Review of limit loads of structures containing defects

Int. J. Pres. Ves. & Piping 32 (1988) 197-327

Review of Limit Loads of Structures Containing Defects

A. G. Mil ler*

Technology Planning and Research Division, CEGB Berkeley Nuclear Laboratories, Berkeley, Gloucestershire GLI 3 9PB, UK

(Received 12 August 1987; accepted 9 September 1987)

ABSTRACT

A survey of existing limit loads of structures containing defects is given here. This is of use in performing a two-criterion failure assessment, in evahtating the J or C* parameters by the reference stress approximation, or in evaluating conthmum creep damage using the reference stress. The geometries and loadings considered are (by section number): (2) single-edge notched plates under tension, bending and shear: (3) internal notches in plates under tension, bending and shear; (4) double-edge notched plates under tension, bending and shear; (5) short surface cracks in plates under tension and bending; (6) axisynmwtric notches in round bars under tension and torsion, and chordal cracks in round bars under torsion and bending; (7) general shell structures; (8) surface/penetrating axial defects in cylinders under pressure; (9) surface/ penetrathlg circumferential dejects in cylinders under pressure and bending; (10) penetrating/short surface/axisymmetric surface defects in spheres under pressure: (11) penetrating/surface longitudinal/circumferential defects in pipe bends under pressure or bending; (12) surface defects at cylinder- ~3'finder intersections under pressure; (13) axisymmetric surface defects at sphere-o'linder intersections under pressure and thrust.

NOMENCLATURE

a crack depth b ligament thickness c constraint factor for 2D cases; crack semi-length for 3D cases d staggered crack separation n N/ayt

* Present address: NIl, St Peter's House, Balliol Road, Bootie L20 3LZ, UK. 197

1988 CEGB

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 L r 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 common 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

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

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 maximum 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 ASME 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~ = tan- 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 method 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 component 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% proof 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 -Mn 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-

Rev&w of limit loads of structures containing defects 201

mentally. A common 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

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--~ Mo


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

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 commonly 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,

Rerie,' ~I" limit loads o/'structures containing defects 205

PLone stroin /

/ /

C[2 ,,. / Plane ~( '~ e'e ~" ,/Mises stress , ~ ,e~

, I : /,/" IJ/,,"

, 2 " " 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 ,/3L

Mises plane strain c = 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).

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.

m2 i 1 -1 I !

Fig. 4. Two-moment limited interaction yield surface for shells.

However, as the bending and stretching are rarely significant simultaneously, 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

When

this gives

(~rt) 2 4a-~-2 + =1

N 4M ~yl~Gyl 2

N 4M x/5_- 1_0.618 ay--~=cryt 2 = 2


Using the limited interaction yield surface gives

N 4M o-rt o-yt 2 -

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 hoop direction). This is shown by Ewing's results discussed in Section 13.2.

For thin shells in a membrane 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 maximum when the stress components are in a ratio of 2:1, as in a closed pressurized cylinder:

P= tryt Tresca P 2tryt R = ~ 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 INGLE-EDGE NOTCHED 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 bound 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

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

i

t

a

~. ligament I

p la te I I I i

I I

N - - - - -~ Q X S

M':M+I/2 Na

a

t

y:_l -a ~- : I -~

N Tension M Bending moment Q Mode TT shear S Mode TIT shear

_ 1

Fig. 5. SEN geometry.


2.1 SENB 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

4M(x) _ 1.072(1 - x) z = 1-072y z x > 0.154 O'yt 2

This result is only valid for deep cracks and must be cont inuous with the unnotched beam result:

4M(0) - - = 1

O-yt 2

The value of the validity limit on x is taken f rom Okamura et al. 19

2.1.3 Plane strain Tresca Deep cracks 2

4M = 1"2606(1 -x ) 2 1'2606v 2 x > 0"295

~yt 2

Shallow cracks 2

4M(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

This is an analyt ic approx imat ion to within 0"5% to the values given in Table 1. The results are shown graphical ly in Fig. 6.

2.1.4 Plane strain Mises This is 1"155 times the plane strain Tresca result.

2.2 SENT tension (M= 0, pin loading)

2.2.1 Plane stress Tresca 22

N

tTyt - n (x ) = I-(1 - x) 2 + x23 ~/2 - x

= [1 - 2x + 2x2] 1/2 - X

n~ 1 - 2x y2

n ....~ __ 2

0~

1.2

1.1

1.0 [ I I

c : Z,N % ( t -a l 2

( Tresco )

1.3

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)


This is the same as in the plain beam result for combined tension and bending, with the moment given by the eccentric tensile force on 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'~ .., L t - .+- - 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

If 7 is put equal to unity, the Tresca result is recovered.

Shallow cracks

N = n(x) = 1 -- x -- x 2 x < 0"146

o'r t

This is an approx imat ion to the tabulated results in Ewing and Richards 2z'za agreeing to within 0-15%:

n -+ l -x x--*0

2.2.3 Plane stra& Tresca Deep cracks 22'z3

N - - = n = 1-702{ I-(0-794 - 392 + 0"58763 ,2] 1/2 O'yl

n ~ 0"6303 '2

Shallow cracks z4

where

N - - n (x )

o'yt

- [ -0 -794 - y ]} x> 0.545

y~0

x < 0.545

and

n(x)/> 1 -- x - 1.232x 2 + x 3 - f (x )

n(x) ~f(x) + 22x3(0.545 - x) 2

n ----~ 1 - -x x~0

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 Kumar 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 = -Na 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

1.0

0.8

N O'yt 06

0/*

0.2

0

Fig. 7,

i - - - Io It U--19- o l -

',~ I. l J

P Tresca plane strain ] Pin- ~"~,C,,._ M Mises 1 -. ~ loading

\ \e~' " T Tresco t wtane| \ \ ~ Ist~ess/

~'~ ~" ~,F Experimental J ) ~',,~ "% ~. _F_ PLane st.,s~ onO rres=o plan*

I l I L, I l l F ~ "~"'~I 0.1 0.2 03 0.~, O.S 06 0.7 08 0.9 I0

a l t

The theoretical and experimental variation of yield load with notch length for single- . ~ ~3 edge notched (SEN) specimens (from Ewing and Richards ... . ).

