11/ .1dr,1y

Distribution Category:

Gas Cooled Reactor Technology(UC-77)

ANL-79-58

ARGONNE NATIONAL LABORATORY9700 South Cass Avenue

Argonne, Illinois 60439

CALCULATIONS OF STRESSES IN GCFR CLADDINGUNDER NORMAL OPERATING CONDITIONS

by

Yung Y. Liu, T. C. Hsieh,and M. C. Billone*

Materials Science Division

November 1979

DISCLAIMER

': 1f ". rr. raeli . . '"15t r .39e 1.r 0 .. r ,,r , st st.r pier " r 1 h Ih f 1... r r

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*Northwestern University, Evanston, Illinois

3

TABLE OF CONTENTS

Page

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

I. INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . 7

II. DESIGN PARAMETERS AND OPERATING CONDITIONS . . . . . . . . . . . . 8

III. AN lXTICAL METHODS. . . . . . . . . . . . . . . . . ........ 8

A. LIFE-GCFR . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

B. Thermoelastic Stresses Due to Radial Temperature Gradient. 11

C. Thermoelastic Stresses Due to Axial Temperature Gradient . 13

D. Stresses Due to Swelling Gradient. . . ............. 16

E. Cladding Creep Analysis. . . . . . . . . . . . . . . . . 17

IV. RESULTS . . . . . . . . . . . . . . . . ............ . 17

A. LIFE-GCFR Analysis of Fuel-Cladding Mechanical Interaction 17

B. Thermoelastic Stresses Due to Radial Temperature Gradient. . . 2C

C. Thermoelastic Stresses Due to Axial Temperature Gradient . 24

D. LIFE-GCFR Assessment of Swelling and Creep Effects onCladding Stresses. . . . . . . . . . . . . . . . . . . . . . . 25

V. SUMMARY AND DISCUSSION. . . . . . . . . . . . . . . . . . . . . . . 27

VI. FUTURE 1ORK . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

ACKNOWLEDGMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

APPENDIXDetailed Analysis of GCFR Cladding Thermal Stresses Induced by AnAxial Temperature Gradient . . . . . . . . . . . . . . . . . . . 33

4

LIST OF FIGURES

No. Title Page

1. Steady-state (100% Power) GCFR Fuel-element Gap Size at X/L = 0.375 47

2. Steady-state GCFR Gap Sizes at X/L = 0.125 for 60, 100, and 115%Power Runs . . . . . . . . .. . . . . . . . . . . . . . . . . . . 47

3. GCFR Fuel-element Gap Sizes at X/L = 0.375 for Steady-state andPower-cycling Histories. . . . . . . . . . . . . . . . . . . . . . 47

4. Temperature Dependence in the Stress-free Swelling of 20% Cold-worked 316 Srainless Steel . . . . . . . . . . . . . . . . . . .. 48

5. Thermoelastic Stress Distributions Across the Cladding Thickness;Cladding Outer-surface Temperature = 600*C, q' = 16.4 kW/m, P. =P0 = 9.0 MPa. ................................. 48

6. Thermoelastic Stress Distributions Across the Cladding Thickness;Cladding Outer-surface Temperature = 600*C, q' = 36.1 kW/m, P. =

P0 = 9.0 MPa...-.......................... 48

7. Thermoelastic Stress Distributions Ae:oss the Cladding Thickness;Cladding Outer-surface Temperature = 600*C, q' = 45.9 kW/m, Pi=P0 = 9.0 MPa . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

8. Thermoelastic Stress Distributions Across the Cladding Thickness;Thin Cladding, Cladding Outer-surface Temperature = 600*C, q' =36.1 kW/m, P. = P = 9.0 MPa . . . . . . . . . . . . . . . . . . . 49

9. Thermoelastic Stress Distributions Across the Cladding Thickness;Thick Cladding, Cladding Outer-surface Temperature = 600*C, q' =33.1 kW/m, P = Po = 9.0 MPa . . . . . . . . . . . . . . . . . . 49

10. Thermoelastic Stress Distributions Across the Cladding Thickness;Cladding Outer-surface Temperature = 600*C, q' = 36.1 kW/m, P =

PO = 0 MPa . . . . . . . . . . . . . . ..................- 50

11. Schematic Diagrams of the GCFR Thermal Stress Problems due to anAxial Temperature Gradient . ..........-.......... 50

12. Cladding Hoop- and Axial-stress Distributions with Creep andSwelling Operating .-.-........................ 50

13. Cladding Hoop-stress Distributions with Swelling Suppressed andCreep Operating............................ 51

5

LIST OF FIGURES

No. Title Page

14. Cladding Axial-stress Distributions with Swelling Suppressed andCreep Operating. . . . . . . . . . . . . . . . . . . . . . . . . . 51

15. Cladding Hoop-stress Distributions with Swelling Operating and CreepSuppressed . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

16. Cladding Hoop-stress Distributions with Temperature and SwellingGradients Opposite to Each Other ...................... .. ..... 52

6

LIST OF TABLES

No. Title Page

I . Major Design Parameters of 300 MW(e) GCFR Fuel Assembly . . . . . 9

IT.. Operating Conditions for the Peak-power Rod in the 300 MW(e)GCFR Demonstration Plant Design . . . ; . . . . . . . . . . . . . 10

III. Fuel-Cladding Interfacial Pressure and Cladding Stresses DuringFuel-Cladding Mechanical Interaction (LIFE-GCFR Analysis) . . . . 19

IV* Calculated Cladding Temperature Drop as a Function of Fuel-elementLinear Heat Rating, q', and the Cladding Outer-surface TemperatureTs. . .. . . . . . . . . . . . . . . . . . . . . . . . . . 21

V. claddingg Stress-free and Stress-affected Swelling Calculated forTs = 600*C, and Three Values of q'. . . . . . . . . . . . . . . . 23

VI . Summary of Computed Stress Biaxiality Values (a Iazz). . . . . . 30

A-I . Normalized a for $ = 7/2 41

A-II . Normalized ae for 'P = r. . . . . . . . . . . . . 41

A-III . Normalized o at z* = $ . . . . . . . . . . . . . . . . . . . . . 4

A-IV . Normalized o for 'P = 7/2 . . . . . . . 9.. . . . . . . ... 46

7

CALCULATIONS OF STRESSES IN GCFR CLADDING UNDER NORMALOPERATING CONDITIONS

by

Yung Y. Liu, T. C. Hsieh and M. C. Billone

ABSTRACT

A modified version of the LIFE-III code, LIFE-GCFR, andclassical stress-analysis techniques have been used to calculate

the stresses in the GCFR cladding under normal reactor operating

conditions. Several types of loadings on the cladding that occurduring normal operation have been considered. These include fuel-

cladding mechanical interaction, thermal stresses induced by radial

and axial temperature gradients, and swelling gradient-induced

stresses. The combined and individual effects of these loadings,

as well as the effect of creep on cladding stresses, have been

assessed.

LIFE-GCFR predicts fuel-cladding mechanical interaction in fuelpins operated under several types of reference GCFR power histories.

The swelling of fuel and cladding, the differential thermal expansionbetween fuel and cladding, and the fuel cracking and relocation appear

to be the main factors determining the fuel-cladding contact pressure,for the steady-state, step power-change, and power-cycling operations,

respectively. Creep and swelling affect cladding stresses in an oppo-

site manner. Both thermal and irradiation creep may be beneficial in

relaxing cladding stresses. Swelling, on the other hand, may be of

concern because the swelling-induced stresses generally increase with

burnup. The subtlety in the significance of cladding swelling isthat the resulting cladding stress distribution is sensitive to

whether the cladding temperatures are above or below the peak swelling

temperature of the cladding material. Above the incubation fast

fluence for cladding swelling, a fatigue-type problem may be en-countered if the cladding temperatures swing back and forth across

the peak swelling temperature during power-cycling operation.

Thermal stresses induced by an axial temperature gradient in

the cladding smooth-to-ribbed transition region can be greatlyreduced by an appropriate design of the transition region. Theanalysis provides a quantitative estimate of the transition-regionlength above which the cladding stresses induced by an axial temp-erature gradient become insignificant.

I. INTRODUCTION

The Gas Cooled Fast Reactor (GCFR) fuel-element design incorporates apressure-equalization system (PES)l to prevent the buildup of fission-gaspressure within tie fuel rod. Coolant and plenum pressure are maintained ata constant level of '9.0 MPa (1280 psia) during steady-state operation. Inthe absence of any primary pressure loading on the ribbed GCFR cladding due

to fission gas (assuming PES works as designed), the only other source of

8

internal primary loading during normal operation is due to fuel-cladding mech-

anical interaction (FCMI). Several types of secondary loadings on the GCFR

cladding can arise, however, from both the cladding radial and axial tempera-ture gradients and from the cladding swelling gradient during normal operation.This investigation provides an assessment of the cladding stresses caused bythese various loading mechanisms. A modified version of the LIFE-III code,2

LIFE-GCFR, has been supplemented by classical stress-analysis techniques toperform the investigation.

II. DESIGN PARAMETERS AND OPERATING CONDITIONS

General Atomic Company supplied the design parameters and operatingconditions (Tables I and II) for the reference GCFR fuel element. Steady-stateperformance studies were conducted for fuel elements at 60, 100, and 115% offull-power and neutron-flux values. In addition, because of the fuel-rotation

scheme, histories with step power changes from 60 to 100% (with the possibilityof a 15% overpower) at midburnup were studied by imposing a reactor shutdown

and startup every 250 full-power hours.

For all the above power-history cases studied with LIFE-GCFR in this in-vestigation, five axial sections were used in the analyses: plenum region,

smooth-cladding fuel region, and three ribbed-cladding fuel regions. An effec-tive cladding thickness, teff, has been chosen to represent the mechanicalresponse of ribbed cladding in a one-dimensional stress analysis. This clad-ding effective thickness is defined by

tef = t + hpw , (1)

eff p

where t, h, w, and p are the cladding root thickness, rib height, rib width,and pitch, respectively.

III. ANALYTICAL METHODS

A. LIFE-GCFR

Earlier modifications of LIFE-III to generate LIFE-GCFR have been describedby Billone.3 Briefly, these earlier modifications were mainly related to areasin fuel-element heat-transfer calculations. Sodium coolant properties havebeen replaced by those of the helium gas; and separate empirical Nussult Numbercorrelations were incorporated to account for the difference in heat-transfercharacteristics along the smooth and the ribbed portions of the cladding. Inaddition, only the thermal conductivity of the helium gas is specified in thegap-conductance calculations. Recent modifications of LIFE-GCFR include:

1. an expansion of the maximum number of axial sections (into which afuel element can be divided for modeling purposes) from 6 to 10;

2. the addition of an oxide-fuel primary creep term in the fuel creepconstitutive equation based on the experimental results of Solomon;4

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Table I. Major Design Parameters of 300-MW(e) GCFR Fuel Assembly

Fuel Assembly

Cross section ..........................................

