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Methods for estimating strain in structures - conditions and definitions
Nuclear Engineering and Design 96 (1986) 427-436 North-Holland, Amsterdam
427
METHODS FOR ESTIMATING STRAIN IN STRUCTURES CONDITIONS AND DEFINITIONS B. C H A R A L A M B U S ,
-
K. H A R T Z
Kraftwerk Union, Hammerbacher Str. 12 + 14, 8520 Erlangen, Fed. Rep. Germany
D. B L I N D a n d E. R O O S Staatliche Materialpriifanstalt, Universiti~t Stuttgart, Pfaffenwaldring 32, 7000 Stuttgart 80, Fed. Rep. Germany
Received December 1985
Analyses of the impact of actual earthquakes on industrial plants and experimental investigations simulating earthquakes and pressure surges show that loads can be sustained without damage, which are according to the existing design rules far beyond the allowable limit and which cannot be calculated realistically assuming linear-elastic material behaviour according to plastic deformation. To rely on the existing capability to absorb large amounts of energy, it is necessary to have models for describing their behaviour in the plastic range taking into account the plastic deformation volume and the strain values reached. Especially for the calculation of piping systems under loads with a small probability of occurrence, the main parameters for describing the energy dissipated and the additional strain in comparison to the strain in a straight pipe are described, independent of the calculation method used (simplified or elasto-plastic time integration). In addition, strain categories and the individual strain category responsible for the integrity of the overall system are defined.
1. Introduction The following discussion relates to the application of strain limitation to piping systems. It can be applied analogously to other parts. The decisive design parameters for piping systems are the operational loads. The dimensioning of piping and the concluding analysis are conducted on the basis of postulation of linear-elastic material behaviour in accordance with applicable codes and standards specifying stress limits. Additionally, it is necessary for safety-related systems to ensure that the systems thus designed remain within the specified service limits even in the event of non-operational dynamic loadings with a low probability of occurrence. To date modal analysis has constituted a cost-effective method of design and safety analysis of piping systems for such loadings as occur during external and internal events (EE and IE). The allowable stress limits are selected for this purpose as if the response were largely elastic. Consequently the amplitudes of excitation and the responses of the system are proportional to each other. Analysis for loadings with a low probability of occur-
rence in design with the aid of allowable stresses can have negative effects on operation since the increase in rigidity results in higher stresses and interaction loads in normal operation. Observations of the actual effects of earthquakes and load measurements in system tests have demonstrated that the present design limits for loadings with a low probability of occurrence permit the loadability of components made of ductile steels to be exploited to only a small extent [1,2]. It is therefore expedient to make better use of the material reserves under E E / E I loadings by utilizing the plastic deformation capacity. This entails that linearelastic response of the systems can no longer be postulated. One of the evaluation criteria used in this case is the plastic strain resulting in the system, that is, stress limitation is supplemented by "strain limitation". In this case the plastic strains caused by bending moments and shear forces and by interaction loads in the system are kept below limits which depend on the time histories.
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B. Charalambus et al. / Methods for estimating strain in structures R
Material 1
/ Material 2
as early as at the design stage. This has been reflected in the so-called shape factor incorporated into the codes and standards (TRD, AD) [3]. A number of studies have shown that materials which conform to the requirements of Basic Safety [4,5], possess by virtue of their ductility an inherent safety reserve in their considerable plastification capacities. This correlation is shown in fig. 1 in schematic form for two materials, with the material of higher ductility as expressed in its absorbed energy versus temperature curve capable of absorbing a far higher energy of deformation before the maximum load is reached. This is also shown by results which have been obtained on large specimens in the " H e a v y Component Research Program F K S " (fig. 2). The specimens were tested at the upper shelf of the energy absorbed versus temperature curve (fig. 3a). These notched large tensile test specimens, which have a high degree of constraint in the ligament by virtue of their thickness, reveal an increase
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431
B. Charalambus et aL / Methods for estimating strain in structures
in load-bearing capacity as the upper shelf energy absorbed increases and, in particular, a significant increase in the C O D values. The C O D values of the specimen made of the material of 90 J upper shelf (KS01) are about three times as high as those of the specimen made of the material of 40 J upper shelf (KS07) [8]. Further investigations in the transition range and on upper-shelf energy absorbed [6,7], which by virtue of the evaluation of a large number of tensile and bend test specimens and pipes under internal pressure and bending loads provide statistical confidence, show however that the loading of notched specimens does not necessarily increase with energy absorbed. Under dynamic loading the transition range of ductility is shifted with increasing loading velocity towards higher temperatures (fig. 3b). The use of highductility materials results in the nil-ductility transition temperature shifting towards lower temperatures with increasing upper-shelf absorbed energy (fig. 3c). This means that under accident conditions the high-ductility material remains at the upper shelf level while the less ductile material may be in the transition region which would result in reduced plastic deformability [9].