Review o[ limit loads of structures containing defects 213

2.3.2 Platte strain M ises 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 moment 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'~ - 2M + Nt (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 moment due to the eccentricity of the ligament.

These results may be rewritten in terms of the moment referred to the centre-line of the ligament:

M' = M + Na/2 L~ LGr( t _ a) 2

N/( t -- a)

)'e = (2M') / [ ( t - a) 2] + N/(t - a)

a,(t- a) q(Y~)

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.

214 A. G. Mi l ler

2.4.1 Plane stress Tresca The results may be written

2M + 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 unnotched beam result:

V + -?- =1

with account being taken of the effect of l igament eccentricity:

M---, M + Na/2

As the square root may have either sign, and plastic collapse may occur in either tension or compression, the expression for L r may 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 Mises The analogous results apply. The deep crack validity limits are given by Okamura et al. 19

Fig. 8.

O.1540{ l+N/ [ay( t -a ) ]} if N x> x o = - -


This depth limit agrees with the value OfXo = 0-154 for pure bending given in Section 2.1.2 at the validity limit:

x=0-146 N N

~ = 0-832 ayt ar( t -- a)

- - = 0.974 > 0.5476

This therefore satisfies x 0 = 0.146 < 0-220 and is consistent with the above. Okamura derived lower bounds for shallow cracks. These are shown in Fig. 8.

2.4.3 Plane stra& Tresca 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(ye -- 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 maximum for all values of M/Nt (i.e. the deep crack solution is valid for all M/Nt 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 )

Fig. 9.

Fig. i0.

1.tl

1.6

1 .4

1.2 q(y)

1.0

0.~,

06

2M. Nt Lr= (t-o) 2 cry q(Y)

Upper bound /

Lower bound to q

I I l I 02 0t. 06 08

y : ( t -o ) / ( t .2MIN)

Limit moment and force for SEN plate (from EwingZ'~). Plane strain.

. . . . . Rice's upper bound

. . . . . Shirotor iand Dodd f ield Rice opproximote

expression

, . ,

1.0

O.S m" -= 4 M'

, o-y ( t _o )Z / 0.6

/ . . . . / . = N ,, O.Z. / 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 ""

' \ .~ -" Combined bending and tension for deep-cracked SEN plate in plane strain (from

Nicholson and Paris~S).


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 lane strain M ises This is 1"155 times the plane strain Tresca limit load.

2.4.5 Sha l low cracks in p lane strain (T resca or M ises ) t9"24"28"3

The results are no longer expressible in terms of a single function q(Ye) only, as for deep cracks. Physically the reason is that plastic yielding can spread to the top free surfaces on either side of the notch. Ewing derived an upper bound solution from the shallow crack bending solution. His results are given in Tables 2 and 3.

Table 2 gives an upper bound limit load for the region where the shallow crack solution is appropriate. Table 3 compares these results with the pure bending results for shallow cracks given above, and the values taken from

TABLE 3 SEN Plane Strain Upper Bound for Shallow Cracks

4M Values of 4M (Tresca) or (Mises)

~yt 2 I-I 55ayt 2

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.

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 ORACLE by Parsons, 33 is to ignore the contribution to the bending moment produced by the eccentricity of the tensile force:

Met f = M -- Na

This cannot be rigorously justified, and it should be confirmed that redistributing the moment Na does not cause another part of the structure to be in a more onerous condition than the ligament. This version is used by ORACLE 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 =1 C ------- O.rt2( 1 __ X) 2


Fig. 11.

I 1.07

1.06

~.os a " - I .Or,

~" 1.03 o '- 1.02

E 0 1.01 U

~ Upper bound Lower bound 1.00 , , , A /,lO 0 / i ~ I

90 80 70 60 50 3 20 10 0

SENB with V-notch: plane stress Mises (from Ford and Lianists).

2.6.2 Plane stress M ises 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 rad ians

75 < :~ < 90 c = 1 + 0"229(rc/2 - :~) ~ in rad ians

For :~> 67 both lower and upper bounds are given, and they are both represented by the above fo rmulae to within 0"5%.

This is cont inuous with the deep sharp notch result for ~ = 0 (c = 1.072) and the unnotched bar for ~ = re/2 (c = 1). The depth val idity l imits are not known, except for :~ = 0 and ~ = ~/2.

The results are shown in Fig. 11.

2.6.3 Plane strahl Tresca 2'34

4M c= aytZ(l _ .\.)2

0 < :~ < 3"2 c= 1"2606 (exact)

< :~ < 57"3 c = 1"2606 - 0"0386( :~" - - - ' ~6~ "~^^'" 3.2 0-944 J :~ in radians

This fo rmula represents Ewing's numer ica l results to within 0"3%:

rc - 2:( 57"3 < ~ < 90 c= 1 + (exact) :~ in rad ians

4+rc -2~

220 A. G. Miller

I . I .

~1 ,3 u O

C

e, o

1.1

Fig. 12.

1 9O ;o 4 6% ;o ,'0,'o 21, ,'o

oK

SENB with V-notch: plane strain [from Green').

The depth requirements are: 3"*

1 0

Reciew of limit loads o/" structures containing defects 221

Fig. 13.

1.061.07 J t Upper bound

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 .203 0/, 050 .60 .70 .809 b

b+r SENB with circular root: plane stress Mises (from Ford and LianistS).

For deep notches

0 < u < 0"692

0.692 < u < 1

c = 1 + 0"045u 2 to 0.2%

c = 1.072 0"123r ( r ) 2 + 0"022 ~ 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

Deep cracks

4M c -ayt2(1 -x )2

0

222 A. G. Miller

1.4

1.3

Z, u

o 1.2

c-

O

o~ 1 .1 t - O U

Fig. 14.

i i i i i l i

I I I I I I

.1 .2 . .t. .5 .6 .7 .8 .9 .0 b

b-~r SENB 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 Dodd; 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


A w=lOmm /,,0 mm 40ram A

2mm. di . 45" to ta l notch dr i l l ed IF angle. ho le . 0 .25mm. root

radius.