Active core height.....................................Number of fuel rods.....................................

Number of hanger rods...................., . ........... .

Number of instrumentation rods.........................

Fuel rod spacer type...................................

Number of spacer grids*................................Across-flats ID%*.......................................

Across-flats OD(a) Above core*.................................

(b) Core midplane*..............................

Duct-wall thickness

(a) Above core..................................

(b) Core midplane ...............................Fue -rod pitch*......... . . . . . . . . . . . . . . . . . . . . . . . . . ..

Rod-to-rod gap* ........................................

Rod-to-duct gap* .......................................

Rod-to-duct gap*, % of rod to duct gap.................

Fuel-assembly pitch*...................................

ruel Rod

Length .................................................

Outside diameter*......................................Outside root diameter...................

Inside diameter ........................................

Cladding thickness................................. .

Cladding material............ ...........

Fuel and blanket pellet OD*............................Fuel and blanket pellet-to-cladding gap*...............Fuel length ............................................

Fuel material ..........................................

Fuel material planar smear density,%of theoretical.......... .........

Axial blanket length (each), mm........................Axial blanket material........ .........................

Axial blanket material planar smear density,%of theoretical ..................................

hexagonal3665.0 mm1130.0 mm26461modifiedhexagonal grid10183.0 mm

10J.6 mm188.0 mm

3.8 mm2.5 mm11.05 mi:3.85 mm1.77 mm46.1197.1 mm

2230.0 mm7.46 mm7.20 mm6.44 mm0.38 mm316SS-20% CW6.30 mm0.14 mm1130.0 mmmixed UO2-PuO2

85.5450.0depleted U02

90.0

Surface Roughening

Fraction of active core roughened, effective %.........Roughening height* .....................................Roughening width*......................................Roughening pitch*......................................

* Designates values that have changed.

75.00.13 mm0.45 m1.56 mm

10

Table II. Operating Conditions for the Peak-power Rod in the 300-MW(e)GCFR Demonstration Plant Design

Parameter

Peak Power

Peak Fast Flux

Inlet Coolant Temperature

Outlet Coolant Temperature

System Pressure

Peak Burnup

Irradiation Time

Value

36.1 kW/m (11 kW/ft)

3.45 x 1015 n/cm2-s, 1.95 x 1015 n/cm2 s

353*C (667*F)

538*C (1000 F)

9.0 MPa (1280 psi)

100,000 MWD/MT

750 FPD

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3. the incorporation of a time-, temperature-, and pressure-dependent

fuel crack-healing model developed by Lovejoy and Evans;5 and

4. an update of the stress-free swelling correlation for the 20% cold-worked 316 stainless steel cladding.6

After these modifications were made, LIFE-GCFR was used to perform analysisof the irradiation experiment GB-9.7 The code underpredicted the fuel-pelletdiametral expansion obtained from postirradiation examination (transversemetallography). Examination of the computed fuel inelastic-strain components

reveals that the swelling contribution from solid fission products in the

present fuel-swelling model in LIFE-GCFR may be too low. Based on a recentreview conducted by the General Electric Company,8 the solid fission-product

swelling rate has been increased to 0.8% AV/V per at. % burnup, and the codenow predicts the fuel-pellet diametral expansion in reasonable agreement with

the GB-9 data.

B. Thermoelastic Stresses Due to Radial Temperature Gradient

Although LIFE-GCFR can be used to compute the cladding thermoelasticstresses due to a radial temperature gradient, it is generally expedient to

obtain these stresses directly from classical stress-analysis techniques. Thisis true when none of the cladding inelastic deformation such as creep and swel-ling is considered. As discussed in the following sections, an attempt has

been made to distinguish the effects of the various secondary-loading mechanisms

so that the contribution from each individual secondary-loading mechanism can beassessed. Based on the assumptions of axisymmetry and generalized plane strain,

the three principal thermoelastic stress components, orr' and a0 0 , and 0 zz, due

to a radial temperature distribution, T(r), in a hollow cylinder wall with

inner and outer radii ri and ro, respectively, are given by9

Ea 1 r E Cl C2 )Ez (2a)(r)- rT(r)dr + - -- + IEcz 2 12rE -~1 (arr 1-v r2r 1 + v 1 - 2v r 1 - 2v

Ea 1 r Eaa6(r) = IrT(r)dr - T(r) +

0 1-v r r 1-v

E C C2 ve+ --+, (2b)

l+v 1- 2v r2 1- 2v

and

a (r) =- - T(r) + E - 2vC1 + (1 - v)E , (2c)zz 1 - v (l + v) (1 - 20)

where E, v, and a are the Young's modulus, Poisson's ratio, and thermal-expan-sion coefficient of the cladding, ez is the axial strain component, and C1 and

12

C2 are constants to be determined from the radial boundary conditions. For in-

ternal and external pressure loadings of Pi and Po, the radial boundary condi-

tions are

arr (r)=-P. ; 0rr(r ) = - P. (3)

Substituting Eq. 3 into Eq. 2a, we can express C1 and C2 in terms of the

other parameters, and only cz is left to be determined from the axial boundarycondition. For an open-ended tube (which simulates the GCFR vented fuel clad-

ding), the axial boundary condition becomes

fadA = 0, (4)

where the integration is performed over the cladding area upon which azz acts.

For a constant cladding thermal conductivity kc, the temperature distri-

bution, T(r), across the cladding thickness in Eq. 2 can be written explicitly

as

q1 r0

T(r) = T + In -, (5)2Trk r

c

where Ts is the cladding outer-surface temperature and q' is the fuel-elementlinear heat rating. When the cladding thermal conductivity is linearly tem-perature-dependent, the temperature distribution across the cladding thicknesscan be written as

(q ' /Tr) n(r0/r)

T(r) = T + -/, (6)A1 + A2 Ts + [(Ai + A2 T) + (A2 q'/J)ln(r 0/r)]1/2

where A1 and A2 are the constants in the temperature-dependent cladding thermal-

conductivity relation.2

With representative values of Ts and q' for typical GCFR operating condi-tions, the temperatures at the cladding inner surface, T(ri), calculated Ly Eqs.5 and 6 differ by only a few degrees. (See Section IV. B, Table IV.) Forpractical purposes, Eq. 5 is used in evaluating the integrals in Eq. 2; thisconsiderably simplifies the mathematics.

Another simplifying procedure was tnken regarding the material propertyvalues a, E, and v. Since these properties are only mildly temperature-depen-dent and since the temperature drop across the cladding thickness is %50C orless under typical GCFR operating conditions, they have been approximated byconstant values obtained at an appropriate average cladding temperature.

A computer program has been written to calculate the three principalthermoelastic stress components across the cladding thickness. Once thesestress distributions are obtained, the distributions of the equivalent stress,ae, and the hydrostatic stress, aH, can be found from

13

a l=- [( -a0 0 )2 + (a 0 0 -a )2 + (a -a%) 2 /2 , (7)

and

a = -(a +Ca +i Z). (8)3 rr 00 zz

Stresses ae and aH are important in considering the cladding plasticity and

creep and the stress effect on irradiation-induced void swelling.

C. Thermoelastic Stresses due to Axial Temperature Gradient

The coolant-cladding heat-transfer coefficient is about a factor of twolarger in the ribbed portion of the GCFR fuel element as compared to the co-

efficient in the smooth portion. This difference can induce an axial tem-

perature gradient in the smooth-to-ribbed transition region as steep as theradial temperature gradient. The magnitude of the axial temperature gradientdepends on the geometry of the transition region, the effective fluid-dynamiclength for increasing turbulence, and axial heat transfer.

Similar to the thermoelastic stresses induced by the radial temperaturegradient, the axial temperature gradient also induces thermal stresses in thecladding. However, this problem cannot be examined by the one-dimensionalstress-analysis method described earlier (including LIFE-GCFR) because thatmethod applies only to a radial temperature distribution in a cylindricalgeometry. Classical thin-shell thermoelastic analysis can he applied to theaxial-gradient GCFR problem with reasonable accuracy.1 0 For a cylindricalshell of mean radius R, the displacement equation may be written as

4d u + 4u = 4RaT,

(9)dz*4

where a and T have their usual meaning, u is the radial displacement of a

point on the middle surface of the shell (positive outward), and z* is the ratioof the axial coordinate of the shell, z, to the characteristic length, Q. Thesequantities are

z* z/

and

_R2 eff ,(10)

3(1 -v2)

where v is the Poisson's ratio and teff is used to represent the shell thickness.The radial, circumferential and axial strains (Er, e0,s E) and stresses(arr, a0 0 , azz) are defined by the strain-displacement and constitutive equa-tions as

2Er = - v + 1 d u(R - r) + (1 + v) aT,

R + ( dz*2

e = -,

R

E _

Ea =rr 2

1 - v

Ea0 0 - 21 - v

1 +v

[(s - aT) + v(e - aT)],r -

[ - caT) + v(E - aT)], (12)

a =0.zz

where r is measured from the center of the fuel rod (positive outward).

The homogeneous solution, uh, to Eq. 9 can be written as10

uh= C1 ccs z* sinh z* + C 2 cos z* cosh z*

+ C3 sin z* cosh z* + C4 sin z* sinh z*, (13)

where C1, C2 , C3 , and C 4 are constants determined by the four boundary condi-tions required for a unique solution. The algebra involved in determiningthese constants can become quite complicated, depending on the complexity ofthe boundary conditions.

The ease with which a particular solution, up, can be obtained for Eq. 9depends on the functional form of T = T(z*). For a simple functional form(e.g., polynomial, sinusoidal, or exponential), the determination of up isrelatively straightforward. Since the exact temperature variation in thecladding smooth-to-ribbed transition region is not known and the axial gradientoutside of this region is relatively insignificant, the following sinusoidaltemperature distribution is chosen for analytica, convenience:

T z* < - -Is 2

T -)T + [1 - sinnz*R/6); - <ir +2 22

LT * >- -- '

(14)

14

and

(11)

15

where 6 is the length of the transition region, ATz is the axial temperature

drop in the transition region, Ts is the assumed constant temperature in the

smooth region, and Tr is the assumed constant temperature in the ribbed region(Ts = Tr + ATz). The particular solutions to Eq. 9 for this temperature distri-

bution are

6u =RRaT z* < -

ps s 2k

u = R a T + 1 - sin ( -z*/) _- < z < - 15)PC ~ 2 1 + -(7r9/6)

4 ~ 2

u = R a T z * > .pr r 2

The complete solution to Eq. 9 is found by summing the particular solution(Eq. 15) and the homogeneous solution (Eq. 13). Because the particular solu-

tion is of different functional form for each of the three regions, the completesolution must be written for each region as

6u + u z* < - -

rhs ps26 6

u u + u -- < z < -ht +u 2? - - 2Q (16)

6su + up z* >-

- hr pr2

Each of the homogeneous solutions uhs, uht, and uhr contains four undeter-mined constants, bringing the total number to 12. These are found from theboundary conditions at the ends of the cladding and matching conditions atz* = 6/2Q.