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This can be demonstrated on the basis of static and dynamic studies conducted on smooth and notched specimens made of material 17 M n M o V 6 4 with an upper shelf absorbed energy for the test orientation of 100 J (fig. 4a). Tests were conducted on smooth and also single-notched ( S E N T ) and double-notched ( D E N T ) specimens of notch depths of 0.1 and 0.25 times the section thickness. The notches were made in the base metal (BM), the weld metal (WM) and in the heat-affected zones (HAZ) (fig. 4a). Measurements were taken for the permanent elongation of the specimen after fracture relative to a gauge length of 150 mm (A150) [10,11]. The dynamic tests were conducted at extension rates of 6.5 to 7.5 m / s , equivalent to a strain rate of approx. 30 s-1 relative to the gauge length. This strain rate is far above that expected for E E / I E loadings of less than 1.0 s - ] . The test results for the BM, W M and H A Z specimens are shown in fig. 4b. In spite of the stringent testing and evaluation conditions all measured results both for RT and also for 250°C are well above one percent integral strain although, with a few exceptions, the dynamic values are considerably lower than those obtained in comparable static tests.
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B. Charalarnbus et al. / Methods for estimating strain in structures
432
3. Description of the behaviour of parts in the non-elastic region
The parts have a significant effect on system behaviour in the non-elastic region on account of energy dissipation if large volumes plasticise. Local geometric discontinuities by contrast only affect the magnitude of local strain. The behaviour of the individual piping parts is described and identified in simplified form in the following in a qualitative fashion. The energy input into the system due to excitation (internal or supportbase excitation) is dissipated proportionately under partly plastic deformation in accordance with the hysteresis loop. In the case of dynamic loadings of piping parts, these are primarily subject to loadings due to bending loads non-uniformly distributed over their length. The energy dissipated for each cycle can be formulated as:
ED=frfo,
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It is apparent from the equation above that part of the input energy is dissipated either by very high local plastic deformations or by plastification of large portions of the cross-section of the part. If the intention is to make use of strain limitation, it is necessary to use parts which are designed such that as large volumes as possible undergo plastification under the time-dependent loadings of E E / I E service conditions. The plastification component that can be sustained without damage depends on the geometrical shape of the part and the design strain value considered allowable for the material concerned. It proves beneficial to use only a small number of piping parts in piping systems, such as: - Straight pipe, Elbow, Tee, - Reducer, - Valve. With the exception of valves, of which the parts generally have to perform mechanical functions reliably after completion of the excitation, plastic deformation within certain limits does not impair the function of the other piping parts and is therefore allowable. The effect of plasticizing parts on system behaviour and on strain magnitude is discussed below:
varied until the pipe undergoes plastification, the calculated pattern of plastic strains at the surface of the pipe plotted in fig. 5 is obtained for excitation at the first natural frequency of the pipe. When the amplitude is further increased the strain increases, but the length which undergoes plastification remains approximately constant in the axial direction provided that the bending mode of a higher natural frequency does not give rise to plastification (fig. 5). Determination of the bending moments and shear forces in each cross-section of the pipe is conducted under the postulate that the cross-sections remain plane (Bernoulli's hypothesis). This hypothesis has been experimentally validated for the allowable strains concerned. The moments and forces are derived by integrating the stresses with respect to the cross-section on the basis of the stress-strain curve of the material. Minor geometric discontinuities on the surface of the pipe (such as attachment-welded lugs or small blank nozzles, fig. 6, for the attachment of instrument lines) have an insignificant effect on system behaviour and bending moments or shear forces in the non-elastic region but do cause strain restraint and hence strain concentrations in such areas. It is apparent from this that different factors have to be used for the derivation of moments and forces and for strain analysis if the part differs from the continuous shape of the straight pipe.
Stress- strain pattern in co.tinuos straighl pipe
M $1ress- strain pattern in straight pipe in the area of a disconliuit y (attachment-welded lug)
3.1. Straight pipe If a straight pipe - simply supported or clamped at both ends - is subjected to spontaneous support-base excitation such that the amplitude of the excitation is
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Fig. 6. Schematic patterns of strain due to bending.
B. Charalambus et aL / Methods for estimating strain in structures 3.2. Elbows
the flexibility factor
The elbow is the most flexible part in a piping system. Furthermore higher stresses and strains prevail (for the same loading) in the elbow than in the straight pipe. Under a constant moment along the bend angle only a small portion of the elbow undergoes plastification. The location of plastification and the plastifying volume of the elbow depend on various factors such as
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B. Charalarnbus et aL / Methods for estimating strain in structures
example of the pattern of strains in an elbow. The behaviour of the elbow in the system is affected by numerous parameters. The plastification region of an elbow essentially depends on the nature of the connection to the straight pipe (flanged, welded), the length of the connected continuous straight pipe [13], the bend angle and the internal pressure. These influences on the nonlinear behaviour of the elbow can be considered by means of suitable methods (such as finite element analysis or measurements). The nonlinear behaviour and the associated strain concentration factors of the elbow can thus be predicted and recorded in a file which makes system analysis more economic.
fication on the load/deformation characteristic of both the run pipe and the branch is small. For this reason it might be possible to neglect this effect in elastic-plastic analysis. 3.4. Reducers
The cross-sections of reducers behave in the nonelastic region in the same manner as those of straight pipes. The maximum strain of the reducer is derived by multiplication of the maximum strain of the straight pipe with a geometry factor which has yet to be established.
