffness, E3 = fastener stiffness, constants a = 2/3 and b = 4.2 for bolted graphite/epoxy joints. For

the structure studied in this paper, C = 33.4mm/N (5.85×10-6

in/lb). In the opening direction (Mode I), a stiffness of

kI = E3×Area = 141.8×103 N/mm (809940 lb/in) is used.

1 2

1 1 2 2 1 3 2 3

1 1 1 1

2 2

at t b

Cd n t E nt E nt E nt E

+ = + + +

(2)

B. FEA Model The skin-stringer structure is modeled in 2-D in commercial FEA software ABAQUS (Figure 3). The model is

identical to a double cantilever beam (DCB) except that only the lower beam (skin, 16ply) is loaded while the upper

beam (stringer, 16ply) is free. The disbond (crack) is at the interface between the skin and stringer within the matrix

material. The beams are L = 101.6mm (4in) long. Initial crack length is ao = 60.96mm (2.4in) or a/L = 0.6. The

springs representing the fastener are located at lfast = 63.5mm (2.5in). Plane strain quadrilateral elements reduced

integration with hourglass control is used. Each ply is modeled with one element through the thickness; element

length is 0.254mm (0.01in) in the longitudinal direction. Load is applied at the lower beam at the mid-plane.

Figure 3. FEA Model of Skin-Stringer (upper); Crack Tip and Fastener as Spring (lower)

Crack propagation is modeled using Virtual Crack Closure Technique elements in ABAQUS [8]. Strain energy

release rate for each mode is calculated separately using the nodal forces at the crack tip and displacement behind

the crack tip for the corresponding mode. The crack propagates to the next node when the mixed-mode fracture

criterion (1) is met.

The fastener is modeled with two separate springs acting in two independent directions. In the opening direction

or Mode I, a spring acting only in y-direction with stiffness kI = E3×Area/(t1+t2) is used. In the sliding direction or

Mode II, a spring acting only in x-direction with stiffness kII = 1/C (C is joint compliance/fastener flexibility) is

used. For sensitivity with respect to fastener flexibility, five kII values are used for each load case: kII =

0.02994N/mm; kII+5%; kII+12.5%; kII-5%; and kII-12.5%. The fastener flexibility calculations assume width of

25.4mm (1.00in), or fastener spacing of 4×fastener diameter. No failure points for the springs are defined.

C. Load Conditions For demonstration purpose, only laminate failure using Tsai-Hill first ply failure criterion (3) is used to generate

a failure envelope as a reference for the current study. However, Tsai-Hill criterion does not imply destruction of the

laminate; the structure can generally be further loaded after first ply failure. Other failure modes such as bearing or

pull-through should be considered in general. Also only tensile axial loads and opening bending moments are used

to avoid buckling and contact friction considerations in the fracture analysis. The material properties and laminate

lay-up produced a failure envelop shown in Figure 4. Four load cases shown in Figure 4 are analyzed to generate a

crude respond surface for fastener effectiveness: A) N only; B) M only; C) N:M = 24.2:1; and D) N:M = 110.3:1.

2 2 2

11 11 22 22 12

2 2 2 21

X X Y S

σ σ σ σ σ− + + = (3)

Figure 4. Failure Envelop of Laminate using Tsai-Hill Criterion (normalized to width of 25.4mm)

D. Results and Discussions

Figures 5 to 8 show the results of the crack propagation analyses. Results for where there is no fastener are

plotted for comparison. The normalized equivalent load (P/Pc) is plotted on the x-axis, where P is for a certain

combination of N and M (constant mix ratio) and Pc is the critical load calculated with Tsai-Hill criterion. The

normalized crack length (a/L) is plotted on the y-axis, where ao = 0.6 and lfast = 0.625. If the crack length vs. load

curve is vertical, crack propagation is unstable; if the curve is horizontal, crack propagation is completely arrested.

