improved ratcheting analysis of piping components

Improved ratcheting analysis of piping components

Tasnim Hassan*, Yimin Zhu1, Vernon C. MatzenCenter for Nuclear Power Plant Structures, Equipment and Piping, North Carolina State University, Box 7908, Raleigh, NC 27695-7908, USA

Accepted 22 June 1998

Abstract

It is well known that ratcheting (defined as the accumulation of deformation with cycles) can reduce fatigue life or cause failure of pipingcomponents or systems subjected to seismic or other cyclic loads. This phenomenon is sometime referred to as fatigue-ratcheting, which isyet to be understood clearly. Commercial finite element codes cannot accurately simulate the ratcheting responses recorded in tests on pipingcomponents or systems. One of the reasons for this deficiency has been traced to inadequate constitutive models in the existing analysiscodes. To overcome this deficiency, an improved cyclic plasticity model, composed of the Armstrong–Frederick kinematic hardening ruleand the Drucker–Palgen plastic modulus equation, is incorporated into an ANSYS material model subroutine. The modified ANSYSprogram is verified against three sets of experimental results. The simulations from this modified ANSYS show a significant improvementover the unmodified ANSYS and the ABAQUS codes.q 1998 Elsevier Science Ltd. All rights reserved

Keywords:Cyclic plasticity; Ratcheting experiments; Ratcheting analyses; Piping analyses; ASME code

Nomenclature

a ¼ current center of the yield surface in deviatoric stress spaceE ¼ modulus of elasticityH ¼ plastic modulusN ¼ number of loading cyclesn ¼ Ramberg–Osgood hardening parameterSm ¼ allowable design stress intensitys ¼ deviatoric stress tensora ¼ current center of yield surface in stress spaced«p ¼ plastic strain increment tensor«x ¼ axial strain«xc ¼ amplitude of the prescribed axial strain cycles«xp ¼ maximum axial strain in a cycle« v ¼ circumferential strain« vp ¼ maximum circumferential strain in a cyclej ¼ stress tensorj0 ¼ size of yield surfacejx ¼ axial stressjy ¼ Ramberg–Osgood yield parameterjv ¼ circumferential stress

1. Introduction

A question that continues to generate discussion in the

nuclear piping area concerns the primary mode of failure inpiping components and systems subjected to seismicexcitation—is it collapse as assumed in the 1992 [1] andearlier ASME Codes or is it fatigue as indicated by severalrecent sets of experiments? Another question that needs tobe addressed is how much does ratcheting affect the fatiguelife. The 1995 ASME Boiler and Pressure Vessel Code,Section III [2] made an attempt at incorporating reverseddynamic loading and ratcheting into the Code. This revisionwas based on a set of experiments performed as part of thePiping, Fitting and Dynamic Reliability (PFDR) Programsponsored by the Electric Power Research Institute(EPRI)/General Electric (GE)/ United States NuclearRegulatory Commission (USNRC) [3,4]. The 1995 ASMECode revision has not yet been widely accepted by the engi-neering community.

It has been demonstrated experimentally by the PFDRprogram and others [5] that nuclear power plant pipingcomponents and systems may be loaded into the plasticrange due to transients, seismic loading or other unexpectedevents. During such load cycles, if the loading along anydirection is force controlled, then the strain in that directiontends to ratchet [5–7]. Persistent cycling involving ratchet-ing may result in failure due to fracture or reduction of thefatigue life of the structures [8]. Our understanding aboutratcheting needs to be broadened before resolving the manyquestions about the 1995 revised Code, and being able toincorporate ratcheting in the Code in a rational manner.