3.3. Tees
4. Categorization of strains in parts
The plasticizing volumes of the tee are entered in schematic form in fig. 8 as a function of the directions of loading. The general statement may be made that the energy dissipated in these volumes is small since the volumes are small, whereas the stresses and strains at the surfaces can take on high values. On account of the high strain concentration at the tees the effect of plasti-
Piping systems are generally simulated with the aid of one-dimensional piping elements along their axes. The properties in the radial and circumferential directions are considered by means of postulates (such as that of the cross-sections of the straight pipe remaining plane). Possible patterns of strains in the most important piping parts were explained in section 3. The strains in the geometrically continuous cylindrical wall are, apart from their dependence on the dimensions of the pipe, load-dependent only and can be derived from the bending moments and shear forces. This is only conditionally possible for parts containing geometric discontinuities. In such parts, strains which stem from the geometric discontinuity are present in addition to the strains which maintain equilibrium with the loads prevalent at the time under consideration in accordance with the stress-strain curve (see fig. 6). The magnitude of these strains (and stresses) generally has a set relationship with the strains which are derived from considerations of equilibrium provided that a specific geometry of the part is postulated. Fig. 6 shows the general pattern of strains in the area of a discontinuity in a cylindrical shell. By analogy with the categorization of stresses as primary, secondary and peak stresses it is proposed that strains should be categorized according to their effects. The criteria, however, on which the stress categorization is based cannot be adopted since the strains in the part above the elasticity limit increase to a larger extent than the stresses. The following categorization is proposed:
Moments on run pipe
Moments on branch pipe
L
Fig. 8. Plasticizing zone in tees, lines of equal strain.
Integral strain Integral strain is defined as encompassing all strains of which the average across the wall thickness is ob-
B. Charalambus et al. / Methods for estimating strain in structures
tained from the instantaneous state of equilibrium on the basis of the stress-strain curve of the tensile test. The amount of integral strain is dependent only on the loading and the geometric shape and must be considerably lower than that of proportional strain since the part might otherwise undergo spontaneous failure [2], fig. 5. Failure can, however, occur on repetition of the loading. Secondary strain Consideration of such strain, which corresponds to secondary stress, can be omitted if it is limited in terms of amount and number of cycles. Peak strains Peak strains occur on the surface on both the inside
epl 0.0250
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435
and the outside in the region of local geometric discontinuities in the shell; they are dependent on the magnitude of the loading and exhibit pronounced gradients over the wall thickness. The essential feature of peak strains is that they only give rise to crack initiation, and to failure due to crack propagation, under repeated loading. Mean strain The strains that occur under E E / I E loadings in the non-elastic region generally differ in magnitude in the tensile and compressive directions. This results in a cumulative mean strain occuring at every plasticizing location in the system (fig. 9). Mean strains occur in systems subject to plastic loadings also as a result of the effect of static loads (such as deadweight, operational weight). This applies both to integral strain and to peak strains. Variations in this mean strain remain small during the small number of stress cycle imposed by E E / I E loadings with the result that they need not be taken into consideration in design (cf. fig 9).
0.0150 5. Possibilities for analysis 0.0100
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Under the methods of stress limitation applied to date the analysis of bending moments and shear forces and of interaction loads due to support-base excitation (EE loads) was conducted by means of response spectrum modal analysis (RSMA) and for internal excitation by means of modal time-history integration. Both methods stand out for their ease of handling, low CPU time and generally conservative results. Furthermore the evaluation of the effects of an increase in amplitude for the same frequency spectrum of the loading is possible without having to repeat the analysis. Additionally the effects of different loadings can be determined separately and superposition can be performed algebraically. Nonlinearities (such as clearances, friction, supports) can only be accounted for (inaccurately) with the aid of modal damping. The essential difference between linear-elastic and elastic-plastic analysis lies in the fact that the law of superposition is invalidated. Depending on the postulates (strain concentration factor), the elastic-plastic analysis of a piping system under transient loadings gives both the integral strain and the strain concentration at every point of the structure. Prerequisites for strain analysis are, for supportbase excitation (earthquake, aircraft crash), the specification of the time history of the displacements at sup-
436
B. Charalarnbus et al. / Methods for estimating strain in structures
port locations and, for internal excitation, the specification of the time histories of the fluid dynamic forces at their points at action. The strains thus obtained are strain time histories and also include the mean strain at the considered location (see fig. 9). The elastic strain c o m p o n e n t may be considered or neglected provided that it is ensured that this c o m p o n e n t is also allowed for in the magnitudes of strain considered allowable. Analyses which take into account the non-elastic, i.e. nonlinear region can be conducted for dynamic loadings by integration against time. This applies to both support-base and internal excitation. If more than one point of excitation is present, all are to be postulated as acting simultaneously. The superposition of results for different loadings is not possible. The ranges of strain as shown in fig. 9b are to be derived from the calculated time histories (fig. 9a).
