effect of overload on fatigue crack growth behavior of air frame structure

Effect of overload on fatigue crack growth behavior of air frame structure (fuselage)

Effect of overload on fatigue crack growth behavior of air frame

structure

GUIDEDr. P.K. DASH

BANGALORE AIRCRAFT INDUSTRY LTD.

PRESENTED BY MR. SHISHIR SHETTYUSN NO: 3KB08AE013

PROJECT CARRIED OUT ATBANGALORE AIRCRAFT INDUSTRY (PVT) Ltd.

Abstract• Catastrophic structural failures in many engineering fields

like aircraft, automobile and ships are primarily due to fatigue, where any structure experiences fluctuating loading during service. Its load carrying capacity decreases due to a process known as fatigue. Fatigue damage accumulates during every cycle of loading

• Designing an airframe against fatigue failure under the above assumption requires the “damage tolerance design concept”. In this design concept, a structure is made to tolerate the presence of damage. In other words, presence of a fatigue crack, the airframe retains a certain specified load carrying capability. This load carrying capacity is specified by certifying authorities and is normally taken as the design limit load.

Abstract continued… • Airframe will experience the variable loading during the

service. If a damage is present in the structure in the form of a crack (or one assumes a small crack present in the structure in the damage tolerance design process), then one needs to calculate the fatigue crack growth life. This is essential to properly schedule the inspection intervals to ensure the safety of the structure during its service.

Abstract continued…• In the current project work a segment of fuselage is

considered for the analysis. • Local analysis is carried out at the location of maximum

tensile stress to initiate a crack at the critical location. • Pressurization of the fuselage is one of the critical load cases

considered in the design process.• In the current project work internal pressurization is

considered for the analysis. • The overload in the load spectrum will affect the crack

growth rate in the material. The crack growth rate before and after an over load is calculated.

• Finite element analysis approach is used for the stress analysis.

Problem definition

Effect of overload on fatigue crack growth behavior of a air frame structure considering the segment of fuselage.

Objective

• Stress analysis of segment of fuselage.• Identifying maximum stress location in the fuselage

segment.• Local analysis of stiffened panel at the highest stress

location.• Crack growth calculation in the stiffened panel.• Study of overload on the fatigue crack growth.

Introduction to aircraft structure

• An aircraft is a complex structure, but a very efficient man-made flying machine.

• Aircrafts are generally built-up from the basic components of wings, fuselage, tail

Fuselage• The main body structure is the fuselage to

which all other components are attached. The fuselage contains the cockpit,passenger compartment and cargo compartment.

• The fuselage structure consisting of a thin shell stiffened by longitudinal axial elements (stringers and Longerons) supported by many traverse frames are rings (Bulkheads) along the length.

• The fuselage skin carries the shear stresses produced by torques and transverse forces. It also bears the hoop stresses produced by internal pressures.

Fuselage LoadsThe fuselage will experience a wide range of loads

from a number of sources. The weight of the fuselage structure and payload will cause the fuselage to bend downwards from its support at the wing, putting the top in tension and the bottom in compressionThe larger part of passenger and freighter aircraft is usually pressurized for safety. Internal pressure will generate large bending loads in fuselage frames. The structure in these areas must be reinforced to withstand these loads.

• The most common metals used in aircraft construction are aluminum, magnesium, titanium, steel, and their alloys.

• Traditional metallic materials used in aircraft structures are Aluminum, Titanium and steel alloys.

• In the past three decades applications of advanced fiber composites have rapidly gained momentum.

• To date, some modern military jet fighters already contain composite materials up to 50% of their structural weight.

Aircraft Materials

Selection of aircraft materials depends on initial material cost, manufacturing cost and maintenance cost and structural performance are

• Density (weight) • Stiffness (young’s modulus) • Strength (ultimate and yield strengths) • Durability (fatigue) • Damage tolerance (fracture toughness and crack

growth) • Corrosion

 Material

PropertiesE

GPa(msi)γ 

σt

MPa(ksi)

σy

MPa(ksi)

ρg/cm3

Aluminum2024-T37075-T6

 72(10.5)71(10.3)

 0.330.33

 449(65)538(78)

 324(47)490(71)

 2.78(0.1

0)2.78(0.1

0)TitaniumTi-6Al-4V

 110(16)

 0.31

 925(138)

 869(126)

 4.46(0.1

6)Steel

AISI4340300M

 200(29)200(29)

 0.320.32

 1790(260

)1860(270

)

 1483(21

2)1520(22

0)

 7.8(0.28)7.8(0.28)

Material properties of metals at room temperature of aircraft structure

σy = Tensile Yield strength σt = Tensile ultimate strength

INTRODUCTION TO FATIGUE CRACK GROWTH

If the airframe does not have any fatigue cracks, the load carrying capacity is the design ultimate load.

• During service this critical location must be regularly inspected so that the presence of a crack can be detected before it reaches the size acr. Repair or replacement action can be taken to remove the crack from the structure.

• In order to carry out safety this repair or replacement action the time taken for a crack to grow to its authorities Crack size must be established this is schematically shown in finger

• Hear ‘ai’ is the initial crack length and ‘acr’ is the critical crack length as obtained from figure ,‘H’ is the number of flight hours during which the crack will grow to its critical size.

• Determination of ‘H’ requires a fatigue crack growth analysis under service load spectrum.

• Such an analysis needs fatigue crack growth rate data property under constant amplitude loading in the form of and∆K curve as shown in figure.

• where da/dN is the crack extension per cycle of loading and ∆K is the stress intensity factor range expressed as.

