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Energy-based method for determination of allowable stress—Part II
ht. J. Pres. Ves. & Piping 62 (1995) 95-101 @ 1995 Elsevier Science Limited Printed in Northern Ireland. All rights reserved 030%0161/95/$09.50
ELSEVIER
Energy-based method for determination allowable stress-Part II
of
E. V. Chechin for Probterns
fnsfifute
of Sfrengfh, Academy
of Sciences of Ukraine,
2 Ti~iyy~z~skaya
str., Kiev 2.52014, Ukraine
(Received 24 January 1994;accepted2 February 1994)
An energy-basedmethod for determining allowable stresses,the main link in strength analysis, which for the first time takes account of a basic set of mechanicalproperties of structural materials, namely resourcesof plasticity, hardening (strain and low-temperature) and crack resistance, has been developed instead of the classicalmechanicalmethod involving the material strength characteristicsonly. By taking into account the basic mechanical properties, an increase in allowable stressaccording to the new method is achieved: from fractions of a percent for high-strengthlow-plasticity materialsto 30% at room temperature and up to 50% at cryogenic temperaturesfor plastic hardeningand nonembrittling materials and this allows reduced metal consumptionin structures and machineswithout lowering the level of their reliability. Realization of the energy-basedmethod for determining allowable stresses (calculatedresistances)in the strength analysisopensup prospectsof resolving the problem of making maximumuseof actual strength for all steelsand alloys usedin machinebuilding. This makesthe method of great practical use.
NOTATION
allowable stress taking into consideration strain hardening of the specific material at normal and specific temperatures
If, rlTss Coefficient of strength reserve taking into consideration the material strain hardening; [a,]“, [u]~,~ The allowable stress determined by the energy-based method at normal and cryogenic temperatures @fail,a& Real values of failure stresses at normal and cryogenic temperatures [A#
exe = 2
The second correction factor to the standard allowable stress taking into consideration the resource of energy absorption which is available due to excess in the stress of the yield point over the allowable stress
- [a]s
Correction factor to the standard allowable stress derived from the direct experiment at normal and cryogenic temperatures
tw
ISHM
= 2,
[A+ISHM
DETERMINATION OF ALLOWABLE STRESSES AT ROOM TEMPERATURE In determining allowable stresses with regard to the strain hardenability of structural materials, we have to take into account the fact that in machine building the standard allowable stresses are determined by minimal values of strength characteristics, i.e. the minimal values of
The maxi-
= $$2
mum correction factor for the ideally strain-hardening material at normal and cryogenic temperatures
[Aull= e-t- UO.2)Pl ?zs ’
EAulT = w - d*)PT 1
nT,S
allowable stresses [aIS = min {y;:}
The first correction factor to the standard 95
(Fig. I).
96
E. V. Chechin
600 400 E
l d
200
/ IV,
m, ,-rrr ,rr,
I
1.0
i.5
1.6’ ’
2.0
2.5
gt’“o.2
Fig. 1. Allowable
stress values used in machine building versus (TJu~.~.
allowable loading of a part (structural element) not causing residual deformation. When the latter condition is met, the appearance of the Bauschinger effect during structural service is eliminated. In view of the above, the maximum value of the correction factor to the standard allowable stress is found as the difference between the stress that is equal to the yield strength of the material and the standard allowable stress determined from the yield strength or from the tensile strength. Thus
[Au]y = uo,2 ;” = u”.2(;y - l) In this connection, the most convenient way of presenting them is as a sum of the original (standard) value of allowable stress [a] and the correction factors [AU], and [AU]*, taking into account the reserves of strain-hardening materials, which are proportional to the constituents of the energy absorption of the material: [ale = [u]” + [AU], + [Aal z---f-ulim n
AU1
ns
+
-Au,
n
(1)
where (Tlimis the limiting value of the stress of the material (go,2 or a,) n is the standard strength safety factor, AU, = (TV- u~.~. and Au, = (T~.~[u] (Fig. 2). As can be seen from eqn (1) the coefficient ~1’ is unknown. To find it, let us consider the boundary conditions consisting in the following. We assume that the strength safety factor for a material with any level of hardening cannot be less than unity. Also, the maximum value of allowable stress must not exceed the yield strength (T~.~,which determines the upper limit of
cr
il
0,
ut
Y [Au]yx
=
uo,2
_
z
=
(To.2(nY
(2)
Y -
ut/“@2)s
(3)
nt
To obtain the relationships determining the coefficient ns we use the characteristics of the ideally strain-hardening material (ISHM) in which the uniform strain E, = 1.0, the energy parameter of strain hardening p = PO= 1.0, and the tensile strength a,* = 3uo+ In view of these characteristics and eqn (9) the maximum value of the correction factor for an ideally strainhardening material can be determined with the use of the yield strength (T~.~(see Fig. 2) from the equation
= 3uo.2 - uo.2=- 2uo.2 (4) n”y n”y and with the use of the tensile strength a:’ from the equation
= u:: - 1/3u,* =- 2uo.2 . (5) s nt n: Considering that the maximum value of allowable stress should not exceed the material yield strength, the magnitudes of the maximum correction factors for real and ideal materials should be (see Fig. 2)
ao.2
[VI 0 Fig. 2. Determination. of correction factors to the allowable stressesat normal temperature.
uo.* - [aIs = [Ao]‘~~~. Solving eqns (2) and (4) together, we obtain the design-basis equation for determining the coefficient n;: u,.,(n,
- 1)/n, = 2u,.,/n;
Energy-based
method for determination.