Because the transition-region length 6 is small compared to the claddinglength L, the choice of boundary conditions is somewhat arbitrary as long asthe cladding is free to expand axially. The problem simplifies considerablyif we choose free-end boundary conditions and let the ends b defined byz* = L/2Q; thiin ensures

2d u2 _ du = 0 at z* = L/2Q. (17)

dz* dz*

The other eight relationships are found by matching the displacements andthe first three derivatives of the displacements (i.e., slope, moment, andshear) at z* = 6/L. Notice that the temperature distribution T (Ts + Tr)/2is chosen to be an antisymmetric function of z*. This choice, along with theselection of simple boundary conditions at L/29,, results in a displacementsolution u - Ra(Ts + Tr)/2 and a hoop-stress solution that are also antisym-metric functions of z*. These choices effectively reduce the difficulty of theproblem from solving 12 equations in 12 unknowns to solving 4 equations in 4unknowns.

16

A detailed treatment of the GCFR cladding thermal-stress problems inducedby an axial temperature gradient is given in the appendix. Three cases havebeen considered, and the results are discussed in Section IV.C with emphasis on

the limiting case with a step-function temperature rise in the transition region.

D. Stresses Due to Swelling Gradient

Irradiation-induced void swelling in Type 316 stainless steel has been asubject under study over the past 10 years. A significant amount of data nowexists on a variety of sample steels irradiated at various temperatures and tovarious fast-neutron fluence levels, but essentially under stress-free condi-tions. An empirical correlation for stress-free swelling of the 20% cold-worked Type 316 stainless steel was developed by Bates6 and has the functionalform

E = R(T)f$t + 1In[1 + exp[a(T - , (18)a 1 + exp(aT )

where E is the fractional density change before and after neutron irradiation, $tis the neutron fluence (E > 0.1 Mev), a is a parameter governing the curvature in

the transition region between zero swelling and measurable swelling, T is theincubation fluence below which no appreciable swelling occurs, and R(T) is theswelling rate given by

R(T) = exp(C1 + C2P + C32 4 + C4 + C54), (19)

where s = (T - 500)/100 and C1 - C5 are the empirical constants.

The fractional stress-free volumetric swelling, AV/V0, is related to E by

V ._ (20)

V 1 - E

0

Following the usual assumption that the stress-free swelling is isotropic,

the cladding swelling strain components, eij, in the three principal directionscan be obtained directly from Eq. 20 as

s _ 1 AV (21)eij 3 V0'

Since the cladding stress-free swelling is a function of fluence, whichvaries with neutron exposure, the swelling strains are typically treated infuel-performance modeling codes as incremental quantities and the total amountof swelling is accumulated over each calculational time step. Notice that thefluence and the temperature dependence of stress-free swelling in Eq. 18 areexpressed as decoupled terms. This means that, at a given fluence, the stress-free swelling depends only on Lemperature and the swelling strains may betreated analogously to the thermal-expansion strains, since both terms areassumed to be isotropic. This temperature dependence of the stress-free swel-ling [R(T) in Eq. 18] gives rise to the swelling-induced stresses in the cladding.

17

Irradiation-induced swelling can be affected by the presence of hydrostatic

stress, aH, in the material. An equation for the stress-affected swelling hasbeen derived from very limited data by Batesll and is given by

S = S (1 + PaH), (22)

S H

where S and So are the stress-affected and the stress-free swelling rates

[So = R(T) in Eq. 19], respectively, and P is an empirical constant. Equation22 has not been used in LIFE-GCFR calculations in this investigation because

the data supporting it are presently too scarce. In Section IV.B, results ob-

tained from several illustrative calculations using Eqs. 19 and 22 are presented

to show that the cladding swelling gradient may be modified substantially whenthe stress effect on swelling is taken into account.

E. Cladding Creep Analysis

In LIFE-GCFR, cladding creep analysis is handled using a successive elastic-approximation scheme first developed by Mendelson et al.12 Semiempirical rateequations have been derived from test data for both thermal and irradiation

creep of 20% CW-316SS. These rate equations are functions of the cladding equiv-

alent stress, temperature, and neutron flux. For a given time interval, the

values of these parameters at the end of the previous time step are used as

starting values in the rate equations to calculate the current creep-strainincrements. Successive refinements of the stress-strain calculation are made

through iteration until a specified convergence criterion is satisfied.

Rather than redeveloping the full algorithm for the cladding elastic andinelastic stress analysis, we will use LIFE-GCFR to investigate the separate

effects of swelling and creep on cladding stresses. By simple alterations inthe relevant models, these mechanisms can be either activated or suppressed.For any given GCFR operating conditions, analyses of the following four casesare sufficient to isolate the effects from the individual mechanism:

(1) with both creep and swelling operating,(2) with creep operating and swelling suppressed,

(3) with creep suppressed and swelling operating, and(4) with both creep and swelling suppressed.

In the last case, with both creep and swelling suppressed, one would ex-

pect that under the same thermal conditions, the cladding stresses calculatedby LIFE-GCFR should follow closely those calculated by the thermoelasticformulation given in Section III.B.

IV. RESULTS

A. LIFE-GCFR Analysis of Fuel-Cladding Mechanical Interaction

To determine if any significant loads are imposed on the cladding due tofuel-cladding mechanical interaction, LIFE-GCFR was used to analyze the refer-ence GCFR Demonstration Plant fuel rod under steady-state, step-power changes,and power-cycling operation. Since these cases have been studied3 with anearlier version of LIFE-GCFR, the current and the previous predictions have

18

been compared. These comparisons are shown in Figs. 1-3 for the variation ofthe fuel-cladding radial gap as a function of burnup. The dashed lines onthese figures are the previous predictions,3 whereas the solid lines are thosemade by the current version of LIFE-GCFR. A number of observations as well asour interpretation f 'these figures are given below.

First notice that, in each figure, the initial hot radial gaps remainalmost identical for the corresponding cases (dashed lines vs. solid lines).These results are expected because the initial fuel-rod response under thegiven reactor startup condition is essentially thermoelastic with negligibletime-dependent swelling and creep deformation. Since the recent modificationsto LIFE-GCFR are a 1 related to the time-dependent processes of the fuel andcladding, these modifications should not affect the size of the initial hotradial gap.

The gap size generally decreases with burnup due to fuel swelling untilcladding swelling is initiated. The inflection points on the curves of gapsize versus burnup can be identified with the incubation fluence for claddingstress-free swelling. This can be verified by using the previous and thecurrent values of T (4.0 x 1022 and 7.0 x 1 0 22 n/cm 2 ., E > 0.1 MeV, respectively)and dividing them by the corresponding local neutron-flux values.

The trends observed in Fig. 2 for the steady-state cases at 60, 100, and115% power can be explained by the temperature dependence in the cladding stress-free swelling. Since this temperature dependence is contained exclusively inR(T) in Eq. 18, cladding swelling behavior can be compared by examining thecurve of R(T) versus temperature. Figure 4 shows the temperature dependence ofR(T) for 20% CW - 316 SS between 350 and 700*C. Superimposed on this curve arethe calculated cladding inner- and outer-surface temperatures for the threesteady-state power cases. Under steady-state operations, these cladding tem-peratures stay fairly constant over the entire irradiation period, with a sl eightreduction toward the end of life. Evidently, the swelling rate is lowerin the 60% power case than in both the 100 and 115% power cases. This lowerswelling rate and the lower fast neutron flux in the 60% power case explainswhy it is the only case with gap closure during irradiation.

In Fig. 3, the gap size is smaller in the power-cycling case than in the100% power case. This can be attributed to power-cycling induced fuel crackingand relocation. Physically, fuel cracking and relocation can occur during eachstartup-shutdown cycle due to cycle-induced thermal stresses. In LIFE-GCFR, 2fuel cracking and relocation phenomena are modeled by a reduced-modulus approachin which the Young's modulus and the Poisson's ratio of fuel are reduced whencracking occurs. The effect of these reductions is that the radial displace-ment at the fuel surface becomes larger than that of an untracked fuel as aresult of the increased fuel compliance.

Displacements at fuel surface as well as at cladding inner surface havebeen compared between the steady-state and the power-cycling cases. No notice-able difference has been observed regarding the displacements at the claddinginner surface, which suggests that fuel cracking and relocation alone accountfor the reduction of gap size in the power-cycling case.

Table III summarizes the variations of fuel-cladding interfacial pressureduring the period of fuel-cladding mechanical interaction for the 60% power,

Table III. Fuel-Cladding Interfacial Pressure and Cladding Stresses During

Fuel-Cladding Mechanical Interaction (LIFE-GCFR Analysis)

Gap ClosurePeriod (hr.)

From To

FCMIAxial Location

X/L from Top

Interfacial PressureDuring FCMI, MPa(ksi)

Min. Max.

MaximumCladding Uae,MPa(ksi)

I.D. O.D.

Maximum

Cladding azzMPa(ksi)

I.D. O.D.

60% 16,340 18,000Power

StepPower-

Change(from 60% to115% at13,000 hrs.)

13,01C 13,520

1/8

1/8

9.38(1.36)

8.83(1.28)

15.87(2.30)

36.57 34.57 11.87 14.35(5.30) (5.01) (1.72) (2.08)

34.09 132.62 168.64(4.94) (19.22) (24.44)

72.66 128.0(10.53) (18.55)

7,770 11,080 3/8 8090(1.29)

11.52 -27.39 14.70 -30.08 9.80(1.67) (-3.97) (2.13) (-4.36) (1.42)

Case

Power-Cycling

20

power-cycling, and step power-change cases. These are the only three cases inthis investigation for which LIFE-GCFR predicted gap closure during irradiation.The last case was obtained by a step increase in power Crom the previous steady-state power level of 60% at 13,000 hours to 115%. The 115% power was held for10 hours before it was reduced to 100% for the remainder of the irradiationtime.

Comparatively, the step power change gives the largest fuel-cladding inter-facial pressure of 34.1 MPa (4.9 ksi) during the 10-hour hold at 115% power.However, this interfacial pressure is completely relieved when the gap reopensafter about 500 hours following the reduction of power to 100%. In the 60%-

power case, the gap closes late in life at "16,340 hours and stays closed untilthe end of life (Fig. 2). As discussed previously, this behavior is essentiallydue to the small amount of cladding swelling at the lower cladding temperatures.The interfacial pressure builds up slowly with time and reaches a maximum valueof 15.9 MPa (2.3 ksi) at the end of life.

