2. 198 A. G. Milh'r
r notch root radius
t thickness
u b/(b + r)
x a/t in plate; meridional coordinate in cone
y 1-x
F force
Lr load/limit load for proportional loading
M plate: bending moment/length; cylinder: moment
N plate: tensile force/length
P pressure
Q mode II shear resultant
R radius of sphere or cylinder
S mode III shear resultant
:~ notch angle (0 for sharp crack, ~/2 for plain bar)
/~ semi-angle of circumferential crack in cylinder
7 2/x/3 = 1"155
q fractional ligament thickness in shell (equivalent to y in plate)
p o/(RI) 1/2
ar (G + G)/2
G ultimate tensile strength
G uniaxial yield stress
shear stress
~b meridional angle in shell
1 INTRODUCTION
1.1 Failure analysis
The two-criteria method for assessing defects ~ (called R6 from here on)
provides a method of interpolating between plastic collapse and fracture
governed by linear elastic fracture mechanics. An accurate assessment of
plastic collapse would take into account material hardening, finite strain and
finite deformation effects. Commonly, however, a simpler assessment is
performed using limit analysis, and neglecting these effects. This note gives a
list of available limit analysis solutions for c o m m o n structural geometries.
Limit analysis may also be used for assessing other fracture parameters.
The elastic-plastic parameter J may be assessed by reference stress methods
using the limit load. 2"3 The creep crack growth parameter C* may be
estimated by assuming that the creep stress distribution is similar to the
stress distribution at the limit load.* The reference stress itself is used to
assess continuum damage due to creep. 5
3. Review of limit loads of structures containing defects 199
1.2 Limit analysis
Limit analysis calculates the maximum load that a given structure made of
perfectly plastic material can sustain. The loading is assumed to vary
proportionally with a single factor. The maximum sustainable load is called
the limit load, and when this load is reached the deformations become
unbounded and the structure becomes a mechanism.
The effect of large deformations is not considered in the solutions given
here (except in the case of axial thrust on nozzles in spheres).
Complete solutions are hard to calculate, but bounds may be obtained by
using the two bounding theorems. A lower bound to the limit load is
obtained by a statically admissible stress field satisfying equilibrium and
yield, and an upper bound is obtained by a kinematically admissible strain
rate field satisfying compatibility and the flow rule. Usually a safe estimate
of the load-carrying capacity of a structure is required, and a lower bound is
appropriate. Sometimes, however, R6 is used in an inverse manner to assess
the maximum defect size that would have survived a proof test. Then an
upper bound may be appropriate.
1.3 Geometries considered
Most of the solutions given here are effectively two-dimensional, being
derived from plane strain, plane stress or thin shell assumptions. As far as
possible a uniform notation has been maintained, but the notation has been
repeated for each geometry to avoid confusion. The limit loads have been
made non-dimensional by referring them to the limit load of the unflawed
structure, or to the load given by a uniform stress across the ligament.
The different geometries are considered in order of increasing structural
complexity; that is, plates, cylinders, spheres, pipe-bends, shell/nozzle
intersection. Within each of these geometries different defect geometries are
considered. Where possible, an analytical representation of the results is
given. Where this is not possible, the results are presented graphically for a
range of geometries.
1.4 Experimental verification
Where available a comparison is given between theory and experiment. Care
must be taken that the experiments are indeed governed by plastic collapse.
For small testpieces made from aluminium or mild steel, however, brittle
fracture is demonstrably unimportant if
4. 200 A. G. Miller
where K: is the appropriate critical stress intensity factor, ~y is the yield stress
and a is a characteristic length such as defect size or ligament size. If ligament
fracture or m a x i m u m load is used, then the flow stress should be used to
normalize the result (see Section 1.5). Alternatively, deformation-based
criteria can be used, and in this case the yield stress should be used as
normalization. The deformation (or strain) definition given by A S M E 6 is as
follows:
NB-3213.25 Plastic Analysis-Collapse Load. A plastic analysis may be
used to determine the collapse load for a given combination of loads on a
given structure. The following criterion for determination of the collapse
load shall be used. A load-deflection or load-strain curve is plotted with
load as the ordinate and deflection or strain as the abscissa. The angle that
the linear part of the load-deflection or load-strain curve makes with the
ordinate is called 0. A second straight line, hereafter called the collapse limit
line, is drawn through the origin so that it makes an angle q~= t a n - t(2 tan 0)
with the ordinate. The collapse load is the load at the intersection of the
load-deflection or load-strain curve and the collapse limit line. If this
m e t h o d is used, particular care should be given to ensure that the strains or
deflections that are used are indicative of the load-carrying capacity of the
structure.
Depending on the choice of deformation/strain c o m p o n e n t and location,
the value of the collapse load will vary. Gerdeen v recommended using the
generalized displacement conjugate to the load in order to remove this
ambiguity.
1.5 Material and geometric hardening
Limit analysis ignores the hardening of the material, and so a choice must be
made as to what value of stress to use in the limit solution. To evaluate the L r
parameter for R6 Rev. 3 the 0"2% p r o o f stress t~y should be used. The cut-off
at L~ ax is based on the flow stress (ay + o,)/2, except where higher values may
be justified. For the assessment of C - M n steels given in Appendix 8 of R6
Rev. 3, the S, parameter is evaluated using the flow stress (oy + au)/2 again.
Geometry changes lead to both hardening and softening, for example:
(1) plates under lateral pressure become stiffer as membrane effects arise;
(2) tension produces thinning which may lead to instability;
(3) compression gives rise to buckling (limit point or bifurcation);
(4) meridian line changes in pressurized nozzles increase stability.
1.6 Approximate solutions
For structures where there is no existing limit solution, the solution may be
calculated, using one of the bounding theorems, or determined experi-
5. Rev&w of limit loads of structures containing defects 201
mentally. A c o m m o n way of calculating a lower bound solution for a shell is
to calculate the stresses that would be present if the structure were uncracked
and elastic (which is always possible using finite elements) and then to take
the elastic value of the stress resultants across the cracked section and use the
appropriate limit load expression for a plate in plane stress or strain under
combined tension and bending.
The Tresca plane stress limit solutions depend only on the plane of the
ligament. In perfect plasticity it is permissible to have local discontinuities in
the other components of the stress resultants, and hence their elastic values
may be ignored. If desired, they may be specifically taken into account by the
method in Section 2.12.
In plane stress with the Mises yield criterion, or in plane strain with either
the Tresca or Mises yield criterion, the stress field caused by the defect
extends a distance of order t from the defect. If this is small compared with
the characteristic shell distance (Rt) t/2, then the higher plane strain limit
solution may be more appropriate than the plane stress limit solution. The
Tresca plane stress solution relies on the ligament plane being free to neck
down. If it is constrained from doing this, say by shell curvature, then it will
be in plane strain. R6 Rev. 3 recommends the plane stress solutions in
general.
A less conservative estimate is given for short cracks by taking the stresses
calculated elastically for the cracked body. This is discussed further in
Section 1.8.
1.7 Multiple loading
If it were desired to combine the solutions here with other types of loading
for the same geometry, a conservative estimate may be derived from the
convexity lemma (see, for example, Ref. 8). Consider a set of independent
load parameters p~. Then proportional loading is defined by
Pi = /'PiO
where 2 is a variable andpio is constant in any particular case. There will be a
unique value of 2 = 2 o at which equilibrated plastic flow takes place. There
will be such a value 2 corresponding to every set of ratios pi o. Hence 2o,pi o
defines a yield point loading surface in the multi-dimensional load space.
The convexity lemma states that this yield point loading surface is convex, as
a consequence of the convexity of the yield surface. Hence the planes
through the intersections of the yield point loading surface with the axes
form an inscribed surface, and a lower bound to the limit load is given by the
criterion
Piy
i
6. 202 A. G. Miller
where Ply are the limit loads under a single type of load. The displacement
boundary conditions must be the same in each case. For example, results are
given here for the limit loads of plates under combined tension and bending.
