influence of a central straight crack on the buckling behaviour of thin plates under tension,...

8/12/2019 Influence of a Central Straight Crack on the Buckling Behaviour of Thin Plates Under Tension, Compression or She…

http://slidepdf.com/reader/full/influence-of-a-central-straight-crack-on-the-buckling-behaviour-of-thin-plates 1/15

Influence of a central straight crack on the buckling

behaviour of thin plates under tension, compressionor shear loading

Roberto Brighenti

Received: 25 November 2009 / Accepted: 31 March 2010 / Published online: 17 April 2010

Springer Science+Business Media, B.V. 2010

Abstract Thin-walled structural components, such

as plates and shells, are used in several aerospace,

naval, nuclear power plant, pressure vessels, mechan-

ical and civil structures. Due to their high slender-

ness, the safety assessment of such structural

components requires to carefully assess the buckling

collapse which can strongly limit their bearing

capacity. For very thin plate, buckling collapse can

occur under shear, compression or even under

tension. In the latter case, fracture or plastic failure

can also take place instead of elastic instability. In thepresent paper, the effects of a central straight crack

on the buckling collapse of rectangular elastic thin-

plates—characterized by different boundary condi-

tions, crack length and orientation—under compres-

sion, tension or shear loading are analysed. Accurate

FE numerical parametric analyses have been per-

formed to get the critical load multipliers in such

loading cases. Moreover the effect of crack faces

contact is examined and discussed. Some useful

conclusions related to the sensitivity to cracks of the

buckling loads for thin plates, especially in the caseof shear stresses, are drawn. Cracked plates under

tension are finally considered in order to determine

the most probable collapse mechanism among

fracture, plastic flow or buckling and some failure-

type maps are determined.

Keywords Cracked plate Buckling Fracture Collapse

1 Introduction

The nucleation of defects such as cracks in thin-

walled structures, can often occur in practical appli-

cations due to cyclic loading, corrosive environment,

imperfect welding, etc. As is well-known, flaws can

heavily affects the safety of thin structures since the

common modes of failure such as buckling, plastic

flow or fracture can occur more easily. If the plate

thickness is sufficiently small with respect to others

plate sizes, buckling collapse under shear, compres-

sion or even under tension (when fracture or plastic

collapse does not precede the buckling one) canoccur, and the presence of cracks can remarkably

modify such an ultimate load.

Thin-walled panels are nowadays used in several

application in different engineering fields such as

aerospace, naval, mechanical, power industries, civil

engineering. In order to reduce economic costs, an

accurate evaluation, in accordance with the so-called

damage tolerant design concepts, is crucial for such

structural components.

R. Brighenti (&)

Department of Civil and Environmental Engineering &

Architecture, University of Parma, Viale G.P. Usberti

181/A, Parma 43100, Italy

e-mail: [email protected]

1 3

Int J Mech Mater Des (2010) 6:73–87

DOI 10.1007/s10999-010-9122-6



Geometrical imperfections such as holes, cracks,

corner and so on, can heavily affect the residual

strength of panels (Guz and Dyshel 2004; Paik 2008,

2009) and must be taken into account in the design of

such structural components.

Several investigations have been carried out in

order to determine the buckling load, responsible of the failure phenomena in damaged or undamaged

plates under shear (Alinia and Dastfan 2006; Alinia

et al. 2007a, b, 2008), under compression (Wang

et al. 1994; Matsunaga 1997; Byklum and Amdahl

2002; Bert and Devarakonda 2003; Paik et al. 2005,

Khedmati et al. 2009) or under tension (Zielsdorff

and Carlson 1972; Markstrom and Storakers 1980;

Sih and Lee 1986; Shaw and Huang 1990; Shimizu

and Enomoto 1991; Nageswara 1992; Riks et al.

1992; Friedl et al. 2000; Guz and Dyshel 2001;

Brighenti 2005a; Paik et al. 2005).The influence of imperfections on the buckling

phenomena in plates and shells have recently been

studied. The influence of holes (Shimizu and Enomoto

1991) and cracks on the buckling load in compressed or

tensioned homogeneous (Markstrom and Storakers

1980; Sih and Lee 1986; Shaw and Huang 1990; Riks

et al. 1992; Estekanchi and Vafai 1999; Guz and

Dyshel 2001; Vafai and Estekanchi 1999; Vafai et al.

2002; Dyshel 2002; Brighenti 2005a; Paik et al. 2005)

or composite plates (Barut et al. 1997) and shells

(Alinia et al. 2007a, b) have been examined. OtherAuthors have considered the case of curved panels such

as tubes (Dimarogonas 1981) or cylindrical shells

under membrane loading (Vaziri and Estekanchi 2006;

Vaziri 2007) weakened by cracks.

The buckling collapse in plates is strongly affected

by the presence of a crack which are strictly related to

the crack length and orientation and to the boundary

conditions of the structures being examined. Even if

buckling under tension seems to be unrealistic, it can

easily appears in common situations as a local

phenomenon that develops in regions around thedefects (such as cracks or holes, Markstrom and

Storakers 1980; Sih and Lee 1986; Shaw and Huang

1990; Shimizu and Enomoto 1991; Riks et al. 1992;

Estekanchi and Vafai 1999; Guz and Dyshel 2001;

Vafai et al. 1999, 2002; Dyshel 2002; Brighenti

2005a; Paik et al. 2005).

In the present paper, the sensitivity to crack length

and orientation of the buckling load of rectangular

elastic thin-plates, characterized by different boundary

conditions, under compression, tension or under shear,

is examined.

In particular, the critical buckling load multiplier

is determined for different values of relative crack

length and crack orientation angle measured with

respect to the loading direction, and for different plate

boundary conditions.The results obtained for tensioned cracked plates

are discussed and used to determine fracture-buckling

and plastic-buckling collapse maps which can be

useful to derive the most probable collapse mecha-

nism of such structures. Finally, some conclusions

regarding the sensitivity of buckling failure to the

above mentioned parameters in cracked plates under

compression, tension or shear loading are drawn, and

the effect of crack face contact is discussed.

2 The buckling phenomena in plates

Since buckling collapse can easily take place in thin-

walled structural elements under membrane stresses,

it must carefully be examined in the design process.

In the buckling phenomena, the second-order geo-

metrical effects must be taken into account (Von

Karman plate’s theory, see Timoshenko and Gere

1961) in the well-known fourth-order partial differ-

ential equation which describes the plate’s deflection

through the function w( x, y),

r4w x; yð Þ ¼ 1

D ð N x w; xx x; yð Þ þ 2 N xy w; xy x; yð Þ

þ N y w; yy x; yð ÞÞ ð1Þ

together with the appropriate boundary conditions. In

Eq. 1 the notations (•),ij indicates the partial deriv-

atives with respect to the spatial variables i ¼ x; y; j ¼ x; y (the medium plane of the plate belongs to the x, y

plane), the operator r4 stands for r4 ¼ð Þ; xxxx þ2 ð Þ; xxyy þ ð Þ; yyyy ¼ o

o x4 þ 2 oo x2 y2 þ o

o y4; N x,

N y, N xy are the membrane forces (per unit plate edgelength) acting in the corresponding directions. Fur-

ther D ¼ E t 3

12ð1 m2Þ is the plate flexural rigid-

ity, and t is the plate thickness.