In the power-cycling case, the gap first closes at "'7770 hours, but thenreopens at "11,080 hours. The buildup of interfacial pressure during the gapclosure period is not nearly as large as in the 60%-power case, and the inter-facial pressure decreases after a maximum value of 11.5 MPa (1.7 ksi) isreached at 8300 hours. Again, this behavior can be explained by the power-cycling-induced fuel cracking and reloca-ion during the gap-closure period.(There are about 13 startup and shutdown cycles during that period.) The re-duced fuel modulus due to cracking makes fuel easier to deform in response to

the interfacial pressure than an untracked fuel, hence preventing the pressure

buildup.

B. Thermoelastic Stresses Due to Radial Temperature Gradient

Before presenting results on the thermoelastic stresses generated by a

radial temperature gradient in the cladding, we should examine the calculated

cladding temperature drop (difference between the cladding inner- and outer-surlace temperatures) as a function of fuel-element linear heat rating q' and

the cladding outer-surface temperature Ts.

Table IV gives the cladding temperature drops calculated by Eqs. 5 and 6over a range of representative GCFR operating conditions. Notice that thereare practically negligible differences between the temperature drops (andtherefore the temperatures) calculated by these two equations. Also noticethat, for a given Ts, the change in temperature drop with varying q' is approx-imately linear, from 0.9 C/kW/nm at Ts = 500*C to 0.8C/kW/m at Ts = 700*C.These simple relationships are convenient when cladding temperatures need to beestimated from given values of q' and Ts.

The knowledge of the temperatures at the cladding inner and outer surfacesallows the cladding temperature gradient to be assessed. These temperaturesalso provide a ready indication about the nature of the cladding swellinggradient. The latter concern arises from the possible stress reversal in thecladding late in life when the cladding temperatures swing back and forth acrossthe peak swelling temperature of the cladding (Fig. 4). The possibility of thiscladding-stress reversal behavior is discussed separately in Section IV. D.

Table IV. Calculated Cladding Temperature

Fuel-element Linear Heat Rating

Outer-surface Temperature Ts.

Drop As A Function of

q' and the Cladding

q' kW/m 16.4 19.7 23.0 26.2 29.5 32.8 36.1 39.3 42.6 46.0

(kW/ft) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14)

TS ( C)

Eq.5 15 18 21 24 27 30 33 36 39 42

500 _ _ _ _ __ _ _ _ ____ _ _ _ _

Eq. 6 14.9 17.9 20.9 23.9 26.8 29.8 32.7 35.7 38.6 41.5

Eq. 5 14.6 17.5 20.4 23.3 26.2 29.1 32.0 35.0 37.9 40.8

550

Eq. 6 14.5 17.4 20.2 23.1 2b.0 28.8 31.7 34.6 37.4 40.2

Eq. 5 1l.1 16.9 19.8 22.6 25.4 28.2 31.0 33.9 36.7 39.5600

Eq. 6 14.0 -16.8 19.6 22.4 25.2 27.9 30.7 33.5 36.3 i9.0

Eq. 5 13.7 16.4 19.2 21.9 24.6 27.4 30.1 32.8 35.6 38.3

650

Eq. 6 13.6 16.3 19.0 21.7 24.4 27.1 29.8 32.5 35.2 37.9

Eq. 5 13.3 15.9 18.6 21.3 23.9 26.6 29.2 31.9 34.5 37.2

700

Eq. 6 13.2 15.8 18.5 21.1 23.7 26.4 28.9 31.6 34.2 36.8

22

Figures 5, 6, and 7 show the stress distributions of aeG, arr, ozz, ae,and oH across the cladding thickness for q' equal to 16.4, 36.1, and 45.9 kW/m(5, 11, and 14 kW/ft), respectively. Both the internal and external pressureswere set equal to 9.0 MPa (1280 psi) in these calculations, with Ts = 600*Cand with teff representing the cladding thickness. The general trend in thesestress distributions is such that aee, zz, and aH are all compressive near thecladding inner surface and tensile near the cladding outer surface. The tran-sition from a compressive stress state to a tensile stress state occurs approxi-mately at the midpoint of the cladding thickness. As the radial temperature

drop increases (i.e., as q' increases for constant Ts), the magnitude of all

the stress components also becomes larger.

From the distributions of the cladding temperatures and cH, several illus-trative calculations have been performed on the stress-free and stress-affected

swelling using Eqs. 18 and 22. Table V lists the results of So, and S, and thecorresponding AV/Vo [at 2.24 x 1023 n/cm2 (E > 0.1 MeV)] for three given valuesof q'. In all three cases, the stre. -free swelling is seen to be larger atthe cladding outer surface than at the inner surface, and the swelling gradient

becomes larger as q' increases. This can be explained by the temperature de-pendence of R(T) shown in Fig. 4, which has a peak at 580*C. Since Ts is equalto 600*C in these cases, the cladding temperatures are therefore all above thepeak swelling temperature. Furthermore, since the cladding inner-surface tem-

peratures are further away from 580*C than the cladding outer-surface tempera-tures in these cases, the stress-free swelling at the cladding outer surface

should be larger.

The effect of hydrostatic stress on swelling is that swelling is enhancedin hydrostatic tension and suppressed in hydrostatic compression. From thenature of the hydrostatic-stress distributions shown in Figs. 5-7, it is ap-

parent that the cladding swelling gradient will be enhanced. However, thestress-affected swelling, (AV/Vo)SA, in Table V has been calculated withouttaking into account cladding stress relaxation due to creep. Consequently,(AV/Vo)SA tens to be overestimated. The calculated stress-free swelling,

(AV/Vo)SF, on the other hand, is not subjected to the influence of creep re-laxation. Even in the latter case, appreciable cladding swelling gradient canbe developed during irradiation. The swelling-induced stresses in the claddingare assessed in Section IV. D.

The above results on the cladding thermoplastic stresses and swelling

behavior were obtained using teff for the cladding thickness. Calculationshave also been performed for two cases in which cladding thicknesses at theroot of the rib (thin cladding) and at the top of the rib (thick cladding) wereused. The resulting stress distributions are shown in Figs. 8 and 9, respec-tively, for the thin and the thick cladding. In both cases, Ts and q' were setequal to 600*C and 36.1 kW/m (11 kW/ft). By comparing the stresses in Figs. 8and 9 to those in Fig. 6 (obtained at the same Ts and q' but with teff), we seethat the stresses are higher in the thick cladding (Fig. 9) and lower in thethin cladding (Fig. 8) than those calculated using teff (Fig. 6). Since thethermal stresses depend on the magnitude of the temperature gradient, theseresults can be readily understood by noting that for constant TS, q , and the

cladding inner radius ri, the cladding temperature drop increases with in-creasing cladding thickness. Both stress-free and stress-affected swellingscan also be expected to follow a similar trend in that the thick cladding willhave a larger swelling gradient than in the thin cladding.

Table V. Cladding Stress-free and Stress-affected Swelling

Calculated for TS = 600*C and Three Values of q'

. * 22 2*S, S are in units of % per 10 n/cm

N

q'

kW/m 16.4 (5.0) 36.1 (11.0) 45.9 (14.0)(kW/ft)

Cladding* S(A/ S S S (AV/Vo SF o)SA So SoSF ( So SA

Inner 1.91 1.18 29.36 18.17 1.09 0.21 16.71 3.29 0.72 -0.02 10.98 -0.38Surface 1.98 1.35 30.39 20.75 1.25 0.44 19.15 6.74 0.89 0.15 13.68 2.35

2.04 1.52 31.36 23.38 1.41 0.71 21.62 10.85 1.08 0.40 16.59 6.14

2.10 1.70 32.29 26.05 1.57 1.01 24.06 15.54 1.28 0.72 19.62 10.98

2.16 1.87 33.17 28.73 1.72 1.35 26.44 20.74 1.48 1.10 22.69 16.82

2.21 2.05 33.99 31.42 1.87 1.72 28.70 26.34 1.67 1.53 25.70 23.50

2.27 2.22 34.77 34.10 2.01 2.10 30.82 32.24 1.86 2.01 28.57 30.86

2.31 2.40 35.49 36.77 2.13 2.50 32.77 38.32 2.03 2.52 31.23 38.67

2.36 2.57 36.16 39.41 2.25 2.90 34.51 44.45 2.19 3.04 33.60 46.68

2.40 2.74 36.77 42.02 2.35 3.29 36.04 50.53 2.32 3.56 35.64 54.66

Outer 2.43 2.90 37.34 44.58 2.43 3.68 37.34 56.44 2.43 4.06 37.34 62.38

Surface

24

Finally, a case was studied in which the internal and external pressureshave been set equal to zero. The cladding stresses obtained for this case arethus caused only by the temperature gradient. When these results are comparedto the previous ones (with both temperature gradient and pressures), the rela-tive contribution from temperaturE gradient and pressures on the overall clad-ding stresses can be assessed. Another way of separating the effect of thetemperature gradient from the pressure loadings is to calculate the elasticstresses due to pressure loadings by classical Lame's equations. The elasticstresses can then be subtracted from the overall stresses to obtain the stressescaused by the temperature gradient alone. The results should be the same.

A comparison of stresses in Fig. 10 obtained with zero pressures to thosein Fig. 6 shows that the temperature gradient accounts for a large part of theoverall cladding stresses. Note also that aH becomes less tensile near thecladding outer surface and more compressive near the cladding inner surfacewith equalized pressures being applied than the corresponding aH1 with only thetemperature gradient. This may be beneficial in the sense that both the amountof stress-affected swelling and the swelling gradient will be reduced thehigher the level of the equalized pressures.

C. Thermoelastic Stresses Due to Axial Temperature Gradient

From the formulations derived in Section III. C, the thermoelastic stressesdue to an axial temperature gradient can be calculated. This calculation israther tedious and depends on input from other sources (such as LIFE-GCFR) forthe axial temperature gradient. Without going over the additional mathematicalsteps (given in detail in the appendix), we summarize only the results in thissection.

The three cases considered in the appendix a-e shown schematically in Fig.

11. Case I represents the limiting case with the length of the smooth-to-ribbed

transition region, 6, approaching zero. This giver an upper-bound solution of

the hoop stress ce. Case II gives an approximate :olution of aGe and is onlyof academic interest because unrealistic free-end boL idary conditions have been

assumed at the ends of the transition region. The stro3 results for this case

would not be accurate, but may be used in studying the effect of the transitionlength. Case III represents a realistic solution of a0 6 for the reference GCFR

fuel-element design, and any finite 6 can be incorporated into the analysis.