The functional goal of the fastener is to arrest the crack, which is to make the curve effectively flat. Instead, there

exists a region where the crack is significantly retarded, after which the crack begins to propagate rapidly and the

fastener has “failed” its function. There does not exist a standard way to define the end of the retardation region

(which marks the failure/critical point). For practical purpose, the point where the curve becomes vertical is defined

as the failure point.

Figures 5 to 8 shows that the fastener as crack arrest mechanism will successfully arrest the crack up to critical

load (P/Pc = 1) defined by Tsai-Hill for all load cases considered. In contrast, if the fastener had been absent, the

crack would grow unperturbed below Pc causing higher level structural failure (for load case C, crack will not grow

even without fastener). This also means that it is necessary to have crack arrest mechanism for this type of structure,

and that the fastener design is effective. It should be noted that 1) Tsai-Hill criterion only provides first ply failure

prediction, in general, much higher loads are required to cause final fracture of the laminate; and 2) after first ply

failure, P/Pc = 1, the behavior of the laminate would change and the subsequent behavior predicted by FEA becomes

inaccurate.

For load cases where N is non-zero (Figure 5, 7 and 8), the presence of the fastener has an effect on the crack

initiation load. However, at initiation, the crack tip has not yet reached the fastener and should not have noticeable

effect until the crack reaches the fastener at a/L = 0.625. This could be due to the rotation cause by axial load being

applied off from the crack interface. Since the Mode II spring respond to relative displacement along the x-axis,

significant rotation would induce sufficient displacement to load the spring before the crack tip reaches the spring

itself. Further investigation is needed to understand this discrepancy; enhancement in modeling technique may be

needed to rectify this problem in the future.

Focusing on the effectiveness of the fastener as crack arrest mechanism, temporarily ignoring the first ply

failure predicted by Tsai-Hill criterion, the fastener provides significant crack retardation capability in all load cases.

At the failure point of the crack arrest mechanism, for the loads required to reach the same crack lengths are

increased by 40%, 143%, 42% and 41% for load cases A, B, C and D respectively (horizontal distance between the

reference curve and fastener curves at failure).

The sensitivity of results with respect to variation of Mode II spring stiffness, kII, is low. The variation of arrest

mechanism failure load is at least 7 times smaller than the variation of kII. That is, for every 7% change in kII, the

failure load only changes 1%. This suggests that the crack arrest feature is robust against variations in parameters

that affect kII or fastener flexibility. The sensitivity in load case B, applied moment only, is much smaller than the

other load cases.

Normalized Load vs. Crack Length (N only)

0.6

0.65

0.7

0.75

0.8

0.85

0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

Load (P/Pc)

Cra

ck

Le

ng

th (

a/L

)

KII = 149526

KII = 162343

KII = 170887

KII = 179431

KII = 192248

no fastener

Figure 5. Load vs. Crack Length w/ and w/o Fastener – for A) N only

Normalized Load vs. Crack Length (M only)

0.6

0.65

0.7

0.75

0.8

0.85

0.4 0.6 0.8 1 1.2 1.4

Load (P/Pc)

Cra

ck

Le

ng

th (

a/L

)

KII = 149526

KII = 162343

KII = 170887

KII = 179431

KII = 192248

no fastener

Figure 6. Load vs. Crack Length w/ and w/o Fastener – for B) M only

Normalized Load vs. Crack Length (N:M=24.2:1)

0.6

0.65

0.7

0.75

0.8

0.85

1.5 2 2.5 3 3.5

Load (P/Pc)

Cra

ck

Le

ng

th (

a/L

)

KII = 149526

KII = 162343

KII = 170887

KII = 179431

KII = 192248

no fastener

Figure 7. Load vs. Crack Length w/ and w/o Fastener – for C) N:M = 24.2:1

Normalized Load vs. Crack Length (N:M=110.3:1)

0.6

0.65

0.7

0.75

0.8

0.85

0.8 0.9 1 1.1 1.2 1.3 1.4

Load (P/Pc)

Cra

ck

Le

ng

th (

a/L

)