∆K= ∆σ

Hear• ∆σ= σmax- σmin is the stress range under

constant amplitude loading.• a= half crack length for a center crack .full

crack length for an edge crack• f (a/w) = geometry correction factor.• w=width of the plate.

Effect of overload on fatigue crack growth behaviorThe crack growth characteristics depend to a large extent on this plastically deformed material at the crack tip. The size of crack tip plastic zone depends on the magnitude of external loading of given by the expression.

rp =α Whererp =radius of the plastic zone.

α = a constant depending upon the state of stress(plane stress or plane strain) at the crack tip.Fty= yield strength of the material.• The fatigue crack growth is influenced by the

crack tip plastic zone under constant amplitude loading.

• The crack tip plastic zone size at any crack length depends on the Kmax value.

Constant amplitude loading plastic zone.

Let us compare two load sequences given below to understand the load interaction effects.

single overload cycle. is applied after n- load cycles

Constant amplitude loading plastic zone and over load plastic zone.

• One can see that the current plastic zone size is embedded in a large plastic zone created due to the over load.

• In the case of a load cycle after an overload cycles the current plastic zone remains under the influence of the over load plastic zone.

• As consequences the crack growth rate after the overload cycle is seen to be significantly different than that under constant amplitude loading (without overload effect).

current plastic zone size touches the boundary of the overload plastic zone.

• This difference is crack growth rate is due to a condition known as “load-interaction effect”. This load interaction effect will affect the fatigue crack grow till the current plastic zone size touches the boundary of the overload plastic zone as shown in figure

FINITE ELEMENT ANALYSISThe finite element method (FEM), sometimes referred to as finite element analysis (FEA), is a computational technique used to obtain approximate solutions.Finite Element Analysis software programs The stress analysis of Fuselage of the Transport aircraft has been carried out using MSC NASTRAN MSC PATRAN  MSC PATRANMeshing is done in MSC Patran (pre-post processor)is by using various meshing options.

Element Quality Criteria Once required mesh pattern is got, it is

necessary to check the quality of mesh generated, this can be done using quality checks available in MSC Patran. Elements are checked for quality parameters like war page, aspect ratio, skew and Jacobean. - Check the maximum and minimum interior angles of all elements, Checking for shell normals, Check for free edges, Check for connectivity, Check for duplicates.

MSC NASTRAN

MSC NASTRAN (solver ) is one of the most popular general purpose finite element packages available for structural analysis.

Validation of FEA approach • In this section validation of FEA approach can be

down by the considering the rectangular plate with center hole.

• By varying the hole diameter keeping the plate dimension’s constant for various a/w ratio different scf are obtained from the Eq,

σnominal =

SCF =

• Below fig shows the internal force lines are denser near the hole.

• The boundary conditions are one end is constrained and other end is uniformly distributed load is applied. The boundary conditions are same for all iterations.

• these results are comparing with the standard experimental results (scf vs. a/w) graph shown in below.

Internal force lines are denser near the hole

Stress concentration factor for rectangular plate with central hole.

Geometric configuration and Finite element model of the plate with hole

Geometric modeling is carried out by using PATRAN software .Geometric dimensions of plate with hole fig. All dimensions are in mm.

The finite element mesh generated on each part of the structure using MSC PATRAN. Fig shows the finite element mesh on plate with hole.

Finite element meshing of plate with hole.

Close up view of mesh near the hole

Finite element meshing is carried out near the hole fine meshing is done in this sections where stresses are expected to be more to get good results shown below fig

Loads and boundary conditions

All the edge nodes of plate are constrained in all five degree of freedom (i.e13456) shown in figure. Except loading direction which is Y direction (i.e. 2). At the loading direction(Y direction) UDL (uniformly distributed load) is applied. All the elements along the thickness direction are constrained to avoid the eccentricity due to stiffening members.

Close up view of UDL load along y

Close up view of constrained boundary

Loads and boundary conditions on plate

For iteration 5 Consider hole diameter a=60 mm

σnominal = Where P is applied load= 1000kgFor udl 1000/width of the plate in mm

=1000/200=5Kg/mmA is area of load applied = (width of the plate-hole diameter)*thickness of the plate = (200-60)*2=280 mm2

σnominal = = 3.5714Kg/mm2

σmax= 8.42Kg/mm2 from the FEA results SCF = = = 2.36

Results obtained from the finite element analysis of the plate with hole.• Pre-processing and post-processing

is carried out by using MSC Patran software and Solved by using MSC Nastran (solver) software.

• The response of the plate with hole in terms of stresses due to loads and boundary conditions described in the previous sections are explained in the following sections.

Stress contour for plate with hole

Similarly the fallowing tabulated results are obtained for different a/w ratioComparison of obtain SCF values with SCF value plate with hole.

Number of Iterations 1 2 3 4 5

Radius of hole “r” In mm 10 15 20 25 30

Diameter of hole “a” In mm 20 30 40 50 60

Width of the plate “w” 200 200 200 200 200

a\w ratio 0.1 0.15 0.2 0.25 0.3

Length of the plate “L” in mm 400 400 400 400 400

Thickness “t” In mm 2 2 2 2 2

Stress concentration factor “SCF” 2.72 2.61 2.51 2.43 2.36

σmax 7.50 7.64 7.80 8.09 8.42

σnominal 2.77 2.94 3.12 3.33 3.57

Stress concentration factor obtain“SCF”

2.71 2.60 2.5 2.43 2.36

Comparison of obtain SCF values with SCF values of plate with hole.