However, if the limiting values of the stress resultants under pure tension
(and zero moment) and pure bending (and zero tension) are given by
INI ~< No and IM[ ~< Mo
respectively, then a lower bound to the limit load under combined tension
and bending is given by
INI + [MJ
N--~ M o <~1
1.8 Global and local collapse loads
Conventional limit analysis calculates what may be called the 'global'
collapse load, at which displacements become unbounded. However, in
elastic-plastic structures, the plastic strains at the ligament may become
large long before the global limit is reached, and hence an estimate of the
'local collapse load' at which gross plasticity occurs in the ligament may be
more relevant to ligament fracture.
In the case of through cracks, there is obviously no ligament to yield
before general yield, but it is conceivable that there may be a local instability.
Miller 9 considered surface defects in tension in plates and steels (see Fig. 1)
and concluded that ductile failure occurred at a nominal strain of
kt-at
E--
4 a c
where k is a material constant between 0.4 and 1-5 for the mild steels
considered. If e is greater than the material strain at the flow stress, then the
cut-offat L maxis described by the structure limit load. Ire is less, then the cut-
offshould be taken at a load based on a reduced flow stress. Ire is less than
the yield strain, then the local collapse load should be used with the flow
stress.
Q
2c
Fig. 1. Geometry of surface defect.
7. Review of limit loads of structures containing defects 203
2h
.[
0 0 0 0
@ ® @ @
T
Fig. 2. Geometry of embedded defect.
Ewing '° considered an eccentric defect in mode III (Fig. 2) and
constructed the failure assessment line using a critical crack tip opening
displacement criterion. He concluded that this agreed well with the R6
diagram using the global collapse load, although there was a small dip inside
the diagram at the local collapse load. Bradford tt derived a simplified
plastic line spring model to calculate J for surface defects. Only one
numerical example is given, and in this the global collapse load gives a better
reference stress than does the local collapse load.
Miller t2 reviewed published calculations of J at surface defects for plates
in tension, cylinders with circumferential defects in tension and cylinders
with axial defects under pressure. In all cases the global collapse load gave
better reference stress J estimates than did the local collapse load but the
number of results was small.
R6 recommends the use of the local collapse load, as it is conservative.
There are a large number of test results for L~ ax which show that this
conservatism may be relaxed at ductile instability, but the evidence in the
elasto-plastic r~gime is still limited.
This issue may be resolved by performing a J-integral calculation (as in R6
Rev. 3). The ligament behaviour should be controlled by J. If this is
estimated by the reference stress method, the appropriate limit load to use
remains to be resolved.
1.9 Defect characterization
The existing codes, which give rules for defect characterization (Refs 13, 14
and R6), give rules which are based on LEFM. At present there is little
8. 204 A. G. Miller
information about how defects should be characterized for the purpose of
assessing plastic collapse. However, it can be stated that if the defect size is
increased, the plastic collapse load cannot be increased, so circumscribing a
defect with a bigger effective defect is always conservative.
Miller t5 considered ductile failure test results for a variety of multiple
defect geometries and concluded that the code characterization was always
conservative. For purely ductile failure, net section area was a valid method
to use, and thin or multiple ligaments did not need any special treatment
except for that described in Section 1.8.
The limit solutions available in the literature for notched plates consider
the geometries with a finite root radius, or a V-shaped notch with any given
flank angle. The stress intensity factor is only relevant for sharp, parallel-
sided notches, and in practice defects are characterized for assessment
purposes as being of this form.
1.10 Yield criteria
The most c o m m o n l y used yield criteria are Tresca and von Mises:
Tresca max {]0.2 - 0.3], ]0.3 - atl, 10.1- 0._,1}= %
2
Mises (0.~ + 0. 5 ...[_0.2) __ (0.20.3 + 0"30"1 -'{'-0"10.2) m 0.:,,
or (0.~1+ a~z + a~3) -{022033 + 0.330"11 "~-0.110"22) + 3(0.-~3+ air + 0.~2) = Cry
0.~ principal stresses 0"~j stress components ay uniaxial yield stress
It can be shown that the difference in limit load given by these yield surfaces
is
N/3 3
0"866L~t" = 2 L~I, <~ Lr~< L,~I. ~ < 5 5 L T = l ' 1 5 5 L T
where L r and Lxl are the Tresca and Mises limit load respectively. In practice
this difference is small compared to other factors, and the choice is usually
made on grounds o f convenience.
1.11 Yield criteria for plane stress and plane strain
The yield criteria for plane stress (a 3 = 0) and plane strain (% = 0) are shown
in Fig. 3.
For plane stress the yield surfaces are plotted by putting 0.3 = 0 in the
above yield surfaces, to give a hexagon (Tresca) or an ellipse (Mises) as in
Fig. 3. For plane strain the condition % = 0 implies that
(0.1 + 0.'2)
0"3-- 2
9. Rerie,' ~I" limit loads o/'structures containing defects 205
PLone stroin
/
//
C[2,,. / Plane ~ ( ' ~ e'e
~" ,/Mises stress , ~,e~
, I: /,/¢"
, 2 "
IJ/,,"
"
t /
f#
Fig. 3. Plane stress and strain yield criteria.
and consequently Je~ - 02] is constant. It equals ay for Tresca and 1"155cr, for
Mises. The yield surface is thus two parallel lines as in Fig. 3.
As the plane strain yield surface circumscribes the plane stress yield
surface for both Tresca and Mises, the plane strain limit load is always
higher. Moreover, as the Mises plane strain surface may be obtained by
scaling the Tresca plane strain surface by a factor of 2/x/3, the limit loads are
in the same ratio. This increase in limit load is described by the constraint
factor c:
L
Tresca plane strain c=
LTo
L
Mises plane stress c=
/-'To
Mises plane strain c=
,/3L
2LT~
where L is the appropriate limit load and LT~,is the Tresca plane stress limit
load. Hence the two plane strain constraint factors are the same but the
plane strain Mises limit load is 1-155 times the plane strain Tresca load. It can
be seen from Fig. 3 that
1 ~< c ~ 1.155 Mises plane stress
whereas in plane strain the constraint factor is unbounded. The constraint
factor may also be regarded in tensile cases without bending as the ratio of
the average stress to the yield stress (or 1-155ay for plane strain Mises).
10. 206 A. G. Miller
1.12 Yield criteria for shells
Shell calculations are done using the tensile and bending stress resultants
rather than stresses. As the relationship between the yield criteria for stress
resultants and those for stresses are complicated, simplified yield criteria are
commonly used for shell stress resultants. The shell is in a state of plane
stress, and the commonest criterion is the two-moment limited interaction
yield surface shown in Fig. 4. Hodge 8 shows that
0"618LL ~<LT ~<LL 0"618Lt. ~< L,4 ~< I'155LL
where L:4 = Mises limit load, LT = Tresca limit load and t L = two-moment
limited interaction limit load.
-1
I
m2 1
i !
Fig. 4. Two-momentlimited interaction yield surface for shells.
However, as the bending and stretching are rarely significant simul-
taneously, the approximation is often better than implied by the inequalities.
However, the limits may be achieved in simple loading cases when both
bending and stretching are important. The origin of this factor 0"618 may be
illustrated by considering the case of a plain beam. The results in Section
2.4.1 show that for Tresca plane stress
( ~ r t ) 2 + 4a-~-2=1
When
N 4M
~yl~Gyl 2
this gives
N 4M x/5_- 1_0.618
ay--~=cryt2 = 2
11. Review of limit loads of structures containing defects 207
Using the limited interaction yield surface gives
N 4M
o-rt o-yt2 -
If this yield surface is used without the 0"618 factor being applied, the
absence of simultaneous bending and stretching should always be checked
for. The collapse of a shell under boss loading (point force) provides a
counter example in which bending and stretching arise simultaneously (in
the h o o p direction). This is shown by Ewing's results discussed in Section
13.2.
For thin shells in a m e m b r a n e stress state the Mises and Tresca yield
criteria give the same result for equibiaxial stresses, as in a pressurized
sphere:
P = 2ayt both Tresca and Mises
R
When the two principal stresses are not equal, Mises gives a higher limit
load, the difference being at its m a x i m u m when the stress components are in
a ratio of 2:1, as in a closed pressurized cylinder:
P = tryt
R Tresca P = ~2tryt Mises
If the two principal stresses are of opposite sign, then the Mises/Tresca
ratio reaches 2/,,/3 at (1, - 1,0) and is between 1 and 1-155 for other values.