Furthermore,
L2n, n, - 1
stress-Part
97
II
of the material are found from the following relationships:
from which n,-
of allowable
(6)
,Au],,, _ 2
- b”.’ ,t”‘;‘”
(10)
we solve eqns (3) and (5) together:
nJ.*(% - (TtI(To.2)Iyzt= 2a,.,ld from which n; =
2% n t - ut/u0.2
(7)
We should note that the coefficients PZ;and PZS derived in such a manner for an ideally strain-hardening material, when used to determine the correction factors to the standard allowable stresses of specific structural materials, must be corrected to take into consideration the difference in strain hardening of an ideally strain-hardening material and the specific material and the observance of energy similarity of the compared values. For this we multiply the coefficients n”y and nf by the ratio of the energy parameters characterizing an ideally strainhardening material (&) and the specific material (PJ, considering in this case that PO= 1-O. Then the equations for determination of the first correction factors to the standard allowable stresses, taking into consideration strain hardening of the specific material, acquire the form
In this case the values of the corrections to the standard allowable stresses are proportional to the part of the total energy absorption of the material contained between the levels of stresses co.2 and [a] with a strain equal to G. Therefore, using the energy-based approach to determine the allowable stresses we find their differentiated values, taking into account strain hardening and the reserve of energy absorption of specific structural materials, from the equation
-n,[4”,)P2. (6 , bl’ = min{T + 2n,,(n _ 1j + Y 3 + (at - flO.2)Pl+ - [u]s)P2 (12) Int 2n,l(fl,- uJu~.~) nt (go.2
~0.2)Pl
(co.2
This equation can be represented in simpler form as
[ale= minbl; . (1 + [A41,y. PI + [AL,, . P2>; bl: . (1+ [W,t . PI + LA4.t. Ps)) (13) where
[Au],,, =% =ut- u”.2. -& n”y 0 1
[GLy =2n,l(n, (8)
go.2
- 1) . [u]; ’
- [cl; ny. [a]”
co.2
[A&
[Au],,,= $ = ut ,suo-z.$-jr t 0 1
@t-
=
Y
ut GIL, = 2%/(4- [a]; m2,t = nt - Lals t *
co.2
dJ-0.2).
=
(5
-
~0.2)Pl s nl
In other words, the correction factors considering strain hardening of the material are proportional to its energy absorption related to strain hardening. Furthermore, since with the energy-based approach to determining the allowable stresses for structural materials the ‘usable’ part of the total energy absorption of the material is that one on the level of allowable stress (Wr,] = [a]~,, (see Fig. 2)‘, the secondary correction factors considering the reserves of the energy ablsorption
[a]; ’
go.2
Considering the expression between the round brackets (eqn (13)) as a reduced coefficient in relation to the standard allowable stress, we note that this coefficient is always more than unity and will adopt different values for materials with different parameters p1 and p2 and, consequently, with different resources of the plastic strain of materials and their strain hardening. Moreover the reduced coefficient will also vary from the value of the chosen standard allowable
106
Contents of Volume 61
Numbers
2-3 SPECIAL
An International
127
Guest
Opening 129
ISSUE
FAILURES ‘94 Symposium on Risk, Economy Failure Minimisation and Analysis
and Safety,
Editorial
and Keynote
Address
The future role of nuclear power W. STUMPF (South Africa)
in addressing
global
environmental
problems
Overview 157
A practising engineer’s experiences station boilers J. A. COLLINS (South Africa)
177
Case studies in disaster-a coded E. A. BRADLEY (South Africa)
199
Failure types, consequences R. K. PENNY (South Africa)
,Economy
in overcoming
inherent
design
problems
in thermal
power
approach
and possible
remedies
and Safety
213
The important role of technical C. 0. BAUER (Germany)
229
Risk management in safety K. BURRAGE (UK1
257
The contribution of environmental E. REYNOLDS (South Africa)
283
Design responsibility. Consequences legal requirements C. 0. BAUER (Germany)
303
A failure at a licensed Atomic J. R. DE WET (South Africa)
Failure Analysis
documentation
critical
in product
liability
areas
pressure
Energy
under
to project
vulnerability
civil and criminal
Corporation
facility,
law. Ways
analysed
and means
to conform
with
thematically
and Minimisation
315
Fatigue failure of deck support J. STEYN (South Africa)
beams
on a vibrating
329
Cost-effective prevention of equipment E. DREYER (South Africa)
349
Assessment of critical dragline boom suspension policy R. J. MCKINNELL (South Africa)
failure
screen
in the mining
components:
industry
an essential
for planned
replacement
Energy-based
method for determination
of allowable stress-Part
99
II
toughness of structural materials, are determined from the equations
from which
2nt ntT.s= nt - (~~lu;f.~ *
(20)
[a]e,T
=
min{
~
+
[ ‘a~
11~~2’~’ Y’