(a) (b)

Fig. 15.

I I t mm I

I [~

I

2b[ rrrn

/ \ (c)

2ram

8ram

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.

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,(t - a) - - - - - - -~ (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 and R ichards 16 quote the near ly ident i ca l resu l t :

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 Kumar 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


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 factor. - - - , 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 XSE N --~ ( 1 -+ XCT )

where n(x) - N/%t.

226 A. G. Miller

CT5 1 I

! t , SEN - [

O SEN- - - - -

I Lood L ine

t

Fig. 17. Compact-tension specimen geometry.


n(x) = - (1 + x) + (2 + 2X2) l '2

x--* 1 n ---~y2/4

l>x>0

2.11.2 Plane stress Mises

n(x) = - (~x + 1) + [(Tx 2 + 1)(1 + ;,)] ~/2

2 y = ~ = 1.155

1.072v 2 x ~ 1 n ~ 0.268y 2 = 4

for l>x>0

2.11.3 Plane strain Tresca

n(x) = - (1 + 1"702x) + [-2-702 + 4"599x 2] 1/2

x~ 1 n ~ 0"315y 2 - 1"260)'2 4

These results are shown in Fig. 18.

for 1 > x > 0.090

2.11.4 Plane strain Mises This is 1"155 times the plane strain Tresca limit load.

2.11.5 Kumar et al. 25 give values for the limit loads which, as in the pin- loaded SENT case (Section 2.2.5), are the Tresca plane stress formulae,

Review of limit loads of structures containing defects

o ,7e-

227

N

0.6

O.S

0.~

0.3

0.2

0.1

T fescQ

PLane s t ra in

Ol I I I 0 0.2 0./, 0.6 0.8

X

Fig. 18. Compact-tension specimen limit load.

1.0

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 -2a) 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 +

228 A. G. Miller

Fig. 19.

j Surface defect

/1Lo 1 Plate under multi-axial loads.

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 ~,/3

NI=(N 2+Q2), +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 Na 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.

Review ~/" limit loads o[" structures containing deJec'ts

TABLE 7 Values of L r for Edge-cracked Plate under Tension, Bend and Mode I1 Shear

(a) Table of F expressed in terms of m' and n'

229

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 0"10 1.006 1-029 1"060 1"094 1-127 1'157 1'!80 1"!93 0"20 1"022 1-052 1"086 1'119 1-150 1-176 1"195 1-204 0"30 1.044 1"076 1-109 1'141 1"169 1"191 1"204 1'206 0"40 1.067 1"099 1"130 1"159 1'!82 1"198 1-205 1'197 0-50 1"088 1"119 1"147 1"170 1'188 1"198 1'195 1"172 0"60 1"106 1'133 1"157 1-175 1"185 1"!84 1-167 1"119 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

1"!54 1"144 1"000 1"192 1"164 1"!96 1'156 1'188 1'127 1.164 1"049 1"109 0-892

(b) Table of F expressed in terms of 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 0"10 0"997 1"047 1"090 1"128 1"158 1-180 1"193 1'194 0-20 0-987 1'040 1"087 1'127 1"160 1"185 1"200 1"203 0-30 0'971 1'027 1"077 1'120 1-156 1'184 1-202 1"207 0-40 0-949 t'009 1'063 1-109 1'148 1"179 1"199 1"205 0-50 0-922 0"986 1"044 1'094 1"137 1"170 1-191 1-198 0"60 0-892 0"962 1-024 1'079 1"124 1"159 1"181 1-184 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

1-121 1"076 1"178 1-135 1"190 1'146 1'194 1"146 1"191 1"122 1'177 1"106

1.000

The notation is as follows: t= plate thickness; a=crack depth (a/t>~ 0-4); m = bending moment parameter = M/I'26M' for M' = ay(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 specia l case o f zero moment (referred to the l igament centre- l ine,

Ewing 44 has ca lcu la ted a more accurate so lu t ion for deep cracks):

+ 1.03 = 1

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 Swingler 43 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 ly ( / - - a)/2 The upper bound is given in Table 8 and Fig. 22, along with the minimum

-1

Fig. 21.

kN/N1 Nominal Nominal

Per Bound

wet" Bound

QIQ 1

~ NO~lna[ -1

x 0.83

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.

232 A. G. Miller

Upper Bound (Notched Canti lever)

Bound

M/M

I

8/w 2

Nominc]| Lower Bound (M2/M2~.Q2/Q2=1)

Lower Bound

Fig. 22.

0"6

O.t,.

0"2

\ \

I 2/. ~l 0 I I I I I 02 or, 06 o-8 1

Q/Q1

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 common 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:

~ = b 2 +~ b 2 +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 of the crack on the collapse

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 N1 = a , ( t - a), S l = a , ( t - a) /2 An upper bound (which cannot be exact) is given by

NS

This is shown in Fig. 23.

S= S t (N) 2 N 1S'~ z

=0

Fig. 23.

N/N 1 0-5

~ U p p e r Bound - ~ \ \~ \ \

Lower Bound / \ (N/N 1)2+(sIs1 )2= 1 \\

I I I I I I I f I 0'5 s/s, Combined tension and mode III shear (from Ewing and Swingler'~)).

234 A. G. Miller

t

Fig. 24.

" , Line of sidegrooves

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)

Fig. 25.

i i i I i i i i %

%

0.9 \

\ 0.5 ~.......-~U n i vet so I s ing le- h inge

\ 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

a l t

Collapse loads for ungrooved single-edge inclined notch specimens (from Ewing~5).


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:

. fa , 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

t a

e

h N

Centre-cracked plate in tension

plate width of thickness crack width or depth crack eccentricity (see Fig. 26) plate length force/width or thickness

M Crock

M

Fig. 26. Geometry of eccentric crack under multi-axial loading.

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, and plane strain Tresca

N=ay( t -a ) (Ref. 16)

This is compared with experimental results in Fig. 27, taken from Willoughby.17

This result is not valid for short plates (h _.2a(t -a) . This is always satisfied if h/t > l /x/2 = 0"707.

The results for short plates from Ainsworth's lower bound method are shown in Table 9 for plane stress and Mises yield criterion.