The solution of Eq. 1 can analytically be obtained

only for some simple plate geometries, boundary

conditions and membrane or shear load distributions,

such in the case of a simply supported compressed

uncracked rectangular plate. The buckling Euler

stress under compression, rE,c, and the buckling

74 R. Brighenti

1 3



Euler stress under shear, sE , for a rectangular plate,

can conveniently be expressed as follows (Timo-

shenko and Gere 1961):

rE ;c ¼ k c p2 D

W 2 t ; sE ¼ k s

p2 D

W 2 t ð2Þ

where k c and k s are the minimum value attained bythe functions k c = k c(W *) and k s = k s(W *)

(W * = W / L being the dimensionless plate aspect

ratio), under compression and under shear, respec-

tively (Figs. 1, 2).

The buckling load multipliers for cracked plates

under compression, tension or shear loading can be

usefully expressed in dimensionless form as follows:

k ¼ r

0

rE ;c \0; kþ ¼

rþ0

rE ;c [ 0; ks ¼

s0

sE j j[0

ð3Þ

It can be observed that the tension buckling stress

multiplier rþ0 in an uncracked plate is usually greater

than the corresponding buckling stress multiplier

under compression, r0 , and therefore buckling col-

lapse in tensioned plates is considered to be an

unrealistic phenomenon. The same conclusion cannot

generally drawn for a flawed plates. As a matter of

fact cracked plates can easily buckle, even for

relatively low values of the applied tension load,due to the compression effects induced around the

flawed area. Such a failure under tension appears as a

local phenomenon localized around the defect, pro-

duced by the transversal compressive stresses which

develop in the material.

On the other hand, structural component subjected

to shear force develops an equal amount of tensile

and compressive principal stresses in the linear

elastic stage. By increasing the shear force up to a

critical value, the buckling phenomena takes place

due to the compressive stresses that develop along thecompressive principal stress direction, and large out-

of-plane displacements in such a direction appear

before collapse.

The described phenomenon is heavily affected by

the influence of the global and local boundaries

effects (crack edges can change their geometry with

deformation or deflection), and this fact complicates

the problem of buckling load estimation, which

usually can be quantified only by mean of approx-

imate methods.

In the present paper several linear bucklinganalyses have been performed by using the FE

method, for different values of the above mentioned

parameters in order to quantify their effects.

The effects of cracks on the buckling behaviour

of plates are shown to not be univocally detri-

mental in members under compression or shear

(sometimes cracks are detrimental to the bearing

plate capacity, and sometimes they are not), while

they always determine a reduction of the critical

load (related to the buckling collapse) in tensioned

members.Moreover it must be taken into account as cracks

can also be responsible of fracture-like collapse under

tension or shear which can occur when the Stress-

Intensity Factor (SIF, Broek 1982) reaches a critical

value K IC (critical Stress-Intensity Factor, i.e. the

fracture toughness of the material under study) at

given environmental conditions such as the operating

temperature. The critical condition for fracture-like

failure can be written as follows:

X

Y

Z

(a)

X

Y

Z

(b)

τ 0

θ

τ 0

τ 0

τ 0

σ 0 σ

0

θ

Fig. 2 Central cracked thin-plate under an unidirectional

normal stress r0 (a) and under a tangential stress s0 (b)

θ

2 L

2 W

t

2 a

X

Y

Z

L

W

Fig. 1 Central cracked thin-plate: geometrical parameters

Influence of a central straight crack 75

1 3



K eqðW ; a; h; r0; s0Þ ¼ K IC ð4Þ

where K eq is the equivalent Stress-Intensity Factor,

which is a function of the elementary Mode I (K I ),

Mode II (K II ) and Mode III (K III ) SIFs, in the general

case of mixed mode of fracture. Elementary modes of

fracture depend on the plate aspect ratio W * = W / L ,the relative crack length a* = a / W , the crack orien-

tation h and the remote applied stresses r0, s0.

In order to consider all the possible failure types,

especially in tensioned thin panels, fracture collapse,

plastic collapse or buckling collapse must be assumed

to potentially occur, and the lowest load level that

produces material collapse—and the corresponding

failure mechanism—must be determined for struc-

tural safety assessment.

3 Buckling in cracked rectangular panels

In the present study the influence of a straight central

crack in a rectangular thin panel under compression,

tension or shear load is analysed (Fig. 2). Different

boundary conditions are assumed for the plate

(Fig. 3): (a) all plate edges simply supported, (b)

two opposite edges simply supported, (c) all plate

edges clamped and (d) two opposite edges clamped.

The plate is assumed to have a rectangular shape with

length, width and thickness equal to 2 L , 2W and t ,respectively. Such a quantities have been kept

constant for all the examined cases. A through-the-

thickness straight crack, characterised by a length 2a

and orientation h (with respect to the direction

orthogonal to the loading direction and considered

as positive if counter-clockwise), is present in the

centre of the plate (Fig. 1). In particular, some

relative sizes of the above geometrical parameters

are assumed as constant: the plate aspect ratio

W * = W / L is assumed to be equal to 1/2, the relative

plate thickness is assumed to be equal to

t ¼ t =W ¼ 1=200. On the other hand, the relativecrack length a* = a / W a ¼ a=W is assumed to be

equal to 0.1, 0.2, 0.3, 0.4, 0.5, and the crack

orientation angle h is assumed to be equal to 0,

15, 30, 45, 60, 75, 90. It must be noted that, in

the case of compression or tension load, the buckling

phenomenon does not depend on the sign of the crack

orientation angle, i.e. the buckling load factor mul-

tiplier is the same for a crack angle equal to ±h

(Fig. 1) since the minimum value of the angle formed

by the crack and by the principal direction of

compressive stress is the same in both cases.On the other hand, for plates under shear, the

buckling load factor multiplier depends on the sign of

the crack orientation angle since the minimum value

of the angle formed by the crack and by the principal

compressive stress direction is different for the two

angles ±h. For such a reason, the following crack

orientation angles have been considered in the case of

shear loading: h = -75, -60, -45, -30 0, 15 ,

30 45, 60, 75, 90. Furthermore it can be noted as

only the cases characterized by h = 0, ±90 are

independent to the sign of the crack orientation angle.The plate material being examined is supposed to

be linear elastic and isotropic, characterized by the

Young modulus equal to E = 70000 MPa and the

Poisson’s ratio equal to m = 0.3 (i.e. the material

represents an aluminium alloy). The effect of the

Poisson’s ratio on the buckling in compressed and

tensioned plates has been considered in Brighenti

(2005a, 2009).