After a correction similar to that which was carried out for the same factors under conditions of normal temperature, we use the equations obtained to determine the factors $? and tip in the correction factor calculations, taking into account low-temperature hardening of specific materials. The first corrections to the standard allowable stresses calculated by yield and tensile strengths, which take into consideration the strain hardening of the material at low temperature, are determined in a similar manner to eqns (8) and (9:
+ (d2
- [~l”,)PZ nY
1J(KA + bZ* - HW pt 1J(K,S) I. 4 pT
(25)
These equations, which are determined by the energy-based method at normal temperature as in the case of allowable stresses, can be represented in a more simple form: [aleyT = min{[a]; . [l + ([K&,
. PT
+ Pd*,, *Pmm lb]; *[l + ([GIJ *PT +
@&
. p%?(K,Zi)])
(26)
where T
[*q
= (UT - &2MT 1,t ntT,s
[Gil
=
(22)
The secondary correction factors, which taken into consideration the hardening of the material due to the increase in its yield strength when deformed under low-temperature conditions, are determined as in the case of eqns (10) and (11):
(24) In order to take into consideration the change in fracture toughness (crack resistance) of materials at reduce operating temperatures, it is necessary to introduce the coefficient PJ’, which is equal to the ratio of the critical values of the J-integral at low and normal temperatures, or the coefficient PZ, which is equal to the ratio aT/& of critical crack opening displacement. For high-strength materials it is necessary to use the coefficient pz, which is equal to the ratio of the values of KIC at low and normal temperatures. The definition of these criteria is regulated by GOST.3 Therefore, using the energy-based approach, the differentiated values of allowable stresses at low temperatures taking into account temperature hardening and the change in fracture
(Tt 2n,l(n,
[j& =
-
-
&
a,.,/d.,>
. [u]; ’
a?-- aL h,l(n,
- a,/G2)
. [GlT ’
d.2 - [aI% P4J = nt’ [a]s * Y
Here too the allowable stresses determined by the energy-based method take into account the basic mechanical properties of structural materials and the greater the strain hardening at cryogenic temperature and the yield strength of the material in comparison with its value at normal temperature, the higher the allowable stresses will be. In determining the allowable stresses for welded joints it is necessary, in addition to multiplying the terms of the allowable stresses by the coefficient of strength of the welded joint, to replace the coefficient fiJ(ps, pK) by the coefficient PFWeld= fi~we’d/JIC, or /32we’d= ScT,we’d/SC, or T weld = KT,weld PK. lc I&. It is obvious that in the case of a change in mechanical properties, e.g. due to mechanical thermal treatment or from exposure to high temperatures, the allowable stresses must be determined using equations similar to those given above. This will make it possible to follow the
100
E. V. Chechin
energy-based concept of safety in calculating the strength of machine parts and structural elements. It should also be noted that for cryogenic structures tested for strength under normal temperature conditions the selected level of allowable stresses at cryogenic temperatures must provide them with an adequate strength margin. The validity of the developed energy-based method for the determination of allowable stresses which can be used for the calculation of static strength of any products has been sufficiently reliably confirmed by the results of experimental investigations of full-sized vessels and by their large-sized welded models. These investigations have been carried out under conditions as close as possible to the real ones in static or cyclic loading by internal pressure.4a The experimental vessels were made according to the standard process or with special defects (incomplete fusion, surface notches of varying sharpness) to provide similarity of processes in the production of full-sized and experimental vessels. The tests were made using methods developed in the Institute for Problems of Strength of the Academy of Sciences of Ukraine.4 Table 1 gives certain test results for vessels (in addition to data given earlier4-8) for materials of different types and compares the correction
Table
Type of metal
1. Calculated
and experimental
Vessel dimensions
(mm)
factors to the standard allowable stresses determined by calculation and experiment. The correction factors [Aa]l.z and [Au]T.~ to the standard allowable stresses were found using eqns (8)-(lo), (11) and (21)-(24). Their values were compared with the correction factors obtained from direct experiment. In the latter case the corrections to the allowable stresses were found by using the relationships [A#~P
=
$?!!
-
[Aa]T,exp = $!