0308-0161/98/$ - see front matterq 1998 Elsevier Science Ltd. All rights reservedPII: S0308-0161(98)00070-2

* Corresponding author. Tel.: +1-919-515-8123; Fax: +1-919-515-5301;E-mail: [email protected]

1 Also: Black and Veatch, 11401 Lamar, P3, Overland Park, KS 66211,USA.

International Journal of Pressure Vessels and Piping 75 (1998) 643–652

IPVP 1860

Furthermore, in the 1995 ASME Code Section NB3227.6, an allowable stress of 3Sm in primary plus secondarystress intensity is intended to ensure shakedown, defined asthe event when ratcheting ceases to occur after a few cycles.Acker et al. [9] demonstrated through a set of experimentalresults on elbows that this criterion does not alwaysguarantee shakedown. When the primary plus secondarystress intensity exceeds 3Sm, a limit of 5% of the strain isimposed by the Code (NB 3228.4). The basis of thisstrain limit is not clear. An analysis tool which can simulatecyclic responses of structures can be extremely useful inreviewing and revising these ASME Code criteriarationally.

It has been demonstrated that the ratcheting response ofmaterials is significantly influenced by the stress history[6,7,10–12], which, in a structure, depends on the externalload as well as on the geometry. Including all theseparameters experimentally is cost prohibitive. Hence,Code revisions must also be based on analysis which cansimulate ratcheting accurately for different load patterns andgeometries.

Following a review of numerous test and analysis resultson piping components and systems [5,13], we have con-cluded that: (a) ratcheting is a significant contributing factorin the failure of components and systems subjected to highlevel seismic excitation; (b) existing commercial finiteelement codes cannot simulate ratcheting in structures;and (c) the reason for this deficiency can be traced toinadequacies in the plasticity constitutive models used.Motivated by the need for an analysis tool which canaccurately simulate ratcheting in piping components (e.g.straight pipes, elbows and branch pipes) we have attemptedto improve the performance of ANSYS [14], a widely usedfinite element program, by incorporating into it an improvedplasticity model. Because of the complexity of the problem,we chose to begin our study by looking at the behavior ofstraight pipes subjected to cyclic bending and internalpressure.

We took the following approach in our research: first, weidentified a constitutive model which can simulate the ratch-eting response of materials subjected to a somewhat similarloading history as experienced by critical points in a straightpipe; second, we implemented this constitutive model intothe ANSYS code; and, third, we validated the modified codeby correlating the analysis results with those from tests onstraight pipes in cyclic bending and internal pressure.Hassan et al. [7] demonstrated for a series of biaxial tests,with constant internal pressure and axial strain controlcycles, that the Drucker–Palgen plastic modulus equation[15] with the Armstrong–Frederick kinematic hardeningrule [16] in modified forms produced very good predictions.As the stress histories in these material tests are close tothose in a straight pipe subjected to cyclic bending at con-stant internal pressure, we implemented the above modelinto ANSYS. In this paper we discuss briefly the theoriesimplemented into ANSYS and present the simulations

obtained from the modified ANSYS code. The simulationsare found to be promising.

2. Theory

A brief review of the cyclic plasticity theory used andhow it was incorporated into ANSYS will be presentedfirst. The constitutive model assumes cyclically stablematerial behavior andJ2-type plasticity in which the yieldsurface is represented by

f (j ¹ a) ¼32(s¹ a)·(s¹ a)

� �1=2

¼ j0 (1)

wherej is the stress tensor,s is the deviatoric stress tensor,a is the current center of the yield surface in stress space,anda is the center in the deviatoric stress space. The size(j0) and shape of the yield surface remain unchanged duringplastic loading increments. The parameterj0 is determinedfrom the linear part of the stable hysteresis loops obtainedfrom a uniaxial test.

The total strain increment is assumed to be decomposedinto an elastic part and a plastic part. The incrementalstress–plastic strain relationship (flow rule) is given by:

d«p ¼1H

dfdj

·dj

� �dfdj

(2)

whereH is the plastic modulus which is evaluated using theDrucker–Palgen Model [15]. In this modelH is assumed tobe a function of the second invariantJ2 of the deviatoricstress tensor. In our calculations, we evaluatedH using thefollowing equation [17]

H ¼37

nE

3J2

j2y

! n¹ 12

26643775

¹ 1

(3)

whereE, n, jy are material parameters evaluated by fittingthe Ramberg–Osgood equation to the upper part of a stablehysteresis loop.