KII = 149526

KII = 162343

KII = 170887

KII = 179431

KII = 192248

no fastener

Figure 8. Load vs. Crack Length w/ and w/o Fastener – for D) N:M = 110.3:1

A crude failure envelope is constructed for the fastener arrest mechanism using various load combinations

shown as pink squares in Figure 9. The Tsai-Hill failure envelope is shown as thick blue curve for reference. It is

immediately noticeable that envelope can be decomposed into two parts: one primarily driven by applied moment

(red curve to the left) and another driven by tensile loading (blue curve to the right). The presence of the fastener

restricts the Mode I opening of the crack, effectively eliminates propagation by Mode I fracture in all load

combinations. While all crack propagation beyond the fastener is driven by Mode II fracture, the mechanics through

which the applied moment and tensile load achieve Mode II fracture at the crack tip are different, hence the

partitioned failure envelope. While tensile loads cause Mode II fracture by displacing the lower laminate and leaving

the upper laminate unloaded, applied moments cause Mode II fracture through transverse shear developed from

bending. Another noticeable feature of the failure envelope is that addition of tensile load to a primarily moment

loaded case actually enhances the effective strength of the arrest mechanism, while adding moment to a primarily

tension loaded case has very little effect on the tensile strength. This behavior is different from failure envelopes

developed using traditional strength of material approach (e.g. distortional energy). Also, since no failure load for

the springs are defined, it is mathematically possible to have crack propagation at unrealistically high loads.

Figure 9. Failure Envelope for Fastener Crack Arrest Mechanism (normalized to width of 25.4mm)

III. Reliability of Damage Tolerant Composite Structure Probabilistic Problem

The probabilistic analysis aims to demonstrate the methodology used to evaluate the reliability of a damage

tolerant structure, with proper inspection and repair program. The skin-stringer disbond problem is suitable for such

approach because the initial crack, or disbond, is dictated by impact damage occurrence, which is a probabilistic

event. It should be cautioned that the accuracy of any probabilistic method is highly dependent on the accuracy and

applicability of the data, and ability for the analytical model to capture real world physics. Engineering judgment

and extensive considerations must be made with every assumption taken, and the interpretation of results.

A. Probabilistic Description of Problem The probabilistic method is based on Monte Carlo simulation of structure lives, with distributions for variables

such as initial strength, damage time, damage size, loads, etc. The following input data are used as the input to the

RELACS software in [2]:

Design Cases: Subsonic flight in turbulent atmosphere

Damage Types: Skin-stringer disbond

Inspection Types: Visual (Method 1) and Instrumental (Method 2)

Repair Types: Field (Method 1) and Facility (Method 2)

It was assumed that the temperature and aging do not affect the residual strength; it was assumed that repairs are

capable of completely restoring original strength.

The equation for the cumulative frequency of thunderstorm gust load occurrence per life may be expressed as

[9]:

12.32 11.8( ) 22000 78000Z Z

ZF z e e

− −= + (4)

where Z = gust load / RDL. This equation has been obtained from the equation used in [9] assuming a life of 50000

hours, which corresponds to a safety factor 1.5 and limit load exceedance frequency of 0.5 times per life. The

probability that the load exceeds the strength per life (probability of failure) is:

1 exp[ ( )]f Z

P F z= − − (5)

Damage Size Exceedance Curve

0.1000

1.0000

10.0000

0.0 20.0 40.0 60.0 80.0

Damage Size, mm

Exceed

an

ces p

er

life

Disbond

Figure 10. Damage Exceedance Data

The exceedance data of damage occurrence shown in Figure 10 is taken from an FAA report [10] and

recalculated for 50000 flight hours and typical fin panel area. It should be noted that the damage statistics for

fuselage panel would be different from that of the vertical fin. The vertical fin data is used only for demonstration of

procedures.