_ [a]”
where gfail and a& are the actual failure stresses of the tested vessels and ~1, is the standard strength margin by the tensile strength, and equal to 2.4 for steels, 3-O for titanium alloys and 3.5 for aluminum alloys. The experimental results (Table 1) indicate significant unutilized strength reserves of shell structures which have been produced from materials which show plastic, strain hardening and which are nonembrittling materials at low temperatures. Therefore, the determination of allowable stresses by the energy-based method provides an increase of their values compared to the standard ones from zero (for high-strength steels and titanium alloys) up to 33% at normal
factors to standard
allowable
Failure stress @@a)
Defect (% of wall thickness) 0)
296 615 530 748 501 865 1102 1499 429 402 522 1100 1260 969 1449 361 452
26 6 20 13 Valve imitator 35.6 34.1 35.6 -
12Kh18NlOT
293 71
D =420, L=870, tz8.0
03Kh13AG19
D =420, L=870, tz8.0
EP810
D=380, L=850,t=2.0
304 A516 A517” VTS-lkt
D = 384, L = 1220, t = 12.8 D =384, L= 1220, t=13.5 D = 378, L = 1220, t = 12-8 D = 360, t = 3.2 (sphere)
VT6S
D = 490, t = 4.3 (sphere)
AMg6
D =450, L=SOO, t=2.6
293 77 293 77 293 77 293 293 293 293 77 293 77 293 20
(28)
t
Test temperature WI
D = 315, t = 3.7 (sphere)
(27)
t
values of the correction
lKhlSN9T
[o-]”
stresses
Mca,c
blexp
w>
(%I
17 46 10 22 8 14 0 1.5 33 3 i 15b 0 14h 8 6
” The calculation for the materials was made using data taken from Ref. 9. bData without taking into considerationthe changein crack resistanceand texture hardening of titanium alloys.
31 173 24 88 54 82 18 59 58 11 -1 34 54 17 78 5 31
Energy-based
-1n P,
method
for determination
35
%
Fig. 4. Curves of allowable stressvalues determined by the standardsprevailing in machine building, [a]‘, and by the energy-basedmethod at normal temperatures, [orle, and at cryogenic temperature, [alevT.
temperature and up to 50% at cryogenic temperatures for nickel-chromium noncorrosive steels) and allows a corresponding reduction in metal consumption in the manufacture of structures and machines. The plots in Fig. 4 clearly demonstrate the advantage of energybased stresses over the force-based method. CONCLUSIONS The energy-based method for determining allowable stresses analytically gives maxirnum use of the actual strength properties of structural materials. It takes into consideration a combination of their basic mechanical properties, making it possible to decrease the metal consumption of structures, especially plastic strain-hardening materials, without reducing their reliability. The application of this method eliminates a main disadvantage of existing standards of design for strength in machine building, such as establishment of deterministic safety factors for structural materials of different types and enables allowable stresses to be determined adequate to real strength of materials. The validity of the method has been confirmed at normal and cryogenic temperatures with tests on large models and full-scale welded pressure vessels produced from materials of different types
of allowable
stress-Part
II
101
as the most critical structures of machine building. Due to the development of this new method, the determination of allowable stresses has been scientifically substantiated, enabling the metalproducing industry, the machine building branch of industry and construction engineering to apply greater strength reserves for a wide variety of strain-hardening materials and to reduce its metal consumption. REFERENCES 1. Gulyaev, A. P., Selectionof the type of steelfor machine parts (basic statements).Metalloved. Term. Obrab. Met., 1 (1983) 54-9. 2. Chechin, E. V., Energy-basedmethod for determination of allowable stresses-Part I. Znt. J. Pres. Ves. & Piping (in press). 3. GGST 25.50685.Methods of Mechanical Testsof Metais. Determination of the Characteristicsof Crack Resistance (Fracture Toughness) in Static Loading. Standards, Moscow 1985.[in Russian]. 4. Strizhalo, V. A., Filin, N. V., Kuranov, B. A. et al., The Strength of Materials and Structures at Cryogenic Temperatures.Naukova Dumka, Kiev, Ukraine, 1988. 5. Chechin, E. V., Potapov, I. K., Kes’yan, P. N. et al., Determination of the safety margin of vesselson welded models at normal and low temperatures. Automat. Svarka, 1 (1981)27-30. [in Russian]. 6. Chechin, E. V., The influence of designand production factors on the strength and deformability of thin-walled elements of vesselsunder low-temperature conditions. Cosmic Research in Ukraine, 15 (1981) 14-23 (in Russian). 7. Muratov, V. M., Kopysitskaya, L. N., Chechin, E. V. & Sharshukov, G. K., Low-cycle fatigue of weld joints in objects for cryogenic technology. Automat. Svarka, 11 (1985) 12-16. 8. Muratov, V. M., Kopysitskaya, L. N. & Chechin, E. V., Evaluation of low-cycle strength of cryogenic equipment. In The Strength of Materials and Structures at Low-Temperatures Naukova Dumka, Kiev, Ukraine, 1990,pp. 161-7 [in Russian]. 9. Royer, C. P. & Folfe, S. T., The influence of the strain hardening exponent and stress concentration on the character of rupture of pressurevessels.Teor. Osnouy Znzh. Raschetov (Tr. Amer.