A hardening rule determines the evolution of the yieldsurface in stress space during plastic loading increments—itcan be kinematic, isotropic or a combination of the two. Weuse the Armstrong–Frederick kinematic hardening rule inthe following form [7]:

da ¼ dm[(1¹ k)(j ¹ a) ¹ ka] (4)

wherek is a model parameter determined using a biaxialratcheting experiment and dm is determined using the con-sistency condition. This constitutive model was incorpo-rated into a customized version of ANSYS through a usersubroutine. A Newton–Raphson iteration scheme was usedfor nonlinear material response calculations.

The model parameterk in the kinematic hardening rule in

644 T. Hassan et al./International Journal of Pressure Vessels and Piping 75 (1998) 643–652

Eq. (4) is used to model the evolution of the yield surfacedue to ratcheting. During plastic loading the yield surfaceevolves through translation, size change and shape change[18,19]. None of the existing plasticity models has yetsuccessfully implemented all three yield surface evolutionfeatures. A simple method of modeling all these features forratcheting simulation was attempted by Hassan et al. [7]through the incorporation of a model parameterk in Eq.(4). This modeling scheme is not universally applicablebut can be tuned to simulate a class of ratcheting responsesas demonstrated by Corona et al. [12], and results based onthis approach are presented in this paper.

3. Verification

Test data sets from three types of tests on straight pipeswere used for verification: (1) symmetric axial strain cycling

at constant internal pressure [7]; (2) symmetric shear straincycling at constant tension [20]; and (3) displacement con-trolled cyclic bending at constant internal pressure [21,22].The first two sets are at the material level, i.e. the state ofstress, except near the ends, is everywhere the same. The

Fig. 1. (a) Test specimen and loading; (b) finite element model for constantpressure–cyclic axial strain test.

Fig. 2. Axial strain cycling at constant internal pressure. (a), (b) Measured axial stress–strain and axial–circumferential strain responses [7];(c), (d) simulationsby modified ANSYS code.

645T. Hassan et al./International Journal of Pressure Vessels and Piping 75 (1998) 643–652

last set, a cyclic bending test, is at the structural level and thestate of stress varies from point to point.

Since the first set of tests was performed on cyclicallystabilized specimens of carbon steel (CS) 1026 (see Hassanet al. [7] for details), the assumption of cyclically stabilizedmaterial for this data set was appropriate. The cyclic shearstrain and bending tests were, respectively, conducted onpipes of CS 1020, which exhibits cyclic softening, and stain-less steel (SS) 304, which exhibits cyclic hardening, and sothe assumption of cyclically stabilized material is notstrictly valid for these cases. However, we do not currentlyhave a cyclic plasticity model which can accurately simulateratcheting of cyclically softening or hardening materials[11]. Before carrying out this more complex step of devel-oping such a model, we examined the performance of thecurrent modified ANSYS code in simulating ratcheting inthese cyclic shear and bending tests.

3.1. Symmetric axial strain cycling at constant pressure

This set of tests involved axial strain symmetric cycling ata prescribed constant internal pressure [7]. A schematicdrawing of the test specimen and loading is shown inFig. 1a. The reduced portion of the specimen had a 10(25.4 mm) outer diameter with a wall thickness of 0.050(1.27 mm). A sketch of the loading path prescribed on thespecimen is shown in the inset of Fig. 2. The specimens inthis set of tests were stabilized prior to the ratcheting experi-ments by prescribing 12 axial strain cycles of amplitude 1%(see Hassan and Kyriakides [6] for detail). The stablehysteresis loop from this stabilization step is shown in thefigure by a dashed line.

