Determination of stress intensity factor for a crack emanating from a hole in

International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –

6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 2, March - April (2013) © IAEME

373

DETERMINATION OF STRESS INTENSITY FACTOR FOR A CRACK

EMANATING FROM A HOLE IN A PRESSURIZED CYLINDER USING

DISPLACEMENT EXTRAPOLATION METHOD

AKASH.D.A (1)

, Anand.A (2)

, G.V.GNANENDRA REDDY(3)

, SUDEV.L.J

(4)

(1)

Department of Mechanical Engineering, SJCIT Chickaballapur, 562101 (2) Department of Mechanical Engineering, Vidyavardhaka College of engineering, Mysore, 570002

(3) Department of Mechanical Engineering, SJCIT Chickaballapur, 562101 (4)

Department of Mechanical Engineering, Vidyavardhaka College of engineering, Mysore, 570002

ABSTRACT

The machine elements with cylindrical profile such as pressure vessel, cylindrical

shells, which have been used extensively as the structural configuration in aerospace and

shipping industries needs to be leak proof. But however it’s not possible to fabricate 100%

leak proof pressure vessel / cylindrical shell as the industrial materials do not have uniform

composition. Thus defects or cracks are inevitable in their substructure, also during their

service life a crack may initiate on an internal/external boundary of circular cylinder which

has important influence on stress distribution in the structure. Hence the structural

assessments of hallow cylinders ranging from thick walled pressure vessel to thin walled

pipes has to be carried out, that in-turn relay’s on availability of Stress Intensity Factor (

S.I.F) for fracture analysis. The magnitude of the S.I.F determines the propagation of crack.

In this paper, Considerable effort has been devoted for computation of the S.I.F of crack

emanating from a hole in pressurized cylinder. The objective of this work is to determine SIF

(Plane Strain) for a crack emanating from a hole in a Pressurised cylinder using Finite

Element Method (FEM). From this study it was observed that the value of SIF rises suddenly

when the crack tip is near to the hole and it stabilises as the crack tip move far from the hole.

The SIF values evaluated for different crack length using FEM is normalised with the

analytical values obtained from theoretical equation with respect to (a/D) ratio which

provides important information for subsequent studies such as the crack growth rate

determination and prediction of residual strength with plane strain and plane stress

conditions.

INTERNATIONAL JOURNAL OF MECHANICAL ENGINEERING

AND TECHNOLOGY (IJMET)

ISSN 0976 – 6340 (Print)

ISSN 0976 – 6359 (Online)

Volume 4, Issue 2, March - April (2013), pp. 373-382

© IAEME: www.iaeme.com/ijmet.asp Journal Impact Factor (2013): 5.7731 (Calculated by GISI) www.jifactor.com

IJMET

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374

Keywords: Fracture Mechanics, S.I.F and crack emanating from Hole

1. INTRODUCTION

Pressure vessel and cylindrical shells have been used extensively as the structural

configuration in aerospace and shipping industries needs to be leak proof. However new

advancement in computers has made Finite Element Analysis (FEA) a practical tool in the

study of pressure vessels [1], especially in determining stresses in local areas such as

penetrations and service holes. The most likely places for crack initiating and development of

cracks are the service holes. Due to the high stress concentration in this area cracks may grow in

time, leading to a loss of strength and the reduction of the lifetime of the product as shown in

Figure1. If the structure is concerned with different loading, the crack behaviour must be assessed

in order to avoid catastrophic failures. For this, the knowledge of the crack size, service stress,

material properties and Stress Intensity Factor (SIF) is required [2]. Hence the structural

assessments of hallow cylinders /shell ranging from thick walled pressure vessel to thin

walled pipes has to be done, that in-turn relay’s on availability of stress intensity factor for

fracture and fatigue analysis. Thus it has been recognized that the stress intensity factor is an

important parameter to determine the safety of a cracked component, but the basic practical

problem a designer faces, is to make a decision to opt the method for determining stress

intensity factors. It is not easy to strike a balance between the accuracy of the method, time

required to get a solution, and cost. Numerous equations for stress intensity factors are

available in the literature [1 – 6]. These factors represent various geometries and loading

conditions of fundamental importance in the prediction of structural failure of cracked

cylindrical bodies. In all there are probably more than 600 formulas for calculating K values

for different crack configurations, body geometries, and loading situations.