D. 96, 4 (1974) 54-61.
Obshch. Znzh.-Mokh.),
Ser.
ELSEVIER
Energy-based method for determination allowable stress-Part II
of
E. V. Chechin for Probterns
fnsfifute
of Sfrengfh, Academy
of Sciences of Ukraine,
2 Ti~iyy~z~skaya
str., Kiev 2.52014, Ukraine
(Received 24 January 1994;accepted2 February 1994)
An energy-basedmethod for determining allowable stresses,the main link in strength analysis, which for the first time takes account of a basic set of mechanicalproperties of structural materials, namely resourcesof plasticity, hardening (strain and low-temperature) and crack resistance, has been developed instead of the classicalmechanicalmethod involving the material strength characteristicsonly. By taking into account the basic mechanical properties, an increase in allowable stressaccording to the new method is achieved: from fractions of a percent for high-strengthlow-plasticity materialsto 30% at room temperature and up to 50% at cryogenic temperaturesfor plastic hardeningand nonembrittling materials and this allows reduced metal consumptionin structures and machineswithout lowering the level of their reliability. Realization of the energy-basedmethod for determining allowable stresses (calculatedresistances)in the strength analysisopensup prospectsof resolving the problem of making maximumuseof actual strength for all steelsand alloys usedin machinebuilding. This makesthe method of great practical use.
NOTATION
allowable stress taking into consideration strain hardening of the specific material at normal and specific temperatures
If, rlTss Coefficient of strength reserve taking into consideration the material strain hardening; [a,]“, [u]~,~ The allowable stress determined by the energy-based method at normal and cryogenic temperatures @fail,a& Real values of failure stresses at normal and cryogenic temperatures [A#
exe = 2
The second correction factor to the standard allowable stress taking into consideration the resource of energy absorption which is available due to excess in the stress of the yield point over the allowable stress
- [a]s
Correction factor to the standard allowable stress derived from the direct experiment at normal and cryogenic temperatures
tw
ISHM
= 2,
[A+ISHM
DETERMINATION OF ALLOWABLE STRESSES AT ROOM TEMPERATURE In determining allowable stresses with regard to the strain hardenability of structural materials, we have to take into account the fact that in machine building the standard allowable stresses are determined by minimal values of strength characteristics, i.e. the minimal values of
The maxi-
= $$2
mum correction factor for the ideally strain-hardening material at normal and cryogenic temperatures
[Aull= e-t- UO.2)Pl ?zs ’
EAulT = w - d*)PT 1
nT,S
allowable stresses [aIS = min {y;:}
The first correction factor to the standard 95
(Fig. I).
96
E. V. Chechin
600 400 E
l d
200
/ IV,
m, ,-rrr ,rr,
I
1.0
i.5
1.6’ ’
2.0
2.5
gt’“o.2
Fig. 1. Allowable
stress values used in machine building versus (TJu~.~.
allowable loading of a part (structural element) not causing residual deformation. When the latter condition is met, the appearance of the Bauschinger effect during structural service is eliminated. In view of the above, the maximum value of the correction factor to the standard allowable stress is found as the difference between the stress that is equal to the yield strength of the material and the standard allowable stress determined from the yield strength or from the tensile strength. Thus
[Au]y = uo,2 ;” = u”.2(;y - l) In this connection, the most convenient way of presenting them is as a sum of the original (standard) value of allowable stress [a] and the correction factors [AU], and [AU]*, taking into account the reserves of strain-hardening materials, which are proportional to the constituents of the energy absorption of the material: [ale = [u]” + [AU], + [Aal z---f-ulim n
AU1
ns
+
-Au,
n
(1)
where (Tlimis the limiting value of the stress of the material (go,2 or a,) n is the standard strength safety factor, AU, = (TV- u~.~. and Au, = (T~.~[u] (Fig. 2). As can be seen from eqn (1) the coefficient ~1’ is unknown. To find it, let us consider the boundary conditions consisting in the following. We assume that the strength safety factor for a material with any level of hardening cannot be less than unity. Also, the maximum value of allowable stress must not exceed the yield strength (T~.~,which determines the upper limit of
cr
il
0,
ut
Y [Au]yx
=
uo,2
_
z
=
(To.2(nY
(2)
Y -
ut/“@2)s
(3)
nt
To obtain the relationships determining the coefficient ns we use the characteristics of the ideally strain-hardening material (ISHM) in which the uniform strain E, = 1.0, the energy parameter of strain hardening p = PO= 1.0, and the tensile strength a,* = 3uo+ In view of these characteristics and eqn (9) the maximum value of the correction factor for an ideally strainhardening material can be determined with the use of the yield strength (T~.~(see Fig. 2) from the equation
= 3uo.2 - uo.2=- 2uo.2 (4) n”y n”y and with the use of the tensile strength a:’ from the equation
= u:: - 1/3u,* =- 2uo.2 . (5) s nt n: Considering that the maximum value of allowable stress should not exceed the material yield strength, the magnitudes of the maximum correction factors for real and ideal materials should be (see Fig. 2)
ao.2
[VI 0 Fig. 2. Determination. of correction factors to the allowable stressesat normal temperature.
uo.* - [aIs = [Ao]‘~~~. Solving eqns (2) and (4) together, we obtain the design-basis equation for determining the coefficient n;: u,.,(n,
- 1)/n, = 2u,.,/n;
Energy-based
method for determination.
