First wall stress analysis for a tokamak reactor

Nuclear Engineering and Design 42 (1977) 391-407 © North-Holland Publishing Company

391

FIRST WALL STRESS ANALYSIS FOR A TOKAMAK REACTOR *

Carl K. YOUNGDAHL Argonne National Laboratory, Argonne, 1160439, USA

Received 14 February 1977

An outline of the general design considerations for the first wall is given, and stress analysis studies which contributed to the conceptual design of the first wall of a fusion reactor are described.

1. Introduction

An integrated conceptual design has been developed at Argonne National Laboratory for a Tokamak experimental power reactor (EPR). The design basis performance objectives of this fusion reactor are (1) to operate for ten years with a plant capacity factor of 50% [the plant capacity factor is defined as the product of the duty cycle (75%) and the plant availability (67%)]; and (2) to operate under conditions such that net electrical power production is feasible. The design of the EPR was based upon technology that is consistent with a mid-to-late 1980s operation date. A detailed description of this conceptual design is presented in ref. [ 1] with a brief summary provided in ref. [2]; a perspective view of the reactor is shown in fig. 1. The purpose of this paper is to describe the scoping stress analyses that were performed to evaluate the merits of various first wall structural concepts and to determine appropriate dimensions for the chosen configuration based on its performance objective. Proceeding outward from the plasma axis (fig. 2), the principal elements of the reactor are the plasma, first wall, blanket, shield, and magnets. The first wall is required (1) to maintain the vacuum and other environmental conditions required for the plasma, and (2) to survive the severe radiation and cyclic thermal loadings imposed by the plasma for the desired life* Work performed under the auspices of the Energy Research and Development Administration.

time of the reactor. The major part of the neutron radiation is absorbed in the blanket, where the resultant thermal energy is transferred to the coolant and ultimately used for the power cycle of the reactor. The function of the shield is to further attenuate the neutron flux to protect the magnets. The blanket and shield are assembled from massive blocks with internal coolant passages, the blanket blocks being made of stainless steel and the shield blocks being made of stainless steel, boron carbide, graphite, and lead mortar. All structural elements in the first wall, blanket, and shield are made of type 316 stainless steel, and pressurized water is used for the coolant. The EPR operates in a pulsed mode at a frequency of one burn cycle per minute. In the ANL EPR design, the first wall is composed of a vacuum vessel and detachable coolant panels. The vacuum vessel supports the mechanical loads on the first wall, while the detachable coolant panels, mounted on the inside of the vessel on sliding supports, withstand the high heat flux deposited on the plasmaexposed surface. The free-standing vacuum vessel approximates a toroidal shell of 6.25 m major radius and 2.4 m minor radius (see fig. 3). It is constructed from 16 circular cylindrical segments of 2 cm-thick stainless steel plate with reinforcing rings and spars. The two reinforcing rings per segment are 1 m apart and straddle the vacuum ports. The joints between segments are covered by formed bands which house the coolant manifold piping. Coolant channels, bonded to the vessel, remove the heat generated in the vessel wall. The vessel weight is carried by a three-point per

392

C K. Youngdahl / Stress analysis for a Tokamak reactor OHMIC HEATING AND EQUILIBRIUM FIELD COILS

SHIELD

SUPPORT CYLINDER

INNER BLANKET/SHIELD

,;,

Fig. 1. EPR reactor.

segment, Toiler/slide pad-type support. The detachable coolant panels are made of 0.5 cmthick stainless steel plates from 1 to 2 m 2 in area, depending on location in the toms. The surface facing the plasma is coated with 100-200/~m of beryllium to control impurity contamination of the plasma by stainless steel. Coolant channels are bonded to the other surface to remove heat generated in the panel from surface and volumetric radiation fields. In addition to extensive radiation damage, the coolant panels are subjected to severe thermal fatigue produced by the cyclic heat deposition that results from the cyclic nature of the plasma burn.