For the limiting case with 6 approaching zero, the maximum tensile hoopstress occurs in the ribbed portion of the cladding and is simply given by

(aGO)max = EcATz/2. With the values ATz = 70 C, E = 1.56 x 105 MPa (2.26 x 104ksi), and a = 1.84 x 10-5/ C, the maximum tensile hoop stress is equal to 100.7MPa (14.6 ksi).

The results obtained for Case IT are given in Tables I, II, and III inthe append .x. These tables show the variation of the normalized hoop stress,ae0 /(E AT,./2), with , where $ is the nondimensional half-length of the transi-tion region, 6/29. The normalized hoop stress decreases rapidly as $ increases,and for 6 > 7T, the hoop stress becomes insignificant.

For Case III, the results are presented in Table IV in the appendix. Thesolution gives a maximum tensile hoop stress in the region 0 < z* < 6/2Q and amaximum compressive hoop stress in the region -6/2Q < z* < 0. For example, if

25

= 6/2Q = T/2, the maximum normalized tensile hoop stress occurs at z* = 0.8

and has a value of 0.2 (which corresponds to an actual tensile hoop stress of

20.1 MPa).

D. LIFE-GCFR Assessment of Swelling and Creep Effects on Cladding Stresses

As mentioned in Section III. D, the effects of creep and swelling on clad-ding stresses can be separated by simple alterations of the relevant models in

LIFE-GCFR. In practice, these alterations have been achieved by multiplyingthe increments of creep strain and/or swelling strain with a small number and

thus effectively suppressing either of the two components. The existing models2

in LIFE-GCFR for cladding (20% CW-316 SS) thermal and irradiation creep and theupdated stress-free swelling correlation (Eq. 18) were used in the analysis.

For comparison, all analyse] were performed under the following conditions:

115% power, $m (peak neutron flux) = 1.95 x 1015 n/cm2s, and irradiation time

= 18,000 hours. The results presented below were all taken at the same axial

position on the fuel element, X/L = 3/8 from the coolant inlet end.

Figure 12 shows the cladding ae0 and azz as a function of irradiation time

with both creep and swelling operating. During reactor startup, the cladding

stresses are essentially thermoelastic and both a0 e and azz are tensile near

the cladding outer surface and compressive near the cladding inner surface.These agree with the results obtained from the thermoelastic stress analysis

shown in Figs. 5-7. The reduction in magnitude of both a 60 and azz during the

first 6000 hours can be attributed to creep relaxation. No swelling effect is

expected during this period because the neutron fluence is still below the in-

cubation fluence for the onset of cladding stress-free swelling.

Ac about 7200 hours, the fuel-cladding gap closes and the interfacialpressure from fuel-cladding mechanical interaction starts to build up. The

increase in interfacial pressure loading on the cladding during the gap closure

period accounts for the rising of a0 e and azz at the cladding outer surface.

This can be attributed to the bending action associated with the interfacial

pressure which produces maximum tensile stress at the cladding outer surface.During the final stage of the gap closure period, the stresses are also affected

by cladding swelling because the incubation fluence for the onset of swellingat the given m is reached at 9000 hours.

The swelling effect on the stresses shows up more clearly after the gapreopens at 9200 hours. If there is no swelling, aee and QzZ should reduce in

magnitude due to creep relaxation after the gap reopens. Instead, both a66and QzZ at cladding inner and outer surfaces continue to increase in magnitude,

even upon removal of the interfacial pressure. These stresses finally leveloff at 11,000 hours and stay fairly constant for another 7000 hours until theend of life.

A large part of the overall cladding stresses late in life is induced bythe cladding swelling gradient. This can be seen in Figs. 13 and 14, whichshow the stress-time histories for ae and azz with only creep operating. Theportions of the curves up to 7200 hours are identical to those shown in Fig. 12;this can be expected since there is no contribution from cladding swellingbelow the incubation fluence. After the gap closes at 7200 hours, however, it

26

remains closed until the end of life because swelling has been suppressed,. In

this case, with the existence of interfacial pressure throughout the prolonged

gap-closure period, the tensile Gee and azz at the cladding outer surface are

still smaller than the corresponding stresses shown in Fig. 12 with a reopenedgap late in life and swelling operating.

The reason Gee and ozz are tensile at the cladding outer surface and com-

pressive at the cladding inner surface after swelling becomes significant canbe explained by the nature of the cladding swelling gradient. Under the givenoperating conditions, the calculated cladding temperatures are below the peakswelling temperature shown in Fig. 4. This results in a cladding swellingeradient that follows the cladding temperature, gradient, with more swellingoccurring near the cladding inner surface than near the outer surface. Since

the cladding-swelling strains may be treated analogously to the thermal-expan-sion strains, such a swelling gradient can be expected to induce a similarstress distribution to that due 4o the radial temperature gradient.

In a later case, the fuel-element operating conditions have been adjusted

so that the cladding temperatures are above the peak swelling temperature.This gives a swelling gradient opposite to the temperature gradient. A stress

reversal can be induced in which aee and azz become tensile at the claddinginner -urface and compressive at the outer surface, in direct contrast to the

trench served in Fig. 12.

Figure 15 shows the stress-time history of aee and Gzz with only swellingoperating. A small number has been used as a multiplier to artificially sup-press both thermal and irradiation creep. The LIFE-GCFR analysis was success-ful up to the first few hundred hours of irradiation time until the executionwas terminated due to excessive running time. The problem occurred because of

the very small creep-strain increments computed for a given time step. Thisnecessitates an excessive number of small time steps to he taken before the

specified limit on the machine computing time is exceeded. However, sincecladding creep has been suppressed, the initial thermoelastic stresses may beextrapolated to the point when gap closure occurs. After that, the stresseswill increase in magnitude initially from the buildup of interfacial pressureand later from the cladding-swelling gradient.

The results from LIFE-GCR analysis with both creep and swelling suppressedare similar to those shown in Fig. 15, except for the swelling contribution. Asimilar problem of early run termination was also encountered because of thevery small creep-strain increments computed for each time step.

The stress reversal mentioned earlier can be seen by comparing Figs. 12and 16. The latter figure was obtained by choosing the cladding outer surfacetemperature at 590*C such that the entire cladding temperature range exceedsthe peak swelling temperature. This produces a swelling gradient opposite tothe temperature gradient, and notice the drastic change in the cladding stresses.However, the magnitude of these stresses reduces continuously late in life fromcladding creep relaxation, which is not seen in the previous case (Fig. 12).

27

V. SUMMARY AND DISCUSSION

The highlights from LIFE-GCFR analyses of fuel-cladding mechanical inter-

action under steady-state, step power-change and power-cycling operation can be

summarized as follows: Under the given conditions in steady-state operation upto 18,000 hours, the fuel-cladding gap stays open throughout the entire irradia-

tion time in both the 100%- and 115%-power cases, resulting in no fuel-claddingmechanical interaction. In the 60%-power case, the gap closes at '16,500 hours,

mainly due to the low cladding temperatures which cause relatively low cladding

swelling.

The 60% power, step power change, and power cycling are the only threecases in which gap closure is predicted during irradiation. The buildup offuel-cladding interfacial pressure during the gap-closure period reaches the

highest value of 34.1 MPa (4.9 ksi) in the step power-change case, but thispressure is completely relieved when The gap reopens after "500 hours following

the reduction of power to the nomiui cull-power level. Interfacial pressurebuildup over the extended gap-closure period in power-cycling operation is theleast [maximum of 11.5 MPa (1.7 ksi)] of the three cases. Fuel cracking andrelocation associated with each startup and shutdown cycle are believed to be

the main causes responsible for this behavior. From the viewpoint of cladding

failure, a more cautious approach is probably required regarding any maneuvering

that can cause local reactivity changes, e.g., control-rod movements and fuelshuffling, because these can induce a rather sharp power change in the nearby

fuel assemblies. Similar problems have been encountered in Light Water Reactors,and the preconditioning treatment may be necessary in preventing fuel-element

failures from power ramps.

LIFE-GCFR analysis of fuel-cladding mechanical interaction does not includetwo aspects which are of great importance to fuel-element failure analysis. Oneaspect is related to cladding ridging formed preferentially at the fuel-pelletends. The other aspect is related to pellet cracking, which can cause concen-trated load on the cladding in the vicinity of the fuel cracks. Both of theseare highly localized phenomena and cannot be modeled adequately by LIFE-GCFR.Finite-element techniques may be used to analyze these problems.

The results obtained in this investigation regarding cladding swelling arebased on the current empirical correlation for the 20% cold-worked 316 stainlesssteel. The data base, the theoretical foundation, and the data-analysis pro-cedure for that empirical correlation have been examined recently. 13 The un-certainty factor of the empirical correlation was unusually high (up to 400%),suggesting a need to reevaluate the current practice in developing the empiricalstress-free swelling correlation.

In Section IV. B, illustrative examples were given to show that the clad-ding-swelling behavior can be modified substantially by the presense of hydro-static stress. Few data exist, however, to enable a systematic examination ofthe hydrostatic-stress effect on swelling. Bates and Gilbertl tested Type 316SS tubular specimens under different levels of internal pressures during irradi-ation and used an equation whose form is identical to Eq. 22 in correlating thedata. The important observation from the test results of Bates and Gilbert isthat a stress-enhancement effect on swelling is seen only in relatively high-fluence specimens (ft > 1.97 x 1022 n/cm2). Furthermore, this enhancement

28

effect is seen only when the hoop stresses are below the material elastic pro-portional limit. Data in the low stress range can be correlated acceptably byEq. 22 with a linear stress dependence. For hoop stresses higher than theelastic proportional limit, there is a decrease in swelling. Bates and Gilbertattribute this decrease to the rapid increase in dislocation density. Any ap-plication of Eq. 22 should therefore account for the possible decrease in swel-ling at high values of stresses.

The above discussion is appropriate for Bates-Gilbert experiments in whichthe Type 316 SS tubular specimens are under a tensile state of stress due tointernal pressurization. However, note that Eq. 22 has not been checked againstdata from experiments in which the stress state is compressive. Since a sub-stantial portion of the fuel-element cladding will experience hydrostatic com-pression (Figs. 5-7), additional experiments should be conducted to investigatethe effect of hydrostatic compression on swelling. Judging from the nature ofcladding-stress distributions during fuel-element irradiation and the signifi-cant stress effect on swelling observed to date, these types of experiments arehighly recommended.