By plotting the obtained scf and standard scf results shown in figure we conclude that the Fem approach is valid

STRESS ANALYSIS OF FUSELAGE SEGMENT

As the aircraft reaches higher altitude the atmospheric pressure will keep decreasing. Therefore as the aircraft fly at higher altitudes, fuselage (passenger cabin) will be pressurized for the passenger comfort. Then pressure inside the fuselage will be equivalent to the sea level pressure. For the current analysis the internal pressure is the differential pressure introduced inside the fuselage cabin which is considered as one of the critical load case.

Standard atmospheric pressure chart

Geometric configuration of the fuselage• A segment of the fuselage is

considered in the current study. The structural components of the fuselage are skin, bulkhead and longerons.

• Geometric modeling is carried out by using CATIA V5 software .

• Geometric dimensions and CAD model of fuselage figure and individual component of the fuselage shown below. All dimensions are in mm

Geometric configuration of the fuselage Skin

The above figure shows the skin dimensions. Skin has the thickness of 2 mm. The skin houses rest of the components like Bulkheads, Longerons, it is clear that the rivets which are along the fuselage holds the skin with longeron and the rivets which are in circumference to the fuselage holds the bulkhead, Distance between the longeron rivets are maintained by the 15 degree angle along the fuselage circumference and distance between the circumference rivets are 450 mm, diameter of the rivet used is 5mm, pitch of the rivet is 30mm.

Geometric configuration of the fuselage Bulkhead

Bulkhead is also known as frame. Bulkhead is a stiffening member in circumferential direction in the fuselage structure. There are seven bulkheads in the fuselage segment considered. All the dimensions of the bulkheads are shown in fig

CAD Model of the Bulkhead

Geometric configuration of the fuselage Longeron (stringer)

Longerons are also known as stringers which run in longitudinal direction in the fuselage structure. There are 24 longerons in the fuselage segment, which are 15 degree angle along the fuselage circumference to the each other.

CAD Model of the Longeron

Finite element model of the fuselage

Finite element meshing is carried out for all the components of the fuselage such that there is a node present at the point where riveting need to be simulated and fine meshing is carried out at the critical sections where stresses are expected to be more.The following figures show the details about the finite element mesh generated on each part of the structure using MSC PATRAN. Figure shows the finite element mesh on fuselage.

Finite element Mesh of the fuselage

Finite Element model of the Skin

Finite element Mesh of the fuselage skin

Close up view of mesh on the skin with beam elements as rivets

Riveting is simulated by selecting the node on the skin and the corresponding node on the other component and created a beam element between them.

Finite Element model of the Longeron (Stringers)

Finite element mesh on Longeron

Close-up view of Finite element mesh on stringer.

Finite Element model of the Bulkhead

Finite element mesh on Bulkhead

Close-up view of bulkhead with stringer cut-out (mouse hole)

Fastening (riveting) Using Beam Elements in the FEM of the Fuselage.The rivets are used as the fasteners in the

assembly of the component of the fuselage structure such as skin, tear strap, longeron and bulkhead. The meshing on these structural components is carefully generated such that there is a node present at the point where riveting is to be carried out. The riveting process is completed by creating beam element between the nodes by selecting the node on the skin and the corresponding node on the other component. The pitch of the rivet is 25mm. Diameter of the rivet is 5mm.

Rivets used to assemble all components of fuselage

The figure shows the beam elements which are indicated in red color connects all the components of the stiffened panel and acts as the rivets.

stress analysis of segment of fuselage.

The stress analysis of the fuselage segment is carried out by applying a differential pressure of 0.0413826 MPa (6 psi). This differential pressure is introduced as internal pressure in the fuselage segment.

Loads and boundary conditions of fuselageAll the edge nodes of fuselage segment at both the ends are constrained in all six degree of freedom (i.e123456) and 0.0413826 MPa pressure is applied to the internal pressure shown in figure below.

Pressurization on fuselage

Results obtained from the finite element analysis of the fuselage.

Displacement contour of the fuselage.

where white color showing minimum magnitude of displacement while red color showing maximum magnitude of displacement. In fuselage section displacement is maximum at the skin because of the stiffener members like stringers and bulkheads present in longitudinal and transfer direction of the fuselage. Unstiffened area present in between the stringer and bulkhead gets maximum displacement.For this problem maximum magnitude of displacement is 0.924 mm.

Close up view of Displacement contour of the fuselage

displacement contour of the stiffened fuselage.

Stress contour of the stiffened fuselage.

Stress contour for fuselage

shows the stress contour on fuselage from global analysis results. It is clear that the maximum stress on bulkhead is at stringer cut-out (mouse cut-out) and this maximum stress is uniform in all the stringer cut-outs. The magnitude of maximum tensile stress is 68.4738MPa. In the bulkhead the maximum stress will be at the bulkhead cut-out (mouse hole) which is shown in figure and the maximum stress locations are the probable locations for crack initiation. Invariably these locations will be at stringer cut-out locations in the bulkhead.

Close up view of Stress contour for fuselage

Similarly the fallowing tabulated results are obtained for different pressurization loads

maximum magnitude of Displacement and Stress contour in the fuselage for different load cases.

SR.No Pressure in psi

Pressure in MPa

Max Displacement in mm

Max stress contour in MPa

1 6 0.04138 0.924 68.4738

2 8 0.05517 1.23 91.3311

3 10 0.06896 1.54 113.796

4 16 0.11034 2.46 182.466

5 18 0.12413 2.77 205.029

6 20 0.13793 3.08 233.478

LOCAL ANALYSIS OF THE STIFFENED PANEL

From the finite element analysis carried out on the fuselage segment, the maximum stress location was identified which is explained in the previous section. Based on the maximum stress location a local analysis is carried out by considering a stiffened panel near the maximum stress location.