Since the buckling load factor multiplier in

rectangular plates is heavily affected by the plate

aspect ratio W * = W / L , quantified through the coef-ficients k c and k s (see Eq. 2), the study must be

conducted for a specific value of such a parameter. In

Fig. 4 the values attained by the coefficients k c and k sare represented for a plate aspect ratios equal to

W * = 1/10, 1/5, 1/4, 1/3,1/2, 1/1.

As can be noted, the coefficient k c is always equal

to 4.0 for all the considered values of W * only for

simply supported rectangular plates under compres-

sion (it occurs for plate aspect ratio such that the plate

(a) (b)

(c) (d)

Fig. 3 Boundary conditions assumed for cracked thin-plates

under uniform stress: simply supported edges (a), two opposite

supported edges (b), all clamped edges (c) and two opposite

clamped edges (d)

76 R. Brighenti

1 3



size in one direction is a multiple of the size in the

other direction), while it changes for other boundary

conditions. In the case of shear load, the coefficient k s

is always affected by the plate’s aspect ratio and theboundary conditions.

In the present study, the plate aspect ratio is

assumed to be equal to W * = 1/2 (as a representative

case): for such a W * value, the coefficients k c and k sare equal to k c ffi 4:00; 0:23; 7:86 and 0:97 and k s ffi6:54; 0:81; 10:25 and 1:77 in the case of four sup-

ported edges, two opposite supported edges, four

clamped edges and for two opposite clamped edges,

respectively.

Figure 5 shows the critical mode shapes corre-

sponding to the first and second buckling load

multipliers for rectangular cracked plates under

compression (Fig. 5a, b), tension (Fig. 5c, d) andshear loading (Fig. 5e, f). Note that, in the case of

plates under compression or under shear, the out of

plane displacements involve the whole plate, while

the out of plane displacements in the case of tension

are localized in a narrow area located around the

flaw.

The present study is conducted under the hypoth-

esis that no contact between the crack faces occurs

during the pre-buckling loading process, i.e. by

0.11.0 0.2 0.4 0.6 0.8

Plate aspect ratio, W*=W/L

0

2

4

6

8

10

12

14

B u c k l i n g f a c t o r l o a d m u l t i p l i e r , k c

Boundary conditions4 supported egdes

2 supported egdes

4 clamped egdes

2 clamped egdes

ν = 0.3

(a)

0.11.0 0.2 0.4 0.6 0.8

Plate aspect ratio, W*=W/L

0

2

4

6

8

10

12

14

B u c k l i n g f a c t o r

l o a d m u l t i p l i e r , k s

ν = 0.3

(b)Fig. 4 Buckling factors

load multipliers for some

values of the plate aspect

ratio W *, for uncracked

rectangular plates under

compression (a) and under

shear (b), and for different

boundary conditions

Fig. 5 Buckling mode

deflection shapes under

compression (first (a) and

second mode (b)), under

tension (first (c) and secondmode (d)) and under shear(first (e) and second mode

(f )) for W * = 0.5,

a* = 0.5, h = 0


1 3



assuming that the crack is always open. In other

words, the buckling FE analyses have been per-

formed under the assumption of a fully open crack,

and no crack faces contact is assumed to occur

everywhere along the crack edges during the loading

process. In order to validate such hypothesis, some

fully geometrical non-linear analyses—under shearloading (the case of tension and compression have

been considered in Brighenti 2005b)—with possible

contact and friction between the crack faces are also

performed to investigate such an effect on the

buckling phenomena.

Moreover, in order to assess the numerical

convergence of the FE models, the final pattern of

the mesh used for the different buckling analyses has

been selected after several h-convergence tests in

which the mesh density and elements shapes have

been varied. The buckling load of an uncracked plate,discretised with a regular mesh (square 8-noded

elements) with different element densities, and the

final adopted mesh of the cracked plate with closed

crack faces (i.e. ‘‘welded’’ together), have been

compared with theoretical results. The relative error

between the buckling loads, obtained by using the

regular mesh or the adopted mesh for the plate having

a closed crack, and the analytical values has been

found to be less than about 0.8% in both cases.

The obtained results for compressed and tensioned

plates have extensively been reported in Brighenti(2009), as dimensionless buckling load multipliers

(evaluated according to Eq. 3, e.g. by using the

theoretical buckling stress as the reference stress). In

the following, some further results related to cracked

plates under shear are presented.

4 Discussion of the obtained results

The discussion of the sensitivity analysis with respect

to the mentioned variable parameters is conducted byusing some graphical representations, as is described

in the following.

The effect of the crack size is presented in Fig. 6,

where the buckling load multipliers are displayed

against the relative crack length a* for different crack

orientation h, in the case of two (left column in

Fig. 6) and four (right column in Fig. 6) edge

supported plates, for a plate material with m = 0.3.

The compressive load multiplier k- for a given crack

orientation angle h decreases by increasing the

relative crack length a* (a maximum decrease of

about 16% can be appreciated in the range

0.1 B a* B0.5) for a two opposite supported edge

plate (Fig. 6a), while it increases (in absolute value)

by increasing the relative crack length a* (an increase

of about 10–12% can be appreciated in the range0.1 B a* B0.5) for a simple supported edge plate,

especially for relatively low values of the angle h

(h\&60) (Fig. 6b).

Such an unexpected behaviour (i.e. a buckling load

for flawed plates greater than that for the correspond-

ing unflawed ones, kj j ¼ r0

rE ;c

[ 1), can be

justified by considering that the crack acts as a sort of

obstacle with respect to the development of free

deflection waves, as usually occurs in buckled

compressed plates.

Under tension, all the curves show a similar trendirrespectively of the crack orientation h, i.e. the

effects of the crack length are similar for different

angles (Fig. 6c, d). The most important remark is

that, for plates under tension, the crack is responsible

of a reduction of the critical stress with respect to the

case of an unflawed plate. In particular, cracks

orthogonal to the loading direction (h = 0) appear

to be the most dangerous. Results by Riks et al.

(1992) are also reported in Fig. 6d. As can be noted,

the agreement with the present results is good.

In Fig. 6e, f, the case of cracked plates under shearloading is considered for a plate with two opposite

supported edges (Fig. 6e) and for plate with simple

supported edges (Fig. 6f). The crack orientation angle

is made to vary from -90 up to ?90, as discussed

above. It can be observed that, as in the case of two

opposite supported edge plate (Fig. 6e), the curves

decrease by increasing the relative crack length and,

for some crack orientations (such as h ?30, ?45,

?60, ?75, ?90, -75), a local minimum can be

appreciated. A decrease up to about 23%, with

respect to uncracked plates, can be observed forh = 15, 30. Finally, the case of a simply supported

plate is reported in Fig. 6f: a systematic decrease (up

to about 40%) of the buckling load factor multiplier

can be observed for plates characterized by h = 45,

i.e. for a crack parallel to the minimum principal

(compressive) stress direction produced under shear.