Two types of inspection are assumed: pre-flight (Field) type and regular maintenance (Facility) type. Field

inspection is made every flight; facility inspection is made every 1500 flights. The Log-Odds (6) Probability of

damage detection (POD) used by Lin et al [1] is used, where a is the damage size. Parameters α and β are shown in

Table I. The POD for visual delamination detection assumes that first the indications of damage are found visually

and then tap hammer method is applied.

ln( )

ln( )( )

1

a

a

ePOD a

e

α β

α β

+

+=

+ (6)

Table I – Log-Odds Parameters for POD for Different Inspection Methods

Inspection Method αααα β

Visual (Field) -4.05 2.86

NDI (Facility) -0.55 2.86

Failure loads for the fastener arrest mechanism shown in Figure 9 for each load cases are used. These do NOT

represent the failure loads for the structure as other failure modes are completely ignored. The fastener arrest

mechanism is the sole consideration.

B. Results

Table II – Reliability of the Fastener Arrest Mechanism for Individual Load Cases

Load Case Failure Load (×Tsai-Hill criterion) POF (per life)

A 1.190 2.53×10-4

B 1.208 1.88×10-4

C 2.751 3.03×10-15

D 1.322 2.92×10-5

The probabilities of failure (POF) of the fastener arrest mechanism for each of the 4 load cases are summarized

in Table II. Assuming that the structure has only 4 load cases as above, neglecting all other failure modes, the

combined POF of the fastener arrest mechanism would be 4.70×10-4

per life (or 9.4×10-9

per flight hour) calculated

by equation (7). This value, given the limited considerations, is appropriately close to the standard POF of 1×10-9

per

flight hour for commercial aircraft. However, the failure of the fastener arrest mechanism generally would not cause

catastrophic failure of the structure or aircraft. Therefore, the above results represent a sufficiently safe design.

( )_

1 1n

ii load casePOF POF

== − −∏ (7)

In general, all load cases (typically thousands of load cases) shall be considered to evaluate the overall

reliability. A structural response surface (design curves) that encompasses all the load cases needs to be constructed.

The response surface only needs to be constructed once. All load cases can be analyzed against the response surface

using the above approach for the overall reliability of the structure. The fact that there could be thousands of load

cases does not necessarily increase the overall POF significantly. This is because the overall POF is dominated by

the largest POF components (e.g. load cases A and B); the other POF components (e.g. load cases C and D) make

virtually no impact on the overall POF regardless of how many load cases are there.

IV. Development of Analytical Method

The analytical method needed to analyze the 1-D split-beam delamination with fastener is composed of two

parts: 1) crack tip fracture analysis that calculates strain energy release rates of the delamination tip; and 2)

nonlinear beam analysis that calculates the force equilibrium of the split-beam system with the arrest mechanism.

A. Crack Tip Fracture Analysis

Since delaminations in composites are self-similar, it is important not only to calculate the total strain energy

release rate, but also to determine the individual components of the strain energy release rates. In the 1-D case, both

Mode I SERR, GI, and Mode II SERR, GII, must be determined in order to predict fracture. Wang and Qiao [11-13],

proposed an elegant closed-form solution to the mode decomposed strain energy release rates of a shear deformable

bi-material split-beam interface. Wang and Qiao’s solution provides the added advantages over the analytical

solution proposed by Davidson et al. [14] in that deformability is included and that mode decomposition does not

require an additional mode mix parameter (Ω) which is obtained via FEA or experiments. Figure 11 shows the

schematic of the crack tip element. Important equations and solution are summarized in the following.

Figure 11. Bi-layer Beam System under Generic Loadings

The axial, shear and bending stiffness coefficients of the beam under plane strain condition are given in (8),

where κ is the shear correction coefficient (5/6 for rectangular section).

( )

3

, ,1 12 1

xi i xi ii i xzi i i

xzi zxi xzi zxi

E h E hC B G h Dκ

ν ν ν ν= = =

− − (8)

Global equilibrium conditions yield the following.

( ) ( )

( )

1 2 10 20

1 2 10 20

1 2 1 21 2 1 10 20 10

2 2

T

T

T T

N N N N N

Q x Q x Q Q Q

h h h hM M N M M N Q x M x

+ = + =

+ = + =

+ ++ + = + + + =

(9)

From conventional beam theory, the resultant forces in beam 1 are given by (10).

