Local limit load solutions for surface cracks in plates

and cylinders using finite element analysis

I. Sattari-Far*, P. Dillstrom

Det Norske Veritas, Consulting, P.O. Box 30234, Stockholm SE 104-25, Sweden

Received 24 March 2003; revised 27 November 2003; accepted 28 November 2003

Abstract

By using non-linear finite element (FE) analysis, limit load solutions of plates and cylinders containing surface cracks are determined. The

study covers both shallow and deep cracks with different crack length/crack depth ratios under different loading types. The crack

configurations consist of semi-elliptical surface cracks with a=t ¼ 0:20; 0.40, 0.60, 0.80 and l=a ¼ 2; 5, 10 in plates and cylinders. Also

studied are plates containing infinite surface cracks with a=t ¼ 0:00; 0.20, 0.40, 0.60 and 0.80. The cracked plates are subjected to pure

tension, pure bending and combined tension and bending. The cracked cylinders are subjected to internal pressure. Based on FE results

obtained from this study new limit load solutions are developed for these crack configurations.

q 2004 Elsevier Ltd. All rights reserved.

Keywords: Limit loads; Finite element method; Surface cracks in plates; Axial surface cracks in cylinders

1. Introduction

Information on the plastic limit load of a cracked body is

important in a structural integrity assessment. The plastic

limit loads can be determined using a number of methods.

Miller [1] published a comprehensive reference on limit

load solutions in different crack geometries, mostly based

on analytical analyses. The analytical limit load solutions

are generally shown to be overly conservative for

components with part-through-thickness defects. Wil-

loughby and Davey [2] presented analytical solutions of

limit loads in cracked plates subjected to tension and

bending loads. They gave two solutions, based on different

assumptions about the boundary conditions of the problem.

In their so-called pin-joint solution, it is assumed that the

applied tensile load causes bending in the cracked section. If

it is assumed that there is some rotational restraint and the

bending effect of the applied tensile load is carried

externally, i.e. not by the cracked section, it yields a higher

value of limit load. This is called the rigid-restraint limit

load. These two solutions are expressed in the following

forms.

For the pin-joint solution:

2

3

sb

sf

þ 2zsm

sf

þsm

sf

� �2

¼ ð1 2 zÞ2 ð1Þ

For the rigid-restraint solution:

2

3

sb

sf

þsm

sf

� �2

¼ ð1 2 zÞ2 ð2Þ

where sm is the applied tensile or membrane stress, sb the

applied bending stress and sf the flow strength of the

material. The parameter z is a crack geometry factor, which

converts a three-dimensional (3D) surface crack to a long

extended 2D crack, and is defined as:

z ¼

al

tðl þ 2tÞfor 2W . l þ 2t

al

2tWfor 2W , l þ 2t

8>><>>: ð3Þ

where a is the crack depth, l the crack length, t the plate

thickness and 2W the plate width (see Fig. 1).

Sattari-Far [3] studied limit loads in plates containing

surface cracks using finite element (FE) analysis. He

showed that the pin-joint solution is in general overly

conservative, and the rigid-restraint solution is overly

conservative for deep cracks under bending. Based on

0308-0161/$ - see front matter q 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijpvp.2003.11.015

International Journal of Pressure Vessels and Piping 81 (2004) 57–66

www.elsevier.com/locate/ijpvp

* Corresponding author. Tel.: þ46-8587-94077; fax: þ46-8651-7043.

E-mail address: iradj.sattari.far@dnv.com (I. Sattari-Far).

a limited number of FE analyses, he presented Eq. (4) as a

limit load solution for components containing surface

cracks of a=t # 0:7 under tensile and bending loads.

2

31 2 20

a

l

� �0:75

j3

" #sb

sf

þsm

sf

� �2

¼ ð1 2 jÞ2 ð4Þ

This solution released some over-conservatism in the

rigid-restraint solution expressed in Eq. (2).

The objective of this study is to develop limit load

solutions in plates and cylinders containing shallow and

deep surface cracks under different loading conditions. Non-

linear FE calculations are used to develop new limit load

solutions for these crack configurations.