Results by Alinia et al. (2007b) are also reported for a

simply supported plate with a crack orientation equal

to h = -45, 0, 45: note that the obtained

78 R. Brighenti

1 3



dimensionless buckling values are not directly com-

parable since Alinia et al.’s results have been obtained

for plates characterized by W * = 1. Nevertheless, the

trends shown by the Alinia’s dimensionless buckling

load are similar to the present results for the corre-

sponding crack orientation angles considered.

In Fig. 7 the same graphical representation of the

dimensionless buckling values is reported in the case

of two (left column in Fig. 7) and four (right columnin Fig. 7) edge clamped plates, for a plate material

with m = 0.3.

The compressive load multiplier k-, for a given

crack orientation angle h decreases by increasing the

relative crack length a* (a maximum decrease of about

15–16% can be observed in the range 0.1 B a* B0.5,

Fig. 7a) for a two opposite clamped edge plate.

On the other hand, the parameter under study

increases (in absolute value) up to about 18–20% for

|h| B &45 while it decreases up to about 10–12%

for |h|[&45, by increasing the relative crack

length a* in the range 0.1 B a* B0.5.

Further, for these considered boundary conditions

being considered, the curves related to plates under

tension show a similar trend, irrespectively of the

crack orientation h.

Panels under shear load are finally considered in

Fig. 7e, f. Forplateshavingtwoopposite clamped edges,a decreasing up to about 14–15% of the corresponding

load multiplier ks, with respect to the unflawed plate, can

be appreciated by increasing the crack length. Further,

note that, for cracks characterized by a* B %0.3 and

30 B h B 90, the buckling load factor multiplier is

approximately equal to 1.0, i.e. the effect of the crack in

such cases is practically negligible.

In Fig. 7f the case of a four clamped edge plate

under shear is examined. Similarly to the case

-1.02

-1.00

-0.98

-0.96

-0.94

-0.92

-0.90

-0.88

-0.86

-0.84

B u c k l i n g

l o a d m u l t i p l i e r , λ

−

θ = 0°

θ = 15°

θ = 30°

θ = 45°

θ = 60°

θ = 75°

θ = 90°

ν = 0.3 (a)

-1.10

-1.08

-1.06

-1.04

-1.02

-1.00

-0.98 ν = 0.3 (b)

100

1000

B u c

k l i n g l o a d m u l t i p l i e r , λ

+ ν = 0.3 (c)

1

10

100 ν = 0.3 (d)

Riks et al., 1992

Relative crack length, a*=a/W

0.75

0.80

0.85

0.90

0.95

1.00

1.05

B u c k l i n g l o a d m u l t i p l i e r , λ

s

θ= 0°

θ= 15°

θ= 30°

θ= 45°

θ= 60°

θ= 75°

θ= 90°

θ=-15°

θ=-30°

θ=-45°

θ=-60°

θ=-75°

ν = 0.3 (e)

0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5

Relative crack length,a*=a/W

0.60

0.70

0.80

0.90

1.00 ν = 0.3 (f)

Alinia et. al., 2007b

W *=1

θ = -45

θ = 0

θ = +45

Fig. 6 Buckling load

multipliers for two

supported edge cracked

plates under compression

(a), tension (c) and shear (e)

and for four supported edge

cracked plates under

compression (b), tension (d)

and shear (f ) against the

relative crack length a*, in

the case m = 0.3. Results by

Riks et al. (1992) are

reported in the tension case

(d) and by Alinia et al.

(2007b) are reported in the

shear case (f )


1 3



reported in Fig. 6f (four supported edge plate), the

crack tends to produce a decrease of the buckling

load multiplier ks up to about 40% for a* = 0.5 and

h = 45. On the other hand, for 60 B h B -15, a

slightly improvement of the buckling load with

respect to the uncracked plate can be observed for

relatively small cracks, 0.1 B a* B 0.3. Results by

Alinia et al. (2007b) are also reported in Fig. 7f for

plates with W * = 0.1 and h = -45, 0, 45. As in

the previous comparisons, the trends shown by thedimensionless buckling load are similar to those of

the present results for the corresponding crack

orientation angles considered.

In Fig. 8 the critical stress multiplier for cracked

plates under shear is plotted against the crack

orientation angle h, for different relative crack length

a*, in the case of a simply supported plate (Fig. 8a)

and a four clamped edge plate (Fig. 8b). It can be

observed that, in the range 15 B h B 45, a

significant reduction of the buckling stresses of

panels having crack lengths a* C 0.3 occurs. On

the other hand, the reduction of the buckling stresses

is very limited for -15 B h B 0, independently of

the relative crack length a* when a* B 0.2, whatever

the value of the angle h is. Results by Alinia et al.

(2007b) are also reported in Fig. 8a. Even if such

Authors consider plates characterised by W * = 1, the

trend shown by the curves is in agreement with that

of the present results.Dimensionless buckling load factor multipliers

under shear, ks, are reported in Table 1 for all the

considered cases.

A summary of the obtained results is graphically

displayed in Fig. 9, for compressed, tensioned and

plates under shear, where the buckling load multipli-

ers (k-, k? and ks) are represented by mean of

buckling sensitivity maps (iso-buckling load contours

represented in the domain (a*, h)), for plates with two

ν = 0.3

(a)

-1.04

-1.00

-0.96

-0.92

-0.88

-0.84

B u c k l i n g

l o a d m u l t i p l i e r , λ

−θ = 0°

θ = 15°

θ = 30°

θ = 45°

θ = 60°

θ = 75°

θ = 90°

ν = 0.3

(b)

-1.20

-1.16

-1.12

-1.08

-1.04

-1.00

-0.96

-0.92

-0.88

10

100

1000


+ ν = 0.3

(c)

1

10

100 ν = 0.3

(d)


0.85

0.90

0.95

1.00

1.05


s

θ = 0°

θ = 15°

θ = 30°

θ = 45°

θ = 60°

θ = 75°

θ = 90°

θ =-15°

θ =-30°

θ =-45°

θ =-60°

θ =-75°

ν = 0.3 (e)

0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5


0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

1.05 ν = 0.3

(f)

Alinia et. al., 2007b

W *=1

θ = -45

θ = 0

θ = +45


multipliers for two clamped

edge cracked plates under

compression (a), tension (c)

and shear (e) and for four

clamped edge cracked

plates under compression

(b), tension (d) and shear (f )

against the relative crack

length a*, in the casem = 0.3. Results by Alinia

et al. (2007b) are reported

in the shear case (f )

80 R. Brighenti

1 3



opposite supported edges under compression

(Fig. 9a) and under tension (Fig. 9b) and for plates

with all supported edges under compression (Fig. 9c)

and under tension (Fig. 9d), in the case m = 0.3. As

can be observed, the buckling behaviour undercompression is affected by the plate boundary

conditions, while it is not affected for plates under

tension. Figs 9e, f, are related to the case of plates

under shear for all supported edges (Fig. 9e) and for

all clamped edges (Fig. 9f): it can be observed that

the iso-buckling curves approximately present an

anti-symmetrical pattern with respect to the axis

h = 0. Further, note that, for h[ 0, the relative

crack length a* heavily affects the buckling load

multipliers ks (the curves are closed to each other),

whereas such an influence is less pronounced forh\0 for both boundary conditions.