( )( ) ( )

( )( ) ( )

1

1 21

2

21

2 2

1 2 2 1 2

2 1 2 1 2 1 2

1 2

2 1 2 1 2 1

1

2 2

1 1

2

2

2

2

C M T N T

C M T

C M T N T

M

N

N A M A N

h hQ A Q

D

hM A M A N

D C

D D h D DA

D D D D D h h

D DA

C D D h h D

η

ξ ξ

η η

ξ ξ ξ ξ

ξ

η ξ

η ξ

= +

= + −

= − + −

+ +=

+ + +

+=

+ + +

(10)

( )

1 2

1 2

1 2 2

1 2 2

2 2

1 1

4

h h

D D

h h h

C C D

ξ

η

= +

+= + +

The three concentrated crack tip forces required at the crack tip by equilibrium conditions are given by (11).

( )

( )10 1 10 1

10 1

, 0

0

= − = − =

= − =

C C

C

N N N Q Q Q x

M M M x (11)

For the shear deformable bi-layer beam model, the restraint on the rotation at the crack tip is released. As a

result, the concentrated bending moment is eliminated.

( )

( ) ( )( )( )

( )

1

1

1 2 1 2 1 1 2

1 2 1 2 1

1

1

2,

2 2

2

2

2

2

− = = − − +

+

+ + + =+ +

+=

+

C C

M N h NN Q Q k M

h

B B D D D h hk

D D B B h

M h Nc

h

ξ η

ξ η

η ξ

η ξ

ξ

ξ η

(12)

The strain energy release rates for Mode I and II under effective loadings M, N, Q in (11) are explicitly obtained

as (13).

( )

2

1

1 2

2

1

1 1 1

2 2

1

2

I

II

h NG Q k M

B B

G M Nh

ξ ηξ µ

= + + +

= −+

(13)

The crack tip fracture analysis is compared to conventional beam theory in the cases of DCB (with transverse

shear) and ENF (14) specimen. The results are shown in Figure 12 and 13. The material and geometric properties

used are: E = 10.99×106, d = 0.1, b = 1.0, GC = 3.0. For both cases, it is demonstrated that beam theory over-predicts

the load bearing capabilities of the specimen at short crack length; beam theory under-predicts the strain energy

release rates at any given load. This is particularly important because accuracy at short crack length is needed to

predict the effectiveness of the arrest mechanism; the effective crack length is extremely short as the crack tip passes

and leaves the arrest mechanism.

2 2 2

, 2 3 2

2 2

, 2 3

121

3

9

16

CIC DCB

CIIC ENF

P a dG

Eb d a

P aG

Eb d

= +

=

(14)

P vs crack length (DCB)

0

50

100

150

200

250

300

350

0 0.5 1 1.5 2 2.5 3

Crack Length

P

Beam Theory

Qiao

Figure 12. Crack Tip Fracture Analysis (Qiao) vs. Beam Theory – DCB

P vs crack length (ENF)

0

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2 2.5 3

Crack Length

P

Beam Theory

Qiao

Figure 13. Crack Tip Fracture Analysis (Qiao) vs. Beam Theory - ENF

B. Nonlinear Beam Analysis – Split-Beam with Fastener

Geometrically nonlinear beam is used to model the split-beam and the interaction with the arrest mechanism,

due to the relatively large deflection at near failure loads. The solution by Awtar et al. [15] is used here. Equation set

(15) summarizes the solution to the beam’s end deflection for beams loaded in tension (an analogous solution for

compression is available with trigonometric terms). The variable k = p1/2

, where p is the normalized axial load, is

real for tension and imaginary for compression.