2. Crack geometries and loading

Plates and cylinders containing surface cracks subjected

to different loading types are considered. The crack

geometries cover both finite semi-elliptical and infinitely

long extended surface cracks with a depth variation from 0

to 80% of the wall thickness. The loading types consisted of

pure tensile, pure bending and combined tensile and

bending for the plates, and internal pressure for the

cylinders. For the plates, the tensile loading is applied by

a uniformly distributed tension stress on a surface remote

from the crack plane. The bending load is applied by a

linearly distributed stress with a maximum bending stress sb

on a surface remote from the crack plane. For semi-elliptical

cracks in plates, the applied bending stresses are such that

they cause tension on the crack-ligament. For plates with

infinite surface cracks, applied bending loads of both

tension and compressive stresses on the crack faces are

considered. The different crack configurations, 3D semi-

elliptical and 2D infinite surface cracks, studied in this

report are as below.

For surface cracks in plates:

Semi-elliptical cracks: a=t ¼ 0:20; 0.40, 0.60, 0.80

with l=a ¼ 2; 5, 10

Infinite cracks: a=t ¼ 0:00; 0.20, 0.40, 0.60, 0.80

Loading type: pure tension, pure bending and

combined

For internal surface cracks in cylinders:

Axial semi-elliptical cracks: a=t ¼ 0:20; 0.40, 0.60,

0.80 with l=a ¼ 2; 5,10

Loading type: internal pressure

Limit load solutions are usually estimated for defects in

non-work-hardening materials. Adjustments to allow for the

work-hardening capacity of real materials are then made by

replacing the yield strength by the flow strength of the

material. The flow strength sf is generally accepted to be the

mean value of the yield strength sY and ultimate strength

sU obtained in a conventional tensile test. For the analysis

of the 2D cases, an elastic–perfectly plastic material model

is used. For the analysis of the 3D cases, to reduce the

computing time and to eliminate convergence difficulties

close to the limit loads, a material model with a very weak

hardening effect is used.

Tension and/or bending stresses are applied on the plate

ends as shown in Fig. 1. The plate length is chosen large

enough compared with the plate width and thickness so that

the remote applied stresses do not cause any disturbances on

the crack plane. The geometrical and material data used in

this study are as below:

Plate thickness, t : 50 mm

Ratio a=t for plates: 0.0–0.8

Plate width, 2W : 1.5 ðl þ 2tÞ

Nomenclature

2H plate length

2W plate width and cylinder length

a crack depth

E elastic modulus

Et plastic tangent modulus

l crack length

Lr measure of proximity to plastic collapse in the

R6-method

p internal pressure in cylinder

Ri cylinder inner radius

Rm cylinder mean radius

t plate and cylinder thickness

z equivalent crack depth

sb bending stress

sbL limit load in pure bending

sf flow strength

sL limit load

sm membrane stress

smL limit load in pure tension

sU ultimate strength

sY yield strength

Fig. 1. Schematic on the crack geometry in a plate and the stress distribution

due to tensile and bending loads.

I. Sattari-Far, P. Dillstrom / International Journal of Pressure Vessels and Piping 81 (2004) 57–6658

Plate length, 2H : six times the plate width

Cylinder inner radius, Ri : 500 mm

Cylinder thickness, t : 50 mm

Ratio a=t for cylinders: 0.2–0.8

Cylinder length: 1.5 ðl þ 2tÞ

Elastic modulus, E : 200 GPa

Plastic modulus, Et : E=10; 000

Yield strength, sY : 300 MPa

The analyses are performed for monotonically increasing

loads up to the limit load.

3. Finite element modelling

The general purpose finite element method (FEM)

program ABAQUS [4] was used for the computations

reported in this study. The pre-processor of the FEM

program ANSYS [5] was used to develop the 3D FEM

models. The material is assumed to be completely or almost

elastic–perfectly plastic, obeying the von Mises flow

criterion with its associated flow rule.

For the 3D analysis of the surface cracks in plates, due

to symmetry in geometry and load, only one quarter of

the cracked plate needed to be modelled. For the analysis

of the axial surface cracks in cylinders, it was assumed

that two cracks were located symmetrically in the

cylinder. The sizes of these cracks in relation to the

dimensions of the cylinder were such that they did not

affect each other, thus one-eighth modelling of the

cracked cylinder was sufficient for the FEM analysis.

The cylindrical model is obtained by mapping a plate

model. The 3D FE models used for the analysis of the

plates and cylinders are shown in Fig. 2. The 3D model

consisted of 2040 twenty-noded solid elements with a

total of 29,820 degrees of freedom.