As stated in Sect. 3, in order to discuss the

influence of the contact between the crack faces, a

fully geometrical non-linear analyses with unilateral

contact between the crack edges is needed. The case

of compressed or tensioned cracked plates is dis-

cussed in Brighenti (2005b), whereas the cases of

simply supported plates under shear with a crack

characterised by h = ±45 are analysed in the

following. The unilateral contact between the crack

faces, for which friction coefficient equal to 0.4 isassumed, is modelled through no-tension elements

located between the crack edges, and a geometrical

non-linear analysis under displacement control is

conducted by imposing a crescent shear-type dis-

placements to the plate edges. An initial imperfection

has been assigned to the plate medium plane: an

initial small fictitious deflection to the undeformed

plate—corresponding approximately to the first buck-

ling mode shape—characterised by a maximum out

of plane distance, evaluated with respect to the x–y

reference plane, equal to (2 9 10-3 2a) is imposed.

The plate edge displacements are made to vary from

zero up to a value corresponding to a shear stress

equal to about 1.2 7 1.4 times the buckling stressvalue of the case without any contact. The two

significant cases a* = 0.5, h = ?45 and a* = 0.5,

h = -45 are used since the crack in such situations

is normal to the maximum principal (tensile) stress

direction and to the minimum principal (compres-

sive) stress direction, respectively, and thus represent

the two extreme cases of a crack that presumably

does not develop and that develops crack faces

contact, respectively.

In Fig. 10, the results of the fully geometrical non-

linear analyses are displayed for such two casesa* = 0.5, h = ?45 (Fig. 10a) and a* = 0.5, h = -

45 (Fig. 10b). Note that the post-buckling behaviour

is stable as the positive slope of the curves s / s0against w /2a shows: this implies that the plate is not

sensible to initial geometric imperfections, and it can

sustain load levels beyond the buckling load, but with

considerable values of the in- and out-of-plane

displacements.

Moreover, the relative displacement measured

normal to the crack (v /2a, in dimensionless form) is

positive (i.e. the crack faces increase their relativedistance) for the case h = ?45 that implies the

absence of contact. In the case of h = -45, the

relative displacement between the crack faces is zero

since it is prevented by the contact when the crack

faces tend to approach to each other. Finally, the

relative displacement between the middle points of the

crack faces, measured in the direction parallel to the

crack (u /2a, in dimensionless form), can be observed to

be practically equal to zero in both situations.

-90 -75 -60 -45 -30 -15 0 15 30 45 60 75 90 -90 -75 -60 -45 -30 -15 0 15 30 45 60 75 90

Crack orientation angle, θ (°)

0.40

0.50

0.60

0.70

0.80

0.90

1.00

B u c k l i n g l o

a d m u l t i p l i e r , λ

s

a* = a/W

0.1

0.2

0.3

0.4

0.5

(a)

Alinia et. al., 2007b - W *=1

a*=0.1

a*=0.2

a*=0.3

a*=0.4

a*=0.5

ν = 0.3

-π /2 π /2π /4-π /4 0

Crack orientation angle, θ (°)

a* = a/W

0.1

0.2

0.3

0.4

0.5 (b)

ν = 0.3

-π /2 π /2π /4-π /4 0Fig. 8 Buckling load

multipliers against the crack

orientation angle h for

different crack length a*, in

the case of four supported

edge cracked plates (a) and

four clamped edge cracked

plates (b). Results by Alinia

et al. (2007b) are reported

in the case (a)


1 3



It should be noted that, under the critical shear

load in the case a* = 0.5 and h = 45 (in such a case

the minimum compressive principal stress direction is

normal to the crack), the crack faces show a relative

displacement (measured at their middle points from a

simple linear static analysis of the plate) which is

equal to about 1.2 9 10-4a. That is to say, a very

small initial distance between the crack faces is

enough to prevent the contact during the loading

process up to the final buckling load.It should be finally pointed out that the realistic

geometric assumption made for the initial distance

between the crack faces, supposed to be greater than

about 1.2 9 10-4a, gives raise to an ‘‘always open

crack’’ up to the buckling load. Even in the case

a* = a / W = 0.5 and h = -45, this condition is

fulfilled, confirming the hypothesis of no occurrence

of contact between the crack faces.

5 Possible collapse mechanisms under tension

It can commonly be observed in practical applica-

tions that cracked plates under tension can fail due to

buckling, plastic or fracture mechanism. Therefore,

the knowledge of the situation that corresponds to

the lowest critical load is mandatory for safety

evaluation.

Table 1 Buckling load multipliers for cracked plates under

shear load with all supported edges (a), two opposite supported

edges (b), four clamped edges (c) and for two opposite clamped

edges (d)

h ks

a* = 0.1 a* = 0.2 a* = 0.3 a* = 0.4 a* = 0.5

(a) sE j j=E ¼ 3:698 105

-75 0.98988 0.96623 0.93146 0.89514 0.85960

-60 0.99238 0.97593 0.95152 0.92665 0.90287

-45 0.99457 0.98437 0.96930 0.95580 0.94653

-30 0.99575 0.98898 0.98020 0.97589 0.97898

-15 0.99513 0.98566 0.97202 0.96043 0.95078

0 0.99258 0.97340 0.94083 0.90133 0.85260

15 0.98905 0.95608 0.89654 0.81955 0.72943

30 0.98576 0.94025 0.85675 0.75065 0.63705

45 0.98387 0.93187 0.83724 0.71971 0.59930

60 0.98378 0.93336 0.84408 0.73364 0.6183875 0.98518 0.94265 0.87140 0.78454 0.68927