2 S I N G L E - E D G E N O T C H E D PLATES
These have been extensively studied. Most work has been done on the plane
stress and plane strain cases rather than finite crack lengths. This is of more
relevance to test specimen geometries than to structures. The plane strain
case can be analysed by slip-line field theory which gives an upper b o u n d
when the solution is not complete. 'Complete' means that a statically
admissible stress field has been extended into the rigid regions adjacent to
the plastic regions. Sharp cracks are considered first. A review of limit loads
for these is given by Haigh and Richards, ~6 and a review of test results is
given by Willoughby} 7 The effect of notch root radius and flank angle is also
considered. These can only reduce the limit load, as material is being
removed compared to the sharp crack geometry.
For elastic material with a .V notch, the power of the stress singularity
alters, and with a rounded root the stress singularity becomes a finite stress
concentration. Hence in neither case is the conventional stress intensity
12. 208 A. G. Miller
factor, strictly speaking, a valid parameter. In practice defects are usually
assessed pessimistically assuming them to be sharp:
a defect length M moment/width
b ligament thickness N force/width
r root radius Q mode II shear force/width
t thickness S mode III shear force/width
u b/(b+r) ay yield stress
x a/t ~t notch angle
y 1-x
The geometry is shown in Fig. 5.
QS
Q
u
N
IM M':M+I/2 Na
i
a
t t
a
y:_l - a~- : I - ~
~. ligament
I
plate
I I N Tension
I i M Bending moment
I I Q Mode TT shear
S Mode TIT shear
---N ~
-- Q
X S
_ ¢1
Fig. 5. SEN geometry.
13. Review of limit loads of structures containing defects 209
2.1 S E N B pure bending ( N = 0)
2.1.1 Plane stress Tresca
4M(x_____~) (1 - x) 2 = y2
= 0 ~< X ~< 1
O'yt 2
2.1.2 Plane stress Mises ~8
Deep cracks
4 M ( x ) _ 1.072(1 - x) z = 1-072y z x > 0.154
O'yt 2
T h i s result is o n l y valid f o r d e e p c r a c k s a n d m u s t be c o n t i n u o u s with the
u n n o t c h e d b e a m result:
4M(0)
- - = 1
O-yt 2
T h e value o f the validity limit o n x is t a k e n f r o m O k a m u r a et al. 19
2.1.3 Plane strain Tresca
D e e p c r a c k s 2°
4M
= 1"2606(1 - x ) 2 1'2606v 2 x > 0"295
~yt 2
Shallow c r a c k s 2
4 M ( x ) = [ 1.261 - 2"72(0"31 - x)2](l - x) z x < 0"295
o'rt 2
= [1 + 1"686x - 2"72x2](1 - x) z
1 - 0"31x x--*0
T h i s is an a n a l y t i c a p p r o x i m a t i o n to within 0"5% to the values given in
T a b l e 1. T h e results are s h o w n g r a p h i c a l l y in Fig. 6.
2.1.4 Plane strain Mises
T h i s is 1"155 times the p l a n e strain T r e s c a result.
2.2 S E N T tension ( M = 0, pin loading)
2.2.1 Plane stress Tresca 22
N
- n(x) = I-(1 - x) 2 + x23 ~/2 - x 0~<x~<l
tTyt
= [1 - 2x + 2 x 2 ] 1/2 - X
n ~ 1 - 2x x~0
y2
n ....~ _ _ y-,0
2
14. c: Z,N ( Tresco )
% (t-al 2
1.3
1.2
1.1
1.0 [ I I
o .1 .2 .3 o / t
Fig. 6. Constraint factor for SEN plate in bending in plane strain.
TABLE !
Limit Moment Plane Strain for Single-edge Notched Plate
(from Ewing 21)
a/t = x c (1 - - x ) ' c a/t = x c (I --x)2c
0"296 1'261 0'625 0"089 1'125 0.934
0'258 1"255 0-691 0"065 1.095 0-956
0.249 1.244 0'739 0"060 1.090 0.963
0'197 1"226 0-791 0'036 1-056 0.981
0.164 1'200 0"839 0"017 1.028 0'993
0-130 1' 169 0"885 0.004 I '008 1'000
0"096 1'133 0'926 0 1 1
x = a / t fractional crack depth.
c = constraint factor:
4M 4M
ayt2(l_x)2 (Tresca) 1.155a/Z(l_x)2 (Mises)
15. Review of limit loads of structures containing defects 211
This is the same as in the plain b e a m result f o r c o m b i n e d tension a n d
bending, with the m o m e n t given by the eccentric tensile force o n the
ligament:
Na
M=
2
2.2.2 Plane stress Mises 22"z3
Deep cracks
N_N_=n(x)=[-[ " )'-- 1'~ 2 _ x ) Z ] l/z f 7--1'~
.., Lt-.+--r-) +7(1 - t?x--~--) x >0"146
= --7(l+y)x+v(l+v)x 2 -- 7x--
7y 2 2
for y - ~ O n~ 1 +7=0.536y2 7 - - - ~ r ~ = 1.155
I f 7 is put equal to unity, the Tresca result is recovered.
Shallow cracks
N
= n(x) = 1 -- x -- x 2 x < 0"146
o'rt
T h i s is an a p p r o x i m a t i o n to the t a b u l a t e d results in E w i n g and
R i c h a r d s 2z'za agreeing to within 0-15%:
n-+l -x x--*0
2.2.3 Plane stra& Tresca
D e e p cracks 22'z3
N
- - = n = 1-702{ I-(0-794 - 392 + 0"58763 ,2] 1/2 - [-0-794 - y]} x > 0.545
O'yl
n ~ 0"6303 '2 y~0
Shallow c r a c k s z4
N
-- n(x) x < 0.545
• o'yt
where
n(x)/> 1 -- x - 1.232x 2 + x 3 - f ( x )
and
n(x) ~ f ( x ) + 22x3(0.545 - x) 2
n ----~ 1 - - x x~0
16. 212 A. G. Miller
The pin-loaded limit forces are shown graphically in Fig. 7. This also
shows the results of plane stress tests on mild steel specimens by Ewing and
Richards.22"23
2.2.4 Plane strain Mises
This is 1"155 times the Tresca plane strain result.
2.2.5
K u m a r et aL 25 give values for the limit loads which are the Tresca plane
stress results renormalized to give the correct result as x ---, 1. They are not the
correct limit load and are not recommended for use. The variation of their
h(n,x) functions with n would be reduced if they were normalized with
respect to the correct limit load as a function of x. (If reference stress theory
were exact, the variation would vanish.)
2.3 SEN tension with restrained rotation (fixed grip)
2.3.1 Plane stress Tresca and Mises, and plane strain Tresca
N
~yt
For Tresca plane stress this result may be derived by putting M = -½Na
into the expressions given in Section 2.4. The negative moment, shallow
crack combined bending and tension solution is not available for the other
cases, however.
This is compared with the pin-loading results in Fig. 7.
Z0
i
1.0 ',~ ---Io It U--19- o l -
I. l J
P Tresca plane strain ] Pin-
0.8
~"~,C,,._ M Mises 1 -. ~ loading
N e~'" T Tresco t wtane|
O'yt 06
~ Ist~ess/
~'~ ~" ~,F • Experimental J )
~',,~ "% ~ . _F_ PLanest.,s~ onO rres=o plan*
0/*
0.2
0 I l I L, I l l F ~ "~"'~I
0.1 0.2 03 0.~, O.S 06 0.7 08 0.9 I0
alt
Fig. 7, The theoretical and experimental variation of yield load with notch length for single-
.
edge notched (SEN) specimens (from Ewing and Richards ....~3 ).
~
17. Review o[ limit loads of structures containing defects 213
2.3.2 Platte strain M i s e s
This is 1"155 times the Tresca plane strain limit load.
