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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: brigh@unipr.it
1 3
Int J Mech Mater Des (2010) 6:73–87
DOI 10.1007/s10999-010-9122-6
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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
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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
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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)
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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
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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
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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 )
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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
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)
Relative crack length, a*=a/W
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
0.65
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
Fig. 7 Buckling load
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 )
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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)
Influence of a central straight crack 81
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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
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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)
Crack orientation, θ (rad)
(d)
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
(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
Crack orientation, θ (rad)
(f)
π/2π/40−π/4−π/2
(e)
Fig. 9 Buckling load
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)
Influence of a central straight crack 83
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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
Relative crack length, a* = a/ W
(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
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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
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)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
Relative crack length, a* = a/ W
(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
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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).
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