[3] [4] [5] [6]
[7]
[8]
[9]
6. Summary and prospects Experience to date demonstrates that ductile materials are capable of absorbing large amounts of energy on account of their capacity for plastic deformation. This property has been exploited only insufficiently in prior design practice. If reliably determined strain data are available for parts (e.g. ref. [14]), the plastic strains that can be accepted for the addressed service conditions ( E E / I E ) given a ductile material condition are larger than those in practice to date and no reduction in safety is sustained. This approach requires the establishment of suitable laws of material behaviour such as are provided by cyclic strength charts [15]. The necessary safety margins have yet to be established.
[10]
[11]
[12]
[13]
[14]
References [15] [1] Pipe blowdown tests with valve closure, HDR Test Series RORB, PHDR Report No. 174/85 (July 1985). [2] P. Mihatsch, B. Charalambus and E. Haas, Comparison of
the results of vibration tests with the results of various analysis methods, Nucl. Engrg. Des. 96 (1986) 445 460, in this issue. E. Siebel, New approaches to stress analysis, VDIZeitschrift 90 (1954) 335-341. K. KuP,,maul, German basis safety concept rules out possibility of catastrohpic failure, Nucl. Eng. Int. (Dec. 1984). K. St~ibler, Introduction to assured safety, VGB Kraftwerkstechnik 60 (1980) 428-437. H.J. Golembiewski and G. Vasoukis, Crack analysis in fully ductile regime The critical stress criterion, in: Proc. 8th SMiRT-Conference, Aug. 1985, Brussels, Vol. G 2/3, pp. 59-64. E. Tenckhoff, G. Vasoukis, ,M. Erve and J. Schmidt, Materials for LWRs in the light of new developments, Nucl. Engrg. Des. 87 (1985) 215 223. K. Kul3maul, Fortschritte und Entwichlungstendenzen beim Einsatz hochz~iher und hochfester Werkstoffe, VGB-Ehrenkolloquium: Werkstofftechnik und Betriebserfahrungen, 19. Juni 1985, Mannheim. K. Kul?,maul und E. Roos, Bruchmechanik und Sicherheitskonzepte, Anwendung der Bruchmechanik, Haus der Technik Essen, 27. Februar 1985. D. Sturm, W. Stoppler and J. Scheidermaier, Behaviour of dynamically stressed pipes with circumferential defects under internal pressure and flexural loading, paper delivered at the llth MPA Seminar, Stuttgart, Paper No. 1 (1985). D. Sturm, R. Zirn and R. Haas, Behaviour of piping material WB 35 under dynamic loading in a high-speed tension test, paper delivered at the l l t h MPA Seminar, Stuttgart, Paper No. 3 (1985). W.L. Greenstreet, Experimental study of plastic responses of pipe elbows, Oak Ridge National Laboratory, Contract No. W-7405-enp-26 (February 1978). H. Diem and K.U. Miiller, RL test stand experiments on the deformation behaviour of pipe elbows and pipe elbow connecting cross-sections. K.A. Peters and K.-A. Busch, A method for the derivation of strains due to external/internal event loadings, Nucl. Engrg. Des. 96 (1986) 435-438, in this issue. K. Kessler, G. Vasoukis, D. Blind and K. Maile, Determination of load cycle-dependent characteristics for a design concept based on limited plastic strains, Nucl. Engrg. Des. 96 (1986) 461-473, in this issue.
427
METHODS FOR ESTIMATING STRAIN IN STRUCTURES CONDITIONS AND DEFINITIONS B. C H A R A L A M B U S ,
-
K. H A R T Z
Kraftwerk Union, Hammerbacher Str. 12 + 14, 8520 Erlangen, Fed. Rep. Germany
D. B L I N D a n d E. R O O S Staatliche Materialpriifanstalt, Universiti~t Stuttgart, Pfaffenwaldring 32, 7000 Stuttgart 80, Fed. Rep. Germany
Received December 1985
Analyses of the impact of actual earthquakes on industrial plants and experimental investigations simulating earthquakes and pressure surges show that loads can be sustained without damage, which are according to the existing design rules far beyond the allowable limit and which cannot be calculated realistically assuming linear-elastic material behaviour according to plastic deformation. To rely on the existing capability to absorb large amounts of energy, it is necessary to have models for describing their behaviour in the plastic range taking into account the plastic deformation volume and the strain values reached. Especially for the calculation of piping systems under loads with a small probability of occurrence, the main parameters for describing the energy dissipated and the additional strain in comparison to the strain in a straight pipe are described, independent of the calculation method used (simplified or elasto-plastic time integration). In addition, strain categories and the individual strain category responsible for the integrity of the overall system are defined.