2.4 SEN combined tension and bending
This case may be derived from a transformation of the pin-loaded results, by
a method suggested by Ewing. 22"23 Equivalent results are given by Rice 26
and Shiratori and Dodd. 2~ Proportional loading is assumed. The results are
only valid for deep cracks. The signs are positive for forces and m o m e n t that
tend to open the crack. The effect of crack closure has been ignored:
applied load
Lr = limit load (in R6 Rev. 3 notation with limit load based on or)
( t -- a) N
3'~ - 2 M + N t (for M = 0, y, = 3')
(2M + Nt)q(y~) ),2 N
Lr = crr(t -- a) z q(Y~) n(y~) n(.v) = --tryt
where n is the appropriate function (Tresca or Mises plane stress or plane
strain) taken from Section 2.2 for the pin-loaded case and ),e is the effective
fractional ligament thickness as defined above.
It follows from this that in all cases the results in Section 2.1 obey
4M
°'rt 2 * 2nO') as ) , ~ 0
That is, the tensile force for very deep cracks is governed by the m o m e n t due
to the eccentricity of the ligament.
These results may be rewritten in terms of the m o m e n t referred to the
centre-line of the ligament:
M' = M + Na/2 L~ LGr(t _ a) 2 a,(t- a) q(Y~)
N/( t -- a)
)'e = ( 2 M ' ) / [ ( t - a) 2] + N / ( t - a)
This shows that only the stresses referred to the ligament affect the limit load.
The thickness t has no effect, provided that the crack is sufficiently deep. The
criterion for sufficient depth will now depend on the ratio N / M , and this
must be considered separately for each case. For Tresca plane stress the deep
crack solution is always valid. For Mises plane stress the shallow crack
solution is unknown. For Tresca or Mises plane strain the shallow crack
solution is discussed in Section 2.4.5.
18. 214 A. G. M i l l e r
2.4.1 Plane stress Tresca
The results m a y be written
2 M + Na + [(2M + Na) 2 + N2(t - a) 2] t2
Lr = O'y(t -- a) 2 0< x < 1
This is identical to eqn A2.4.4 in R6 Rev. 2. It is identical to the u n n o t c h e d
b e a m result:
V + - ? - =1
with a c c o u n t being taken o f the effect o f ligament eccentricity:
M---, M + Na/2
As the square root m a y have either sign, and plastic collapse m a y occur in
either tension or compression, the expression for L r m a y be rewritten
I2M + Na[ + E(2M + Na) " + NZ(t - a) z] 1/2
t r=
a~(t -- a) z
where now the positive square root sign is always taken.
2.4.2 Plane stress M i s e s
The a n a l o g o u s results apply. The deep crack validity limits are given by
O k a m u r a et al. 19
O.1540{l+N/[ay(t-a)]} if N
x> xo = - - <0"5475
1 + O'1540{N/[ay(t - a)]} ay(t - a)
N
x > x o < 0-220 if - - > 0"5475
o'y(t - a)
M !
cry (t -o )z .--.'_--_" Upper bound
Ewing a n d Richords
1.2-
" Lower b o u n d
Okomuro etal.
- ~"~."~=~'~ / o/t = 0 22
" V. ol t = 0.1
N~g.. o/t = 055
~N~. a / t = 0.0
%
V I I I 1 I I 1 I 1 I i I 1 t I I i I~ I
-lO -0.5 0.5 1.0
M= = M q.1/2No
Fig. 8. Limit moment and force for SEN plate (from Okamura et al. 1 9 ). Plane stress Mises.
19. Review of limit loads of structures containing defects 215
This depth limit agrees with the value OfXo = 0-154 for pure bending given in
Section 2.1.2 at the validity limit:
N N
x=0-146 ~ = 0-832 - - = 0.974 > 0.5476
ayt ar( t -- a)
This therefore satisfies x 0 = 0.146 < 0-220 and is consistent with the above.
O k a m u r a derived lower bounds for shallow cracks. These are shown in
Fig. 8.
2.4.3 Plane stra& T r e s c a 24
The analogous results may be rewritten for deep cracks (where 'deep' will
be defined later) in terms of x = a/t:
~deep' cracks (in terms of Ye), i.e. bending-dominated
q(ye) = 0.794 - .re + [-(0.794 - 3,)2 + 0.588y2] 1/2 y, < 0"455
"shallow' cracks (in terms of yo), i.e. tension-dominated
q(y~) ~< v~ > 0"455
y~ -- (re + 0"232)(1 -- ye) 2 "
q(Y~) > y2
Ye -- 0', + 0"232)(1 --y,)2 + 22(1 - - ) ' e ) 3 ( y e -- 0"455) 2
These expressions are shown in Fig. 9.
The crack depth limit is given by
N 6M
x>0-4 x>0 M=0 x>0-295 N=0
l 12
The transitional value of 0"4 is the m a x i m u m for all values o f M / N t (i.e. the
deep crack solution is valid for all M / N t if x > 0.4).
Ewing's expressions were developed for the positive tension, positive
bending quadrant. The solution for all sign combinations is shown in Fig. 10
for deep cracks.
An alternative representation of the bending-dominated r6gime is given
by Shiratori and Miyoshi: 29
m " = 1-26 + 0.521 n" - 0-739(n") 2 0 ~< n" ~ 0"551
where
4M' N
a,(t - a) 2 a,(l - a)
20. 1.tl
2 M . Nt
1.6
L r = (t-o) 2 cry q(Y)
1.4
1.2
q(y)
1.0
Upper bound /
0.~,
Lower bound
06 to q
I I l I
02 0t. 06 08
y: (t-o)/ (t.2MIN)
Fig. 9. Limit moment and force for SEN plate (from EwingZ'~). Plane strain.
..... Rice's upper bound
..... S h i r o t o r i a n d Dodd field
Rice opproximote
expression
,.,
1.0
O.S m " -= 4 M'
/
, 0.6 o-y (t_o)Z
/ ,,/ O.Z. . . . . / . n ~* =
- N
/ O.Z Cry ( t - o )
/
-~.l-o.s -o.~'-o.'4--o'..Zo2 o z o.~ o.~ o.a/o n"
,,/
i,~ -0.6
-0.II ""
'.~ -"
Fig. i0. Combined bending and tension for deep-cracked SEN plate in plane strain (from
Nicholson and Paris~S).
21. Review o f limit loads o f structures containing defects 217
TABLE 2
SEN Plane Strain Upper Bound for Shallow Cracks
(from Ewing 24)
N
Values of ~ (Tresca) or - - (Mises)
o'yt 1-155oyt
6M/Nt
a/t 0.5 1 2 4 8
0'05 0"826 0-702 0'520
0-10 0-794 0-672 0.495 0"307 0-167
0"15 0-752 0"633 0.463
0"20 NA 0"584 0-424 0"260 0.141
0"25 0.527 0"379
0"30 0"464 0-330 0'200 0"109
0-35 0-396 0"279 NA NA
NA: not applicable.
2.4.4 P l a n e s t r a i n M i s e s
This is 1"155 times the plane strain Tresca limit load.
2.4.5 S h a l l o w c r a c k s in p l a n e s t r a i n ( T r e s c a or M i s e s ) t9"24"28"3°
T h e results are no longer expressible in terms o f a single f u n c t i o n q(Ye) only,
as for d e e p cracks. Physically the r e a s o n is that plastic yielding can spread to
the top free surfaces on either side o f the notch. Ewing derived an u p p e r
b o u n d solution f r o m the shallow c r a c k b e n d i n g solution. His results are
given in Tables 2 and 3.
T a b l e 2 gives an u p p e r b o u n d limit load for the region w h e r e the shallow
crack solution is a p p r o p r i a t e . T a b l e 3 c o m p a r e s these results with the p u r e
b e n d i n g results for shallow cracks given above, and the values t a k e n f r o m
TABLE 3
SEN Plane Strain Upper Bound for Shallow Cracks
4M
Values of 4M (Tresca) or (Mises)
~yt2 I-I 55ayt2
a/t a b a/t a b alt a b
O"10 0'89 0"92 0"20 0"75 0-79 0.30 0-58 0-62
a Taken from Table 2 with 6M/Nt = 8.
b Taken from Table I with 6M/Nt = zc.