Plates containing infinite edge surface cracks were

studied using 2D FE analyses. Limit load solutions for

both plane-stress and plane-strain conditions are obtained

for different loading cases. For cases with compressive

bending stresses on the crack faces, contact elements were

used on the crack surfaces. The 2D model consisted of 1586

eight-noded plane-stress or plane-strain elements with a

total of 9704 degrees of freedom.

Effects of the material modelling, element type and small

versus large strain formulation were studied by analysis of a

plate containing a surface crack of a=t ¼ 0:60 and l=a ¼ 5

under pure tension. It was observed that it would be difficult

to reach the limit load using the large strain theory in the

analysis. The deviations of the limit loads obtained from the

material models with Et=E ¼ 0 and 1/10,000 were less than

1%. The report prepared by Dillstrom and Sattari-Far [6]

gives more details on this pre-study.

4. Definition of limit load

Conventional limit load analysis calculates the global or

net-section limit load, at which displacements become

unbounded. This corresponds to the maximum load-bearing

capacity of a cracked component. A global limit load

solution may be used for assessment of components

containing through-thickness cracks. For part-through-

thickness cracks, the limit load may be taken conservatively

as a local limit load, which is the load needed to cause

plasticity to spread across the remaining ligament of the

crack plane. The fracture assessment procedure in the R6-

method [7,8] recommends the use of a local limit load in

engineering fracture assessment of cracked components. In

determination of limit loads by using FE calculations, one

important question is how to define the limit load. To study

this and to define an unambiguous definition of the limit

load, the following study was performed.

The development of plasticity in a cracked body depends

mainly on crack geometry, loading type and the imposed

boundary conditions. This development is studied by using a

2D FE analysis of a plate containing an infinite edge crack

of a=t ¼ 0:40 under pure tension. 2D analyses to develop

limit load solutions can be performed either under a plane-

stress or a plane-strain (deformation) condition. Fig. 3

shows the spreading of the plastic zone in the FE model at

two load levels under pure tension, when plane-strain

conditions are assumed. It is observed that the development

of the plastic zone occurs in a direction inclined (about 458)

Fig. 2. Three-dimensional finite element model of a surface crack in a plate.

I. Sattari-Far, P. Dillstrom / International Journal of Pressure Vessels and Piping 81 (2004) 57–66 59

to the crack plane. For a crack plate under plane-strain

conditions, a thickness yielding and not a crack-ligament

yielding is reached at the limit load. Based on this analysis, a

value of smL=sY ¼ 0:565 may be allocated for the plane-

strain limit load of this crack configuration, which is

substantially higher than smL=sY ¼ 0:362 obtained from the

plane-stress analysis.

The development of plasticity in a cracked plate

containing a finite surface crack under pure bending is

studied by using a 3D FE analysis. The crack is semi-

elliptical with a=t ¼ 0:60 and l=a ¼ 5: Fig. 4 shows the

spreading of the plastic zone at three different load levels.

It is observed that at a load level of sb=sY ¼ 1:22; some

sort of thickness yielding is obtained, while some parts of

the ligament are still in elastic conditions. At a load

level of sb=sY ¼ 1:26; a local crack-ligament yielding is

reached, but the load level can still be increased. Finally

at a load level of sb=sY ¼ 1:36; the maximum load-

bearing capacity of the plate is reached, and the plasticity

is spread entirely on the crack plane section. The variation

of the crack-mouth-opening-displacement (CMOD) versus

the applied loads for this case is shown in Fig. 5. It is

observed that a local thickness yielding is likely to occur

before crack-ligament yielding in crack configurations of

finite surface types. The load level corresponding to a

local thickness yielding may be substantially lower than

the load level that causes a local crack-ligament yielding.

For such crack configurations, the load level that causes

a local crack-ligament yielding somewhere along the

crack front, as illustrated in Fig. 5, may be a reason-

able determination of the limit load of the component.

Thus, for cracked bodies containing partially penetrated

surface cracks:

The limit load is defined as the load needed to cause a

local crack-ligament yielding somewhere along the crack

front.