Typical responses of axial stress–axial strain and axialstrain–circumferential strain are shown in Fig. 2a and b,respectively. In this test the internal pressure was 1000 psi(6.9 MPa), which resulted in a circumferential stress (jv) of9655 psi (66.6 MPa) (based on the thin-wall assumption),and the amplitude of the axial strain cycles («xc) was 0.5%.The circumferential stress shown in the figure(j̄v) is thenormalized stress obtained by dividing the circumferentialstress by the yield plateau stress [39 440 psi (271.9 MPa)] ofthe uniaxial monotonic curve. In this test the material ratch-ets in the circumferential direction as shown in Fig. 2b. Wehave plotted in Fig. 3 the maximum circumferential strain ineach cycle («vp) as a function of the number of cycles,N. Wesee that the rate of ratcheting (increment in strain per cycle)can be approximated by a straight-line behavior that isexpected for a material that is cyclically stable.

The finite element model used in the analysis was com-prised of a single Shell 43 (four node flat plate) element asshown in Fig. 1b. Boundary conditions and prescribed loadsare as shown in the figure. At the top of the model weprescribed a constant stress to represent the constantcircumferential stress, and on the right side we prescribeddisplacement increments to represent the axial strains in thetest.

The material parameters in Eq. (3) [E,n,jy] were obtainedby fitting the Ramberg–Osgood equation to the upper partof a uniaxial, stable hysteresis curve similar to the oneshown in Fig. 2a by a dashed line. The value ofk inEq. (4) was evaluated by matching the ratcheting rateobtained from the constitutive model to the ratcheting ratein the test in Fig. 2. The material constants for CS 1026 areshown in Table 1.

Calculated results for this test using the modified ANSYScode are plotted in Figs 2 and 3. Comparing Fig. 2b and d,we see that the circumferential strain ratcheting response isreproduced very well. This is clearly demonstrated in Fig. 3[modified ANSYS results denoted by ANSYS(M)]. Alsoshown in Fig. 3 are results from the unmodified ANSYScode [denoted by ANSYS(U)], where we see that, for thisanalysis, the response ceases to ratchet after the secondcycle. This deficiency is mainly due to the kinematic hard-ening rule (Prager’s rule [23]) used in unmodified ANSYS.(These results were obtained from ANSYS 5.1 [14].)

The same set of constants was then used to simulate theratcheting rate for four other experiments using the modifiedANSYS code. The results of the rate of ratcheting are shownin Fig. 4 in two sets. In the first set (Fig. 4a), the internalpressure was kept constant(j̄v ¼ 0:24) whereas the ampli-tudes of the axial strain cycles in the three tests were varied(«xc ¼ 0.4, 0.5 and 0.65%). In the second set, the internalpressure was varied(j̄v ¼ 0:122, 0.178, 0.245 and 0.357)while the axial strain amplitude was 0.5% in all four tests.The performance of the modified ANSYS in simulating therate of ratcheting is the same as was obtained by the con-stitutive model [7]. These results demonstrate that the newcyclic plasticity model has been incorporated in the ANSYScode properly.

Fig. 3. Comparison of the rates of circumferential ratcheting from unmo-dified (U) and modified (M) ANSYS to test results [7].

Table 1Constitutive model parameters for CS 1026

Material E, ksi(GPa)

n jy, ksi(MPa)

j0, ksi(MPa)

k

CS 1026 26 320 5 17.8 19 0.091(181.5) (123) (131)


3.2. Symmetric shear strain cycling at constant tension

This set of tests involved strain symmetric cycling inshear at a prescribed constant tensile force [20]. Aschematic drawing of the CS 1020 test specimen is shownin Fig. 5a along with its loading. The geometry and dimen-sions of the specimens used in these tests were the same asin the last set of tests. An axial strain–shear strain responsefrom a test is shown in Fig. 6b, where we see that thenominal axial stress due to prescribed constant tensileforce was 17.5 ksi (120.66 MPa) and the amplitude of theprescribed shear strain cycle was 0.8727%. Due to the pre-sence of constant axial stress, ratcheting in this test occurs inthe axial direction. The axial strains,«xp, at the positiveshear strain peaks are plotted against the number of appliedcycles in Fig. 7. After the initial transient, the rate of ratch-eting (increment in strain per cycle) is almost constant aswas found in the previous section. In this set of tests the CS1020 exhibited cyclic softening in shear stress (not shown).Hence, it appears that the cyclic softening in shear had littleeffect on the ratcheting rate in the axial direction.