Here the objective of the work is to determine S.I.F for a crack emanating from a hole in

a pressurized cylinder.

Fig.1: Larger crack formed by the link-up of fatigue cracks at adjacent rivets.

2. FRACTURE MECHANICS

Fracture mechanics involves a study of the presence of the cracks on overall properties

and behaviour of the engineering component. The process of fracture may be initiated at defect

locations like micro-cracks, voids, and the cavities at the grain boundaries. These defects can

lead to the formation of a crack due to the rupture and disentanglement of molecules, rupture of

atomic bonds or dislocation slip [3].

Cracked body can be subjected to three modes of loads as shown in Figure 2. In some

cases, body may experience combination of the three modes:



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1. Opening mode: The principal load is applied normal to the crack surfaces, which tends

to open the crack. This is also referred as Mode I loading (Figure 2a).

2. In-plane shear mode: This mode corresponds to in-plane shear loading which tends to

slide

One crack surface with respect to the other. This is also referred as Mode II loading

(Figure2b).

3. Out-of-plane shear mode: This is the tearing and anti-plane shear mode where the crack

surfaces move relative to one another and parallel to the leading edge of the crack (Figure 2c).

(a) (b) (c)

Fig. 2: Three modes of loading that can be applied to a crack

The Stress Intensity Factor (SIF) is one the most important parameters in fracture

mechanics analysis. It defines the stress field close to the crack tip and provides fundamental

information of how the crack is going to propagate. In this study, A typical and practical point

matching technique, called Displacement Extrapolation Method (DEM) is chosen for the

numerical analysis method. Plane strain assumption is valid for very thin-walled structures; the

evaluation of S.I.F (KI) by Displacement Extrapolation Method (DEM) is as discussed bellow for

plane strain condition.

The stress intensity factors at a crack for a linear elastic fracture mechanics analysis may

be computed using the KCALC command. The analysis uses a fit of the nodal displacements in

the vicinity of the crack. The actual displacements at and near a crack for linear elastic materials

are

)1(22

11k

r

G

Ku ++=

π (1)

)k1(2

r

G2

Kv 1 +

π+= (2)

π+=

2

r

G

K2w 111 (3)



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Where:

u, v, w = displacements in a local Cartesian coordinate system as shown in figure 3

r, θ = coordinates in a local cylindrical coordinate system as shown in figure 3.

G = shear modulus

vK

+=

1

ν In Plane Stress (4)

K= 3 - 4 In Plane strain…………………………..(5)

� = Poisson's ratio

For Mode-1, SIF at crack tip is expressed as

r

V

k1

G2K1

∆

+π= (6)

Where ∆v, are the motions of one crack face with respect to the other.

Then A and B are determined so that

BrAr

V+= (7)

At points J and K.

Next, let r approach 0

Ar

Vlim 0r =→ (8)

Fig. 3: Nodes Used for the Approximate Crack-Tip Displacements for Full crack Model

Thus, Equation 5 becomes:

mmmm

N

k

GAK

211

22

+= π (9)

3. OBJECTIVE OF WORK AND METHODOLOGY

The objective of this work is to determine S.I.F for a longitudinal crack emanating from a

hole in a pressurized cylinder as shown in figure 4. The objective is achieved by developing a

model of a cylinder with hole and a through crack using CATIA V5 software .The CATIA model

is imported to ANSYS.The FE model is meshed using 8-node quadrilateral doubly curved

SHELL 93 elements in the pre-processor of the ANSYS software . Further as a part of the finite

element work, a mesh sensitivity Study was conducted.



377

A shell with a longitudinal crack was meshed using three different mesh densities.

Mainly, the area around the imperfection was modelled with a finer mesh. Further the crack

tip singular elements were created using KSCON command. For this model there are 36

singular elements around the crack tip and the radius of the first row elements is ∆a (Where ∆a =

a/100).The model is then solved (Static Analysis) by subjecting it to an internal pressure of 1MPa

load with appropriate boundary conditions. Then the S.I.F is evaluated in general postprocessor

by using KCALC command.