5. Limit loads of cracked plates

Limit loads of plates containing infinite edge cracks are

obtained using 2D FE analysis. Both plane-stress and plane-

strain conditions are studied. Table 1 summarises infor-

mation on the crack configurations and normalised limit

loads obtained from this analysis. The crack depths vary

from a=t ¼ 0 (plates with no cracks) to deep cracks with

a=t ¼ 0:80: Five different loading types are studied,

including pure tension, pure bending and combined tension

and bending. For bending loads, two different cases are

considered. In the first case, the bending load causes tension

stress on the crack side of the plate, and is denoted as

bending (þ ) in Table 1. In the second case, the bending load

causes compressive stress on the crack side of the plate, and

is denoted as bending (2 ) in Table 1. The combined loading

is performed assuming equal tension and bending stresses

with proportional increase under loading. As expected the

sign of bending load does not affect the limit loads in the

uncracked plates, but does for the cracked plates. It is

observed that the plane-stress analysis yields limit load

Fig. 3. Development of the plastic zone size at two different load levels in a

plate with an infinite edge crack of a=t ¼ 0:40 under pure tension and plane

deformation conditions.

I. Sattari-Far, P. Dillstrom / International Journal of Pressure Vessels and Piping 81 (2004) 57–6660

Fig. 4. Development of the plastic zone size at three different load levels in a plate with a surface crack of a=t ¼ 0:60 and l=a ¼ 5 under pure bending (see also

Fig. 5).

Fig. 5. Variation of the applied load and CMOD in a plate with a surface crack under pure bending (see also Fig. 4). Also illustrated is ligament yielding in

definition of the local limit load.

I. Sattari-Far, P. Dillstrom / International Journal of Pressure Vessels and Piping 81 (2004) 57–66 61

values, which are substantially lower than those from plane-

strain analysis.

Limit loads in plates containing finite surface cracks

are obtained by using 3D FE analysis. The crack

configurations have semi-elliptical shape with depths

varying from shallow cracks of a=t ¼ 0:20 to deep cracks

of a=t ¼ 0:80: For each crack depth, three different crack

lengths of l=a ¼ 2; 5 and 10 are studied. The loading

types include pure tension, pure bending and proportion-

ally combined tension and bending, applied on the plate

end remote from the crack plane. The limit load is the

load that causes a local crack-ligament yielding in the

crack plane, before the plasticity extensively spreads in

the model. The maximum plasticity in the FE models at

the allocated limit loads was of order less than 1%, with

exception for elements along the crack front. In general,

it is observed that the 3D analysis of finite surface cracks

yields limit load values, which are substantially higher

than that from 2D analysis, especially for deep cracks.

For instance, for a plate with a surface crack of

a=t ¼ 0:80 and l=a ¼ 10 under pure bending, the 3D

analysis gives the limit load sbL=sY to be 0.940, while

the 2D plane-strain analysis of an infinite crack of

a=t ¼ 0:80 gives sbL=sY ¼ 0:088:

Fig. 6 shows variations of limit loads in cracked

plates under pure tension as a function of crack depth for

different l=a values. Fig. 6(a) shows this variation as a

function of a=t; and Fig. 6(b) as a function of the

parameter z; evaluated from Eq. (3). Also given in Fig.

6(b) are the corresponding solutions from Eqs. (1), (2)

and (4). It is observed that for shallow cracks ða=t # 0:2Þ;

the effect of the crack is immaterial, and the plate has a

limit load value effectively equal to the limit load for the

plate without a crack. Fig. 7 shows the corresponding

results for plates under pure bending, and Fig. 8 the

corresponding results for proportionally combined load-

ing. The same trends as in Fig. 6 are observed in Figs. 7

and 8. It is also observed that Eqs. (1) and (2) are overly

conservative for deep cracks, and Eq. (4) is valid only up

to z ¼ 0:6:

Effects of combined loading are studied in more detail

for two crack configurations with a=t ¼ 0:40 and 0.80

Table 1

Limit load solutions in plates containing infinite surface cracks under

different loading types obtained from 2D finite element analyses

Case a=t Loading sL=sY

PS PD

P2D-a0-T 0.00 Tension 1.000 1.150

P2D-a0-B1 0.00 Bending (þ ) 1.325 1.500

P2D-a0-B2 0.00 Bending (2) 1.325 –

P2D-a0-TB1 0.00 Combined (þ) 0.598 –

P2D-a0-TB2 0.00 Combined (2) 0.596 –

P2D-a05-T 0.05 Tension 0.950 –

P2D-a1-T 0.10 Tension 0.890 –

P2D-a2-T 0.20 Tension 0.725 0.905

P2D-a2-B1 0.20 Bending (þ ) 1.030 1.365

P2D-a2-B2 0.20 Bending (2) 1.325 –

P2D-a2-TB1 0.20 Combined (þ) 0.490 –

P2D-a2-TB2 0.20 Combined (2) 0.596 –

P2D-a4-T 0.40 Tension 0.370 0.565

P2D-a4-B1 0.40 Bending (þ ) 0.582 0.788

P2D-a4-B2 0.40 Bending (2) 1.327 –

P2D-a4-TB1 0.40 Combined (þ) 0.242 –

P2D-a4-TB2 0.40 Combined (2) 0.596 –

P2D-a6-T 0.60 Tension 0.136 0.208

P2D-a6-B1 0.60 Bending (þ ) 0.260 0.352

P2D-a6-B2 0.60 Bending (2) 1.330 –

P2D-a6-TB1 0.60 Combined (þ) 0.092 –

P2D-a6-TB2 0.60 Combined (2) 0.595 –

P2D-a8-T 0.80 Tension 0.027 0.039

P2D-a8-B1 0.80 Bending (þ ) 0.065 0.088

P2D-a8-B2 0.80 Bending (2) 1.330 –

P2D-a8-TB1 0.80 Combined (þ) 0.019 –

P2D-a8-TB2 0.80 Combined (2) 0.452 –

PS stands for plane-stress and PD for plane deformation.

Fig. 6. Limit loads in plates containing finite semi-elliptical surface cracks

under pure tension.

I. Sattari-Far, P. Dillstrom / International Journal of Pressure Vessels and Piping 81 (2004) 57–6662

having l=a ¼ 5: The results are presented in Table 2,

where normalised limit loads corresponding to five

different loading types are given for these two crack

configurations. The combined loading was applied in

three different manners. In the first manner, tension and

bending stresses were increased proportionally under

loading, denoted as combined (prop.) in Table 2. In the

second manner, a fixed tension stress was applied to the

model, and the bending stress increased monotonically

until the limit load was achieved, denoted as combined

(fixed sm). The fixed values of the tension stress were

0:50sY for the shallow crack ða=t ¼ 0:40Þ and 0:30sY for

the deep one. In the third manner, a fixed bending stress

was applied to the model, and the tension stress increased

monotonically until the limit load was achieved, denoted

as combined (fixed sb). The fixed values of the bending

stress were 0:50sY for the shallow crack and 0:30sY for

the deep one. The results are also shown in Fig. 9, where

equations of the yield loci obtained from curve fitting of

the results are shown.

6. Limit loads of cracked cylinders

Cylinders containing internal axial surface cracks are

studied to determine the limit loads. The cracked cylinders

are subjected to internal pressure, with no crack face

pressure. The applied internal pressure is monotonically

increased in the FE model until a local crack-ligament

Table 2

Limit load solutions in plates containing finite surface cracks under

combined tension and bending ðl=a ¼ 5Þ

Case a=t z Loading ðsm þ sbÞL=sY

P-a40-l5-T 0.40 0.200 Pure tension 0.93 þ 0.00

P-a40-l5-B 0.40 0.200 Pure bending 0.00 þ 1.39

P-a40-l5-TB0 0.40 0.200 Combined (prop.) 0.68 þ 0.68

P-a40-l5-TB1 0.40 0.200 Combined (fixed sm) 0.50 þ 1.00

P-a40-l5-TB2 0.40 0.200 Combined (fixed sb) 0.81 þ 0.50

P-a80-l5-T 0.80 0.533 Pure tension 0.71 þ 0.00

P-a80-l5-B 0.80 0.533 Pure bending 0.00 þ 1.03

P-a80-l5-TB0 0.80 0.533 Combined (prop.) 0.50 þ 0.50

P-a80-l5-TB1 0.80 0.533 Combined (fixed sm) 0.30 þ 0.87

P-a80-l5-TB2 0.80 0.533 Combined (fixed sb) 0.61 þ 0.30

Fig. 7. Limit loads in plates containing finite semi-elliptical surface cracks

under pure bending.

Fig. 8. Limit loads in plates containing surface cracks under proportionally

combined tension and bending.