The finite element model used in the calculation, shownin Fig. 5b, represents the central portion of the specimenwhere stresses can be assumed to be uniform. The model iscomposed of one ring of 32 Shell 43 (four node flat plate)elements around the circumference. For boundary con-ditions, the nodes at one end were constrained against trans-lation in the axial and in the circumferential directions, butwere allowed to translate freely in the radial direction. Atthe other end, rotations were specified to simulate the pre-scribed oscillating shearing strain of the specimen. Theupper part of a stable axial stress–strain hysteresis curvefrom Hassan and Kyriakides [6] was used to determinethe constants in Eq. (3). In Fig. 7, the ratcheting rate

obtained from the constitutive model (dashed line) wasmatched to the stable ratcheting rate in the test (solid circles)to determine the value ofk in Eq. (4). The material para-meters determined for CS 1020 are shown in Table 2.

Calculated results for this test using both the modified andunmodified ANSYS codes are plotted in Figs 6 and 7. Wesee that neither the unmodified nor the modified ANSYScode is able to simulate ratcheting in the first cycle accu-rately. This may be related to the fact that the transientresponse of the material (due to cyclic softening for thismaterial) has not been modeled and the initial anisotropyin the cold-drawn CS 1020 has not been considered in theanalysis. The rate of ratcheting in the remaining cycles isreproduced quite accurately with the modified code. Theunmodified ANSYS code yielded results in which the rateof ratcheting is overpredicted up to the fifth cycle, and thengradually decreases towards shakedown. This deficiency ismainly due to the kinematic hardening rule (Prager’s rule[23]) used by ANSYS. In Fig. 7 we see, by comparing theresults from a FORTRAN program containing the constitu-tive model (but not the finite element method) with themodified ANSYS results, that the incorporation of themodel into the ANSYS user subroutine introduces negligi-ble error for this analysis.

The same set of constants was then used to simulate theratcheting rate for three other experiments in the set. Each ofthese tests was subjected to a different constant tensile forcebut the same shearing strain cycle of amplitude 0.8727%.The results of the rate of ratcheting from these tests areplotted in Fig. 8 where we see that, as the nominal axialstress increases, the rate of ratcheting also increases. Thepredictions of the model are plotted in the same figure. Thesimulations forjx ¼ 17.5 ksi (120.66 MPa) show nearly thesame rate of ratcheting as the experiment since these were

Fig. 4. Test results [7] and simulations of the rate of circumferential ratcheting by modified ANSYS. (a) Constant internal pressure, and (b) constantstrainamplitude.

Fig. 5. (a) Test specimen and loading; (b) finite element model for constanttension–cyclic shear strain tests (boundary conditions and loading notshown).

Table 2Constitutive model parameters for CS 1020


n jy, ksi(MPa)

j0, ksi(MPa)

k

CS 1020 25 125 3.9 27 35 0.091(173.2) (186) (241)


the data used to obtain the material constants. The predic-tion for the two cases with lower axial stresses are also quitegood. For the higher axial stress test, however, the analysisunderpredicts the rate of ratcheting. The reason for this maybe that the kinematic hardening rule we use in our plasticitymodel assumes that both the shape and size of the yieldsurface remain unchanged during plastic deformation.When plastic deformation becomes significant, this assump-tion is no longer valid. To further improve the simulation forthis type of test, we would need to incorporate a materialmodel which accommodates changes in both the shape andsize of the yield surface.