3.1.2 Plane strain Mises This is 1'155 times the plane strain Tresca result.

3.2 Eccentric crack under tension and bending

3.2.1 A lower bound 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

Reriew O/ limit loads t?]'struc'tures containing defects 237

1.2

1.0 , , , .

N 0.8 ~N~ ~ !!el 0.6

0..

0.2

0 0.2 0.4 0.6 0.5 1.0

a / t

Fig. 2"/. Centre-cracked panels in tension (from Willoughby I ~). All data from Table 9. , A533B steel; /~, 316 stainless steel plate; A, 316 stainless steel weld; (3, low alloy steel.

Alternatively, for

M - ae M (t 2 - a 2 - 4e 2)

(a) Nt >" t(t - a-----) and N-t ~< 8et

M -ae M (t 2 - - a 2 - - 4e 2) or (b) Nt

zt~

238 A. G. Miller

d o o o d c~ c~ i

i 0 I

zi;~

/ /$o i~

.= ~' ~ 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 !

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.

t.0

"t/'%

o.s

b I Ih = 1.5

f ~ I LE

@

I ~ = , I , , , j J 0 .$ l .O

bzl b I

Fig. 29. Eccentric defect under mode I | | loading.

240 A. G. Miller

I I

I I I I I

t t t

Fig. 30.

I I I I I I I I

I " " -~1 ^ I j 1 i I i I

t I t t t I t t t Array of eccentric defects under mode I loading.

I I

I I

" "1 I I I I

0.5-

11500 " ~

I

0 0.25 0.5

o l t

_11

2c

I t

Fig. 31. Local collapse load for central embedded elliptical defect in plate in tension.


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

where

Lr m (a~ + 3/4a~ + 3r2) t~z

O'y (with a Mises shear term)

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.

242 A. G. Miller

4 DOUBLE-EDGE NOTCHED (DEN) PLATES

t thickness b l igament thickness ( t -2a) a crack depth u b/ (b+r) x a/t ~ notch flank angle r notch root radius c constraint factor

The geometry is shown in Fig. 32.

4.1 Bending



4M c - - = I

ay(t -- 2a) 2

4M C- -

ay(t - 2a) 2

This is not known but must satisfy 1 0"168

o'y(t -- 2a) 2

For shallow cracks this must be cont inuous with the unnotched bar result (e= 1).

4.1.4 Plane stra& Mises This is 1"155 times the plane strain Tresca limit load.

4.2 Tension

Unlike the SENT geometry, there is no moment due to the eccentricity of the tensile force.

4.2.1 Plane stress Tresca 49

N C.~-

O'y(t - 2a)

4.2.2 Plane stress Mises 49

N C=

ay( t - 2a)

c = 1"155

c = 1 + l '08x

=1 O


N

MA- 0 "

I

I

I Q h Q

I

I ,C I I t

M

N Fig. 32. DEN geometry. N, Tension: M, bending moment; Q, mode II shear.

The deep crack solution is exact. The shallow crack solution and the transitional value of x are approximations.

4.2.3 Plane strain Tresea 50

N C- -

try(t - 2a)

e = 1 + In (1 _--2~x)l - x 0 < x < 0.442

rc c = I + }- = 2.57 0.442 < x < 0-5

x-- ,O c - , l + x

The deep crack result comes from the Prandtl fan slipline field.


244 A. G. Miller

4.2.5 Kumar et aL 2s give formulae which are not the correct limit loads. These results are not recommended for use.

4.3 Combined tension and bending

The general solution for combined tension and bending is not known. A lower bound solution, exact for Tresca plane stress, may be obtained by applying the Tresca plane stress solution for plain beams to the reduced section width:

[2M[ + [4M 2 + N2(t - 2a) 2]~ ' L r = ay(t - - 2a) 2

For a given ligament thickness and ligament load, the single-edge notched solution will be a lower bound to the double-edge notched case.

4.4 Bending: effect of notch angle


4.4.2 Plane stress Mises Not known but 1 ~

1.1, l !

,..1.3 0

"6 ,.2

e.

o I.I

Review o f limit loads ~/" structures containing d~/bcts 245

Fig. 33.

1 ! I

90 eO ?0 6~0 5'0 z:o 3~0 zo I=0

DENB with V-notch: plane strain (from Green2).

4.5 Bending: effect of notch root radius


4M ay(t - 2a) 2

=1


Not known but 1 pC > 1-30 x > xo(r/b ) > 0"115

The exact values o f x o are unknown. The results are shown in Fig. 34.

246 A. G. Miller

1.3

u

1 .2

C

0

C 0

Fig. 34.

m

1 0 .1 .2 .3 ,t. .5 .6 .7 .8 .g 1.0

b

b+r

DENB with circular root: plane strain (from Green-').


A discussion of combined tension and bending in plane strain for large root radii and deep cracks is given by Dodd and Shiratori. 3~

4.6 Tension: effect of notch angle

4.6.1 Plane stress Tresca s~

N e - -1

ay(t - 2a)

Hill showed that the constraint factor is independent of notch shape.

4.6.2 Plane stress Mises 49

< 70"5 shallow deep

c~ > 70-5 shallow deep

N C=

ay(t -- 2a)

c = 1 + l'08x c = 1-155

0


Fig. 35.

1.15

f o 1.10

N ros C 0

(..)

upper bound

bound

1.0 90 80 70 60 S 4 30 20 I0 0

=,c=notch ang le ,~

DENT with V-notch: plane stress Mises (from Ford and Lianis~a).

4.6.3 Plane strain Tresca s

N

ay(t - 2a)

shal low c=l+In 1 -x

deep c = 1 + r~/2 - :~

e =/2 -=- - 1 x < x o = 2e=/2 _= _ 1

x > x o

The transit ion occurs when the two expressions are equal, and so the smaller constraint factor a lways applies. Ewing 52 generalizes this result to al low for r > 0. The results are identical with the sharp crack results for ~ = 0, for both deep and shal low cracks, and with the unnotched bar results for :~ = re/2.