Furthermore,
L2n, n, - 1
stress-Part
97
II
of the material are found from the following relationships:
from which n,-
of allowable
(6)
,Au],,, _ 2
- b”.’ ,t”‘;‘”
(10)
we solve eqns (3) and (5) together:
nJ.*(% - (TtI(To.2)Iyzt= 2a,.,ld from which n; =
2% n t - ut/u0.2
(7)
We should note that the coefficients PZ;and PZS derived in such a manner for an ideally strain-hardening material, when used to determine the correction factors to the standard allowable stresses of specific structural materials, must be corrected to take into consideration the difference in strain hardening of an ideally strain-hardening material and the specific material and the observance of energy similarity of the compared values. For this we multiply the coefficients n”y and nf by the ratio of the energy parameters characterizing an ideally strainhardening material (&) and the specific material (PJ, considering in this case that PO= 1-O. Then the equations for determination of the first correction factors to the standard allowable stresses, taking into consideration strain hardening of the specific material, acquire the form
In this case the values of the corrections to the standard allowable stresses are proportional to the part of the total energy absorption of the material contained between the levels of stresses co.2 and [a] with a strain equal to G. Therefore, using the energy-based approach to determine the allowable stresses we find their differentiated values, taking into account strain hardening and the reserve of energy absorption of specific structural materials, from the equation
-n,[4”,)P2. (6 , bl’ = min{T + 2n,,(n _ 1j + Y 3 + (at - flO.2)Pl+ - [u]s)P2 (12) Int 2n,l(fl,- uJu~.~) nt (go.2
~0.2)Pl
(co.2
This equation can be represented in simpler form as
[ale= minbl; . (1 + [A41,y. PI + [AL,, . P2>; bl: . (1+ [W,t . PI + LA4.t. Ps)) (13) where
[Au],,, =% =ut- u”.2. -& n”y 0 1
[GLy =2n,l(n, (8)
go.2
- 1) . [u]; ’
- [cl; ny. [a]”
co.2
[A&
[Au],,,= $ = ut ,suo-z.$-jr t 0 1
@t-
=
Y
ut GIL, = 2%/(4- [a]; m2,t = nt - Lals t *
co.2
dJ-0.2).
=
(5
-
~0.2)Pl s nl
In other words, the correction factors considering strain hardening of the material are proportional to its energy absorption related to strain hardening. Furthermore, since with the energy-based approach to determining the allowable stresses for structural materials the ‘usable’ part of the total energy absorption of the material is that one on the level of allowable stress (Wr,] = [a]~,, (see Fig. 2)‘, the secondary correction factors considering the reserves of the energy ablsorption
[a]; ’
go.2
Considering the expression between the round brackets (eqn (13)) as a reduced coefficient in relation to the standard allowable stress, we note that this coefficient is always more than unity and will adopt different values for materials with different parameters p1 and p2 and, consequently, with different resources of the plastic strain of materials and their strain hardening. Moreover the reduced coefficient will also vary from the value of the chosen standard allowable
106
Contents of Volume 61
Numbers
2-3 SPECIAL
An International
127
Guest
Opening 129
ISSUE
FAILURES ‘94 Symposium on Risk, Economy Failure Minimisation and Analysis
and Safety,
Editorial
and Keynote
Address
The future role of nuclear power W. STUMPF (South Africa)
in addressing
global
environmental
problems
Overview 157
A practising engineer’s experiences station boilers J. A. COLLINS (South Africa)
177
Case studies in disaster-a coded E. A. BRADLEY (South Africa)
199
Failure types, consequences R. K. PENNY (South Africa)
,Economy
in overcoming
inherent
design
problems
in thermal
power
approach
and possible
remedies
and Safety
213
The important role of technical C. 0. BAUER (Germany)
229
Risk management in safety K. BURRAGE (UK1
257
The contribution of environmental E. REYNOLDS (South Africa)
283
Design responsibility. Consequences legal requirements C. 0. BAUER (Germany)
303
A failure at a licensed Atomic J. R. DE WET (South Africa)
Failure Analysis
documentation
critical
in product
liability
areas
pressure
Energy
under
to project
vulnerability
civil and criminal
Corporation
facility,
law. Ways
analysed
and means
to conform
with
thematically
and Minimisation
315
Fatigue failure of deck support J. STEYN (South Africa)
beams
on a vibrating
329
Cost-effective prevention of equipment E. DREYER (South Africa)
349
Assessment of critical dragline boom suspension policy R. J. MCKINNELL (South Africa)
failure
screen
in the mining
components:
industry
an essential
for planned
replacement
Energy-based
method for determination
of allowable stress-Part
99
II
toughness of structural materials, are determined from the equations
from which
2nt ntT.s= nt - (~~lu;f.~ *
(20)
[a]e,T
=
min{
~
+
[ ‘a~
11~~2’~’ Y’