3.3. Cyclic bending at constant internal pressure

Two sets of test data were selected from Gau [21] andScavuzzo [22] to verify the modified ANSYS code at the

structural level. These tests involve four-point displacementcontrolled cyclic bending of straight pipes at constant inter-nal pressure. The specimens were 10 (25.4 mm), schedule40, SS 304 pipes 480 (1.22 m) long. A schematic drawing oftheir test set-up is shown in Fig. 9. Recorded responses ofaxial strain and circumferential strain at A (see Fig. 9) in thefirst test are shown in Fig. 10 (all measured data from thetests have been digitized from the references cited). In thistest the constant internal pressure was 2000 psi (13.8 MPa)and the cyclic displacements at the rigid spreader bar wereprescribed with amplitudes of60.250 (6.35 mm) [5 cycles],61.00 (25.4 mm) [50 cycles],6 0.50 (12.7 mm) [50 cycles],61.00 (25.4 mm) [12 cycles] and61.50 (38.1 mm) [15cycles]. Due to the presence of constant internal pressurethe circumferential strain in this test ratchets with progres-sive cycles.

In the simulation with ANSYS, because of the double

Fig. 6. Axial ratcheting in constant tension and cyclic shear strain. (a) Modified ANSYS; (b) test [20]; and (c) unmodified ANSYS.

Fig. 7. Comparison of the rates of axial ratcheting from unmodified andmodified ANSYS, constitutive model and test [20] with constant tensionjx ¼ 17.5 ksi (120.66 MPa) and cyclic shear strain of amplitudegc ¼

0.8727%.

Fig. 8. Predictions of the rate of ratcheting by the modified ANSYS for fourtests [20] with constant tensions ofjx ¼ 10 (69.0), 13.5 (93.1), 17.5 (120.7)and 20 (137.9) ksi (MPa) and cyclic shear strain of amplitudegc ¼

0.8727%.


symmetry of the problem, we modeled only one quarter ofthe pipe (see Fig. 11). In the finite element model, we usedeight Shell 43 elements around the half-circumference, 10elements along the length between the mid cross-section andthe loading point (A to B in Fig. 9), and eight elements fromthe loading point to the support (B to C in Fig. 9).

To develop an acceptable finite element model for thecyclic bending problem, we evaluated the ratchetingresponses of straight pipes using both flat elements (Shell43) and curved shell elements (Shell 93). For the samenumber of elements the ratcheting responses from the twomodels of flat element and curved shell element were veryclose to each other, but the CPU time for the model with flatelements was about 40% of that of the curve shell elementmodel. As the main objective of this research is to develop aconstitutive model for ratcheting simulation, we performedthe remaining analyses using flat plate elements.

The material tests needed to evaluate the constants inEqs. (3) and (4) were not available. We used a stablehysteresis curve of SS 304 from Alameel [24] to determinethe constants in Eq. (3). The part of the stable hysteresis

loop fitted to evaluate the model parameters in Eq. (3) ishighlighted by a thicker line in Fig. 12. A biaxial ratchetingexperiment to determinek in Eq. (4) was not availableeither. We fitted the ratcheting rate from ANSYS to theratcheting rate from the first set of Gau’s test (see Fig. 13)to determinek. The evaluated model parameters are given inTable 3.

In the analyses with unmodified ANSYS and ABAQUS[25], bilinear models with kinematic hardening rules [23,26]were used. In the analyses with ABAQUS, Gau [21] usedtwo different values forEt, the elastic–plastic modulus inhis bilinear model: 500 ksi (3.44 GPa) and 5500 ksi(37.5 GPa). In Fig. 12, we see that the hysteresis loop pro-duced by the smaller bilinear modulus is more realistic thanthat from the higher modulus. Hence, in the analysis withunmodified ANSYS, we used 500 ksi as the modulus in thebilinear fit.

The unmodified ANSYS and the ABAQUS with lowerEt

greatly overpredicted the rate of ratcheting as seen in Fig.13. One of the reasons for the overprediction is the kine-matic hardening rule used in the codes. The rule used in theunmodified ANSYS is Prager’s rule [23], and in ABAQUSit is the Prager–Ziegler rule [26]. Both of these have been

Fig. 9. Four-point bending test set-up used by Gau [21] for testing straightpipes in cyclic bending and constant internal pressure.