Stiffened panels are the most generic structural elements in an airframe. The stiffened panel consists of Skin BulkheadLongerons (stringer)and Fasteners (rivets).

Introduction to stiffened panel

Geometric configuration of the stiffened panel

Geometric modeling is carried out by using CATIA V5 software .Geometric dimensions and CAD model of fuselage and individual component of the stiffened panel are shown below. All dimensions are in mm.

Detailed view of skin

• The above shows the skin dimensions considered for the local analysis.

• Skin has the thickness of 2 mm. The skin houses rest of the components like Bulkheads, Longerons, which are assembled by riveting process,

• It is clear that the rivets which are in columns holds the bulkhead, distance between the rows is 450mm and the diameter of the rivet used is 5mm pitch of the rivet is 25mm.The below figure shows the CAD model of the skin with rivet holes.

CAD Model of skin with rivet hole

Geometric configuration of the stiffened panel BulkheadBulkhead is also known as frame. Bulkhead is a stiffening member in circumferential direction in the fuselage structure. There are three bulkheads in this stiffened panel. All the dimensions of the bulkheads are shown in figure.

Cross sectional view (Bottom view) of the bulkhead

CAD Model of the Bulkhead

Finite element model of the stiffened panelshows the finite element mesh on skin. The skin houses rest of the components like bulkheads.

Finite element Mesh on skin

Close up view of mesh on the skin with beam elements as rivets

Finite element mesh of the stiffened panel Bulkhead (Frame)

Bulkhead is also known as Frame. The bulkhead has Z cross-section. The bulkheads are placed on top of the skin and riveted onto the skin.

Finite element mesh on Bulkhead

Close-up view of bulkhead with stringer cut-out (mouse hole)

Rivets used to assemble all components of stiffened panel

Complete finite element mesh on stiffened panel

Finite element model summaryFuselageTotal number of Grid points =211312Total number of Beam elements=19508Total number of Quad elements=121680Total number of Tria elements=31104  

Stiffened panelTotal number of Grid points = 232558Total number of Beam elements= 214Total number of Quad elements= 217564Total number of Tria elements= 26708

Local analysis at maximum stress location.

The maximum stress location and the magnitude of maximum stress are identified from the global analysis of the fuselage segment. As described in the previous section the maximum tensile stress is near the bulkhead cut out region a local model representing the highest stress location is considered for the local analysis. Stiffened panels with three bulkheads are considered for the local analysis.

Loads in the local modelA differential pressure of 9 psi (0.06206MPa) is considered for the current case. Due to this internal pressurization of fuselage (passenger cabin) the hoop stress will be developed in the fuselage structure. The tensile loads at the edge of the panel corresponding to pressurization will be considered for the linear static analysis of the panel.

Hoop stress is given by

σ hoop = ---Eq 1Where Cabin pressure (p)=9 psi=0.06206 MPa Radius of curvature of fuselage(r) = 1500 mm Thickness of skin (t) = 2 mm After substitution of these values in the above eq we will get σ hoop = 4.74525 Kg/mm2

=46.55 MPaWe know that

σ hoop =

Above equation can be written as P = σ hoop *A ---

Eq 2

Uniformly distributed tensile load is applied on either side of the stiffened panel in Y axial direction.Load on the skinHere

Ps=Load on skin σ hoop =4.74525 Kg/mm2

A=Cross sectional area of skin in mm2

i.e. Width *Thickness(1000*2)=2000Substituting these values in the Eq 2 we getPs=9490.5 KgPs=93101.805NUniformly distributed load on skin will be Ps =9490.5 /1000 =9.4905Kg/mm

Load on BulkheadHere Pb =load on Bulkhead in Kg

σ hoop =4.74525 Kg/mm2

A =Cross sectional area of each Bulkhead in mm2

i.e. Width *Thickness, (L1+L2+L3)*tb i.e. (18.5+68.5+18)*1.5=232mm2

Substituting the values in (Eq 2) we get Pb =1100.898 Kg

Pb =10799.8093N Uniformly distributed load on Bulkhead will be Pb = 1100.898 /116 =9.4905 Kg/mm

Similarly for the pressure 12 psi=0.082737154 MPaLoad on BulkheadPb =1467.864 KgPb =14399.746 N Load on Skin Ps =12654 Kg, Ps =124135.74 N

All the edge nodes of stiffened panel are constrained in all five degree of freedom (i.e13456) except loading direction which is Y direction (i.e. 2). All the elements along the thickness direction are constrained to avoid the eccentricity due to stiffening members.

Loads and boundary conditions of stiffened panel

Loads and boundary conditions stiffened panel

Results obtained from the finite element analysis of the stiffened panel

Displacement contour of the stiffened panel

Stress contour of the stiffened panel

Stress contour for skin

Stress counter for Bulkhead

. It is clear that the maximum stress on bulkhead is at stringer cut-out (mouse cut-out) and this maximum stress is uniform in all the stringer cut-outs. The magnitude of maximum tensile stress is 1.29 kg/mm2

which is more than the stresses in all other components of the stiffened panel. In the bulkhead the maximum stress will be at the stringer cut-out (mouse hole)maximum stress locations are the probable locations for crack initiation. Invariably these locations will be at stringer cut-out locations in the bulkhead

Stress counter stiffened panel

From the stress analysis of the stiffened panel it can be observed that a crack will get initiated from the maximum stress location. There are two structural elements at the rivet location near the high stress location. Crack will either get initiated from the bulkhead at stringer cut out or from the nearby rivet location from the rivet hole. Figure shows the rivet force near the high stress location is 84.1kg and 83.2kg.