22. 218 A. G. Miller
Table 2 are slightly lower than the values taken from Table 1, as they should
be.
2.5 SEN approximate solutions for combined tension and bending
2.5.1
R6 Rev. 2, eqn A2.4.5, gives an empirically modified version of the Tresca
plane stress result:
II'5M + Nal + [-(l'5M + Na) 2 + N2(t - - a ) 2 ] 1/2
Lr = o'y(t - - a ) 2
This expression is no longer recommended. It is 6% non-conservative
under pure bending compared to Tresca plane strain but conservative under
combined tension/bending.
2.5.2
The classical plate formulae are pessimistic because they assume that the
ends are free. An approximation sometimes made, 3t'32 or in O R A C L E by
Parsons, 33 is to ignore the contribution to the bending moment produced by
the eccentricity of the tensile force:
Met f = M -- ½Na
This cannot be rigorously justified, and it should be confirmed that
redistributing the moment ½Na does not cause another part of the structure
to be in a more onerous condition than the ligament. This version is used by
O R A C L E for both the Tresca plane stress formula and with the R6 Rev. 2
modification of this.
BS PD6493 uses the Tresca plane stress version of this approximation.
2.5.3
Chel132 gives an approximate solution for plane strain which is equivalent to
the solution here for deep cracks, and is based on a conservative
approximation to the pin-loading SENT results when a/t < 0"545.
2.6 SENB pure bending: effect of notch angle
2.6.1 Plane stress Tresca
The constraint factor is unity, independent of notch angle 2:~:
4M
O.rt2(1 __ X) 2 = 1
C -------
23. Review of limit loads of structures containing defects 219
~
I 1.07
1.06 Upper bound
Lower bound
~.os
a
"- I .Or,
~" 1.03
o
'- 1.02
E
0 1.01
U
1.00 , , , A /,lO /
0 i ~ I
90 80 70 60 5 0 3 20 10 0
Fig. 11. SENB with V-notch: plane stress Mises (from Ford and Lianists).
2.6.2 P l a n e stress M i s e s t s
For deep cracks
4M
C = crrt2( 1 _ x) 2
4/(,,/3)
0 < ~ < 67 ° c = 1 + 2/(,,/3) = 1.072 (exact)
67 ° < :~ < 75 ° c = 1"173 - 0"0859~ :~ in r a d i a n s
75 ° < :~ < 90 ° c = 1 + 0"229(rc/2 - :~) ~ in r a d i a n s
F o r :~> 67 ° b o t h l o w e r a n d u p p e r b o u n d s are given, a n d t h e y are b o t h
r e p r e s e n t e d b y the a b o v e f o r m u l a e to w i t h i n 0"5%.
T h i s is c o n t i n u o u s with the d e e p s h a r p n o t c h result f o r ~ = 0 (c = 1.072)
a n d the u n n o t c h e d b a r f o r ~ = re/2 (c = 1). T h e d e p t h v a l i d i t y limits are n o t
k n o w n , e x c e p t f o r :~ = 0 a n d ~ = ~/2.
T h e results are s h o w n in Fig. 11.
2.6.3 Plane strahl Tresca 2°'34
4M
c = aytZ(l _ ..)2
0 < :~ < 3"2 ° c = 1"2606 (exact)
3.2 ° < :~ < 57"3 ° c = 1"2606 - 0 " 0 3 8 6 :~" - - -"°~° ~ 6"J
( 0-944 ^ ' ~
'^ :~ in r a d i a n s
T h i s f o r m u l a r e p r e s e n t s E w i n g ' s n u m e r i c a l results to w i t h i n 0 " 3 % :
rc - 2:(
57"3 ° < ~ < 90 ° c= 1 + (exact) :~ in r a d i a n s
4+rc-2~
24. 220 A. G. Miller
I.I.
~1,3
u
O
C
e,
o
1.1
1
9O ;o 4 6% ;o ,'0,'o 21, ,'o
oK
Fig. 12. SENB with V-notch: plane strain [from Green'°).
T h e d e p t h r e q u i r e m e n t s are: 3"*
1
0<:~<57.3 ~' - - > 1 . 4 2 3 - 0 . 1 2 4 : ~ 2 :t in r a d i a n s
1-.
1 e ~2-~ -- 1
57-3 ° < ~ < 90 ~ ~ > l -~ ~ in r a d i a n s
1 -x 2 + ~z/2 -
T h i s is slightly m o r e stringent t h a n G r e e n ' s d e p t h r e q u i r e m e n t s .
T h e s h a l l o w c r a c k s o l u t i o n is not k n o w n . T h e s o l u t i o n here is c o n t i n u o u s
with the d e e p c r a c k s o l u t i o n at :~ = 0 (c = 1.2606) a n d with the u n n o t c h e d b a r
s o l u t i o n at :t = re/2 ( c = 1). T h e results are s h o w n in Fig. 12.
Dietrich a n d Szczepinski 35 give the c o m p l e t e slipline field for :t = 60: a n d
their c o n s t r a i n t f a c t o r is the s a m e as a b o v e .
2.6.4 Plane strain Mises
T h i s is 1"155 times the plain strain T r e s c a limit load.
2.7 P u r e b e n d i n g : effect of notch root radius
2.7.1 Phme stress Tresca
4M
= 1 ahvays
t? -- O.y12 ( 1 -- .V) 2
2.7.2 Phme stress Mises Is
n o t c h r o o t radius r
ligament b = t - 2a
b+r
4M
C = ayt2 (1 -- X) 2
25. Reciew o f limit loads o/" structures containing defects 221
1.061.07 J
t Upper b o u n d
I. 05 Lower bound... /j/
u
0 1.0¢
"c 1.03
'6 1.oz
ut 1.01
g
U 1.00 / I .I I I I I I 1.10
0.1 0 . 2 0 3 0/, 0 5 0 . 6 0 . 7 0 . 8 0 9
b
b+r
Fig. 13. S E N B with circular root: plane stress Mises (from Ford a n d LianistS).
For deep notches
0 < u < 0"692 c = 1 + 0"045u 2 to 0.2%
0.692 < u < 1 c = 1.072 0"123r
+ 0"022 ( r ) 2
~ to 0"6%
Both lower and upper bounds are given, and are represented by these
formulae to the stated accuracy.
This merges continuously with the deep sharp notch solution at r = 0
(u = l, c = 1"072) and the unnotched bar solution at r = ~ (u = 0). The depth
limits are not known, except for u - - 0 and u = 1. The results are shown in
Fig. 13.
2.7.3 Plane strain Tresca 2°
4M
c-ayt2(1-x)2
Deep cracks
0<u<0"64 c=1+0"155u to 1.5%
0.64<u<1 c=0.811+0.450u to 1"5%
Both expressions are representations of upper bounds to the stated
accuracy. They merge continuously with the deep sharp crack solution at
TABLE 4
Pure Bending in Plane Strain
r/b Critical Constraint
factor c
0 3"2 ° 1-261
_1_
32 1 I-3 ° 1.243
±
16 17.6 ° 1-227
26. 222 A. G. Miller
1.4 i i i i i l i
1.3
Z,
u
o 1.2
c-
O
o~ 1.1
t-
O
U
I I I I I I
.1 .2 . .t. .5 .6 .0
.7 .8 .9
b
b-~r
Fig. 14. S E N B with circular root: plane strain (from Green2°).
r = 0 (c = 1.261) and the unnotched bar solution at r = oc. The depth validity
limits are not known, except for u = 0 and u = 1. The results are shown in
Fig. 14.
Ewing 34 studied the effect of g > 0 simultaneously with r > 0. For any
given rib the solution is independent of ~, provided that ~ is less than some
critical angle which depends on rib. The values of this critical angle and the
corresponding constraint factors are shown in Table 4.
2.7.4 Plane strain Mises
This gives a limit load 1.155 times the Tresca plane strain limit load.
2.8 Tension: effect of notch root radius and notch angle for fixed grip loading
Notch root radius and flank angle have no effect. The constraint factor is
always unity.