1. Introduction The following discussion relates to the application of strain limitation to piping systems. It can be applied analogously to other parts. The decisive design parameters for piping systems are the operational loads. The dimensioning of piping and the concluding analysis are conducted on the basis of postulation of linear-elastic material behaviour in accordance with applicable codes and standards specifying stress limits. Additionally, it is necessary for safety-related systems to ensure that the systems thus designed remain within the specified service limits even in the event of non-operational dynamic loadings with a low probability of occurrence. To date modal analysis has constituted a cost-effective method of design and safety analysis of piping systems for such loadings as occur during external and internal events (EE and IE). The allowable stress limits are selected for this purpose as if the response were largely elastic. Consequently the amplitudes of excitation and the responses of the system are proportional to each other. Analysis for loadings with a low probability of occur-
rence in design with the aid of allowable stresses can have negative effects on operation since the increase in rigidity results in higher stresses and interaction loads in normal operation. Observations of the actual effects of earthquakes and load measurements in system tests have demonstrated that the present design limits for loadings with a low probability of occurrence permit the loadability of components made of ductile steels to be exploited to only a small extent [1,2]. It is therefore expedient to make better use of the material reserves under E E / E I loadings by utilizing the plastic deformation capacity. This entails that linearelastic response of the systems can no longer be postulated. One of the evaluation criteria used in this case is the plastic strain resulting in the system, that is, stress limitation is supplemented by "strain limitation". In this case the plastic strains caused by bending moments and shear forces and by interaction loads in the system are kept below limits which depend on the time histories.
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428
B. Charalambus et al. / Methods for estimating strain in structures R
Material 1
/ Material 2
as early as at the design stage. This has been reflected in the so-called shape factor incorporated into the codes and standards (TRD, AD) [3]. A number of studies have shown that materials which conform to the requirements of Basic Safety [4,5], possess by virtue of their ductility an inherent safety reserve in their considerable plastification capacities. This correlation is shown in fig. 1 in schematic form for two materials, with the material of higher ductility as expressed in its absorbed energy versus temperature curve capable of absorbing a far higher energy of deformation before the maximum load is reached. This is also shown by results which have been obtained on large specimens in the " H e a v y Component Research Program F K S " (fig. 2). The specimens were tested at the upper shelf of the energy absorbed versus temperature curve (fig. 3a). These notched large tensile test specimens, which have a high degree of constraint in the ligament by virtue of their thickness, reveal an increase
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[
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B. Charalambus et aL
/ Methods for estimating strain in structures
429
300
22NIMoCr370~S01 transverse) 22NiMoCr37 mod.OKS07 transverseJ 20MnMoNIS~KS14 ~ransverse ~_.
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430
Section A - A B
B
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B=60
Notch location
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specimens
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25
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10
SENT,a=1.5 mm,stat. SENT,a=1.5 mm,dyn. SENT,a=2.5 ram,star. SENT,a----2.5 mm,clyn. DENT,a=1.25 rnm,stat. DENT,a=1.25 rnm,dyn.
• = Unnotched, dyn. o = SENT,a=1.Smm,stcrt. • = SENT,a=1.5rnm,dyn.
8
I
[]
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I
.m
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15
0 tO ~-
<~
10
0
0 0
50
100
150
200
250
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300
0
50
100
150
200
250
Temperature/ °C
300
0
50
100
150
200
250
Temperature/ °C
Fig. 4. (a, top) Specimen geometry and notch locations. (b) Comparison of integral strain values after fracture of dynamically and statically tested specimen.
431
B. Charalambus et aL / Methods for estimating strain in structures
in load-bearing capacity as the upper shelf energy absorbed increases and, in particular, a significant increase in the C O D values. The C O D values of the specimen made of the material of 90 J upper shelf (KS01) are about three times as high as those of the specimen made of the material of 40 J upper shelf (KS07) [8]. Further investigations in the transition range and on upper-shelf energy absorbed [6,7], which by virtue of the evaluation of a large number of tensile and bend test specimens and pipes under internal pressure and bending loads provide statistical confidence, show however that the loading of notched specimens does not necessarily increase with energy absorbed. Under dynamic loading the transition range of ductility is shifted with increasing loading velocity towards higher temperatures (fig. 3b). The use of highductility materials results in the nil-ductility transition temperature shifting towards lower temperatures with increasing upper-shelf absorbed energy (fig. 3c). This means that under accident conditions the high-ductility material remains at the upper shelf level while the less ductile material may be in the transition region which would result in reduced plastic deformability [9].