Rivet force near the high stress location

Local analysis at maximum stress location with considering the rivet

hole.

There are two structural elements at the rivet location near the high stress location, the rivet near the high stress location are removed by creating the hole on bulkhead and skin same as the rivet dimensions and applying the rivet force near the high stress location. All other loads and boundary conditions stiffened panel is same as shown in the above

Close up view of hole near the high stress location on the skin

Results obtained from the finite element analysis of the stiffened panel

with considering the rivet hole.

Displacement contour of the stiffened panel

Stress contour of the stiffened panelStress distribution on skin

. Stress contour for skin

• Above shows the stress contour on the skin from local analysis results. It is clear that the maximum stress on skin is at the rivet hole location. The magnitude of maximum tensile stress is 38.4kg/mm2

• which is more than the stresses in all other components of the stiffened panel. In the bulkhead the maximum stress will be at the skin rivet hole which is shown above

• The maximum stress locations are the probable locations for crack initiation. Invariably these locations will be at rivet locations in the skin. Skin is the critical stress locations for the crack initiation.

Maximum stress at the rivet hole

Stress distribution on bulkhead

Stress counter for Bulkhead

Stress counter stiffened panel

Validation of FEM approach for stress intensity factor (SIF)

calculationGeometry, Loads and boundary conditions of unstiffened panel

Geometry of the unstiffened panel

Loads and boundary conditions of unstiffened panel

Fine element mesh at the center of skin near the crack

Close up view of fine mesh at the center of skin near the crack

Consider crack length, 2a=10 mm1. SIF calculation by Theoretical method  KI = * f (Eq (a)Where

=P/A= =2.5Kg/mm2 =5 mm f (= 1.001165 which is calculated by using Eq

f ( =Where = Crack length in mm f ( =Correction factor b=Width of the plate (200 mm)

Substituting above values in Eq(a) .SIF value will be KI theoretical =3.077334 MPa

2. SIF calculation by Analytical method(FEM)

Nodes and Elements near the crack tip

Strain energy relies rate is calculated by Eq G=

For relative displacement adding the displacement of nodes 8608 and 9211 in T2 direction. The displacement is obtained in the f06 file created by the MSC Nastran (solver) software shown in table.

POINT ID.

TYPE T1 T2 T3 R1 R2 R3

8608 G 6.075589E-04 1.981864E-03 0.0 0.0 0.0 7.221852E-04

9211 G 6.075589E-04 1.981864E-03 0.0 0.0 0.0 7.221852E-04

D I S P L A C E M E N T V E C T O R For Unstiffened Panel

For the relative displacement (1.981864E-03+1.981864E-03) = 0.00396 mmFor Forces at the crack tip in kg or N, adding any one side of elements (Elm 8395, 8396 or Elm8595, 8596) forces acting on the crack tip in T2 direction. The Forces at the crack tip is obtained in the f06 file created by the MSC Nastran (solver) software shown in table

POINT-ID ELEMENT-ID SOURCE T1 T2 T3

8808 8395 QUAD4 -2.395632E+00 -6.864131E+00 0.0

8808 8396 QUAD4 +2.395632E+00 +6.652291E+00 0.0

8808 8595 QUAD4 -2.395632E+00 +6.864131E+00 0.0

8808 8596 QUAD4 +2.395632E+00 -6.652291E+00 0.0

G R I D P O I N T F O R C E B A L A N C E For Unstiffened Panel

For Forces at the crack tip F=(6.864131E+00+6.652291E+00)=13.51642 Kg.Where

F = 13.51Kg =132.533NU = 0.00396 mm𝜟c= 1 mm T = 2 mm

Substitute all values in above Eq then G= 0.13120 MPa

Now Analytical SIF is calculated by Eq KI fem=

Where E=7000kg/mm2=68670 MPa

Substituting G and E values in Eq KI fem=3.0423 MPa

The above calculation is carried for different crack length considering a known load. A stress intensity factor value calculated by FEM and stress intensity values calculated by theoretical method for un-stiffened panel is tabulated.

SR.No Crack length ”2a”

in mm

Kfem in MPa√m Kth without

considering the C.F in MPa√m

Correction factor

C.F

Kth with

considering the C.F in MPa√m

%error

1 10 3.042304 3.073753 1.001165 3.077334 1.138

2 20 4.367924 4.346943 1.004824 4.367913 0

3 30 5.406574 5.323896 1.011259 5.383838 0.422

4 40 6.318190 6.147506 1.020810 6.275436 0.681

5 50 7.166964 6.873120 1.033890 7.106050 0.857

6 60 7.991123 7.529126 1.051012 70913202 0.984

7 70 8.816894 8.132386 1.072820 8.724586 1.056

8 80 9.668487 8.693886 1.100134 9.564440 1.080

9 90 10.572659 9.221259 1.134024 10.457129 1.104

10 100 11.545625 9.720060 1.175919 11.430003 1.010

Comparison of analytical (FEM) SIF values with theoretical SIF value for un-stiffened panel for 1 mm element size.