After a correction similar to that which was carried out for the same factors under conditions of normal temperature, we use the equations obtained to determine the factors $? and tip in the correction factor calculations, taking into account low-temperature hardening of specific materials. The first corrections to the standard allowable stresses calculated by yield and tensile strengths, which take into consideration the strain hardening of the material at low temperature, are determined in a similar manner to eqns (8) and (9:
+ (d2
- [~l”,)PZ nY
1J(KA + bZ* - HW pt 1J(K,S) I. 4 pT
(25)
These equations, which are determined by the energy-based method at normal temperature as in the case of allowable stresses, can be represented in a more simple form: [aleyT = min{[a]; . [l + ([K&,
. PT
+ Pd*,, *Pmm lb]; *[l + ([GIJ *PT +
@&
. p%?(K,Zi)])
(26)
where T
[*q
= (UT - &2MT 1,t ntT,s
[Gil
=
(22)
The secondary correction factors, which taken into consideration the hardening of the material due to the increase in its yield strength when deformed under low-temperature conditions, are determined as in the case of eqns (10) and (11):
(24) In order to take into consideration the change in fracture toughness (crack resistance) of materials at reduce operating temperatures, it is necessary to introduce the coefficient PJ’, which is equal to the ratio of the critical values of the J-integral at low and normal temperatures, or the coefficient PZ, which is equal to the ratio aT/& of critical crack opening displacement. For high-strength materials it is necessary to use the coefficient pz, which is equal to the ratio of the values of KIC at low and normal temperatures. The definition of these criteria is regulated by GOST.3 Therefore, using the energy-based approach, the differentiated values of allowable stresses at low temperatures taking into account temperature hardening and the change in fracture
(Tt 2n,l(n,
[j& =
-
-
&
a,.,/d.,>
. [u]; ’
a?-- aL h,l(n,
- a,/G2)
. [GlT ’
d.2 - [aI% P4J = nt’ [a]s * Y
Here too the allowable stresses determined by the energy-based method take into account the basic mechanical properties of structural materials and the greater the strain hardening at cryogenic temperature and the yield strength of the material in comparison with its value at normal temperature, the higher the allowable stresses will be. In determining the allowable stresses for welded joints it is necessary, in addition to multiplying the terms of the allowable stresses by the coefficient of strength of the welded joint, to replace the coefficient fiJ(ps, pK) by the coefficient PFWeld= fi~we’d/JIC, or /32we’d= ScT,we’d/SC, or T weld = KT,weld PK. lc I&. It is obvious that in the case of a change in mechanical properties, e.g. due to mechanical thermal treatment or from exposure to high temperatures, the allowable stresses must be determined using equations similar to those given above. This will make it possible to follow the
100
E. V. Chechin
energy-based concept of safety in calculating the strength of machine parts and structural elements. It should also be noted that for cryogenic structures tested for strength under normal temperature conditions the selected level of allowable stresses at cryogenic temperatures must provide them with an adequate strength margin. The validity of the developed energy-based method for the determination of allowable stresses which can be used for the calculation of static strength of any products has been sufficiently reliably confirmed by the results of experimental investigations of full-sized vessels and by their large-sized welded models. These investigations have been carried out under conditions as close as possible to the real ones in static or cyclic loading by internal pressure.4a The experimental vessels were made according to the standard process or with special defects (incomplete fusion, surface notches of varying sharpness) to provide similarity of processes in the production of full-sized and experimental vessels. The tests were made using methods developed in the Institute for Problems of Strength of the Academy of Sciences of Ukraine.4 Table 1 gives certain test results for vessels (in addition to data given earlier4-8) for materials of different types and compares the correction
Table
Type of metal
1. Calculated
and experimental
Vessel dimensions
(mm)
factors to the standard allowable stresses determined by calculation and experiment. The correction factors [Aa]l.z and [Au]T.~ to the standard allowable stresses were found using eqns (8)-(lo), (11) and (21)-(24). Their values were compared with the correction factors obtained from direct experiment. In the latter case the corrections to the allowable stresses were found by using the relationships [A#~P
=
$?!!
-
[Aa]T,exp = $!