The results in Section IV.D indicate that both thermal and irradiationcreep may be beneficial in relaxing cladding stresses. Swelling, on the otherhand, may cause problems because it generally increases the magnitude of thecladding secondary stresses. The subtlety in the significance of cladding swel-ling is that the resulting cladding-stress distributions are sensitive to whetherthe cladding temperatures are above or below the peak swelling temperature ofthe cladding material. Above the incubation fast fluence for cladding swelling,a fatigue-type problem may be encountered if the cladding temperatures swingback and forth across the peak swelling temperature during power-cycling operation.

Creep and swelling appear to affect cladding stresses in an oppositemanner, but their effects cannot be easily quantified.* In Fig. 12, aQe and azzlevel off late in life, indicating that a balance is reached between creep-stress relaxation and the swelling-induced stress changes. This is not thecase in Fig. 16, however, where creep relaxation tends to be overriding andthe cladding stresses continue to decrease toward the end of life. The reasonthat a66 and azz shown in Figs. 13 and 14 (without swelling) also stay fairlyconstant after gap closure late in life involves a different kind of balanceamong the fuel swelling and creep relaxations of both fuel and cladding. Thisis in contrast to the case shown in Fig. 12, where the balance is only betweencladding-creep relaxation and cladding swelling. The stresses obtained withswelling suppressed are also representative of the stresses in the low-swellingcladding materials such as the advanced alloys.

The formulation derived in Section III. C and the appendix may be adoptedto calculate the cladding thermal stresses caused by an axial temperaturegradient in the smooth-to-ribbed transition region. For the case with a 70Caxial temperature drop in the transition region, the maximum tensile hoop stressis less than 100.7 MPa (14.6 ksi). This hoop stress represents an additionalloading on the cladding when there is a sudden change of cladding profile inthe transition region. This additional loading can be greatly reduced, however,by a gradual change of cladding profile in the transition region. The resultsgiven in the appendix show that, for this specific case, if the length of thetransition region is on the order of 3 cm or more, the hoop stresses become in-significant.

29

Since one of the main purposes of this investigation is to provide inputto the experimental Cladding Development Program (CDP) at ANL, the relevant

items are discussed in the following:

Stress biaxiality, defined as the hoop-stress to the axial-stress ratio,has an important bearing on the creep-rupture life of specimens in biaxialcreep-rupture tests. The 1:1 stress biaxiality represents a more damagingtype loading (shorter creep-rupture life) on tubular specimens than the 2:1stress biaxiality at the same hoop-stress level.14 The 1:1 biaxiality hasoften been referred to as the cladding stress state during fuel-cladding mech-anical interaction. However, no strong evidence can be found to support sucha statement. The deduction of cladding local stress state during fuel-claddingmechanical interaction from postirradiation cladding length and diameter measure-ments is not adequate because of the integral effect along the fuel-elementlength (i.e., varying axial profiles of power and neutron flux, contact andnoncontact regions between the fuel and cladding, etc.). While the exact stress

state in the cladding during fuel-cladding mechanical interaction is not known,the multiaxial, multi-specimen test apparatus currently under construction atANL allows the variation of stress biaxiality through a combination of internalpressurization and axial load. This apparatus therefore may be used to bracketthe stress-biaxiality values resulting from fuel-cladding mechanical interation.

The stress-biaxiality values obtained in this investigation are summarizedin Table 6. This table clearly shows that even without fuel-cladding mechanicalinteraction, the cladding stress biaxiality obtained from the radial temperaturegradient and the equalized pressure loadings is close to 1:1. These calculatedstress-biaxiality values should be properly interpreted, however, recognizingthat none of the localized mechanical interaction has been included in theanalysis.

Ideally, it would be desirable if the cladding temperature and swellinggradients expected during fuel-element irradiation can be simulated in theout-of-pile tests. The experimenters may want to investigate the possibilityof adding a momen4-type loading at the specimen ends, because the stress-coupleinduced by the moment simulates the stress distribution caused by the radialtemperature gradient. Also, since the sense of the stress-couple depends on thesense of the moment, the swelling gradient which is opposite to the temperaturegradient can be simulated by simply reversing the sense of the previous momentloading.

VI. FUTURE WORK

The fuel and cladding behavior models in LIFE-GCFR will be refined, andimprovements will be made as work progresses. Work has already started in in-vestigating the means of replacing the current semiempirical models on fission-gas release and fuel swelling in LIFE-GCFR by the mechanistic models GRASS-SST1 5

end its associated fast-running version FASTGRASS.1 6 After the replacement iscompleted, LIFE-GCFR will be used to study the GCFR fuel-element behavior underthe reference GCFR operating conditions. The verification of LIFE-GCFk againstthe GB-9,7 GB-10,1 7 and F-5 18 irradiation data will be pursued in the near future,recognizing that certain nonprototypicalities exct in both fuel-element designas well as in the irradiation conditions of thcre irradiation experiments.

30

Table VI. Summary of Calculated Stress Biaxiality Values (a 0 /a zz)

Fuel-Cladding

Mechanical a azzInteraction

Case (yes/no) I.D. O.D.

Thermoelastic (radial NO 1.11 0.88

temperature gradient)*

60% Power** YES 3.08 2.41

Step-Power Change** YES 1.83 1.32

Power-Cycling** YES 0.91 1.50

* Fr3I Figure 5

** From Table III

Finite-element computer programs ADINA9 and ADINAT20 are planned to beused for analyzing the behavior of the ribbed cladding in detail. Initially,the focus will be on examining the ribbed cladding behavior during the biaxialcreep-rupture test, and the effect of stress biaxiality will be included. Aswork progresses, these programs will also be applied in studying the localized

mechanical interactions between the fuel-pellet ends and the cladding and be-

tween fuel-pellet cracks and the cladding.

ACKNOWLEDGEMENT

We would like to thank J. Rest and R. W. Weeks for their support and en-

couragement; M. Weber for editing the manuscript; and D. Borst and E. S. Zitko

for typing the manuscript.

31

REFERENCES

1. R.J. Campana, Pressure Equalization System for Gas-Cooled Fast BreederReactor Fuel Eler-nts, Nuci. Tech. 12, Oct. 1971, p. 185-193.

2. M.C. Billone et al., LIFE-III, Fuel Element Performance Code, ERDA-77-56,July 1977.

3. M.C. Billone, LIFE-GCFR Studies of GCFR Demonstration Plant Fuel-RodPerformance, Internal Memo to J.T. Madell (ANL), May 3, 1977.

4. A.A. Solomon, Relationship Between Primary and Steady-State Creep of U0 2at Elevated Temperature and Under Neutron Irradiation, in Deformation ofCeramic Materials, ed. by R.C. Brandt and R.E. Tressler, 1974, p. 313.

5. W.S. Lovejoy and S.K. Evans, A Crack Healing Correlation PredictingRecovery of Fracture Strength in LMFBR Fuel, Trans. Am. Nucl. Soc., 23,174(1976).

6. J.F. Bates, Updated Design Equation for Swelling of 20%CW AISI 3Z6 SS,HEDL-TME 78-3, 1978.

7. R.V. Strain, C.W. Renfro and L.A. Neimark, Postirradiation Examinationsof the GB-9 Element, ANL-8067, Oct. 1976.

8. Oxide Fuel Development Quarterly Progress Report for the Period EndingSeptember 30, 1977, WARD-OX-3045-34, Jan. 1978, p. 8-46.

9. S. Timoshenko and J.N. Goodier, Theory of Elasticity, 2nd Ed., McGraw-Hill, 1951, p. 408.

10. S.S. Manson, Thermal Stress and Low Cycle Fatigue, McGraw-Hill, 1966,p. 30- 32.

11. J.F. Bates and E.R. Gilbert, Experimental Evidence for Stress EnhancedSwelling, J. Nucl. Mtls., 59 (1976), 95-102.

12. A. Mendelson et al., A General Approach for the Practical Solution ofCreep Problems, J. Basic Eng., Dec. 1955, 585-598.

13. Y.Y. Liu et al., Advanced Fuels Development Program, Quarterly ProgressReport for the Period October - December 1978, p. 171-177. ANL-AFP-63 (1979).

14. A. Purohit, Y.Y. Liu, and R.T. Acharya, Postirradiation Biaxial Creep-Rupture Tests of Ribbed GCFR Cladding, Trans. Am. Nucl. Soc. 32, 272,(1979).

15. J. Rest, GRASS-SST: A Comprehensive, Mechanistic Model for the Predictionof Fission-Gas Behavior in U02-Base Fuels During Steady-State and TransientConditions, ANL-78-53, June 1978.

32

16. J. Rest and S. Gahl, The Mechanistic Prediction of Fission-Gas BehaviorDuring In-Cell Transient Heating Tests on LWR Fuel Using the GRASS-SSTand FAFTGRASS Computer Codes, Proc. of the 5th Inter. Conf. on StructuralMechanics in Reactor Technology, Berlin, W. Germany, Aug. 13-17, 1979,Vol. C, C 1/6.

17. D.L. Johnson, Postirradiation Examination of the GCFR GB-b0 Element, tobe published as at: ANL report.

18. D.L. Johnson, private communication.

19. K.T. Bathe, ADINA, A Finite Element Program for Automatic Dynamic Incre-mental Nonlinear Analysis, Massachusetts Institute of Technology, MITReport 82448-1, May 1977.

20. K.J. Bathe, ADINAT, A Finite Element Program for Automatic Dynamic Incre-mental Nonlinear Analysis of Temperature, Massachusetts Institute ofTechnology, MIT Report 82448-5, May 1977.

v33

APPENDIX

Detailed Analysis of GCFR Cladding Thermal

Stresses Induced by an Axial Temperature Gradient

34

NOMENCLATURE

Cladding midwall radius

Radial displacement at the cladding midwall radius (positive outward).

Radial coordinate.

z : Axial coordinate.

z*: Nondimensionalized axial coordinate, z/9

6 : Length of the transition region.

L : Cladding length.

t eff: Effective cladding thickness, t + (hw)/p.

Cladding root thickness.

Cladding rib height.

Cladding rib width.

Cladding pitch.

Characteristic length, [(R2t2 )/3(l - v2 )]l/4

Cladding thermal-expansion coefficient.

Cladding Young's modulus.

Cladding Poisson's ratio.

Temperature.

T : Temperatures in the smooth and the ribbr-

ed portions of the

cladding.

AT : Axial temperature drop in the smooth-to-ribbed transition region,z T -T.

r

uhs, uht, uhr

Up, upt, upr

ur, ut, us

Homogeneous solutions to the governing differentialequation, respectively, for the smooth, transition, andribbed regions of the cladding.

Particular solutions to the governing differentialequation, respectively, for the smooth, transition, andribbed regions of the cladding.

Complete solutions to the governing differential equation,respectively, for the smooth, transition, and ribbed regionsof the cladding.

h :"

w :"

P :'

a :

E :

v :"

T :"

T ,s

t":

.