SR.No Crack length ”2a”

in mm

Kfem in MPa√m Kth without

considering the C.F in MPa√m

Correction factor

C.F

Kth with

considering the C.F in MPa√m

%error

1 10 3.076609 3.073753 1.001165 3.077334 0.023

2 20 4.391599 4.346943 1.004824 4.367913 0.542

3 30 5.426225 5.323896 1.011259 5.383838 0.787

4 40 6.335258 6.147506 1.020810 6.275436 0.953

5 50 7.183796 6.873120 1.033890 7.106050 1.094

6 60 8.007325 7.529126 1.051012 70913202 1.189

7 70 8.833447 8.132386 1.072820 8.724586 1.247

8 80 9.686039 8.693886 1.100134 9.564440 1.271

9 90 10.587531 9.221259 1.134024 10.457129 1.247

10 100 11.565942 9.720060 1.175919 11.430003 1.189

Comparison of analytical (FEM) SIF values with theoretical SIF value for un-stiffened panel for 0.5 mm element size

Comparison of Theoretical SIF value with analytical SIF value

0 10 20 30 40 50 60 70 80 90 100 1100

1

2

3

4

5

6

7

8

9

10

11

12

13

14

SIF by theoretical in MPa√m

Linear (SIF by theoretical in MPa√m)

SIF by analytical in MPa√m for 1 mm element size

Linear (SIF by analytical in MPa√m for 1 mm element size)

SIF by analytical in MPa√m for 0.5 mm element size

Linear (SIF by analytical in MPa√m for 0.5 mm element size)

CRACK LENGTH a in mm

SIF

IN

MPa

√m

Methodology of finding SIF values for un-stiffened panel using FEM was extended to get SIF values for stiffened panel.Where red fringes shows the maximum displacement which is at the center of the panel.

Displacement contour for un-stiffened panel with center crack

FATIGUE CRACK GROWTH CALCULATIONSCalculation of stress intensity factor (SIF) for 100 mm crack in the stiffened panel for constant amplitude loading.

Considering a crack length of 100 mm in the skin at high stress region SIF is calculated.

The maximum load corresponding to 9 PSI which is L=9490.5kg ≈ 9.4905kg/mm uniformly distributed load is applied at the remote edge of the panel.

Loads and boundary conditions of stiffened panel

All the edge nodes of stiffened panel are constrained in all five degree of freedom (i.e13456) except loading direction which is Y direction (i.e. 2). All the elements along the thickness direction are constrained to avoid the eccentricity due to stiffening members. All loads and boundary conditions of stiffened panel is shown in the belowfig

Loads and boundary conditions of stiffened panel

Stress contour stiffened panel

Close up view of Stress contour in the stiffened panel for 100 mm crack

Considering the maximum stress in Z1 direction and max-principal failure theory, the max stress occurred at the skin region is 51.00 Mpa.

SIF calculation by Analytical method (FEM)

Nodes and Elements near the crack tip

Nodes and elements ID shown in above figure are near the crack tip at which maximum stress are acting. Strain energy relies rate is calculated by Equation

G=

For relative displacement subtracting the displacement of nodes 106071 and 233376 in T2 direction. The displacement is obtained in the f06 file created by the MSC Nastran (solver) software shown in table

D I S P L A C E M E N T V E C T O R

POINT ID. TYPE T2

106071 G 0.4655189

233376 G 0.4791793

For the relative displacement (0.4655189-0.4791793) = 0.0136604 mm

For Forces at the crack tip in kg or N, adding any one side of elements (Elm 151247, 151248 or Elm153247, 153248) forces acting on the crack tip in T2 direction. The Forces at the crack tip is obtained in the f06 file created by the MSC Nastran (solver) software shown in tableG R I D P O I N T F O R C E B A L A N

C EPOINT-ID ELEMENT-ID SOURCE T2

104444 151247 QUAD4 20.77951

104444 151248 QUAD4 23.20860

104444 153247 QUAD4 23.97819

104444 153248 QUAD4 20.00991

For Forces at the crack tip F=(20.77951+23.20860)=43.98811 Kg.

WhereF = 43.988 Kg U = 0.0136604 mm𝜟c= 0.5 mm T = 2 mm

Substitute all values in Eq ‘A’ then G= 0.30044759 Kg/mm

Now Analytical SIF is calculated by equation.

KI fem=Where

E=7310kg/mm2

Substituting G and E values in above equation

KI fem=14.53824748 MPa

Calculation of crack growth rate for 100 mm crack in the stiffened panel

The crack growth rate is calculated or obtained through vs k curve from the 𝜟respective material. Therefore to obtain the (crack growth rate) one should first calculate the k𝜟 effective.𝜟keffective is calculated by using the Eq B and Eq C (Ref: “The practical use of fracture mechanics” by David Broek).𝜟Keffective= K𝜟 max- K𝜟 opening ---MPa Eq B𝜟Kopening= K𝜟 max(0.5×0.4R) R→0 ---MPa Eq C

Where 𝜟Kmax = maximum stress intensity factor𝜟Kmax = minimum stress intensity factor R = = 0 because 𝝈minimum→0

We know that Kmax =14.53 MPa. Kmin=0.0Substituting the values in Eq ‘C’ we get Kopening=7.27 MPa Substituting the values in Eq ‘B’ we get Keffective= 7.26 MPaFrom graph shown in below Crack growth rate curve we get the crack growth rate per cycle

For Keffective =7.27 MPa,we get 5×10-5 mm/cycle  To growth a crack of 1mm it requires 20,000 cycles . After 20,000 cycles the crack size will be 101mm which is considered for the next analysis and overload cycle is applied.

Calculation of stress intensity factor (SIF) for 101 mm crack in the stiffened panel with overload is applied.Considering a crack length 0f 101 mm on

the skin and the overload corresponding to 12 PSI which is L=12654kg ≈ 12.654kg/mm uniformly distributed load is applied at the remote edge of the panel.