y 1250
Elasticmaterial
~Kn 6
This can be demonstrated on the basis of static and dynamic studies conducted on smooth and notched specimens made of material 17 M n M o V 6 4 with an upper shelf absorbed energy for the test orientation of 100 J (fig. 4a). Tests were conducted on smooth and also single-notched ( S E N T ) and double-notched ( D E N T ) specimens of notch depths of 0.1 and 0.25 times the section thickness. The notches were made in the base metal (BM), the weld metal (WM) and in the heat-affected zones (HAZ) (fig. 4a). Measurements were taken for the permanent elongation of the specimen after fracture relative to a gauge length of 150 mm (A150) [10,11]. The dynamic tests were conducted at extension rates of 6.5 to 7.5 m / s , equivalent to a strain rate of approx. 30 s-1 relative to the gauge length. This strain rate is far above that expected for E E / I E loadings of less than 1.0 s - ] . The test results for the BM, W M and H A Z specimens are shown in fig. 4b. In spite of the stringent testing and evaluation conditions all measured results both for RT and also for 250°C are well above one percent integral strain although, with a few exceptions, the dynamic values are considerably lower than those obtained in comparable static tests.
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B. Charalarnbus et al. / Methods for estimating strain in structures
432
3. Description of the behaviour of parts in the non-elastic region
The parts have a significant effect on system behaviour in the non-elastic region on account of energy dissipation if large volumes plasticise. Local geometric discontinuities by contrast only affect the magnitude of local strain. The behaviour of the individual piping parts is described and identified in simplified form in the following in a qualitative fashion. The energy input into the system due to excitation (internal or supportbase excitation) is dissipated proportionately under partly plastic deformation in accordance with the hysteresis loop. In the case of dynamic loadings of piping parts, these are primarily subject to loadings due to bending loads non-uniformly distributed over their length. The energy dissipated for each cycle can be formulated as:
ED=frfo,
de dV.
It is apparent from the equation above that part of the input energy is dissipated either by very high local plastic deformations or by plastification of large portions of the cross-section of the part. If the intention is to make use of strain limitation, it is necessary to use parts which are designed such that as large volumes as possible undergo plastification under the time-dependent loadings of E E / I E service conditions. The plastification component that can be sustained without damage depends on the geometrical shape of the part and the design strain value considered allowable for the material concerned. It proves beneficial to use only a small number of piping parts in piping systems, such as: - Straight pipe, Elbow, Tee, - Reducer, - Valve. With the exception of valves, of which the parts generally have to perform mechanical functions reliably after completion of the excitation, plastic deformation within certain limits does not impair the function of the other piping parts and is therefore allowable. The effect of plasticizing parts on system behaviour and on strain magnitude is discussed below:
varied until the pipe undergoes plastification, the calculated pattern of plastic strains at the surface of the pipe plotted in fig. 5 is obtained for excitation at the first natural frequency of the pipe. When the amplitude is further increased the strain increases, but the length which undergoes plastification remains approximately constant in the axial direction provided that the bending mode of a higher natural frequency does not give rise to plastification (fig. 5). Determination of the bending moments and shear forces in each cross-section of the pipe is conducted under the postulate that the cross-sections remain plane (Bernoulli's hypothesis). This hypothesis has been experimentally validated for the allowable strains concerned. The moments and forces are derived by integrating the stresses with respect to the cross-section on the basis of the stress-strain curve of the material. Minor geometric discontinuities on the surface of the pipe (such as attachment-welded lugs or small blank nozzles, fig. 6, for the attachment of instrument lines) have an insignificant effect on system behaviour and bending moments or shear forces in the non-elastic region but do cause strain restraint and hence strain concentrations in such areas. It is apparent from this that different factors have to be used for the derivation of moments and forces and for strain analysis if the part differs from the continuous shape of the straight pipe.
Stress- strain pattern in co.tinuos straighl pipe
M $1ress- strain pattern in straight pipe in the area of a disconliuit y (attachment-welded lug)
3.1. Straight pipe If a straight pipe - simply supported or clamped at both ends - is subjected to spontaneous support-base excitation such that the amplitude of the excitation is
M
Fig. 6. Schematic patterns of strain due to bending.
B. Charalambus et aL / Methods for estimating strain in structures 3.2. Elbows
the flexibility factor
The elbow is the most flexible part in a piping system. Furthermore higher stresses and strains prevail (for the same loading) in the elbow than in the straight pipe. Under a constant moment along the bend angle only a small portion of the elbow undergoes plastification. The location of plastification and the plastifying volume of the elbow depend on various factors such as
X
tR
r2~/1 _ v2 " One possible method of deriving bending moments and shear forces employs the nonlinear load/deformation characteristics which have to be provided by theoretical or experimental analysis [12]. Fig. 7 shows an
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/ Outside
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- 300
0o
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90o
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I
I
i
180° 0 o
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Central of elbow . . . . .
180° ,~0 o
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Fig. 7. Plasticizing zone in elbows.