2.9 Combined tension and bending: effect of notch root radius and notch angle
Complete solutions for this are not available. For deep cracks lower and
upper bounds (sometimes widely different) are given for:
(i) large angle wedges by Shiratori and Dodd; 27
(ii) small angle wedges by Shiratori and D odd; 36
(iii) large radius circular notches by Dodd and Shiratori; 3~
(iv) small radius circular notches by Shiratori and Dodd. 38
Finite-element and experimental results are given by Shiratori and
Dodd. 39
27. Review of limit loads of structures containing defects 223
A w=lOmm
/,,0 m m 40ram A
2mm. di . 45" total notch
drilled IF angle.
hole. 0 . 2 5 m m . root
radius.
(a) (b)
I
I t mm I
[~
I
2ram
I 8ram
2b[ rrrn
/
(c)
Fig. 15. Three-point bend geometry. (al and (b) Charpy test geometries considered by Green
and Hundy; 4° (c) Charpy and lzod geometries considered by Ewing. 3"~
2.10 Three-point bending (Charpy test)
In three-point bending, there is a non-zero (discontinuous) shear force at the
minimum section, which alters the limit moment from the pure bending
value, with zero shear. Pure bending is obtained in a four-point bending test.
The Charpy test is a three-point bending geometry. Only plane strain Tresca
is considered here. (Plane strain Mises will give 1.155 times the Tresca limit
load.)
2.10.1
Green and Hundry 4° considered the two Charpy test geometries shown
in Figs 15(a) and (b), and showed that
4M
ayt2 = 1-21(1 - x ) 2 x>0-18
The reduction in the critical depth due to the presence of shear is similar to
that described in the more general treatment ofcombined bending and shear
given in Section 2.13.
28. 224 A. G. Miller
TABLE 5
Three-point Bending Constraint Factors in Plane Strain
r l b c r 1 b c
0 22 0 1"224 0-25 22 0 1'218
0-5 1"251 0-5 1"245
1"0 1-287 1"0 1"281
20 0 1"216 20 0 1'210
0"5 t'243 0'5 l'238
1-0 1"279 1"0 1"274
4M 4M
c = - - - - - - -a)
a,(t ~ (Tresca) 1"155~rr(t a) 2 (Mises)
-
r, root radius; l, half span; b, half indenter width; t, thickness = 10.
2.10.2
Haigh a n d R i c h a r d s 16 q u o t e the nearly identical result:
4M
O.yt2 = 1"22(1 - X) 2 X > 0"18
2.10.3
Ewing 34 considered the geometry shown in Fig. 15(c) and calculated the
effect of notch root radius r and indenter radius b (approximating the
indenter by a flat punch). The results are shown in Table 5, with
4M
ayt" = c(1 - x) 2
For zero indenter width this agrees with the above results.
2.10.4
K u m a r et al. 25 give the result as the pure bending solution, with no
allowance for shear. Similarly, they give the Mises plane stress solution as
being the pure bending solution. Therefore these results are not
recommended.
TABLE 6
Three-point Bending Constraint Factors in Plane Strain for Shallow
Cracks
a/t c a/t c a/t c
0 1'12 0"08 1'190 0.13 1"211
0"03 1-152 0"10 1'199 0-15 1"215
0"05 1"170 0.177 1"218
29. Review of limit loads of structures containing defects 225
1-3
0
U
o
1'2
- /"////
o
/
1-I /
/
/
/
/
/
10 I I I
0'06 0-12 0'18
a/t
Fig. 16. Three-point bend constraint f a c t o r . - - - , Four-point bend; , three-point bend.
2.10.5
The shallow crack solution (a/t < 0"18) is given by Matsoukas et al. 41 The
constraint factor c is given in Table 6 and compared with the four-point bend
result in Fig. 16. The span is given by l = 2t (see Fig. 15). In the smooth bar
limit, a / t - , O , the constraint factor c tends to the value of 1"12, in agreement
with Green. 4z
2.11 Compact-tension specimen
The limit load for the compact-tension specimen may be calculated from the
pin-loaded SENT results by a transformation given by Ewing and
Richards 22 and Haigh and Richards. x6 The geometry is shown in Fig. 17.
The transformation is
_...1
r/sE N gnCT XSEN --~ ½( 1 -+ XCT)
where n(x) - N / % t .
30. 226 A. G. Miller
CT5 1
I
!t
[ , SEN -
O SEN-----
I Lood Line
t
Fig. 17. Compact-tension specimen geometry.
2.11.1 Plane stress Tresca
n(x) = - ( 1 + x) + (2 + 2X2) l '2 l>x>0
x--* 1 n ---~y2/4
2.11.2 Plane stress Mises
n(x) = - ( ~ x + 1) + [(Tx 2 + 1)(1 + ;,)] ~/2 for l > x > 0
2
y= ~ = 1.155
1.072v 2
x~ 1 n ~ 0.268y 2 = 4
2.11.3 Plane strain Tresca
n(x) = - ( 1 + 1"702x) + [-2-702 + 4"599x 2] 1/2 for 1 > x > 0.090
x~ 1 n ~ 0"315y 2 - 1"260)'2
4
These results are s h o w n in Fig. 18.
2.11.4 Plane strain Mises
This is 1"155 times the plane strain Tresca limit load.
2.11.5
K u m a r et al. 25 give values for the limit loads which, as in the p i n - l o a d e d
S E N T case ( S e c t i o n 2.2.5), are the T r e s c a p l a n e stress f o r m u l a e ,
31. Review of limit loads of structures containing defects 227
o,7e-
0.6
O.S
TfescQ
PLane s t r a i n
0.~
N
0.3
0.2
0.1
Ol I I I
0 0.2 0./, 0.6 0.8 1.0
X
Fig. 18. Compact-tension specimen limit load.
renormalized to give the correct value as x --* 1. They are not the correct limit
loads in general, and are not recommended for use.
2.12 SEN multiaxial tension, with bending and shear
2.12.1
Jeans derived a lower bound expression (quoted by Ewing and Swingler*a):
M moment
N tensile force
ah out-of-plane stress (uniform across section)
Q mode II shear force
b ligament thickness ( t - a ) for SEN (or ( t - 2 a ) for DEN)
The geometry is shown in Fig. 19. The result is useful in cases where the
elastic stresses are available, and it avoids having to choose between the
plane stress and plane strain solution:
+-7 +
32. 228 A. G. Miller
j Surfacedefect
/1Lo
Fig. 19. Plate under multi-axial loads.
1
These expressions are based on a "nominal' Mises yield criterion, and do
not satisfy the boundary conditions at the back surface when Q #- 0. They are
a valid lower bound for double-edge notched plates, or with additional
support at the back surface.
2.12.2
Ewing (pers. comm.) gives a modification of Section 2.12.1 to allow for back
surface interaction with shear present:
0"¢=4 +(P-+L + h
N I = ( N 2+¼Q2),
~,/3
+2_Q
The plane strain case (as opposed to specified out-of-plane stresses) is
considered in Section 2.13.
2.13 Combined tension, bending and mode I1 shear
2.13.1
Ewing (pers. comm.) has derived an approximate solution for deep cracks in
plane strain under combined tension, bending and shear. When shear is
absent, the deep crack solution is valid when a/t > 0.4, but the validity limit is
not known in general. It is assumed that the tensile force acts along the
centre-line of the ligament. If it acts along the centre-line of the plate, an
extra bending moment of ½Na must be included. The solution is given in
Table 7 and Fig. 20 for the Tresca yield criterion. In the Mises case the limit
load should be multiplied by 1.155. An alternative (lower bound) solution is
given in Section 2.12.