Stress contour of the stiffened panelThe stress distribution is shown below in the fig 8.6 at a crack length of 2a=101mm

Stress counter in the stiffened panel for 100 crack

Considering the maximum stress in Z1 direction and max-principal failure theory. The max stress occurred at the skin region is 68.20 Mpa.

Similarly F = 58.8225 Kg U = 0.0182676 mm𝜟c= 0.5 mmT = 2 mmG= 0.302213863 Kg/mmE=7310kg/mm2

KI fem=19.44129471 MPaKmax =19.44129471 MPa.

Kmin=0.0Kopening= 9.720647354 MPa Keffective= 9.720647354 MPa

Crack growth rate We get 5×10-4 mm/cycle

With 1 cycle of load the crack growth increment is 0.0005mm. After the application of one overload cycle with crack increment of 0.0005 mm the total crack size will be 101.0005mm. With the crack length of 101mm another iteration is carried out with the load of 9psi.

Calculation of stress intensity factor (SIF) for 101 mm crack with 9psi load.

Considering a crack length of 101 mm in the skin at high stress region SIF is calculated. The maximum load corresponding to 9 PSI which is L=9490.5kg ≈ 9.4905kg/mm uniformly distributed load is applied at the remote edge of the panel. Stress contour of the stiffened panelThe stress distribution is shown in the below fig 8.8 with the crack length of 2a=101mm and 9psi load

Considering the maximum stress in Z1 direction and max-principal failure theory, the max stress occurred at the skin region is 68.20 Mpa.

Stress counter stiffened panel for 101 mm crack

Similarly F = 44.11688 Kg U = 0.0137006 mm𝜟c= 0.5 mm T = 2 mmG= 0.302213863 Kg/mmKI fem=14.58091865 MPaE=7310kg/mm2

Kmax =14.58091865MPa. Kmin=0.0

Kopening=7.290459325 MPa Keffective= 7.290459325 MPa

we get 5×10-5 mm/cycle

Over load plastic zone size calculations

When one single high stress is interspersed in a constant amplitude history, the crack growth immediately after the “overload” is much slower than before the overload. After a period of very slow growth immediately following the overload, gradually the original growth rates are resumed. This phenomenon is known as “retardation”. The loading pattern used for the calculation of overload effect is shown in the following figure.

A typical load spectrum with an over load

The overload plastic zone size is given by the following Equation

Rpc =Rpo=Where

Rpo= over load plastic zone sizeKmax= maximum SIF value = yield strength

Considering Fty=345 Mpa And Kmax=19.44129471MpaSubstituting the values in Equation.

Rpo= = 5.05396592×10-4mm.

Followed by the 101.0005mm crack and 9psi load Kmax=14.58091865Mpa

Rpc= = 2.842835404×10-4mm.

Where Rpc= current plastic zone size

Calculations for the Retardation factor

crack growth retardation because of the overload plastic zone size the retardation factor is calculated using the following equation.

ØR= here Where

ØR= Retardation factorai = current crack length ao = after the overload crack length wheeler parametric value =1.4

ØR== 0.170600296.

Calculating the crack growth rate with in the overload region

As the crack grows within the overload plastic size the effect of over load on the crack growth rate also varies. Therefore the crack growth rate is calculated by dividing the overload region by four parts each one having region as 1.26349148×10-4mm.

four divisions considered with in the overload plastic zone size

1st region.To calculate crack growth rate in the 1st region following data are consideredLoad of 9psi with crack size now become

ai= 101+(1.26349148×10-

4×1) =101.0001263mm

And ao, Rpc ,Rpo will be as follows ao=

101.0005+(1.26349148×10-4×1) =101.0006263mmRpc=2.842835404×10-4 mmRpo=(1.26349148×10-4×3) =3.79047444×10-4 mm

Retardation factor for first region from EquationØR1= here

ØR1=ØR1 ==0.205890101

Therefore total overload plastic zone size is 5×10-5 by multiplying the retardation factor we get the da/dN crack growth size for the 1st region is calculated.

da/dN= 5×× ØR1

da/dN=5.05396592×10-4×0.205890101da/dN=1.040561554×10-4 mm/cycle

To grow the 1mm crack it required 9610 cycles.

2nd region.To calculate crack growth rate in the 2nd region considering the following data Load of 9psi with crack size now become

ai= 101+(1.26349148×10-4×2) =101.000252mm

And ao, Rpc ,Rpo will be ao

=101.0005+(1.26349148×10-4×2) =101.000752mmRpc=2.842835404×10-4mm,Rpo=(1.26349148×10-4×2) =2.52698296×10-4mm,

Retardation factor for second region from Equation

ØR2= here

ØR2=ØR2==0.256181299

Therefore total overload plastic zone size is 5×10-5 by multiplying the retardation factor we get the da/dN crack growth size for 2nd region is calculated.

da/dN = 5××ØR2

da/dN = 5.05396592×10-

4×0.256181299da/dN =1.28090×10-4 mm/cycle

To grow the 1mm crack it required 7807 cycles.

3rd region.To calculate crack growth rate in the 3rd region we taken fallowing data Load 9psi with crack size now become

ai= 101+(1.26349148×10-

4×3) =101.000378mm,

And ao, Rpc ,Rpo become ao

=101.0005+(1.26349148×10-4×3) =101.000878Rpc=2.842835404×10-4mm,Rpo=(1.26349148×10-4×1) = 1.26349148×10-4,

The overload plastic zone size after the second region is 1.26×10-4 is less than the current plastic zone size. Therefore the assumed 3rd region for the calculation of retardation factor need not to be considered. shown in table.