90 0.98740 0.95486 0.90467 0.84781 0.78621

(b) sE j j=E ¼ 4:590 106

-75 0.99962 0.90872 0.88071 0.89005 0.88071

-60 0.99833 0.90849 0.87766 0.88254 0.86730

-45 0.99644 0.90276 0.86600 0.86204 0.83719

-30 0.99446 0.89635 0.85252 0.83781 0.80038

-15 0.99325 0.89219 0.84351 0.82129 0.77440

0 0.99350 0.89276 0.84451 0.82301 0.77637

15 0.99515 0.89316 0.84959 0.83715 0.80291

30 0.99739 0.90001 0.86210 0.86226 0.83964

45 0.99925 0.90589 0.87402 0.88286 0.86948

60 1.00034 0.90903 0.88031 0.89379 0.88516

75 1.00059 0.90966 0.88177 0.89668 0.88990

90 1.00034 0.90894 0.88019 0.89419 0.88694

(c) sE j j=E ¼ 5:791 105

-75 1.00344 0.98633 0.93487 0.87554 0.81289

-60 1.00581 1.00098 0.97052 0.93792 0.91135

-45 1.00754 1.00996 0.98777 0.96245 0.94500

-30 1.00828 1.01671 1.00094 0.98239 0.97430

-15 1.00754 1.02086 1.01178 1.00211 1.00506

0 1.00581 1.01691 1.00322 0.98727 0.97580

15 1.00352 1.00126 0.96314 0.91512 0.86051

30 1.00112 0.97743 0.90195 0.81067 0.71659

45 0.99943 0.95517 0.84677 0.72447 0.61284

60 0.99866 0.94438 0.82166 0.68781 0.57132

75 0.99923 0.94958 0.83609 0.70878 0.59332

90 1.00097 0.96664 0.88192 0.78180 0.68035

Table 1 continued

h ks

a* = 0.1 a* = 0.2 a* = 0.3 a* = 0.4 a* = 0.5

(d) sE j j=E ¼ 9:999 106

-75 0.99882 0.99601 0.99450 0.98287 0.97378

-60 0.99745 0.99117 0.98498 0.96898 0.95638

-45 0.99561 0.98426 0.97011 0.94447 0.92035

-30 0.99390 0.97757 0.95494 0.91802 0.87915

-15 0.99311 0.97438 0.94753 0.90467 0.85755

0 0.99364 0.97644 0.95219 0.91284 0.87030

15 0.99523 0.98246 0.96536 0.93521 0.90434

30 0.99579 0.98864 0.97864 0.95578 0.93150

45 0.99721 0.99436 0.98864 0.96864 0.94435

60 0.99864 0.99721 0.99436 0.97578 0.95007

75 0.99864 0.99864 0.99721 0.98150 0.96007

90

0.99864 0.99721 0.99721 0.98578 0.97236

82 R. Brighenti

1 3



To discriminate which failure mechanism between

buckling or fracture occur first, a collapse function

F col,1 can be introduced. By considering the following

inequality:

rcða; h; K eq; K IC Þ[kþða; hÞ rE ;c ð5Þ

which defines the condition for buckling failure, the

collapse function F col,1 can be written as follows:

0.1

0.2

0.3

0.4

0.5

R

e l a t i v e c r a c k l e n g t h ,

a * = a / W

(a)

π/2π/40

(b)

π/2π/40

Crack orientation, θ (rad)

0.1

0.2

0.3

0.4

0.5

R e l a t i v e c r a c k l e n g t h ,

a * = a / W

(c)


(d)


0.1

0.2

0.3

0.4

0.5

R e l a t i v e c r a c k l e n g t h ,

a * = a / W

(e)

π/2π/40−π/4−π/2

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5


(f)

π/2π/40−π/4−π/2

(e)


multipliers sensitivity maps

for plates with all supported

edges in compression (a) or

in tension (b) and for plates

with all clamped edges in

compression (c) or in

tension (d). Buckling load

multipliers sensitivity maps

for plates under shear with

all supported edges (e) or

with all clamped edges (f )

(a)

0E+000 2E-003 4E-003

Dimensionless out- and in-planedisplacements, w/2a and u/2a, v/2a

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

D i m e n s i o n l e s s s h

e a r s t r e s s ,

τ / τ 0 = τ / ( λ

s |

τ Ε

|

)

u / 2a

v / 2a

w / 2a

a* = 0.5

ν = 0.3

θ = + 45°

w

u

v

τ

τ

τ

τ

(b)

0E+000 2E-004 4E-004 6E-004 8E-004

Dimensionless out- and in-planedisplacements, w/2a and u/2a, v/2a

0.0

0.2

0.4

0.6

0.8

1.0

1.2u / 2a

v / 2a

w / 2a

a* = 0.5

ν = 0.3

θ = - 45°

w

vuτ

τ

τ

τ

(c) (d)

Fig. 10 Load paths

obtained by fully

geometrical non-linear FEanalyses: dimensionless

shear stress againstdimensionless

characteristics plate out- (w /

2a) and in-plane (u /2a, v /

2a) displacements in

cracked plates witha* = 0.5, v = 0.3 and

h = 45 (a) and h = -45

(b). The corresponding

deformed patterns at the end

of the loading process are

displayed for cracked plates

with h = 45 (c) and h =

-45 (d)


1 3



F col;1ða; hÞ ¼ r2c ða; h; K eq; K IC Þ kþ2ða; hÞ

r2E ;c [ 0 ð6Þ

The two possible collapse mechanism are thus

identified by the following cases:

F col;1ða

; h; bÞ ¼ [ 0 buckling collapse

\0 fracture collapse

ð7Þ

The limit case at which the load level can produce

both buckling or fracture collapse at the same time

can be identified by the condition F col,1(a*, h, b) = 0

or explicitly:

F col;1ða; h; bÞ ¼ b2

p a kþ2 K 2

I þ K 2 II

ð8Þ

where the mixed mode of fracture (I ? II) has been

considered, and the coefficient b is introduced to

define the ratio between the critical Stress-IntensityFactor and the Euler buckling stress, b = K IC / rE,c.

By analysing a cracked plate under tension with

four supported edges, the regions corresponding

to buckling collapse in the domain X = (0.1 B

a* B 0.5; 0 B h B p) can explicitly be identified

once the value of the parameter b is assumed. In

Fig. 11, the regions corresponding to buckling col-

lapse (shaded regions, F col;1ða; hÞ[ 0) are displayed

for two different fracture toughness parameter, to

which correspond such two cases: (a) b = 6[m1/2],

(b) b = 25[m1/2

]. The latter value corresponds to thecase of standard aluminium alloy at room tempera-

ture for which K IC % 39.0 [MPa m1/2].

The different collapse regions can be identified as

follows: long cracks (a* C %0.3), oriented with

respect to the loading direction with angles greater

than about p /4, give buckling-type collapse, while

cracks nearly parallel to the loading direction do

not produce buckling collapse. In other words, the

shortest is the crack length, the smaller must be the

orientation angle h to produce elastic instability.

The approximate boundary (dashed line) between the

main shaded area (buckling collapse region) and thewhite area (fracture collapse region) in Fig. 11b can

be described by the equation h0(a*) = -0.923 ?