33. Review ~/" limit loads o[" structures containing deJec'ts 229
TABLE 7
Values of L r for Edge-cracked Plate under Tension, Bend and Mode I1 Shear
(a) Table o f F expressed in terms o f m' and n'
0"0 O. 1 t)'2 0"3 0"4 0"5 0"6 0"7 0"8 0"9 I'0
0"0 1.000 1"005 1'019 1"040 1"067 1"094 1-121 I'143 1"!54 1"144 1"000
0"10 1.006 1-029 1"060 1"094 1-127 1'157 1'!80 1"!93 1"192 1"164
0"20 1"022 1-052 1"086 1'119 1-150 1-176 1"195 1-204 1"!96 1'156
0"30 1.044 1"076 1-109 1'141 1"169 1"191 1"204 1'206 1'188 1'127
0"40 1.067 1"099 1"130 1"159 1'!82 1"198 1-205 1'197 1.164 1"049
0-50 1"088 1"119 1"147 1"170 1'188 1"198 1'195 1"172 1"109
0"60 1"106 1'133 1"157 1-175 1"185 1"!84 1-167 1"119 0-892
0"70 1"117 1'140 1"158 1-168 1-168 1"152 1-108 0-971
0-80 1-119 1"136 1'146 1"145 1"127 1"080 0-874
0'90 1"103 1"112 1"109 1"085 1-018
I '00 1.000
(b) Table o f F expressed in terms o f re' and q'
0"0 0"1 0.2 0"3 0"4 0"5 0"6 0"7 0"8 0"9 1"0
0"0 1"000 1-046 1-085 1"116 1'138 1"151 1"154 1"145 1-121 1"076 1.000
0"10 0"997 1"047 1"090 1"128 1"158 1-180 1"193 1'194 1"178 1-135
0-20 0-987 1'040 1"087 1'127 1"160 1"185 1"200 1"203 1"190 1'146
0-30 0'971 1'027 1"077 1'120 1-156 1'184 1-202 1"207 1'194 1"146
0-40 0-949 t'009 1'063 1-109 1'148 1"179 1"199 1"205 1"191 1"122
0-50 0-922 0"986 1"044 1'094 1"137 1"170 1-191 1-198 1'177
0"60 0-892 0"962 1-024 1'079 1"124 1"159 1"181 1-184 1"106
0"70 0"868 0-943 1-009 1'066 1-114 1.149 1"169 1"148
0"80 0"874 0-944 1"008 1"065 1"I10 1"141 1'119
0-90 0"925 0"978 1'031 1"031 1'079 1-141
1'00 I '000
The notation is as follows: t = plate thickness; a = c r a c k depth (a/t>~ 0-4); m = bending
moment parameter = M/I'26M' for M' = a y ( t - a)-'/4; n = tension parameter = N/'N' for N' =
a,(t - a); q = shear parameter = Q/Q' for Q' = ay(t - a)/2; m' = m/r, n' = n/r, q' = q/r for r =
(nil + n 2 + q2)t/2; Lr = rF(m', n', q').
2.13.2
In the special case o f zero m o m e n t (referred to the l i g a m e n t centre-line,
E w i n g 44 h a s c a l c u l a t e d a m o r e a c c u r a t e s o l u t i o n f o r d e e p c r a c k s ) :
+ 1.03 = 1
34. 230 A. G. Miller
1 i , , = i , !
q=O
• 0.2
=E
.= o.s 0.6
IE
l0
E
, ,".. ,'x., X -
0 0.5
n = NIN e
Fig. 20. Plastic yield loci for fixed values of the mode |l shear parameter, q = Q/Q'. Here M,
N and Q denote moment, tension and shear in a combination ensuring collapse. M ' =
a) /4, N' = ay(t-- a), Q' = ~ y ( t - a)/2.
For Tresca plane strain
N1 = ay(t - a) Q I = ay(t - a)/2
(Q mode II shear resultant)
This is believed to be accurate to 2%, and can lie 17% inside the nominal
criterion:
+ =1
The yield surfaces are shown in Fig. 21. The depth validity limits are
unknown, except for Q = O.
The Mises limit load is 15% higher.
2.13.3
In the special case of zero tension, Ewing and Swingler43 have calculated
both lower and upper bounds. The lower bound is within 5% of the nominal
criterion:
+ =1
Mt = O'y(/-- a)2/4 (Tresca) QI =" f l y ( / - - a)/2
The upper bound is given in Table 8 and Fig. 22, along with the minimum
35. kN/N1
Nominal
Nominal
Per Bound
wet" Bound
~
-1 QIQ 1
NO~lna[ x 0.83
-1
Fig. 21. Yield criterion for combined tension and mode II shear. ~'~
TABLE 8
Combined Bending and Mode II Shear
(from Ewing and Swingler 43)
Q/Q= Upper bound Depth limit Upper bound Lower bound
M/M t a/t M/MI a M/M t
0.000 1.261 0.297 1.289 1.000
0.050 1.248 0.271 1.279 0.999
0.100 !-232 0-245 1.266 0-997
0.150 1.214 0.218 1-251 0.992
0-200 1.194 0.191 1.233 0.986
0.250 1.171 0.164 !-212 0.978
0-300 I. 145 0.138 I. 189 0.968
0"350 I.I 15 0-112 1.162 0.956
0.400 1.082 0-087 I. 132 0.941
0.450 1.045 0-064 1-098 0.923
0-500 1.003 0-043 1.060 0.902
0.550 0-956 0.025 1.017 0.875
0.600 0.904 0.011 0.969 0.842
0.650 0-846 0.003 0-915 0-797
0.700 0.781 0.854 0.736
0.750 0-709 0.786 0-657
0.800 0-627 0-708 b 0-561
0-850 0.536 0-618 0.447
0-900 0.434 0.509 0-315
0.950 0.3 ! 9 0.362 0.166
1-000 0-189 0.000 0-000
°These results apply to a notch at a cantilevered end.
For Q/Qt >0'8, the results are an upper bound only and cannot be exact.
36. 232 A. G. Miller
Upper B o u n d
(Notched Cantilever)
Bound
M/M Nominc]| L o w e r B o u n d
(M2/M2~.Q2/Q2=1)
I
Lower Bound
8/w 2
0"6
O.t,.
0"2
0
I I I
2/.
I I I
~l
02 or, 06 o-8 1
Q/Q1
Fig. 22. Combined bending and mode II shear (from Ewing and Swingler43).
crack depth. The upper bound is potentially exact for Q/Q ~ > 0-803 (i.e. the
slipline field is statically admissible, but it has not been constructed in full).
When Q = 0, the solution coincides with that given in Section 2.1.3.
If the notch is at a cantilever position, then the limit moment is higher, and
is also shown in Table 8. The depth validity limits for these results have not
been calculated.
The Mises load is a factor of 1"155 greater.
2.13.4
A c o m m o n approximate solution for combined tension, bending and mode
II shear is to generalize Section 2.4.1 to include shear in a Tresca yield
criterion:
~° = b2 +~ b2 +L b~ ) + 7 Lr=--~y
This is similar to Section 2.12.1 with a h = 0, Tresca shear instead of Mises
shear, and an amelioration allowed for the effect o f the crack on the collapse
37. Reriew of limit loads of structures containing defects 233
moment. As in Section 2.12.2, there is no free surface shear correction. When
compared with Section 2.13.1 over the region N > 0, M > 0, Q > 0:
Lr(2.13.1)
0-79 < < 1-17
Lr(2.13.4)
Hence the approximation is conservative (ignoring a 2% error) if a Mises
yield criterion is assumed.
2.14 Combined tension and mode III shear (plane strain)
Ewing and Swingler 43 have calculated both lower and upper bounds for the
Tresca case, with fixed grip loading (tensile force acting along the centre-line
of the ligament).
The nominal yield criterion is a true lower bound, in contrast to the mode
II case described in Section 2.13.2:
S Mode III shear stress resultant:
+ (;:7 = 1 N 1 = a , ( t - a), S l = a,(t - a)/2
An upper bound (which cannot be exact) is given by
N<S S= S t
(N) 2 N 1¢S'~ z
N>S =0
This is shown in Fig. 23.
~ U p p e r Bound
- ~~
N/N1
0-5
Lower Bound /
(N/N1)2+(sIs1)2= 1
I I I I I I I f I
0'5
s/s,
Fig. 23. Combined tension and mode III shear (from Ewing and Swingler'~)).
38. 234 A. G. Miller
t
", Line of sidegrooves
Fig.24. Geometry of inclined notch.