4.6.4 Plane strain Mises This is 1-155 times the plane strain Tresca limit load.

4.7 Tension: effect o f notch root radius

4.7.1 Plane stress Tresca 5~

N e= =1

ay(t - 2a)

Hill showed that the constraint factor is independent of notch shape.

4.7.2 Plane stress Mises 5t Deep cracks

b/2r < 1"071 e = 1 + 0"226b/(b + 2r)

b/2r > 1"071 c = 1"155 - O.080r/b

These are upper bounds accurate to 1"8%. Ford and Lianis ~8 give similar

248 A. G. Miller

Fig. 36.

1.15 Hil l 's upper b

o / /~ . Lower. 1.10

e- 1.05 o

1.0 I 0 0.1 0.2 0.3 0.4 O.S 0.6 0.7 O.S 0.9 1.0

b br

DENT with circular root: plane stress Mises (from Ford and LianisLS).

results. They merge with the deep sharp notch solution at r=0 (c= 1"155) and with the unnotched bar solution at r = vc. The depth validity limits are not known, except at r = 0 and r = c~. The results are shown in Fig. 36, with both upper and lower bounds.

4.7.3 Plane strain Tresca 5z

N c - /. = min [7z/2, In (1 + b/2r)]

ay(t - 2a)

deep notches (exact)

t > b(2e z - 1) - 2r(e ~ - 1) 2

small b/r:/. = In (1 + b/2r)< ~/2 angle at notch that has yielded

c = (1 + 2r/b)ln(1 + b/2r)

This result was given by Hill: 53

large b/r: In(1 + b/2r) > 7t/2 c = 1 + ~/2 -- 2r/b(e ~" 2 _ 1 - ~/2)

shallow notches (approximate)

t < b(2e z - I) - 2r(e z - I) 2

( 2r 2rt"l'/2 (~_r r ) [ b b( 2r 2rt'~] '/2 c= 1+ b b2 j + 1+ In l+2r -2r 1+ b t,-'J_]

When r=0 the solution is identical to that of Ewing and Hill 5 for sharp notches, both deep and shallow. For unnotched bars

r= m Z=0 tc=b The deep notch solution gives c+ 1. The shallow notch solution limit


depends on the order in which the limit is taken, but gives c ~ 1 if r--, oc is taken last. Ewing 52 also gives the effect of ~ > 0.

4.7.4 P lane s t ra in M ises This is 1"155 times the Tresca plane strain limit load.

4.8 Staggered notches under tension with restrained rotation

Connors s4 gives an approximate solution for the geometry shown in Fig. 37, with the notation:

a = crack depth (cracks of equal depth)

d = crack separation

t = plate thickness

N 0 = fly/

Ny = limit load

Ny = min (Ni, NH)

ar id 2 + (t -- 2a) z] N~ = [3d2 + (t - 2a) 2] t/2 using the Mises yield criterion

N, is the pin-loaded SENT result from Section 2.2. The experimental results are shown in Fig. 38 for the limit load based on a deformation criterion.

For d = 0 this reduces to the Tresca plane stress DENT solution (despite the use of the Mises yield criterion).

4.9 Combined bending and mode II shear

This has been calculated for deep cracks in plane strain by Ewing and Swingler. 43 The results also apply to the case where the cracks are at a cantilevered end. Lower and upper bounds are shown in Table 10 and Fig. 39. The 'nominal' yield criterion is also shown:

+ =1

M1 = ~,(t -- 2a)2/4 (Tresca) Q1 = cry(t - 2a)/2

This is within 5% of the lower bound. The upper bound is potentially exact, but is an incomplete solution. In the

zero shear limit, it agrees with the solution given in Section 4.1.3. The depth validity limits are also shown in Table 10. They are reduced by shear.

The Mises limit load is greater by a factor of 1.155.

Fig. 37.

d L =-I

i i I I ' ..... ( _ _ ~a .....

tr I- -/- T ',, / Location of defects

Typical specimen with staggered defects (from Connors54).

Fig. 38.

o

Z

0 0

0

1.0

.9

.?

+t .5

.,

}- i

I.O

.~ .= d No '/'Jt

Olt =0.25

t t ~ ,,:o.37+

;_ ~ ,,=0~

~ at t =0.7

i I I I 2.0 3 .0 ~.0 5 .0

Relative separation of defects, dlt

Experimental results for bars with multiple defects and theoretical predictions (from Connors54). - - , Theoretical predictions.

Review of limit loads of structures containing defects

Upper Bound

251

M/MI I

8 /~ 2

Nominal Lower Bound .(M21MI 2 O21012--1)

Lower Bound

Fig. 39.

0-6

o., - \ '1 \ 0.2 !- \\ j 2/~ 1

o i i I ] i 0-2 0"~ 06 0-8 I

Q/Q1 Combined bending and mode II shear (from Ewing and Swingler~3).

TABLE 10 Combined Bending and Mode II Shear

(from Ewing and Swingler 43)

O/O, Upper Depth Lower Q,"Q I Upper Depth Lower bound limit bound bound limit bound M/M t 2a/t M/M 1 M/M t 2a/'t AI/M t

0-000 0"050 0"100 0-150 0'200 0'250 0-300 0-350 0"400 0"450 0-500

1'380 0"336 1"000 0-550 1-020 1-358 0"314 0"999 0'600 0"970 1'334 0"290 0"997 0"650 0"915 1-308 0'266 0-992 0-700 0-854 1-281 0-241 0'986 0-750 0"786 I "251 0-215 0'978 0-800 0"708 1"220 0-188 0"968 0-850 0-618 I' 185 0" 160 0"956 0'900 0-509 1"149 0"132 0"941 0"950 0"362 I" 109 0' 104 0"923 ! "000 0-000 I "067 0.077 0'902

0-051 0'028 0"011 0-001

0"875 0"842 0.797 0"736 0-657 0-56 I 0.447 0"315 0"166 0"000

252 A. G. Miller

M

_.U n

Fig. 40.

l Iut't , t b N ) / M

I . . . . . 1 + Asymmetric double-edge notched plate.

4.10 Combined tension, bending, mode 11 shear and out-of-plane stress

The solution given by Jeans, 43 quoted in Section 2.12.1, is a true lower bound for this case (with uniform out-of-plane tensile stress across the section).