4.204 a*[rad], that is to say, the buckling occurs

for points for which h\ h0(a*). A secondary minor

buckling collapse area can be identified at the top left

of Fig. 11a, b: such a secondary area becomes more

significant by increasing the ratio b that corresponds

to an increase of the fracture toughness K IC . Such a

region corresponds to crack configurations character-

ized by cracks with small length and oriented nearly

parallel to the loading direction.On the other hand, for high value of the fracture

toughness K IC and low values of the yield stress r y, as

usually occur at elevated temperatures, plastic col-

lapse can easily occur in tensioned thin plates. A

simple estimation of the plastic failure stress r p(obtained by considering the reduced plate section

produced by the crack, measured normal to the

loading direction) can be obtained by assuming an

elastic-perfectly plastic material behaviour :

r pða; h; r yÞ ¼ ð1 a cos hÞ r y ð9Þ

Similarly to the previous case, a buckling-plastic

collapse function F col,2 can be introduced.

From the inequality:

Relative crack length, a* = a/ W

0.2

0.4

0.6

0.8

1.0

1.2

1.4

C r a c k o r i e n t a t i o n a n g l e ,

θ

( r a d ) (a)K IC / σ E,c = ~ 6 m 1/2

0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5


(b)β = K IC / σ E,c =~25 m1/2

π/2

π/4

π/8

3π/8

0

Fig. 11 Regions of

buckling collapse (shaded

areas) and of fracturecollapse (white areas) for

four different fracturetoughness K IC /compressive

critical stress for

compressed uncracked four

supported edge plate ratio

84 R. Brighenti

1 3



r pða; h; r yÞ[ kþða; hÞ rE ;c ð10Þ

which defines the buckling failure, the collapsefunction F col,2 can be written as follows:

F col;2ða; hÞ ¼ r2 pða; h; r yÞ kþ2ða; hÞ r2

E ;c [0

ð11Þ

Finally, two possible collapse mechanism under

consideration can be identified through the following

inequalities:

F col;2ða; h; cÞ ¼ [ 0 buckling collapse

\0 plastic collapse

ð12Þ

The limit case at which the load level can produceeither buckling or plastic collapse at the same time

can be identified by the condition F col;2ða; h; cÞ ¼ 0,

or explicitly:

F col;2ða; h; cÞ ¼ c2 ð1 a cos hÞ kþ2 ð13Þ

where the dimensionless ratio c between the yield

stress r y and the buckling compressive critical stress

rE,c for uncracked plates is shown, c = r y / rE,c [-].

By considering the case of a cracked plate under

tension with four supported edges, the regions

corresponding to buckling collapse in the domainX = (0.1 B a* B 0.5; 0 B h B p) can be explicitly

obtained by knowing the value of the parameter c.

As an example, the 6nnn series (Magnesium–

Silicon) aluminium alloys are considered: for such

alloys, the yield stress can vary from about 60 MPa

(e.g. for AA 6063 T4) up to about 280 MPa (e.g.

for AA 6082 T6). In Fig. 12 the regions correspond-

ing to buckling collapse (shaded regions,

F col;2ða; h; m; cÞ[ 0) are displayed for two different

yield stresses to which correspond the following

values of the parameter c: (a) c = 38 [-] (r y =

60 MPa; rE,c = 1.58 MPa), (b) c = 177 [-] (r y =280 MPa; rE,c = 1.58 MPa).

The location of the buckling collapse regions in

Fig. 12a, b (shaded areas) is similar to that of Fig. 11

for both the considered values of the parameter c: long

cracks (a* C %0.25–0.30), mainly oriented trans-

versely with respect to the loading direction

(0 B h B &p /4), give us buckling-type collapse,

while cracks nearly parallel to the loading direction

can more easily produce plastic collapse. The shortest

is the crack length, the smaller must be the orientation

angle h to produce elastic instability. It can be finallyobserved as the values of the yield stress do not

significantly affect the size and shape of the buckling

region.

By comparing Fig. 11 with Fig.12, it can be

deduced as fracture and plastic collapse easily occur

for similar crack configurations (i.e. for similar values

of the crack length and crack orientation). In practical

cases, the identification of the most dangerous failure

mechanism, i.e. the mechanism which occurs for the

lowest load level, depends on the values assumed for

the mechanical material parameters, such as thefracture toughness, and for the yield stress at given

environmental conditions.

6 Conclusions

In the present paper, the effect of a crack on the

buckling behaviour of variously restrained


0.2

0.4

0.6

0.8

1.0

1.2

1.4

C r a c k o r i e n t a t

i o n a n g l e , θ

( r a d ) (a)γ = σ y / σ E,c = 38 [-]

0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5


(b)γ = σ y / σ E,c = 177 [-]π/2

π/4

π/8

3π/8

0

Fig. 12 Regions of

buckling collapse (shaded

areas) and of plastic

collapse (white areas) for

two different yield stress r y / compressive critical stress

for compressed uncracked

four supported edge plateratio c


1 3



rectangular plates under compression, tension or

shear loading has been examined.

The influence of several geometrical and mechan-

ical parameters on the buckling load has been

analysed by the Finite Element Method (FEM): in

particular, the effects of relative crack length, crack

orientation, boundary conditions on the bucklingphenomena have quantitatively been examined for

the three different mentioned load cases.

The obtained results have shown that the crack

effects on the buckling phenomena under compres-

sive or shearing stress heavily depends on the plate

boundary conditions, while they are almost indepen-

dent of such conditions in the case of tension. On the

other hand, crack length slightly reduces the buckling

load in compressed plates with all supported or

clamped edges, while an improvement with respect to

the uncracked case (up to about 10%) can be found incompressed plates when cracks are transversal to the

loading direction. The critical load multipliers k?

under tension are always higher than the correspond-

ing compressive ones, and cracks have the detrimen-

tal effect to reduce the buckling critical load with

respect to the undamaged case. In such situations,

buckling loads tend to decrease rapidly by increasing

the crack length and by decreasing the crack orien-

tation angle up to the most dangerous case, corre-

sponding to cracks lying orthogonal to the loading

direction (h = 0).The behaviour of cracked plates under shear has

also been discussed: cracks tend to decrease the

buckling load for all the considered boundary condi-

tions, with some exceptions in the case of not

too long cracks (a* B 0.2 7 0.3) characterized by

-60 B h B -15, for which a slight improvement

of the buckling load (with respect to the uncracked

plate) can be observed (i.e. ks[1).

In order to analyse the effect of crack closure on

the buckling behaviour of plates under shear, some

fully geometrical non-linear analyses have beenperformed and discussed. It has been shown as a

very small initial distance between the crack faces is

enough to guarantee the absence of contact during the

loading process up to the critical buckling load under

shear. This validates the use of the linear buckling

analyses to assess the collapse load of thin cracked

plates under shear.