I. Sattari-Far, P. Dillstrom / International Journal of Pressure Vessels and Piping 81 (2004) 57–66 63

yielding in the crack plane is achieved. In all cases studied

here, the cylinder inner radius is assumed to be 10 times the

cylinder thickness. Limit loads of the cracked cylinders are

obtained using 3D FE analysis. The crack configurations

have semi-elliptical shape with depths varying from shallow

cracks of a=t ¼ 0:20 to deep cracks of a=t ¼ 0:80: For each

crack depth, three different crack lengths of l=a ¼ 2; 5 and

10 are studied. The cracked cylinders are subjected to

internal pressure. No pressure is applied on the crack

surfaces.

For cylinders under internal pressure, the hoop stress is in

general not a pure membrane stress but varying along the

cylinder thickness. However, for thin-walled cylinders with

Ri=t $ 10; this variation is relatively small (less than 10%

the maximum hoop stress), and the hoop stress component

sh may be assumed constant along the wall of the cylinder

having a magnitude expressed in Eq. (5).

sh ¼Rm

tp ð5Þ

Here, Rm is the mean radius and t the thickness of the

cylinder and p the applied internal pressure.

Depending on the boundary conditions imposed on the

cylinder, internal pressure may cause two different stress

states in the wall of the cylinder. For a closed cylinder, the

internal pressure causes a biaxial stress state (consisting of

the hoop and axial stress components) on the wall of the

cylinder. For an open thin-walled cylinder under internal

pressure, the stress state is uniaxial, consisting of a

membrane hoop stress in the wall of the cylinder. The

effects of these stress states on the limit loads shL (the limit

hoops stress applying on the cracked section) are studied for

four different crack depths having the same value of l=a ¼ 5:

The results are as follows:

a=t ¼ 0:20 : shL=sY ¼ 1:022 for open cylinder, and

shL=sY ¼ 1:134 for closed cylinder.

a=t ¼ 0:40 : shL=sY ¼ 1:013 for open cylinder, and

shL=sY ¼ 1:060 for closed cylinder.

a=t ¼ 0:60 : shL=sY ¼ 0:966 for open cylinder, and

shL=sY ¼ 0:976 for closed cylinder.

a=t ¼ 0:80 : shL=sY ¼ 0:850 for open cylinder, and

shL=sY ¼ 0:866 for closed cylinder.

It is observed that the closed cylinders (biaxial stress

states) yield higher limit loads. As the main purpose of this

study is to develop limit load solutions for cracked bodies

under uniaxial loading, the presented limit load values of

this crack geometry are determined under assumption that

the cracked cylinders are open and only the hoop stress

component contributes to the limit load. The FE model is

subjected to an increasing internal pressure, until a ligament

yielding as stated in Section 4 is obtained. Eq. (5) is used in

determination of the limit load values.

Fig. 10 shows variations of limit loads as a function of

crack depth of different l=a values in open cracked cylinders

under internal pressure. Fig. 10(a) shows this variation as a

function of a=t, and Fig. 10(b) as a function of the parameter

z; evaluated from Eq. (3). Also given in Fig. 10(b) are the

corresponding solutions from Willoughby and Davey [2]

and Sattari-Far. It is observed that for shallow cracks ða=t #

0:2Þ; the effect of the crack is immaterial, and the cracked

cylinder has a limit load effectively equal to the limit load

for the cylinder without crack. It is also observed the both

the Willoughby and Davey and Sattari-Far solutions give

overly conservative limit load values for intermediate and

deep cracks. Comparison between results, presented in Figs.

6 and 10, indicates that the limit load of a surface crack in a

cylinder under hoop stress is higher that the limit load of a

plate containing the same crack geometry under pure

tension. This is mainly due to the fact that the cylindrical

geometry causes a rotational restraint on the crack plane.

7. Developments of new limit load solutions

7.1. Surface cracks in plates

Considering the results presented in Figs. 6–8, it is

observed that the presented solutions, Eqs. (1)–(4) are

Fig. 9. Limit load solutions in plates containing surface cracks of a=t ¼ 0:40

and 0.80 under different combined loads (see also Table 2).

I. Sattari-Far, P. Dillstrom / International Journal of Pressure Vessels and Piping 81 (2004) 57–6664

overly conservative, especially for bending. It is also

observed that the conservatism of these equations increases

with increasing crack depth. Thus, the following form is

assumed to govern the limit load solutions of these crack

configurations.