35

NOMENCLATURE (Contd.)

Er E , E

ar, a , 6z

: Principal strain components, respectively, in the claddingradial, circumferential, and axial directions.

: Principal stress components, respectively, in the claddingradial, circumferential, and axial directions.

36

I. INTRODUCTION

The GCFR cladding thermal stresses induced by an axial temperature gradient

in the cladding smooth-to-ribbed transition region are treated fully here. Three

cases are considered in the analysis, and these cases are shown schematically in

Fig. 11 (main text). Case I is the limiting case with the length of the transi-tion region, 6, approaching zero. This gives an upper-bound solution on the

hoop stress aee. Case II gives an approximate solution of aee and is of aca-

demic interest because unrealistic free-end conditions are assumed at the ends

of the transition region. The stress results for this case would not be accu-

rate, but may be used in studying the effects of the transition length.

Case III represents a realistic solution of aee for reference GCFR fuel-elementdesign, and any finite 6 can be incorporated into the analysis.

II. ANALYSIS

The governing differential equations, the compatibilitythe constitutive equations for the above three cases are the

in Ref. 10 (main text).

relations, and

same and are given

Differential Equation:

4d u

4 + 4u = 4RaTdz*

(A-1)

Compatibility Relations:

2u 1 d u

Er = -r+ 2 2 (R - r) + (1 + v)aTSdz*

U(A-2)

E = - (E + e ) -+(1 )aTz 1. - v r 0 1- v

Constitutive Equations:

EC = 2[(er-r 1- 2 [(c

1 - v

Ev= [(e -

e 1 2 e

aT) + v(e0 - aT)]

aT) + v(e - aT)]r (A-3)

a =0z

The homogeneous solution, u1 , to Eq. A-1 can be written as

37

uh = C1 cos z* sinh z* + C2 cos z* cosh z*

+ C3 sin z* cosh z* + C4 sin z* sinh z*,

(A-4)

where C1, C2 , C3 , and C4 are constants to be determined from the four boundaryconditions required fur a unique solution. The ease with which a particularsolution, up, can be obtained for Eq. A-i depends on the functional form ofT = T(z*). Since the exact temperature variation in the transition region isnot known and the axial temperature gradient outside this region is relatively

insignificant, the following sinusoidal temperature distribution is chosen for

analytical convenience:

TS

< - 6

Tz _Tz* 6T + A2 (1 - sin TZ*) ; - -- < -r 2 62 2

Tr

(A-5)

6S 2Z

With the above temperature distribution, it can be shown that the particu-lar solutions to Eq. A-1 are

RaTs

26

Rt fT +ATz 4[1 sin( rz*/6)

2 4 + ( r9/6)

- 6 <-2

' 22- -2

RaTr

6z* > -

2Q

The complete solution to Eq. A-1 is found by summing the particular solu-tions, Eq. A-6, and the homogeneous solutions, Eq. A-4, for each of the threeregions.

u + RaThs s

<~6z* <-. --2Q

Uht Rcz ~r+~[z 4 sin (rriz*/6)1}uht + Ra Tr + 4 4+ ht I r 24 + (7rZ/6)4

6 <6"- < z* < a (A-7)

u1 + RaT.r r6

z* > --2

The homogeneous solutions uhs, uht, and uhr each contain four undeterminedconstants, bringing the total number to 12. These constants can be found fromthe four boundary conditions at the ends of the cladding and matching condi-tions at z* = +6/2t. Since these boundary conditions are different for thethree cases, they will be given separately in the following:

U =P

(A-6)

T =

u =

38

Case I (Limiting case for thermal stresses in GCFF cladding induced byan axial temperature gradient)

A. Boundary Conditions

At z = z* = 0, the displacement, slope, moment, and shear must beequal in the smooth and the ribbed portions of the cladding. This entails

d2u d2ur s

Ur us dz* 2 dz*2

(A-8)

du du d3u d3ur s r sd dz* le3 A3

dzR

The free-end conditions at z' = L/2Q give

d3u

s = 0

dz* 3

d~u5 3=0

dz*0

at z* = L/2k

at z* = - L/2%.

B. Displacement Solution for L/2 >> 1

The displacement solutions that satisfy Eqs. A-1, 8, and 9 forL/2 >> 1 are

ATz -zu = Ra(Tr +-,-e cos z*) for z* > 0

ATu = Ro(T - .z e cos z*) for z* < 0s s 2

(A-10)

C. Stress Solution

From Eqs. A-10 and -3, the stresses in the smooth region (z* < 0)are

Ea T1 v 2 R r2 Re sin z*

1 - v 9

z 1 -v 2 R-r*ag = 2 ( cos z* + v 2 R sin z*)e.1 - v 2 2

(A-li)

d2ur2

dz*

and

d2u

dz*2

(A-9)

A I

dz7

39

For the ribbed region (z* > 0), the stresses are

z R - r -z*a = Re sin z*r 1- 2 ,2

1 - v Q A-2(A-l2)

a _ z 2 cos z* + v R- R sin z*)e-1- v

As z* approaches 0,

EaoAT

(a )ma = z , A-13)0 max. 2

and the maximum tensile hoop stress occurs in the ribbed portion of the clad-

ding.

D. Sample Problem

Using the values,

R = 0.341 cm t = 0.038 cm5 eff

E = 1.56 x 105 MPa \v= 0.304

T = 500 C T = 570*Cg

AT = 70*C a = 1.84 x 10-5/ICz

yields k = 0.08862 cm

(a0)max = 100.4 MPa

Case II (Intermediate solution to study the effect of the transition length)

A. Boundary Conditions

Assuming free-end conditions at z*= 6/2Q gives

2 2d2u du = 0 at z* = + - (A-14)

dz* dz*

B. Displacement Solution

From Eqs. A-4 and A-6, the general displacement solution in thiscase is given by

u = C 1 cos z* sinh z* + C2 cos z* cosh z* + C3 sin z* cosh z*

f AT 4sin(rz*/)(A-1')+ C4 sin z* sinh z* + Ra Tr + 2 [1-. 4+ (in 6 4*/

4 + (nR/S)

40

C. Evaluation of Constants

Because the assumed temperature distribution is antisymmetric inz*, the displacement solution will also be antisymmetric in z*. Two of thefour constants in the general solution to Eq. A-l are therefore equal to zero.That is,

and

C2 = C4 = 0.

Applying free-end conditions at * = 6/2Z gives:

Cl = Ra(AT /2)C'1 z 1

C3 = Ra(AT /2)C',

(A-16)

(A-17)

where

C' = (#1)2 [1 + i(-r) 4 ]l11 2 [l+(242

C' = -(22 1 + (-Tr) 4 ~3 2 )4 2i

cos $ cosh $ -sin sinh $sinh 2i - sin 2i 1

(A-18)cos $ cosh i + sin i sinh '( )

sinh 2i - sin 2$ '

and= 6/2z.

D. Stress Solution

From Eqs. A-2 and A-3, the hoop stress at the midradius of thecladding (r = R) is

a6 = E(-u -c T).8 R

From Eqs. A-15, A-17, and A.--19,

(A-19)

EFa - C cosz* sinh z* + C3 sin z* cosh z* +

AT 4(r/6) s naR sin 26 J

ATa6 =EEa-# C' cos z* sinh z* + C3 sin z* cosh z* +

4(Tr/24) sin .

4+(/2$)4 2 i

E. Sample Problem

Two equations have been obtained from Eq. A-20 forS= Tr/2 and = 7r:

or

(A-20)

(A-21)

4t

AT

r = Ea 2[0.2 sin z* + 0.1594(cos z* sinh z* - sin z* cosh z*)]

for $' = /2, andATz 2-

cue= Ea Z2[1.538 x 112 sin z* + 1.0657 x 10 2 (cos z* sinh z*

+ sin z* cosh z*)] for ' = i.

(A-22)

(A-23)

Using the same values of E, a, and ATz as before, we have performed calculationswith Eqs. A-22 and A-23 over a range of discrete values of z*. Tables I and

II give the variations of the normalized aA as a Function of z* for $ = 7T/2 and

71, respectively. Notice that at the same z*, the normalized o0 are smaller for

= T than for $ = 7/2. If the evaluation of the normalized 00 is made at z* =

6/2, = $, the following equation can be used:

a6 _ 1Ea(ATz/2) 1 + 4(2)/i)4 1 2(31)2 sinh 2$ + sin 24'1[1- Tr sinh2'- sin 2 '

Table I. Normalized

00.20.40.60.81.01.21.4ir/2

Table II.

ao for = rr/2

a /(EacTz/2)

00.03890.07110.09000.08920.0625

0046-0.. 89-0.200

Normalized a for $' =

a /(EaATz /2)

00.41.21.47/21.82.22.42.6

00.01610.03250.03600.04210.04010.02510.0114

-0.0912-0.108

Table III gives the variation of the normalized a0 with z* = . Again, thenormalized ae is seen to decrease rather rapidly as 4 increases. Note that the

(A-24)

42

absolute values of these stress results are not accurate because of the assumedfree-end boundary conditions. A realistic solution is provided below.

Table III. Normalized a at z* = $

a /(EtATz/2)

3w/221r4 n

-0.200-0.108-0.0523-0.0302-0. 00775

Case III. (Realistic solution to the problem for any transition length)

A. Differential Equations

The governing differential equation is the same as before, butit is necessary to write one for each of the three regions.

d4us + 4u =

dz*4

d4u4 + 4u =

dz* 4

d4ut

+ 4u =dz*4 r

4RaT

4RaTt;

4RaTr

B. General Displacement Solution

The general displacement solution is the same asis also necessary to write one for each of the three regions.

before, but it

us = Cs 1 cos z* sinh z* + Cs2 cos z* cosh z* + Cs 3 sin z* cosh z*

+ C sin z* sinh z* + RaTs;

ur = Cr1 cos z* sinh z* + Cr2 cos z* cosh z* + Cr3 sin z* cosh z*

+ Cr4 sin z* sinh z* + RaTr

ut Ctl cos z* sinh z* + Ct2 cos z* cosh z* + C0t3 sin z* cosh z*

rAT+ C sin z* sinh z* + Ra T + [ 4 sin( z*/6)

t4 r [ 4

(A-25)

(A-26)

43

C. Boundary Conditions

Assuming free-end conditions at the physical ends of the clad-ding gives:

d2ud~us

2dz* z* = -L/2

2ud u

r

dz*2 z* = L/2!C

d3ud~u_s

3dz* z* = -L/2Q

3d u

r

dz*3 z* = L/2R

= 0;

= 0.