180 °
433
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,@-° 3 D form start of elbow I
I
I
L
i
4 Da form start of elbow
0 o
180°
434
B. Charalarnbus et aL / Methods for estimating strain in structures
example of the pattern of strains in an elbow. The behaviour of the elbow in the system is affected by numerous parameters. The plastification region of an elbow essentially depends on the nature of the connection to the straight pipe (flanged, welded), the length of the connected continuous straight pipe [13], the bend angle and the internal pressure. These influences on the nonlinear behaviour of the elbow can be considered by means of suitable methods (such as finite element analysis or measurements). The nonlinear behaviour and the associated strain concentration factors of the elbow can thus be predicted and recorded in a file which makes system analysis more economic.
fication on the load/deformation characteristic of both the run pipe and the branch is small. For this reason it might be possible to neglect this effect in elastic-plastic analysis. 3.4. Reducers
The cross-sections of reducers behave in the nonelastic region in the same manner as those of straight pipes. The maximum strain of the reducer is derived by multiplication of the maximum strain of the straight pipe with a geometry factor which has yet to be established.
3.3. Tees
4. Categorization of strains in parts
The plasticizing volumes of the tee are entered in schematic form in fig. 8 as a function of the directions of loading. The general statement may be made that the energy dissipated in these volumes is small since the volumes are small, whereas the stresses and strains at the surfaces can take on high values. On account of the high strain concentration at the tees the effect of plasti-
Piping systems are generally simulated with the aid of one-dimensional piping elements along their axes. The properties in the radial and circumferential directions are considered by means of postulates (such as that of the cross-sections of the straight pipe remaining plane). Possible patterns of strains in the most important piping parts were explained in section 3. The strains in the geometrically continuous cylindrical wall are, apart from their dependence on the dimensions of the pipe, load-dependent only and can be derived from the bending moments and shear forces. This is only conditionally possible for parts containing geometric discontinuities. In such parts, strains which stem from the geometric discontinuity are present in addition to the strains which maintain equilibrium with the loads prevalent at the time under consideration in accordance with the stress-strain curve (see fig. 6). The magnitude of these strains (and stresses) generally has a set relationship with the strains which are derived from considerations of equilibrium provided that a specific geometry of the part is postulated. Fig. 6 shows the general pattern of strains in the area of a discontinuity in a cylindrical shell. By analogy with the categorization of stresses as primary, secondary and peak stresses it is proposed that strains should be categorized according to their effects. The criteria, however, on which the stress categorization is based cannot be adopted since the strains in the part above the elasticity limit increase to a larger extent than the stresses. The following categorization is proposed:
Moments on run pipe
Moments on branch pipe
L
Fig. 8. Plasticizing zone in tees, lines of equal strain.
Integral strain Integral strain is defined as encompassing all strains of which the average across the wall thickness is ob-
B. Charalambus et al. / Methods for estimating strain in structures
tained from the instantaneous state of equilibrium on the basis of the stress-strain curve of the tensile test. The amount of integral strain is dependent only on the loading and the geometric shape and must be considerably lower than that of proportional strain since the part might otherwise undergo spontaneous failure [2], fig. 5. Failure can, however, occur on repetition of the loading. Secondary strain Consideration of such strain, which corresponds to secondary stress, can be omitted if it is limited in terms of amount and number of cycles. Peak strains Peak strains occur on the surface on both the inside
epl 0.0250
L
l (_) 0.0200
435
and the outside in the region of local geometric discontinuities in the shell; they are dependent on the magnitude of the loading and exhibit pronounced gradients over the wall thickness. The essential feature of peak strains is that they only give rise to crack initiation, and to failure due to crack propagation, under repeated loading. Mean strain The strains that occur under E E / I E loadings in the non-elastic region generally differ in magnitude in the tensile and compressive directions. This results in a cumulative mean strain occuring at every plasticizing location in the system (fig. 9). Mean strains occur in systems subject to plastic loadings also as a result of the effect of static loads (such as deadweight, operational weight). This applies both to integral strain and to peak strains. Variations in this mean strain remain small during the small number of stress cycle imposed by E E / I E loadings with the result that they need not be taken into consideration in design (cf. fig 9).
0.0150 5. Possibilities for analysis 0.0100
0.0050
0 1.50
2.50
4.50
3.50
5.50 sec 6.50
~ t
0 ~0
T 150,
50
Jilt ,za
0
I/I
-50
-150 0.000
a
4 - - A epl max
0.005
0.010
0.015
rl
1. 0.020
0.025
.~ ~ [ - ]
Fig. 9. Strain patterns in a plasticizing elbow; (a, top) time history of plastic strain, (b, bottom) stress-strain pattern.