The possibility of fracture or plastic flow failure,

instead of buckling collapse, has been finally

examined for tensioned cracked panels. By evaluat-

ing the lowest collapse load for a given crack

configuration and material mechanical parameters,

the regions of buckling or plastic collapse in the

domain of the considered geometrical parameters that

identify the crack have been determined. From the

performed analyses, it can be deduced that thebuckling rupture can occur (instead of fracture or

plastic flow) for long cracks oriented nearly trans-

versal to the loading direction. Such a conclusion can

be different for structural components made of

materials sensitive to environment conditions, such

as the temperature. As a matter of fact, collapse due

to fracture can precede buckling failure even for short

cracks if the environment temperature is below the

ductile-brittle transition temperature (typically shown

by steel alloys).

Acknowledgements The author gratefully acknowledges the

research support for this work provided by the Italian Ministry

for University and Technological and Scientific Research

(MIUR).

References

Alinia, M.M., Dastfan, M.: Behaviour of thin steel plate shear

walls regarding frame members. J. Construct. Steel Res.

62, 730–738 (2006)

Alinia, M.M., Hosseinzadeh, S.A.A., Habashi, H.R.: Numerical

modelling for buckling analysis of cracked shear panels.Thin-Walled Struct. 45, 1058–1067 (2007a)

Alinia, M.M., Hoseinzadeh, S.A.A., Habashi, H.R.: Influence

of central cracks on buckling and postbuckling behaviour

of shear panels. Thin-Walled Struct. 45, 422–431 (2007b)

Alinia, M.M., Hosseinzadeh, S.A.A., Habashi, H.R.: Buckling

and post-buckling strength of shear panels degraded bynear border cracks. J. Construct. Steel Res. 64, 1483–1494

(2008)

Barut, A., Madenci, E., Britt, V.O.: Starnes J.R.: Buckling of a

thin, tension-loaded, composite plate with an inclined

crack. Eng. Fract. Mech. 58, 233–248 (1997)

Bert, C.W., Devarakonda, K.K.: Buckling of rectangular plates

subjected to nonlinearly distributed in-plane loading.

J. Solids Struct. 40, 4097–4106 (2003)

Brighenti, R.: Buckling of cracked thin-plates under tension

and compression. Thin-Walled Struct. 43, 209–224

(2005a)

Brighenti, R.: Numerical buckling analysis of compressed or

tensioned cracked thin-plates. Eng. Struct. 27, 265–276

(2005b)

Brighenti, R.: Buckling sensitivity analysis of cracked thin

plates under membrane tension or compression. Nucl.

Eng. Des. 239, 965–980 (2009)

Broek, D.: Elementary engineering fracture mechanics. Mar-

tinus Nijhoff Publishers, The Hague (1982)

86 R. Brighenti

1 3



Byklum, E., Amdahl, J.: A simplified method for elastic large

deflection analysis of plates and stiffened panels due to

local buckling. Thin-Walled Struct. 40, 925–953 (2002)

Dimarogonas, A.D.: Buckling of rings and tubes with longi-

tudinal cracks. Mech. Res. Commun. 8, 179–186 (1981)

Dyshel, M.S.: Stability and fracture of plates with a central and

an edge crack under tension. Int. Appl. Mech. 38, 472–

476 (2002)

Estekanchi, H.E., Vafai, A.: On the buckling of cylindrical

shells with through cracks under axial loading. Thin-

Walled Struct. 35, 255–274 (1999)

Friedl, N., Rammerstorfer, F.G., Fischer, F.D.: Buckling of

stretched strips. Comput. Struct. 78, 185–190 (2000)

Guz, A.N., Dyshel, MSh: Fracture and buckling of thin panels

with edge crack in tension. Theor. Appl. Fract. Mech. 36,

57–60 (2001)

Guz, A.N., Dyshel, MSh: Stability and residual strength of

panels with straight and curved cracks. Theor. Appl. Fract.

Mech. 41, 95–101 (2004)

Khedmati, M.R., Edalat, P., Javidruzi, M.: Sensitivity analysis

of the elastic buckling of cracked plate elements under

axial compression. Thin-Walled Struct. 47, 522–536(2009)

Markstrom, K., Storakers, B.: Buckling of cracked members

under tension. Int. J. Solids Struct. 16, 217–229 (1980)

Matsunaga, H.: Buckling instabilities of thick elastic plates

subjected to in-plane stresses. Comput. Struct. 62, 205–

214 (1997)

Nageswara, R.: Instability load for cracked configurations in

plate materials. Eng. Fract. Mech. 43, 887–893 (1992)Paik, J.K., Satish Kumar, Y.V., Lee, J.M.: Ultimate strength of

cracked plate elements under axial compression or ten-

sion. Thin-Walled Struct. 43, 237–272 (2005)

Paik, J.K.: Residual ultimate strength of steel plates with lon-

gitudinal cracks under axial compression–experiments.

Ocean Eng. 35, 1775–1783 (2008)

Paik, J.K.: Residual ultimate strength of steel plates with lon-

gitudinal cracks under axial compression—nonlinear

finite element method investigations. Ocean Eng. 36, 266–

276 (2009)

Riks, E., Rankin, C.C., Brogan, F.A.: The buckling behavior of

a central crack in a plate under tension. Eng. Fract. Mech.

43, 529–547 (1992)

Shaw, D., Huang, Y.H.: Buckling behavior of a central cracked

thin plate under tension. Eng. Fract. Mech. 35, 1019–1027

(1990)

Shimizu, S., Enomoto, S.Y.N.: Buckling of plates with a hole

under tension. Thin-Walled Struct. 12, 35–49 (1991)

Sih, G.C., Lee, Y.D.: Tensile and compressive buckling of

plates weakened by cracks. Theor. Appl. Fract. Mech. 6,

129–138 (1986)

Timoshenko, S.P., Gere, J.M.: Theory of elastic stability, 2nd

ed. edn. McGraw-Hill Book Inc., New York (1961)

Vafai, A., Estekanchi, H.E.: A parametric finite element study

of cracked plates and shells. Thin-Walled Struct. 35, 255–

274 (1999)

Vafai, A., Javidruzi, M., Estekanchi, H.E.: Parametric insta-

bility of edge cracked plates. Thin-Walled Struct. 40, 29–44 (2002)

Vaziri, A., Estekanchi, H.E.: Buckling of cracked cylindrical

thin shells under combined internal pressure and axial

compression. Thin-Walled Struct. 44, 141–151 (2006)

Vaziri, A.: On the buckling of cracked composite cylindrical

shells under axial compression. Compos. Struct. 80, 152–

158 (2007)

Wang, C.M., Xiang, Y., Kitipornchai, S.K., Liew, M.: Buck-ling solutions for Mindlin plates of various shapes. Eng.

Struct. 16, 119–127 (1994)

Zielsdorff, G.F., Carlson, R.L.: On the buckling of thin ten-

sioned sheets with cracks and slots. Eng. Fract. Mech. 4,

939–950 (1972)


1 3

influence of a central straight crack on the buckling behaviour of thin plates under tension,...

Documents