(A-27)

The matching conditions at the interfaces are

u = us t

diff dus t

dz* dz*

2d u

s

dz *2

d 3us

dz* 3

6at z* -2d ut

_t

dz*2

d3u= t

dz* 3

} (A-28a)

and

u =ur t

dur dut

dz* dz2

2d u

r

dz* 2

d 2 udz*

~dz*2

6at z* = g

3 3d u d3ur t3 30dz* dz*

Because of antisymmetry about z = z* = 0, it is only necessaryto solve the problem for z* > 0. At z* = 0, one expects that

T +T ATu (0) = Ra s r = Ra(T +

t 2 r 2

d2ut(0) = 0.

dz*2 I

(A-29)

I (A-28b)

and

44

D. Evaluation of Constants

The boundary conditions, Eq. A-27, imply

Ct2 t4=0'

C L -C.4(A-30)r3 r4,

and

Crl r2.

The matching conditions, Eq. A-28, along with the boundary conditions above giveAT

Ctl = R 2 Cti1

ATCt3 = Ra 2 z C ,

AT (A-31)

Crl= Ra,- 2z Crl,

and AT

C4 = Ra C ,r4 2Cr4,

where 2 2

C' _ ( /2$) [2 sin ( - ( ) cos $f]e tl 4 + (n/2)

2

Ct3= - (iT/2$) 2 [2 oos ' + (2)2 sin4 + (n/2$)

2

=r (n/2$ )4 f[cus $P ++( )2 sin ]sin(2 p)e (A-32)4+(rr/2*)

+ [sin 2 sinh 'p- cos2 $ cosh p]'12 sin ' - (2)2 cos

r4= (r/2i) 2 r1(r2'C = 4([sinP/-2$)2 cos$] sin(2p)e

4+(nr/2i)22'

+ [sin2 ' sinh p + cos2 $ cosh ]-[2 cos ' + ()2 n

From these constants and Eq. A-26, the displacement solutions are

ut = Ctl cos z* sinh z* + Ct. sin z* cosh z* +

Ra T + AT [1 - 4 sin(lz*/2)}

and * (A-33)ur = Crl cos z* + Cr4 sin z*)ez + RaT *

45

E. Stress Solution

Substituting Eq. A-2 into Eq. A-3 for a0 and rearranging terms

gives2

a0 =Ea((--T)+ E 2 R-r d 2 .u(A-34)Ra 1-v2 2 dz*2

Previous experience indicates that the second term on the rightside of Eq. A-34 is small compared to the first term. Also, at r = R along thecladding midradius, Eq. A-34 reduces to Eq. A-19. In the sample-problem calcu-lation, Eq. A-19 will be used with the displacement solution provided by Eq.A-33. Substituting Eq. A-33 for ut and us into Eq. A-19 gives

[Eae ]/)= C' cos z* sinh z* i C' sin z* cosh z*Ec(AT /2) t tl t3I

+ [ (n/2$p)4 ] sin nz*(A-35)4 + (ir/2$)4 2i

[ a 2] = (C' cos z* + C' sin z*)eEc(AT/2) r 'rlr4

F. Sample Froblem

Two equatio-is have been derived from Eq. A-35 for t= /2:

[Ea(AT /2) t = [sin z* + (2 cos z* sinh z* - sin z* cosh z*)]e'

and z (A-36)

[Ea](AT /2)r =- sinh (-) [2 cos z* + sin z*] _. (A-37)

Using the same values of E, a, and ATz as before, we have per-formed calculations with Eq. A-36 over a range of discrete values of z*. TableIV gives the variation of the normalized a0 as a function of z* for p = n/2.Comparing the Case II results in Table I to the Case III results in Table IV,one can see that the intermediate solution (Case II analysis) gives a maximumhoop stress that is low by 50%. Therefore, the solution presented for Case IIIis recommended for detailed analysis. Also, for transition length 6 > n9, themaximum value of tensile hoop stress is less than 21 MPa (3000 psi) for ATz=70*C.

46

Table IV. Normalized a0 for $ = Tr/2

a0 /(EazTz/2)

0 00.2 0.0780.4 0.1450.6 0.1890.8 0.1990.9 0.1871.0 0.1621.2 0.06761.4 -0.09747r/2 0.0957

3

2

U I

I 2 3 4 5 6 7 8 9 10 II

BURNUP AT X/L: 0.375 (ot %)

4.I I I I I I I 1 1

- -- REFERENCE I / -- PRESENT WORK /

/m = 3 45 11015 n/cm2~-s

m= 95a1015n/cm2

~ -

1 1 1 1 1 1I 1 I1 1 I

4

Fig. 2.

Steady-state GCFR Gap Sizes at X/L =0.125 for 60, 100, and 115% PowerRuns. ($ = 3.45 x 1015 n/cm2 -s)

m

E

-J

4r

0.10

009

008

0.07

006

005

004

0 03

0 02

0 01

T___F-=T) 0 10

- - - REFERENCE I 100 %/ -- 009-- PRESENT WORK '---

115% -0 07

006

005

.----- 100%9 -0.04

0.03

60% 002

----- -0.01

0 2 4 6 8 10 12 14 16 18 20

TIME (I000h)

-1 1 1 1- - - -REFERENCE I- PRESENT WORK

I 1 1--

STEADY STATE

72 POWER-- CYCLES -

S 2 3 4 5 6 7 8 9 10 II 12

BURNUP AT X /L= 0 375 (ot. %)

0.100.09

0 08

0.06

004

0 03

002

001

Fig. 3.

GCFR Fuel-element Gap Sizes atX/L = 0.375 for Steady-state andPower-cycling Histories. (100%Power, m = 1.95 x 1015 n/cm2 -s).

a

47

J

4r

i

cc

EE~

4

'3

41

a4

Fig. 1.

Steady-state (100% power) GCFRFuel-element Gap Size atX/L = 0.375.

3

2

C

0

2.0

E

0

G3

crC7Z-J.JLW

L3

U,W,LW

ix

340 420 500 580 660 740

IRRADIATION TEMPERATURE (*C)

SI 1 I I:I15 % POWER -

o :100% POWER- :6O0% POWER -

I-...'

50

40-

Fig. 5.

Thermoelastic Stress DistributionsAcross the Cladding Thickness;Cladding Outer-surface Temperature =600*C, q' = 16.4 kW/m, Pi = P = 9.0MPa.

30

20

10

0

I-N,

ook IK ' FT- TYI-TTT is

50

0

-50

-20

-30

-40

-50

--- 6

- r -

La - - -

---

- 60

ID10

4

2

0

-2 ;;

-6

-8

5

0,

U,

_5 I-U,

-1o

Fig. 6.

Thermoelastic Stress DistributionsAcross the Cladding Thickness; CladdingOuter-surface Temperature = 600*C,q' = 36.1 kW/m, P = P =9.0MPa.

0

-15

LL LLO DLL-JID 00

48

Fig. 4.

Temperature Dependence in the Stress-free Swelling of 20% Cold-worked316 Stainless Steel.

1.0F

2~

N

1- -

- UN

- U1

-

- 9 -.

_I

1 1_

-100 --

I5

5

100

50

-_ -

ZZ

gee

- I I I ---- _L .L___1ID 00

Fig. 8.

Thermoelastic Stress Distributions Acrossthe Cladding Thickness; Thin Cladding, CCladding Outer-surface Temperature = N600*C, q' = 36.1 kW/m, p. = p = 9.0 MPa.

0L/,

100

50

0

-50

-100

ID

10

5

0

-5

-10

CnVt)WUN

Fig. 9.

Thermoelastic Stress Distributions Acrossthe Cladding Thickness; Thick Cladding,Cladding Outer-surface Temperature =600*C, q' = 36.1 kW/m, p = p = 9.0 MPa.

_15

ID O DID 00

-10

-15

49

0

Fig. 7.

Thermoelastic Stress Distributions Acrossthe Cladding Thickness; Cladding Outer-surface Temperature = 600 C, q' = 45.9kW/m, p. = p = 9.0 MPa.

0n

U,

ac

-50

-100

15

10

5

0

Ct)-5

- I I I I I III00

100 I I I15

-T--

- -

- cT- 1

N

zrCNUN

I-Co

-50

-Ion

0

- 5 -

501

0

w 4IF

F-W 15

10

5

Fig. 10.

Thermoelastic Stress Distributions Acrossthe Cladding Thickness; Cladding Outer-surface Temperature = 600 C, q' = 36.1kW/m, p. = p = 0 MPa.

u

-5

WU

I-

-10

-100 --

ID 0D

Fig. 11.

Schematic Diagrams of the GCFR Thermal StressProblems due to an Axial Temperature Gradient.

50 CLOSE REOPEN0 GAP GAP

5CLADDING OD

U, V7

-FD sN0G - -za

$ $ Clad(C., Disti

Swel.CLADDING IDo

d-50

-10

2 4 6 8 10 12 14 16 18IRRADIATION TIME (10 3h)

CASE I

CASE I I

Ts

CASE III

<Tr

ITr

L

Fig. 12.

ding Hoop- and Axial-stressributions with Creep andling Operating.

50

100

50

ur0

CL

N,(J

W-

-50 cr r

In

CLOSE

CL ADDING OD

CLADDING ID

--IO

8 10 12 14 16 18

IRRADIATION TIME (103 h)

501

Fig. 14.

Cladding Axial-stress Distributionswith Swelling Suppressed and CreepOperating.

b

I)

J

4

-JU-

CLOSEGAP

CLADDING OD b

CLADDING ID

C7,

-- J

-10

1 LJLiI2 4 6 8 10 12 14 16 18

IRRADIATION TIME (IO 3 h)

5

U Q_ -J

-5

CLADDING ID -10

2 4 6 8 10 12IRRADIATION TIME (IO 3h)

b

CL,

a-00

CDz00

-J(.)

Fig. 15.

Cladding Hoop-stress Distribution with

Swelling Operating and Creep Suppressed.

14 16 18

50

51

m~

Lb%

LV

0W

O

0Q

-U

0

-50

5

U,

aO0

-5 z0ca-J

C.)

2 4 6

50' CLADDING OD - - - EXTRAPOLATIONS ~

a-

CID

LU)U,

I-

CI,

0

O

JU-

Fig. 13.

Cladding Hoop-stress Distributions withSwelling Suppressed and Creep Operating.

-

0

-50

-5

SI I I I ICLOSE REOPEN

-GAP GAP

CL ADDING ID 5

- 00

a0

z

- -I-5IJCLADDING O M-

-10

2 4 6 8 10 12 14 16 18

IRRADIATION TIME (103 h)

Fig. 16.

Cladding Hoop-stress Distribu-tions with Temperature andSwelling Gradients Opposite toEach Other.

52

50

0

V Ji 0

-0

0

0O

00

J

-50