_ [a]”
where gfail and a& are the actual failure stresses of the tested vessels and ~1, is the standard strength margin by the tensile strength, and equal to 2.4 for steels, 3-O for titanium alloys and 3.5 for aluminum alloys. The experimental results (Table 1) indicate significant unutilized strength reserves of shell structures which have been produced from materials which show plastic, strain hardening and which are nonembrittling materials at low temperatures. Therefore, the determination of allowable stresses by the energy-based method provides an increase of their values compared to the standard ones from zero (for high-strength steels and titanium alloys) up to 33% at normal
factors to standard
allowable
Failure stress @@a)
Defect (% of wall thickness) 0)
296 615 530 748 501 865 1102 1499 429 402 522 1100 1260 969 1449 361 452
26 6 20 13 Valve imitator 35.6 34.1 35.6 -
12Kh18NlOT
293 71
D =420, L=870, tz8.0
03Kh13AG19
D =420, L=870, tz8.0
EP810
D=380, L=850,t=2.0
304 A516 A517” VTS-lkt
D = 384, L = 1220, t = 12.8 D =384, L= 1220, t=13.5 D = 378, L = 1220, t = 12-8 D = 360, t = 3.2 (sphere)
VT6S
D = 490, t = 4.3 (sphere)
AMg6
D =450, L=SOO, t=2.6
293 77 293 77 293 77 293 293 293 293 77 293 77 293 20
(28)
t
Test temperature WI
D = 315, t = 3.7 (sphere)
(27)
t
values of the correction
lKhlSN9T
[o-]”
stresses
Mca,c
blexp
w>
(%I
17 46 10 22 8 14 0 1.5 33 3 i 15b 0 14h 8 6
” The calculation for the materials was made using data taken from Ref. 9. bData without taking into considerationthe changein crack resistanceand texture hardening of titanium alloys.
31 173 24 88 54 82 18 59 58 11 -1 34 54 17 78 5 31
Energy-based
-1n P,
method
for determination
35
%
Fig. 4. Curves of allowable stressvalues determined by the standardsprevailing in machine building, [a]‘, and by the energy-basedmethod at normal temperatures, [orle, and at cryogenic temperature, [alevT.
temperature and up to 50% at cryogenic temperatures for nickel-chromium noncorrosive steels) and allows a corresponding reduction in metal consumption in the manufacture of structures and machines. The plots in Fig. 4 clearly demonstrate the advantage of energybased stresses over the force-based method. CONCLUSIONS The energy-based method for determining allowable stresses analytically gives maxirnum use of the actual strength properties of structural materials. It takes into consideration a combination of their basic mechanical properties, making it possible to decrease the metal consumption of structures, especially plastic strain-hardening materials, without reducing their reliability. The application of this method eliminates a main disadvantage of existing standards of design for strength in machine building, such as establishment of deterministic safety factors for structural materials of different types and enables allowable stresses to be determined adequate to real strength of materials. The validity of the method has been confirmed at normal and cryogenic temperatures with tests on large models and full-scale welded pressure vessels produced from materials of different types
of allowable
stress-Part
II
101
as the most critical structures of machine building. Due to the development of this new method, the determination of allowable stresses has been scientifically substantiated, enabling the metalproducing industry, the machine building branch of industry and construction engineering to apply greater strength reserves for a wide variety of strain-hardening materials and to reduce its metal consumption. REFERENCES 1. Gulyaev, A. P., Selectionof the type of steelfor machine parts (basic statements).Metalloved. Term. Obrab. Met., 1 (1983) 54-9. 2. Chechin, E. V., Energy-basedmethod for determination of allowable stresses-Part I. Znt. J. Pres. Ves. & Piping (in press). 3. GGST 25.50685.Methods of Mechanical Testsof Metais. Determination of the Characteristicsof Crack Resistance (Fracture Toughness) in Static Loading. Standards, Moscow 1985.[in Russian]. 4. Strizhalo, V. A., Filin, N. V., Kuranov, B. A. et al., The Strength of Materials and Structures at Cryogenic Temperatures.Naukova Dumka, Kiev, Ukraine, 1988. 5. Chechin, E. V., Potapov, I. K., Kes’yan, P. N. et al., Determination of the safety margin of vesselson welded models at normal and low temperatures. Automat. Svarka, 1 (1981)27-30. [in Russian]. 6. Chechin, E. V., The influence of designand production factors on the strength and deformability of thin-walled elements of vesselsunder low-temperature conditions. Cosmic Research in Ukraine, 15 (1981) 14-23 (in Russian). 7. Muratov, V. M., Kopysitskaya, L. N., Chechin, E. V. & Sharshukov, G. K., Low-cycle fatigue of weld joints in objects for cryogenic technology. Automat. Svarka, 11 (1985) 12-16. 8. Muratov, V. M., Kopysitskaya, L. N. & Chechin, E. V., Evaluation of low-cycle strength of cryogenic equipment. In The Strength of Materials and Structures at Low-Temperatures Naukova Dumka, Kiev, Ukraine, 1990,pp. 161-7 [in Russian]. 9. Royer, C. P. & Folfe, S. T., The influence of the strain hardening exponent and stress concentration on the character of rupture of pressurevessels.Teor. Osnouy Znzh. Raschetov (Tr. Amer.
D. 96, 4 (1974) 54-61.
Obshch. Znzh.-Mokh.),
Ser.