Under the methods of stress limitation applied to date the analysis of bending moments and shear forces and of interaction loads due to support-base excitation (EE loads) was conducted by means of response spectrum modal analysis (RSMA) and for internal excitation by means of modal time-history integration. Both methods stand out for their ease of handling, low CPU time and generally conservative results. Furthermore the evaluation of the effects of an increase in amplitude for the same frequency spectrum of the loading is possible without having to repeat the analysis. Additionally the effects of different loadings can be determined separately and superposition can be performed algebraically. Nonlinearities (such as clearances, friction, supports) can only be accounted for (inaccurately) with the aid of modal damping. The essential difference between linear-elastic and elastic-plastic analysis lies in the fact that the law of superposition is invalidated. Depending on the postulates (strain concentration factor), the elastic-plastic analysis of a piping system under transient loadings gives both the integral strain and the strain concentration at every point of the structure. Prerequisites for strain analysis are, for supportbase excitation (earthquake, aircraft crash), the specification of the time history of the displacements at sup-
436
B. Charalarnbus et al. / Methods for estimating strain in structures
port locations and, for internal excitation, the specification of the time histories of the fluid dynamic forces at their points at action. The strains thus obtained are strain time histories and also include the mean strain at the considered location (see fig. 9). The elastic strain c o m p o n e n t may be considered or neglected provided that it is ensured that this c o m p o n e n t is also allowed for in the magnitudes of strain considered allowable. Analyses which take into account the non-elastic, i.e. nonlinear region can be conducted for dynamic loadings by integration against time. This applies to both support-base and internal excitation. If more than one point of excitation is present, all are to be postulated as acting simultaneously. The superposition of results for different loadings is not possible. The ranges of strain as shown in fig. 9b are to be derived from the calculated time histories (fig. 9a).
[3] [4] [5] [6]
[7]
[8]
[9]
6. Summary and prospects Experience to date demonstrates that ductile materials are capable of absorbing large amounts of energy on account of their capacity for plastic deformation. This property has been exploited only insufficiently in prior design practice. If reliably determined strain data are available for parts (e.g. ref. [14]), the plastic strains that can be accepted for the addressed service conditions ( E E / I E ) given a ductile material condition are larger than those in practice to date and no reduction in safety is sustained. This approach requires the establishment of suitable laws of material behaviour such as are provided by cyclic strength charts [15]. The necessary safety margins have yet to be established.
[10]
[11]
[12]
[13]
[14]
References [15] [1] Pipe blowdown tests with valve closure, HDR Test Series RORB, PHDR Report No. 174/85 (July 1985). [2] P. Mihatsch, B. Charalambus and E. Haas, Comparison of
the results of vibration tests with the results of various analysis methods, Nucl. Engrg. Des. 96 (1986) 445 460, in this issue. E. Siebel, New approaches to stress analysis, VDIZeitschrift 90 (1954) 335-341. K. KuP,,maul, German basis safety concept rules out possibility of catastrohpic failure, Nucl. Eng. Int. (Dec. 1984). K. St~ibler, Introduction to assured safety, VGB Kraftwerkstechnik 60 (1980) 428-437. H.J. Golembiewski and G. Vasoukis, Crack analysis in fully ductile regime The critical stress criterion, in: Proc. 8th SMiRT-Conference, Aug. 1985, Brussels, Vol. G 2/3, pp. 59-64. E. Tenckhoff, G. Vasoukis, ,M. Erve and J. Schmidt, Materials for LWRs in the light of new developments, Nucl. Engrg. Des. 87 (1985) 215 223. K. Kul3maul, Fortschritte und Entwichlungstendenzen beim Einsatz hochz~iher und hochfester Werkstoffe, VGB-Ehrenkolloquium: Werkstofftechnik und Betriebserfahrungen, 19. Juni 1985, Mannheim. K. Kul?,maul und E. Roos, Bruchmechanik und Sicherheitskonzepte, Anwendung der Bruchmechanik, Haus der Technik Essen, 27. Februar 1985. D. Sturm, W. Stoppler and J. Scheidermaier, Behaviour of dynamically stressed pipes with circumferential defects under internal pressure and flexural loading, paper delivered at the llth MPA Seminar, Stuttgart, Paper No. 1 (1985). D. Sturm, R. Zirn and R. Haas, Behaviour of piping material WB 35 under dynamic loading in a high-speed tension test, paper delivered at the l l t h MPA Seminar, Stuttgart, Paper No. 3 (1985). W.L. Greenstreet, Experimental study of plastic responses of pipe elbows, Oak Ridge National Laboratory, Contract No. W-7405-enp-26 (February 1978). H. Diem and K.U. Miiller, RL test stand experiments on the deformation behaviour of pipe elbows and pipe elbow connecting cross-sections. K.A. Peters and K.-A. Busch, A method for the derivation of strains due to external/internal event loadings, Nucl. Engrg. Des. 96 (1986) 435-438, in this issue. K. Kessler, G. Vasoukis, D. Blind and K. Maile, Determination of load cycle-dependent characteristics for a design concept based on limited plastic strains, Nucl. Engrg. Des. 96 (1986) 461-473, in this issue.