The geometry of the meshed test model with crack tip singular elements in ANSYS 12 is

as shown in the Figure 5.The material considered is 304 steel (ASME). The material is assumed

to be linear elastic with young’s modulus of 2.5GPa and poisons ratio 0.3

Where,

D= Diameter of the hole (20mm),

a= Half Crack length

σ=applied hoop stress (Pr/t)

P= Internal pressure 1MPa

t=Thickness of the cylindrical shell

10mm

Fig.4: Geometry of model

(a) (b) (c)

Fig. 5: (a) Finite Element Model meshed with Boundary Condition (b) Zoomed View of

elements near crack tip (c) Zoomed View of Crack Tip Singular Elements



378

4. VALIDATION OF PROPOSED METHODOLOGY

The methodology proposed to determine Mode-1 S.I.F in previous section is validated

using the practical problem “Determination of S.I.F of longitudinal cracks in a pressurized

cylindrical shell” from reference 3.A cylindircal shell with varying longitudinal crack length given

in reference 3 is as shown in Figure 6.

Fig.6: Longitudinal crack in internally pressurized cylinder

Mode I S.I.F (KI) is given by

KI(Theo)= )(f . 1 απσ a ………………………………………………(10)

Where

)(

)07.029.152.01()(f 32

1

Rt

ax

xxx

=

−++=α

The half crack length was varied from 20mm to maximum half crack length of 439.53mm .The

maximum crack length in a given dimensions of cylindrical shell was determined using curvature

parameter β [9]

4 2 )1(12Rt

aν−=β ……………………….(11)

if β =8 for longitudinal cracks , thickness of the cylindrical shell is 10mm and radius of the

cylindrical shell is 1000mm then the maximum crack length for given set of cylindrical shell

dimension is 439.53mm.

The values of S.I.F obtained by the theoretical and FEA is given in table 1.



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Table 1: Mode-1 S.I.F (KI) Using FEA and theoretical KI(Theo) for different Half Crack

lengths (a)

Fig.7: variation of theoretical and FEA values of S.I.F Vs crack length for longitudinal

crack in Pressurized cylinder

From the Fig 7 it is indicated that the results which were obtained by using the finite

element method are in good agreement with theoretical equation for a longitudinal through crack

emanating in internally pressurized cylindrical shell with an average percentage of error 1.53%

which is negligible. Thus the proposed methodology to determine the Mode-1 S.I.F for

longitudinal cracks in pressurised cylindrical shell is validated against a standard Procedure.

Half Crack

Length (a) mm

Mode -I SIF by

FEA

mmMPa

Mode -I SIF

by Analytical

mmMPa % error

20 821.49 851.322 3.50

40 1282.6 1330.62 3.61

60 1749.6 1828.751 4.33

80 2265.4 2352.63 3.71

100 2831.4 2931.63 3.42

120 3448.9 3555.117 2.99

140 4114.8 4221.655 3.42

180 5570.8 5669.381 1.74

220 7168.4 7247.292 1.09

260 8886.5 8933.523 0.53

300 10711 10710.546 0.00

340 12634 12561.166 -0.58

380 14608 14471.544 -0.94

400 15685 15446.58 -1.54

439.532 17798 17398.911 -2.29



380

5. RESULTS AND DISCUSSIONS

For the problem “Determination of SIF for a Crack Emanating from a Hole in a

Pressurized cylinder” the test models containing a through crack emanating from a hole are

meshed and with plane strain condition it was internally pressurized and respective SIF’s are

calculated.Theoritical Mode-1 is calculated using the relation

mm

mm

Na)Theo(K

2eff1 πσ= …………………… (12)

The variation of normalized Stress Intensity Factor (KI/KO) (By Plane Strain Method)

with respect to a/D ratio [actual crack length (a) to the Diameter of the hole (D)] is as shown in

Figure 8. The normalized SIF (KI/KO) is used to obtain the characteristic curve of SIF which

depends only on the geometrical factor and its variation within the given domain (a/D).

It’s observed that as the crack is near to the hole the stress concentration around holes has

a strong influence on the SIF value. For a/D ratio 0.5 there is a steep rise in SIF KI, this is due to

crack is small and the crack tip is near to stress concentration at the hole from which crack in

emanating. As the crack grows further (for a/D ranging from 0.1 to 25) the crack tip moves far

from the stressed areas hence the value of SIF increases the system will fail with increase in crack

length.