Fig. 10. Axial strain and circumferential strain responses in a four-point cyclic bending test [21] with constant internal pressure of 2000 psi (13.8MPa) anddisplacement amplitudes of6 0.250 (6.35 mm) [5 cycles],6 1.00 (25.4 mm) [50 cycles],6 0.50 (12.7 mm) [50 cycles],6 1.00 (25.4 mm) [12 cycles] and61.50 (38.1 mm) [15 cycles] and corresponding analyses results from modified ANSYS.

Table 3Constitutive model parameters for SS 304


n jy, ksi(MPa)

j0, ksi(MPa)

k

SS304 28 500 7.3 32 35 0.040(196.5) (221) (241)


shown to be incapable of simulating biaxial ratcheting[7,27]. The modified ANSYS analysis and the ABAQUSanalysis which uses the larger value ofEt both match therate of ratcheting quite well, with the modified ANSYS codegiving a slightly better correlation. In the modified ANSYScode the improvement is mainly achieved by incorporatinga new kinematic hardening rule. In the ABAQUS analysisthe kinematic hardening rule was not modified, but theimprovement in simulation was obtained in an artificialway by making the material stiffer during plastic deforma-tion. As shown in Fig. 12, the largerEt value results in anunrealistic hysteresis loop. The applicability of this methodof improving ratcheting simulation in structures needsfurther attention. The results from the modified ANSYScode match the rate of ratcheting quite well, but this is notsurprising since this part of the test was used to obtain theconstitutive model parameterk in Eq. (4).

To validate the performance of the modified ANSYScode we first used the material constants in Table 3 to

predict ratcheting over the whole range of the first test(the material parameters were obtained using the first fewcycles of this test). The results are plotted in Fig. 10, wherewe see that the modified ANSYS simulated the circum-ferential ratcheting in the test with good accuracy (refer toFig. 10b and d). The test data and simulations from both theunmodified and modified ANSYS are plotted in Fig. 14, inwhich are shown the maximum circumferential strains ineach cycle as a function of the number of cycles.

The first step of five cycles is in the elastic region, andhence no ratcheting occurs and the predicted results fromboth the unmodified and modified ANSYS are good. In thesecond step the pipe is loaded into the plastic range and, dueto the presence of internal pressure, ratcheting occurs in thecircumferential direction. In the simulation by the unmodi-fied ANSYS code, the rate of ratcheting is greatly overpre-dicted in the first 25 cycles and then it quickly stabilizes toshakedown. On the other hand, the modified ANSYS simu-lated the ratcheting in this step reasonably well. In the sub-sequent cycles, the unmodified ANSYS simulated eithershakedown or negative ratcheting. The modified ANSYSpredicted the increasing ratcheting strain with goodaccuracy.

The same set of constitutive model parameters was thenused to predict the ratcheting responses in the second test.This test had an internal pressure of 3000 psi (20.7 MPa)

Fig. 11. Finite element model for analyses of four-point bending tests.

Fig. 12. Monotonic curve and stable hysteresis loop of SS 304 and modelsof modulus calculation by the Drucker–Palgen equation (D–P) [15] andbilinear model [26].

Fig. 13. Comparison of the test [21] and analyses of the circumferentialstrain ratcheting of a straight pipe subjected to cyclic bending [d ¼ 1.00(25.4 mm)] and constant internal pressure of 2000 psi (13.8 MPa) [part ofsecond step in Fig. 10].

Fig. 14. Comparison of the test [shown in Fig. 10] and analyses results ofthe rate of circumferential strain ratcheting.


and cyclic displacement amplitudes prescribed in steps of 10cycles atd ¼ 0.250 (6.35 mm), 50 cycles at 1.00 (25.4 mm),10 cycles at 1.50 (38.1 mm), 10 cycles at 1.00 (25.4 mm),and then 20 cycles at 1.50 (38.1 mm). A comparison of thetest data and simulations from both the unmodified andmodified ANSYS codes is plotted in Fig. 15, in which themaximum circumferential strains in each cycle are plottedas a function of the number of cycles. The modified ANSYSresults show significant improvement over the unmodifiedANSYS.