4.11 Combined tension and bending: asymmetric notches

The geometry considered is shown in Fig. 40. Ewing (pers. comm.) constructed a Tresca plane stress lower bound solution as follows. With notation as in Fig. 40

Mt = M + Nat /2

is the moment referred to the middle of ligament h I (= t - at). Assuming that the cracks are able to support compressive stress, then

"~ 2 "~ 12 , 2 L r = [2M t + (4M/- + N h~) ]/(ay/-,t)

provided that the compressive region is at least as deep as crack a_,. The height h of this region satisfies

-hay + (h t - h)cTy = N~.om,p~c = N,/L r

so that

2h = h t - N/L ,ay ~> 2a_, (26)

N

a;t

t r bl 'tl

+ :x: ( in contact )

t

dl

d2 J ~c12

Fig. 41. Asymmetric double-edge notched plate alternative solution.


Ifeqn (26) fails, ignore material below some ligament 'b' lying between b~ and b~ - a z, i.e. solve for b and L r the simultaneous equations

L r = {2M" + [4M ''2 + N2b2] t /2} /uyb 2 b = 2a~ + N/Lray

where

a'~ = a z - b I + b M" = M I - N (b I - b)/2

Alternatively, for very deep cracks a2 in compression, assume that only the outside part 'x' is compressed and that the system is equivalent to an internal crack (see Fig. 41). At collapse

N/L r = ay(bt - a2 - x) M/L r = ry[(bl - a2)dt + xd2]

which can be solved for x and Lr as unknowns. This gives an alternative lower bound.

5 SHORT SURFACE CRACKS IN PLATES

5.1 Wide plates in tension

Ligaments in finite length cracks are stronger than ligaments in extended defects, as they derive support from the adjacent uncracked plate. If ligament failure is the subject of concern, then this 'local' limit load goes to zero as the ligament thickness goes to zero. This is in contrast to the 'global' limit load, which would tend to the through-cracked plate limit load (as in Section 3) as the ligament thickness tended to zero. Moreover, for wide plates (width >> defect length) the defect has no effect on the global limit load.

a defect depth (see Fig. 42) 2c defect surface length t plate thickness

W

2c P

Fig. 42. Geometry of surface defect.

254 A. G. Miller

Fig. 43.

/ '~)e f f

1.0 D

0.8

0.~ 0.5

, , , 0 0.2 0 :, 06 0.8 1.0

( l i t

Ligament correction parameter as a function of a/t (from MilneSS).

5.1.1 R6 Rev. 2 recommends that short surface cracks may be treated as extended defects with an equivalent depth given by

__

o 2t(2c+ t) 0.1 < 2--~ < 0.5

O


He proposed that this formula applied to all geometries, not only plates. This transformation is analogous in form to the Battelle formulation in Section 8.2.3, which may be written

a[ 1 - ( l /M) ]

ae = 1 - (a / tM)

where M is a function of both c/t and R/t (R is the cylinder radius):

C 2 '~ 1/2 M- 1 + 1"05 ~-~-,/

CheWs transformation gives a lower limit load for cylinders than the Battelle transformation, as f> M. This transformation is the simplest rational function with the above three properties.

5.1.3 Mattheck et al. 57 propose an expression for ligament yielding based on Dugdale model calculations. Their expression is tr = arM, where

M = (1 - I - 1.9071 1 + 1.5151(a)'16596(/)2

1 52(a']214'O(a'] 3 1 x [ - 0.74 + 3.855 a - 3.825(a)2 - 2.89(a)3

+ 4"356(ay l} [1 - ( t ) "4 1

This equation can be applied for a/c < 0-7; ~r is applied membrane stress. This formula is shown in Fig. 44(a) and compared with their detailed

numerical results. The results are compared with the above expressions from R6 and Chell in Fig. 44(b). It can be seen that in general the Mattheck result is most conservative and the R6 Rev. 2 result is least conservative. The R6 result has been plotted beyond the claimed validity limits for a/t.

5.1.4 Miller 9 derived a semi-empirical model for ductile failure (i.e. at L m~x) of surface defects and concluded that the reference strain at failure was

k t 1 -a / t

4 c a/t

256

l o ' f 0.5

0

A. G. Miller

t .0

ale 0.7 0.5 0.3 o.,-----'~~

O O . 5 1. 0

oil (a)

1.0

o . . . . . . . ~ 1/fl' 0.5 i.O

olt

(b) Fig. 44. Effect of surface crack length (from Mattheck et al:~). (a) - - - , Fitted equation:

, numerical results. (b)'.-, Chell;---, Harrison (R61; - - , present method.

where k is a material constant between 0.4 and 1"5 for mild steel. This strain may be used to give an appropriate stress for use with the limit load.

5.1.5 If this strain (Section 5.1.4) is below the yield strain, then Miller recommended that a simplified line spring model could be used to estimate the load at which the ligament goes plastic. For pin-loading with remote load cry, this is given by

O" x (I a/t)[(~mm + A)(Z%b + Aft) z

GF A(O~bb -t- Aft) where

m~-- m 2c 1 l+v


2[ (i- ~-)z =~i ()

(~.)z

mm

mb= bm

bb

o I o

[ I I I I 0.2 O.t, 0.6 0.8 1.0

OIt

Fig. 45. Spring compliance (SEN).

and ~ is shown in Fig. 45. ~ may be ca lcu lated f rom formulae given by Ewing: 5s

x =- a/t y -- 1 -- x ~ij = nli j

Imm(x)=O'629 44x2( l + 5"474xZ +13"38x4- -32xS + 58"32x6) i f x 0"5

258 A. G. Miller

5.1.6

An alternative geometrical approach suggested by the Welding Institute (pers. comm.) is that the stress is unloaded from the crack on to the nearest part of the cracked structure. The critical load is taken as that which the thinner ligament first yields, using a local fixed grip criterion:

(1 - x)(1 - x + 2R/ t )a m - ab/3 af = 1 - 2x + 2R/ t

R is the radius of curvature of the crack front (c2/a for ellipse).

5.1.7

It is recommended that the global collapse load is used if the criterion given in Section 5.1.4 is satisfied. Otherwise the local collapse load given in Section 5.1.5 should be used.