Table 2: The normalized Stress Intensity Factor (KI/KO) with respect to a/D ratio

Half

Crack

Length (a)

mm

Hole

dia(D)

mm

a/D

Mode -I SIF by

FEA(KI)

mmMPa

aeff

mm

Mode -I SIF by

Theoritical(Ko)

mmMPa

KI / Ko

10 20 0.50 743.96 15 686.51 1.08368

20 20 1.00 853.01 20 792.72 1.076059

40 20 2.00 1085.8 30 970.88 1.118372

60 20 3.00 1310.4 40 1121.07 1.168882

80 20 4.00 1540.1 50 1253.40 1.228742

140 20 7.00 2302.5 80 1585.43 1.452284

180 20 9.00 2881.6 100 1772.57 1.625663

200 20 10.00 3191.3 110 1859.09 1.716596

220 20 11.00 3514.8 120 1941.75 1.810118

260 20 13.00 4200.3 140 2097.33 2.002688

300 20 15.00 4932.1 160 2242.14 2.199727

320 20 16.00 5313.5 170 2311.15 2.299075

380 20 19.00 6519.8 200 2506.79 2.600855

400 20 20.00 6938.3 210 2568.70 2.701098



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Fig.8: Variation of normalized S.I.F for a crack emanating from circular hole in

pressurized cylinder Vs a/D

The Deformed Geometries for crack length (a) of 20 mm is as shown in Figures 9, The

maximum Von-Misses stress is found to be at crack tip.

Fig 9: (a) VonMises Stress Distribution for Pressurised cylinder containing hole dia

D=20mm and crack length a=20mm

6. CONCLUSION

The problem of determining stress intensity factors for a crack emanating from a hole in a

pressurized cylinder is of prime importance in damage tolerance analysis. In the present study

ANSYS12, unified FEA software is chosen. It has the required pre-processing capabilities

for finite element modeling and analysis of cracked shell structures as demonstrated in this

paper.



382

The variation of normalized S.I.F (Ki/Ko) with respect to a/D ratio is used to obtain the

characteristic curve of SIF which depends only on the geometrical factor and its variation within

the given domain (a/D).

The fracture mechanics analysis on “effects of pressure bulging near the crack on the

stress intensity factor” is described in this paper that provides an explanation for the lower

strength of cracked cylinders.

The presented stress intensity factors in this paper are essential to predict

(1) Mixed mode fracture under static, dynamic and sustained loads

(2) Residual strength

(3) Crack growth life under cyclic loading conditions.

However there is a clear need to verify the predictions using experimental

investigations, but the method used in this paper can be utilized for calculating the stress

intensity factor for many other loading cases and many values of the crack length. This provides

important information for subsequent studies, especially for fatigue loads, where stress intensity

factor is necessary for the crack growth rate determination.

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[2] Chuin-Shan Chen, Paul A. Wawrzynek, and Anthony R. Ingraffea, 1999, “Crack Growth

Simulation and Residual Strength Prediction in Airplane Fuselages”, NASA/CR-1999-209115.

[3] “Fracture mechanics-Fundamentals and Applications” by T.L.Anderson published by Taylor

and Francis group 2005

[4] Gustavo V. Guinea, Jaime Planas and Manuel Elices, 2000, “KI Evaluation by the

Displacement Extrapolation Technique”, Engineering Fracture Mechanics 66 (2000), pp. 243-

255.

[5] Miloud Souiyah, A. Muchtar, Abdulnaser Alshoaibi and A.K. Ariffin, 2009, “Finite Element

Analysis of the Crack Propagation for Solid Materials” American Journal of Applied Sciences 6

(7), pp. 1396-1402.

[6] Craig A. Barwell, Lorenz.,A Study of Failure in Small Pressurized Cylindrical Shells

Containing a Crack., NASA/CR-1998-208454

[7] Yao-Chen Li, 1984, “The Finite Element Method By Employing The Singular Element With

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pp. 959-972.

[8] Agne Karlsson and Jan Backlund, 1978, “Summary of SIF Design Graphs for Cracks

Emanating From Circular Holes” International Journal of Fracture, Vol. 14, No. 6

[9] Richard D. Young“Non-linear local bending response and bulging factors for longitudinal

and circumferential cracks in pressurised cylindrical shells”.

[10] Kannan.P, K.Balamurugan and K. Thirunavukkarasu, “Experimental Investigation on the

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