2.15 Inclined notch under tension (plane strain)
The geometry of this is shown in Fig. 24. Ewing 45 has considered this.
2.15.1
Unsidegrooved plane strain Tresca, pin-loading:
F= Btayn(a/t)
where B is thickness in transverse direction, Fis end load, and n(a/t) is shown
in Fig. 25 for c¢= 15 ° and ~ = 30 °.
The solution for deep cracks is exact; the solution for shallow cracks is an
upper bound.
2.15.2
Unsidegrooved plane strain Mises, pin-loading:
F= l'155Btayn(a/t)
i i i I i i i i
%
%
0.9
0.5 ~.......-~U n i vet so I s i n g l e - hinge
upper bound.
0.7
0.6 . ~
b;" ~=15,
~n 0..5 ~,
0,2 ~
0.1
I I I I I I I I
0,I 0,2 0,3 0./. 0.5 0.6 02 0,8 0.9
alt
Fig. 25. Collapse loads for ungrooved single-edge inclined notch specimens (from
Ewing~5).
39. Review of limit loads of structures containing defects 235
2.15.3
Sidegrooved plane strain Tresca, fixed grip loading (load applied through
the centre of the ligament):
~a,B!t -- a)
F = min [ayB (t - a) cosec 2ct
where B' is reduced thickness across sidegrooves.
2.15.4
Sidegrooved plane strain Mises, fixed grip loading:
. f a , B(t-- a)
F= rain
), 1"15arB (t - a) cosec 2~t
3 INTERNAL NOTCHES IN PLATES
Mainly solutions for through-thickness or extended defects are considered
here. No solutions for embedded elliptical defects are known, except for the
limited results given in Section 3.4.
3.1 Centre-cracked plate in tension
t plate width of thickness
a crack width or depth
e crack eccentricity (see Fig. 26)
h plate length
N force/width or thickness
Crock
M M
Fig. 26. Geometry of eccentric crack under multi-axial loading.
40. 236 A. G. Miller
TABLE 9
Centre-cracked Plate in Tension (values given in units of a/'~,)
(plane stress Mises)
a't h/t
0.2 0.4 0.6 >.0.71
o-l 0.650 0-753 0.900 0.900
0.2 0.390 0.654 0.800 0.800
0.3 0-230 0.530 0.646 0.700
0.4 0.145 0.425 0.538 0-600
0.5 0.100 0.312 0-427 0-500
0.6 0.076 0.225 0.338 0.400
0.7 0.065 o. 160 0.270 0-300
0.8 0-049 o. I 17 0-200 0.200
0.9 0.027 0.090 o-I oo o-I oo
3.1.1 Plane stress Tresca aJld Mises, a n d p l a n e strain Tresca
N=ay(t-a) (Ref. 16)
This is c o m p a r e d with experimental results in Fig. 27, taken from
Willoughby.17
This result is not valid for short plates (h << t). H o d g e ~6 d e m o n s t r a t e d that
it was exact for square plates (h = t). A i n s w o r t h (pets. comm.) derived an
a p p r o x i m a t e lower b o u n d solution for the case of a uniform applied stress.
This agreed with the above solution when h 2 > _ . 2 a ( t - a ) . This is always
satisfied if h/t > l / x / 2 = 0"707.
The results for short plates from Ainsworth's lower b o u n d m e t h o d are
s h o w n in Table 9 for plane stress and Mises yield criterion.
3.1.2 Plane strain M i s e s
This is 1'155 times the plane strain Tresca result.
3.2 Eccentric crack under tension and bending
3.2.1
A lower b o u n d solution which reduces to the Tresca plane stress result is, for
M - ae M (t 2 - a z - 4e 2)
(a) Nt >~t(t-a~ and ~/> 8et
M - ae M (t 2 - a 2 - 4e 2)
or(b) Nt <<'t(t-a~ and N-t~< 8et
]M + aN/2l + [(M + aN~2) z + N 2 { ( t 2 - aZ)/4 - ae}] 1 ,
Lr = 2~y[(t 2 - a2)/4 - ae]
42. 238 A. G. Miller
zt~
d o o o d c~ c~
i
i 0 I
zi;~
/ /$oi~
.= ~' ~ o d c~ o o o c~ o --
I i i i
¢M
0
• . . .
o o c~ o o o d o
i i i !
43. Ret,iew of limit loads of structures containing defects 239
For e = 0 and M = 0, the solution reduces to the centre-cracked plate in
tension given in Section 3.1.1.
As e---, 1/2(t- a) the range of validity of the second solution shrinks to
zero. In the limit the first solution agrees with the single-edge notched plate
solution in Section 2.4.1.
3.2.2
BS PD6493 gives local collapse loads for embedded defects. These are based
on elastic-plastic finite-element calculations, with a criterion of 1% strain in
the thinnest ligament (here defined as b). Hence the b/t = 0 result does not
agree with the surface defect results, as described in Section 2.4. The results
are shown in Fig. 28. The geometry is shown in Fig. 26.
3.2.3
R6 Rev. 2 Appendix 247 recommends that an embedded defect should be
treated as two separate surface defects, by bisecting the defect, and assessing
each ligament separately.
No recommendation on load sharing is given. This method is very
conservative, as it ignores the resistance to rotation offered by the other
ligament, and does not allow any load shedding on to the other ligament. As
the defect approaches the surface, the limit load does not change
continuously into the single-edge notched limit load but goes to zero. For the
centre-cracked plate under tension, the R6 Rev. 2 proposal is compared with
the true limit load and some experimental results in Fig. 27. This issue of
ligament failure is similar to that in Section 1.8. Once one ligament has failed,
the defect should be recharacterized and re-assessed.
b I Ih = 1.5
t.0
"t/'%
o.s
f ~ ILE
• • • @
I ~ = , I , , , j J
0.$ l.O
bzl b I
Fig. 29. Eccentric defect under m o d e I | | loading.
44. 240 A. G. Miller
I I I I
I I I I
I I I
I I I
I "" -~1 ^ ""1
I I
I j 1 I
I I
i I I
I
I i I I
t t t t I tt t I t t t
Fig. 30. Array of eccentric defects under mode I loading.
0.5-
11500 " ~
I
0 0.25 0.5
olt
It
_11 •
2c
Fig. 31. Local collapse load for central embedded elliptical defect in plate in tension.
45. Review of limit loads of structures containing defects 241
3.2.4
Ewing ~° has considered an eccentric defect in anti-plane shear, as shown in
Fig. 29. The following quantities were calculated:
q. stress required to spread plasticity across shorter ligament, assuming
strip yielding model
rLE value of v~. estimated from elastic stress resultants
rG stress needed to spread plasticity across both ligaments (constraint
factor is unity)
The same numerical results apply to the mode I tensile analogy shown in Fig.
30, and may be considered as an approximation to the mode I loading of a
single strip with an eccentric defect.
3.3 Eccentric crack under tension, bending and out-of-plane loading
The lower bound solution in Section 3.2 may be generalized to include out-
of-plane tension and shear (but not out-of-plane bending). The geometry is
shown in Fig. 26. Free surface shear stress effects have been ignored (see
Section 2.12). Let
N' = N- 1/26h(t - a) M ' = M + l/2ahae
Then
(a~ + 3/4a~ + 3r2) t~z
Lr m (with a Mises shear term)
O'y
where
M ' + aN'~2 + { ( M ' + aN'~2) 2 + {N')2[(t -' - a2)/4 - ae]} l'z
a~ = 2[(t 2 - aZ)/4 - ae]
a. is the out-of-plane tensile stress and r-" is the sum of the squares of the
shear stresses.
For the assumed stress distribution to be valid
N'>O M'>O
For a~ = r = 0, this reduces to the solution given in Section 3.2.1.
3.4 Embedded elliptical defect in tension
The only results known to the author are those for the local ligament
collapse load for a central elliptical embedded defect in a plate in tension
given by Goerner. 4s A simplified strip yielding model was used, and the
calculated load was the load at which yielding first extended across the
ligament at the thinnest point. The calculations are analogous to those for
surface defects quoted in Section 5.1.3, and the results are shown in Fig. 31.