The above solutions have only considered a single surface defect. They all have the limiting behaviour

C O" m

t o'f

(7 m (I - >> I - - - * I - -

t af t

Therefore it is suggested that in the case of an extended defect of variable depth, the ligament thickness should be taken as an average thickness over a length equal to the plate thickness, and a local fixed grip criterion used:

am- 1 c~ o'f [

where ~ is the average crack depth. This suggestion is provisional only.

5.2 Narrow plates in tension

When the plate width is not much greater than the defect length, the global collapse load limit is reduced by the presence of the defect. For rectangular surface defects in a plate under fixed grip loading, the constraint factor is unity:

F= ayA

F tensile force A net-section area in plane of defect

For non-rectangular defects this will give a lower bound.

Review of lirnit loads of structures containing defects 259

5.2.1 A review of CEGB and Welding Institute test results is given by Miller. 59 The degree of gtrain hardening is observed to increase with increasing ligament thickness.

Miller 59 gives the nominal strain at failure as

kt(t - a) 4ca[ 1 - (2c/w)] "

w plate width k material constant ~0.4-1-5

A lower bound to the final plate failure load after ligament snap-through is given by F= a,A o, where A o is the net-section area without the ligament.

5.2.2 Hasegawa et al. 6 observed a similar increase with ligament thickness of the net-section stress at the onset of crack penetration. They empirically put the ligament stress cr~ and the stress cr~ in the rest of the plate as

~rl = (1 - R)crf (R = reduction of area in tensile test)

tro = tru - (a. - ar)(a/t)

A lower bound to the final plate failure load was given by

F= Aomin la , ,a f+ (a . - O'f)( 1 --a/t~210"9 }_]

which is slightly lower than that given in Section 5.2.1.

5.2.3 Mattheck and Goerner 6t used a strip yielding model where the stress along the crack front and along the ligament centre-line was a u, just before ligament rupture, whereas the stress at the end of the plastic zone at the surface is reduced to the yield stress. Although detailed results were not presented, this will also give an increase of net-section stress with ligament thickness. Agreement with ligament rupture test results was good.

5.2.4 Munz 62 gives a review of surface cracks. He also reports further plate tests of Goering, which agreed with the hardening models of Sections 5.2.2 and 5.2.3, but where the maximum load was underestimated by using flow stress.

5.2.5 The above are concerned mainly with the ductile instability cut-off at Lr max. Miller t2 reviewed published J calculations for surface defects in plates in

260 A. G. Miller

tension. He concluded that using net-section (global) collapse gave better reference stresses fo/" J estimation than using local collapse.

It is recommended, therefore, to use the global collapse load for J estimation. Provided the strain criterion criterion of Section 5.2.1 is satisfied, it may also be used for ductile instability assessment. If it is not satisfied, then a local limit load should be used for ductile instability.

5.3 Plates in bending

There is no generally accepted limit moment for this geometry. The geometrical transformations of Sections 5.1.1 or 5.1.2 might be used. Alternatively, the simplified line spring model used in Section 5.1.5 may be adapted, along with an appropriate ligament yield criterion, such as that given in Section 2.4.3.

6 ROUND BARS

6.1 Bar with axisymmetric sharp notch under tension

R maximum radius b minimum radius F axial force

The Tresca yield criterion is used:

unnotched

deep sharp notch 63'6"~

c- -

or similarly v

F 7~h20"y

4'--

F= ~/9 20"y

- - -=2"85 h R

Reciew o/'limit Ioad~ 01 structures containing de/i'cts 261

C (Shield)

heotet ica l 2.6 -%,,cu rve 2=.

2.&

2.2 ~ f>p. 2.0 ~. "%%.~ 2b Moteriol oluminium 1.8 "'~

16 f u .L /% 0"~ ..

1.4 % ~

1.2 '~'

1.0 J i I I I ~1 , 15 30" 45 G0" 75" 90

(7,.

Fig. 46. Theoretical and experimental values of the constraint factors. , Axial symmetry calculated points: A, experimental ./;r points (yield point); ff], experimental fu.~. points

(ultimate load). (From Szczepinski et al. 65)

Fig. 47 and are slightly less restrictive than those give by Shield. Experimental results are also shown in Fig. 46.

A lower bound that is within 4% of the results in Fig. 46 is

F 3'7:~ c - - ~,rcb2 a - - 2"85 - - (x in radians)

7~

An approximation to the depth requirement is

R 3-88:( -- > 2"94 - - - b ~z

6.3 Bar with axisymmetric round notch under tension

b minimum radius r notch root radius F axial force

Fig. 47.

Rib 3.0, 2.6 2 .6 2. t, 2.2 2.0 1.8 1.6 1.4 1.2 1.0 . -.-.~-

0 30 e 60 90 l z0" lS0 t 180 Notch ang le 20(

Theoretical values of the R/b ratio for V-notched bars (from Szczepinski et al.65).

262 A. G. Miller

TABLE I I Constraint Factors for Axisymmetric Bars with a Round Notch

under Tension

h/r Bridgman 66 Siehe168 S:czepinski et al. 6~

0 I I I I/3 1.078 1.083 1/2 I'115 1.125 I 1-215 1.250 2 1.386 1.500 2"5 1.65 3 1.524 1.750 4 1.649 2.000

Mises yield criterion (except for Szczepinski et al., 65 who uses Tresca):

(l 7~b2o.y d- In + (Bridgman 66) F b

-- ~b2o.y - 1 + ~ (Davidenkov and Spiridinova 67 and Siebel 6s)

These both reduce to a constraint factor of unity for an unnotched bar. The formulae are compared numerically in Table 11. The Davidenkov form is the small b/r limit of the Bridgman formula. At large b/r the constraint factor must be limited by the sharp notch result:

b/r = ~ c = 2"85 Szczepinski gives an approximate deep crack validity limit:

- ->2"95 1-1-68 -1 -=~ b b

Some geometries with longer notches were considered by Szczepinski et al., 65 and the constraint factor is shown in Fig. 48, along with the experimental results. (There is an inconsistency in the description of the geometry in the paper.)