Figure 14. Schematic of Geometrically Nonlinear Beam

( ) ( )( )

( )( )

( )( )

( ) ( )( ) ( )( )( ) ( )( )

( )( ) ( ) ( )( ) ( )( )

3 2 211 12

12 22

2

2 2

11 2

2

12

tanh cosh 1

cosh,

12cosh 1 tanh

cosh

cosh cosh 2 3 sinh 1

2 sinh 2cosh 2

cosh 1 sinh cosh 1 4 cosh 1

− −

= = − −

+ − − −=

− +

− − + − − −= −

y y

x y

k k k

k k k r rf pt

r rmk k

k k k

k k k k kr

k k k

k k k k k kr

δ δδ δ θ

θ θ

( ) ( )( )

( ) ( )( ) ( ) ( )( ) ( ) ( )( )( ) ( )( )

2

2

3 2 2

22 2

4 sinh 2cosh 2

sinh cosh 2 2 cosh cosh 1 2sinh cosh 1

4 sinh 2cosh 2

− +

− + + − − − + −=

− +

k k k k

k k k k k k k k kr

k k k k

(15)

The solution to the end displacement parameters, δx, δy and θ, can then be used to solve for the force and

moment equilibriums with the interactions between the arrest mechanism, as shown in Figure 15. The set of force

and moment equilibrium equations for the whole system are:

( )

1 21 2

1 2

1 21 2

2

2

s s x x

c c y y

beam beam s

t tF k

F k

t tM M M F

θ δ δ

δ δ

+ ≅ − + −

= −

+= + +

(16)

Figure 15. Schematic of a Split-Beam with Fastener System

The interactions between the beams and the fastener can be summarized as follows. The expressions for M1 and

M2 assume that beams 1 and 2 are identical.

1 1

2 2

1 21

1 22

,

4

4

= = −

= − =

+= −

+ = −

s c

s c

s

s

P F F F

P P F F F

t tM F

t tM M F

(17)

An assumption is made such that θ1 = θ2. This assumption is made primarily to make the deformed shapes of the

two beams consistent, so as to avoid significant penetration along the length of the beam. This is a realistic

assumption given the constraints provided by the fastener and the materials on both sides of the fastener.

The system of equations (15-17) is nonlinear; as a result, solution has to be found iteratively. It should be noted

that the hyperbolic and trigonometric terms can be a significant difficulty due to the sensitivity to initial guess.

These functions may have to be approximated by polynomials for better robustness in the results from the nonlinear

system.

V. Conclusion

Analysis of effectiveness of fastener as disbond arrest mechanism in bonded composite structure has been

demonstrated. Using a realistic skin-stringer bonded structure, it was shown that the presence of the fastener is

highly effective in retarding the propagation of a crack in Mode I for all load combinations (N and M) considered.

The failure load of the arrest mechanism is at least 40% higher than the case where no fastener was present. It was

also noticed that the axial portion (N) of the failure load stays constant even with increasing applied moment (M).

This is primarily due to the fact that axial load and moment contribute to two separate fracture modes, Mode I and

Mode II. The details of which may become the subject of investigation in future studies.

The procedures and considerations used to apply the above results to a reliability analysis were also discussed

and demonstrated. It can be used to verify design margins and evaluate reliability of composite bonded structure

using fastener as disbond arrest mechanism. Cautions when using probabilistic methods, such as the consideration of

multiple failure modes and applicability of data, are discussed.

An analytical method to analyze the effectiveness of fastener or other arrest mechanism is being developed.

However, the method has yet to reach maturity. Future work will focus on further developing, refining and

implementing the analytical solution for the effectiveness of fastener crack arrest mechanism.

Acknowledgments

This work was supported by the Federal Aviation Administration (FAA) Research Grant, “Development of

Reliability-Based Damage Tolerant Structural Design Methodology”. Curtis Davies and Larry Ilcewicz were the

FAA grant monitors. The authors wish to thank the FAA Center of Excellence at the University of Washington

(AMTAS) for sponsoring the current research project. The Boeing Company also supported the current work.

Special thanks are given to Gerald Mabson and Eric Cregger of Boeing for their technical advice and guidance.

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