Crack growth calculations  

SIF Number of cycles required for a crack growth increment of

1mm.

Constant amplitude loading (9PSI)

9.4906Kg/mm2 for 100 mm crack

SIF= 14.5382

5 10𝖷 -5cycles 20,000

Overload (without considering the overload

plastic zone size applied)

(12PSI)12.654Kg/mm2

SIF=19.4412 at 101mm crack length

5 10𝖷 -4cycles 2000

After the overload plastic zone size effect

With considering the retardation factor

0.1706002960.170600296 5 10𝖷 𝖷 -5 =

8.5300148 10𝖷 -

6cycles.1,17,233

OBSERVATIONS AND DISCUSSIONS• Damage tolerance design philosophy is

generally used in the aircraft structural design to reduce the weight of the structure.

• Stiffened panel is a generic structural element of the fuselage structure. Therefore it is considered for the current study.

• A FEM approach is followed for the stress analysis of the stiffened panel.

• The internal pressure is one of the main loads that the fuselage needs to hold.

• Stress analysis is carried out to identify the maximum tensile stress location in the stiffened panel.

• A local analysis is carried out at the maximum stress location with the rivet hole representation.

• The crack is initiated from the location of maximum tensile stress.

• MVCCI method is used for calculation of stress intensity factor

• A crack in the skin is initiated with the local model to capture the stress intensity factor

• Stress intensity factor calculations are carried out for various incremental cracks

• When the SIF at the crack tip reaches a value equivalent to the fracture toughness of the material, then the crack will propagate rapidly leading to catastrophic failure of the structure

• A load spectrum consisting of constant load cycles with a over load in-between is considered to study the over load effect on the crack growth rate

• The crack growth rate for constant amplitude load cycles is carried out by considering the crack growth rate data curve (da/dN Vs K)𝜟

• Plastic zone size due to constant amplitude load cycles and over load cycle is calculated.

• The calculations have indicated that the over load will reduce the rate of crack growth due to large plastic zone size near the crack tip.

• The effect of large plastic zone due to over load is estimated by calculating the crack growth retardation factor

• The crack growth retardation factor with a over load reduces the rate of crack growth to 17% than compared to crack growth rate without a over load.

SCOPE FOR FURTHER STUDIES

• The overload effect can be calculated for different load spectrum using the similar approach

• Structural testing of the stiffened panel can be carried out for validating the analytical predictions.

• Crack growth analysis in the stiffened panel with a different skin material can be carried out

• The bi-axial stress field can be considered for the crack growth study in the stiffened panel

REFERENCES1. F. Erdogan and M. Ratwani, International journal of

fracture mechanics, Vol. 6, No.4, December 1970.2. H. Vlieger, 1973, “The residual strength characteristics

of stiffened panels containing fatigue crakes”, engineering fracture mechanics, Vol. 5pp447-477, Pergamon press.

3. H. Vlieger, 1979, “ Application of fracture mechanics of built up structures”, NLR MP79044U.

4. Thomas P. Rich, Mansoor M. Ghassem, David J. Cartwright, “Fracture diagram for crack stiffened panel”, Engineering Fracture Mechanics, Volume 21, Issue 5, 1985, Pages 1005-1017

5. Pir M. Toor “On damage tolerance design of fuselage structure (longitudinal cracks)”, Engineering Fracture Mechanics, Volume 24, Issue 6, 1986, Pages 915-927

6. Pir M. Toor “On damage tolerance design of fuselage structure (circumferential cracks) Engineering fracture mechanics, Volume 26, Issue 5, 1987, Pages 771-782

7. Federal Aviation Administration technical center “ Damage tolerance handbook” Vol. 1 and 2. 1993. 8. T. Swift “Damage tolerance capability”, international journal of fatigue, Volume 16, Issue 1, January 1994, Pages 75-949. J. Schijve “Multiple –site damage in aircraft fuselage

structure” In 10 November 1994.10. T. Swift, 1997, “Damage tolerances analysis of

redundant structure”,AGARD- fracture mechanics design methodology LS-97,pp 5-1:5-34.

11. E.F. Rybicki and M.F. Kanninen, 1997, “A finite element calculation of stress intensity factor by a modified crack closure integral.”, Engineering fracture mechanics, vol. 9, pp. 931-938.

12. Amy L. Cowan “Crack path bifurcation at a tear strap in a pressurized stiffened cylindrical Shell” in August 24, 1999

13. Andrzej Leski, 2006, “Implementation of the virtual crack closure technique in engineering FE calculations”. Finite element analysis and design 43, 2003,261-268.

14. Jaap Schijve, “Fatigue damage in aircraft structures, not wanted, but tolerated?” international journal of fatigue, Volume 31, Issue 6, June 2009, Pages 998-1011

15. X Zhang “Fail-safe design of integral metallic aircraft structures reinforced by bonded crack retarders”. Departments of Aerospace Engineering and Materials, Cranfield University Bedfordshire, in 3rd may 2008.16. Michael F. Ashby and David R. H. Jones “Engineering

materials and an introduction to their properties and applications”, Department of Engineering, University of Ambridge, UK.

17. D.P Rokke and D.J.Cartwright “Compendium of stress intensity factor”, Royal Aircraft Establishment Farnborough and University of Southampton.

18. Michael Chun-Yung Niu “Airframe stress analysis and sizing” Second edition-1999.

19. “The practical use of fracture mechanics” by David Broek Kluwer academic publishers-1988

Thank you