Hoffman and Seeger 69 give a finite-element result:

R r 1 = 2 b - 11"~ - 0"085 c = 1"90

This is bounded above by Ewing's result for r = 0 in Section 6.1 (c = 2), as it should be. It is bounded below by Szczepinski's result for rib = 0-4 (c = 1"65), also as it should be.


c'" I,., , _~ ,~ ,_-t.,~

~ . I Ld Material: 1,5 ~/theoret#cal curve ~ mild steel

IX

~1"1 "~...~n, ,~ . . . .~ , , e 0, , , . . . .

0./* O.G 0.15 1.0 1.2 t.t, 1.6 1.8 2.0 d/b

Fig. 48. Theoretical and experimental values of the constraint factors. O, Axial symmetry calculated points; A, experimentalfyp points; 0, experimentalfu.L points. (From Szczepinski

et aL 65)

6.4 Bar with axisymmetric notch under torsion

b minimum radius T twisting moment

The constraint factor is unity for all notch geometries:

3T

c=/tb3o. ~ - 1 Tresca (Walgh and Mackenzie 7) 3x/3 T

c = _ .---=-=-= 1 Mises 2rcb~cT,

6.5 Round bar with chordal crack under bending

R bar radius a crack depth b ligament depth (2R- a) M L collapse bending moment M o collapse value for uncracked bar T L collapse twisting moment To collapse value for uncracked bar y b/R

Ir v 3 2R Fig. 49.

Crack depth ,.a

Ligament depth, b

t Schematic diagram of a bar containing a chordal crack (from Akhurst and

EwingVn).

264 A. G. Miller

ML

1.0

0.9

0.8

0.7

(P ' 6)"

0.6

O.S

0.4

0.3

0.2

0.1

01 I l = 0 0.1 0)2 0.3 01.4 O'.S 016 0~7 0.' 0.9

al 2R Fig. 50. Lower bound bending moments for p.lastic collapse of chordally cracked bar in bend (from Akhurst and Ewing7~). M L. Bending moment for plastic collapse; Mo, bending

moment of untracked bar at plastic collapse; a, crack depth; R, bar radius.

The geometry is shown in Fig. 49. Akhurst and Ewing v ~ give a lower bound Tresca solution, shown in Fig. 50 and Table 12 for O


TABLE 12 Normalized Collapse Loads for Chordal Crack

a/2R MtjMo (bend) TtJT o (torsion) Tt./To (torsion) lower bound constrained ends free ends

0 1 1 ! 0-05 0.958 0.973 0.964 0-1 0.889 0.928 0.902 0-15 0-810 0.875 0.829 0-2 0-725 0.819 0.749 0-25 0.640 0.762 0.667 0.3 0-556 0.705 0-584 0-35 0.475 0,651 0.504 0-4 0.400 0.598 0-427 0.45 0.329 0.548 0-355 0.5 0-265 0.500 0.288 0'55 0.208 0-452 0.227 0.6 0.158 0.402 0,174 0-65 0.116 0.349 0.128 0.7 0.080 0 0-295 0.088 9 0-75 0-051 6 0-238 0.057 7 0.8 0.0300 0-181 0,033 7 0-85 0-014 8 0.125 0.016 8 0.9 0-005 5 0.072 0 0-006 2 0.95 0,001 0 0-026 9 0,001 1 1 0 0 0

torque is reduced. The results for both cases are shown 12:

uncracked T O = nR3ay/3

6.6.1 Deep crack, constrained:

TL/T o = (0"9 - 0"5y + 0ly2)y 3/2 b < R

6.6.2 Deep crack, unconstrained:

TL/T o = (0"360 - 0.072y)y 5'2 b < R

These approximations are accurate to 1/2%.

6.7 Round bar with chordal crack under combined torsion and bending

Akhurst and Ewing ~ give a lower bound formula:

(M/ML) 2 + (T/TL) 2 = 1

in Fig. 51 and Table

266 A. G. Miller

T L

To

0.9

0.8

0.7

0.6

Fully rigid bar.

of shear = Bar

Ret, iew of limit loads of structures containing defects 267

bending moments. The solutions are usually expressed in terms of a shell parameter p:

c p = (Rt)l/2

where c = characteristic defect length, R = characteristic shell radius and t = thickness.

The simplest shell structures are those with membrane stress solutions (e.g. spheres and cylinders). Considering pressure loading, the limit pressure tends to the membrane solution for the uncracked shell as p --, 0, and to the membrane solution for the cracked shell as p-~ ~ (this will be zero for through-cracked shells). The limit pressure is a non-increasing function ofp.

The tendency for the limit pressure to approach the membrane solution for the cracked shell as p increases may be demonstrated on the assumption that plasticity is confined to a shallow region round the crack. Then in Cartesian coordinates the equilibrium equations controlling are (neglecting terms of order t/R)

~'2M U c~x, ~xj + ~qjN, j = P

where J,- is the curvature tensor. Introducing the dimension variables

x, p2 c2 Xi E l Kij : RKij R- 1 -rcii = Rt

_ 4Mij Nij PR IH i j - - F/ij -- p - O'yl 2 fly/ O'yl

then the equilibrium equations become

1 ~2tnij t- K, jn~j = p

4p 2 c~X i Xj

Since m, n, K and X are all of order unity, as p increases, the size of the bending term is reduced, and hence the pressure tends to the membrane solution. This also demonstrates that p is the relevant parameter when local effects predominate.

Limit loads for some defect-free shell structures are shown in Fig. 52.

7.2 Membrane solution for pressure loading

For an axisymmetric closed shell the membrane solution at any point depends only on the local geometry and is given by

Pro Pro I r c~ 1 __N~=2sin~b No= 2 _ 2 sin ~b r I sm

268 A. G. Miller

e

e~

-=

= =

u i i

Y

= E c.U

"E I

C u~

o -"

.4: i~ i ,

U E

z - : : ..-: :~o~ o" -" " J ~a

~ oo ~ ,,.

o oo_~ ~ ~ ,,

UlV lv

r ~

.%

.=_

r_..

L~

r~ br ,

Review of limit loads of structures

Review of limit loads of structures containing defects

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