2

3ð1 2 zÞA

sb

sf

þ ð1 2 zÞBsm

sf

� �2

¼ ð1 2 zÞ2 ð6Þ

One condition here is that the expression should also

approach the 2D solutions when a=l approaches zero. Fitting

all FE results of cracks in plates under different loading

types to Eq. (6) gives values of exponents A and B to be 1.58

and 1.14, respectively. Thus, the limit load solutions for

cracks in plates under different loading types are rec-

ommended to be:

2

3ð1 2 zÞ1:58 sb

sf

þ ð1 2 zÞ1:14 sm

sf

� �2

¼ ð1 2 zÞ2 ð7Þ

Limit load results of Eq. (7) compared with the FE results

for different loading types as a function of z are shown in

Fig. 11. It is observed that Eq. (7) is in good agreement with

the FE results for all three loading types.

Eq. (7) is valid for cracked plates of 0 # a=t # 0:8 and

2W . l þ 2t:

7.2. Internal axial surface cracks in cylinders

Considering the results presented in Fig. 10, it is

observed that the presented solutions, Eqs. (1)–(4), are

overly conservative. It is also observed that the conserva-

tism of these equations increases with increasing crack

depth. In addition, cracks in cylinders give higher limit load

values than those in plates, mainly due to the rotational

restraints in the cylindrical geometries. Thus, the following

form is assumed to govern the limit load solutions of these

crack configurations under membrane stresses.

ð1 2 zBÞAsm

sf

� �2

¼ ð1 2 zBÞ2 ð8Þ

Fitting the FE results of axial surface cracks in

cylinders under membrane stresses to Eq. (8) gives values

of exponents A and B to be 0.1 and 3.11, respectively.

Fig. 10. Limit loads in cylinders containing internal axial surface cracks

subjected to pure hoop stresses.

Fig. 11. Comparison of Eq. (7) and FE limit loads for surface cracks in

plates under different loading types.

Fig. 12. Comparison of Eqs. (7) and (9) and FE limit loads for axial surface

cracks in cylinders under membrane stress.

I. Sattari-Far, P. Dillstrom / International Journal of Pressure Vessels and Piping 81 (2004) 57–66 65

Thus, the limit load solutions for internal axial surface

cracks in cylinders under membrane stresses are rec-

ommended to be:

ð1 2 z3:11Þ0:1sm

sf

� �2

¼ ð1 2 z3:11Þ2 ð9Þ

Limit load results of Eq. (9) compared with the FE results

and Eq. (7) as a function of z are shown in Fig. 12. It is

observed that Eq. (9) is in good agreement with the FE

results, and Eq. (7), which is a plate solution, gives

conservative results for axial surface cracks in cylinders

under membrane stresses.

Eq. (9) is valid for cylinders containing internal axial

surface cracks of 0 # a=t # 0:8 and 2W . l þ 2t:

7.3. Internal circumferential surface cracks in cylinders

In order to verify the limit load assumptions made in

this study, a comparison is also made against the

analytical solutions for circumferential surface cracks in

cylinders given by Delfin [9]. This comparison is shown

in Fig. 13, and as can be observed, the agreement is very

good.

8. Conclusions

Limit load solutions of plates and cylinders containing

surface cracks are determined using non-linear FE analysis.

The study covers both shallow and deep cracks with

different crack length/crack depth ratios under different

loading types. Based on this study, the following con-

clusions can be made.

† The limit loads of cracked geometries can be determined

by using FE computations with a very low work-

hardening material.

† It is observed that local thickness yielding is likely to

occur before crack-ligament yielding in crack configur-

ations of finite surface types, and the load level

corresponding to a local thickness yielding may be

substantially lower than the load level that causes crack-

ligament yielding.

† For cracked bodies containing partially penetrating

surface cracks, the limit load is defined to be the load

needed to cause plasticity to spread locally somewhere

across the crack-ligament of the crack plane.

† New limit load solutions are developed on the basis of

FEM results for surface cracks in plates and cylinders.

The new solutions significantly reduce the conservatism

observed in some limit load solutions presented in the

literature.

Acknowledgements

This work is sponsored by the Swedish Nuclear Power

Inspection (SKI) and the Swedish Nuclear Power Plant

owners. Their support is greatly appreciated.

References

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Fig. 13. Comparison of Delfin’s solutions [9] and FE limit loads for

circumferential surface cracks in cylinders under pure tension loading.

I. Sattari-Far, P. Dillstrom / International Journal of Pressure Vessels and Piping 81 (2004) 57–6666