The small differences observed in the modified ANSYScorrelations during the initial transient in the second step(see Figs 14 and 15) may be caused by the assumption thatthe material in the analyses is cyclically stable. To improvethe simulation further improvements in constitutive model-ing are needed.

4. Summary, conclusions and future work

Commercial finite element codes (ANSYS 5.1 andABAQUS 4.7) cannot simulate ratcheting in a straightpipe subjected to constant internal pressure and cyclic bend-ing. This deficiency is related to the kinematic hardeningrules incorporated in the codes. In this research a kinematichardening rule based on the Armstrong–Frederick rule isincorporated into ANSYS and the correlations of resultsfor three sets of tests—symmetric axial strain cycles atconstant internal pressure, symmetric shear strain cyclingat constant tension, and cyclic bending at constant internalpressure—show that the modified ANSYS code is able tosimulate ratcheting in straight pipes more accurately thanthe unmodified ANSYS or the ABAQUS codes. Areas offurther improvements to simulate transients more accuratelyare identified. Further improvement in modeling wouldrequire development and incorporation of models for cyclichardening and softening. Change in the shape of the yieldsurface should also be incorporated for more accurate simu-lations. These more difficult tasks will be the subject offuture research.

One difficulty in using component test data presented inthe literature is that the necessary material tests to evaluateall five constitutive model parameters are not available. Toaddress this problem, we are now performing a consistentset of tests at the material level and the structural level forboth straight pipes and elbows. Following the successfulverification of our improved ANSYS code for quasi-staticbehavior at room temperature, we will modify the constitu-tive model to include both time and temperature dependen-cies so that it can be used to simulate the seismic response ofpiping components and systems under operating conditions.

In this research, we attempted to improve the ratchetingsimulation for a specific problem—a straight pipe subjectedto constant internal pressure and cyclic bending. Theresearch described in this paper, while admittedly incom-plete, is, nevertheless, a significant advance in the ability tosimulate ratcheting using finite element analysis. At thispoint, we would like to mention to the readers that theapplicability of the constitutive model used here to anyother problems needs further investigation. In a recentpaper, Corona et al. [12] demonstrated that the model ofthe class used here failed to simulate ratcheting responsesin materials for complex loading histories. Hence, otheravenues of research towards developing improved constitu-tive models should also be pursued. It should be noted that,even though our work used a specific analysis code—ANSYS—it is presumably applicable to any other finiteelement code which allows incorporation of constitutivemodels through a user subroutine.

Finally, with a finite element code capable of simulatingratcheting accurately under realistic operating conditions, itwill be possible to provide a rational basis for the incorpora-tion of ratcheting in the ASME Code. The result will be lessuncertainty in piping designs, and piping systems which areboth more economical and safer.

Acknowledgements

The authors wish to acknowledge the Center for NuclearPower Plant Structures, Equipment and Piping and theDepartment of Civil Engineering at North Carolina StateUniversity for their support of this research. The assistancefrom Dr. Robert Mallett of Mallett Technology, Inc., Dr.Peter Kohnke of ANSYS, Inc., Mr. David Maxham andMr. Robert Gurdal of Framatome Technologies, Dr. EdwardA. Wais of Wais and Associates, Inc. and Mr. M. StephenSills of Duke Engineering & Services are greatlyappreciated.

References

[1] ASME, Boiler and Pressure Vessel Code, Section III, Division 1,Subsection NB-3600 Piping Design, The American Society ofMechanical Engineers, 1992.

[2] ASME, Boiler and Pressure Vessel Code, Section III, Division 1,

Fig. 15. Comparison of the test [21] and analyses results of the rate ofcircumferential strain ratcheting of a straight pipe subjected to cyclic bend-ing {d ¼ 0.250 (6.35 mm) [10 cycles], 1.00 (25.4 mm) [50 cycles], 1.50(38.1 mm) [10 cycles], 1.00 (25.4 mm) [10 cycles], 1.50 (38.1 mm) [20cycles]} at internal pressure of 3000 psi (20.7 MPa).


Subsection NB-3600 Piping Design, The American Society ofMechanical Engineers, 1995.

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