2. General design considerations for first wall The structural design and analysis of the EPR first wall is based on two general guidelines: (1) 'minimum mass' design within the constraints of the required geometric configuration and the applied mechanical and thermal loads; and (2) maximum first wall lifetime for the prescribed cyclic thermal loading and Jr-

radiation-induced changes of material properties. Minimum mass design is analogous to minimum weight design in the aerospace industry, where the payoff is in reduction of propulsion plant requirements. For the EPR, decreasing the mass of the first wall structure reduces the radiation it absorbs, which is accompanied by a reduction of temperature gradients, thermal lag between parts of the structure during thermal cycling, and thermal stresses. Consequently, the size and complexity of the coolant system required for the first wall can be reduced. Moreover, because the structural functions of the first wall require that its maximum allowable temperature be less than that for the blanket, heat removed from the first wall is less useful from the point of view of power plant economics. Decreasing the mass of the first wall also reduces its volume, leaving more room for the blanket and shield within the toroidal field coils. However, minimum mass design also involves trade-offs with fabrication and analysis costs, since it is likely to involve more complex structural configurations and increased hazard of buckling instability. A number of types of structural failure must be

C K. Youngdahl I Stress analysis for a Tokamak reactor

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394

C.K. Youngdahl I Stress analysis for a Tokamak reactor EXPERIMENTALq PORTS (4) I/

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Fig. 3. Vacuum vessel. considered in assuring an adequate lifetime for a first wall structure. Plastic strains resulting from a combination of sustained and cyclic mechanical loads can result in immediate stress failure or low cycle fatigue failure. A sustained mechanical load accompanied by cyclic thermal strains can result in thermal ratcheting, which leads to plastic instability failure or low cycle fatigue failure. High temperature operation under sustained loads can cause creep deformation, resuiting in loss of structural function or creep failure. High cycle fatigue failure may result from low ampli-

tude, high frequency strain variations accompanying thermal cycling. Atmospheric pressure acting on the evacuated first wall structure may cause short-time mechanical buckling or, at elevated temperatures, creep buckling. The long term failure mechanisms are influenced by the effects of strain history and irradiation and thermal environment on material properties. A suitable choice of design restrictions and structural configurations can eliminate some failure mechanisms from consideration, and others can be designed against by a proper choice of structural di-

C K. Youngdahl / Stress analysis for a Tokamak reactor

mensions and configurations and coolant system specifications. The general considerations discussed above contribute to a number of decisions and restrictions for the first wall structural design. In addition, design decisions for other EPR components also affect the first wall structural configuration. Analyses providing quantitative information on the sensitivity of stress levels to structural configurations and dimensions are summarized in subsequent sections of this paper. Some general conclusions and design decisions pertaining to or following from stress analysis of the EPR first wall are given below. (1) The first wall should not be supported directly by the blanket, since temperature differences between the two structures or shifting of the massive blanket blocks could result in failure of the thin-walled vacuum vessel. Support should be provided by S~ruts or hangers with a minimum of constraint to avoid excessive stresses from non-uniform thermal expansion of the vessel and support structure. (2) A ring-reinforced thin-walled vacuum vessel is preferable to a thicker uniform-wall shell from minimum mass design considerations for vessels of EPR size. (3) Reinforcing rings can be made significantly smaller for a given stress level if the vessel is supported from the sides rather than from the top or bottom. (4) Buckling can be prevented by proper design of the vessel wall and its reinforcing rings and stringers. However, further analytical and experimental work is required in this area to assure a reliable design because of the structural complexity of the EPR vacuum vessel. The wall thickness is sized primarily from buckling considerations; therefore, an improvement in the accuracy of the calculation of the buckling load could result in a reduction in the applied safety factor and a significant decrease in first wall mass. (5) Detachable coolant panels should be located inside the vacuum vessel on sliding supports to avoid thermal ratcheting failure and assure a long cyclic fatigue lifetime. The vacuum vessel supports the atmospheric pressure load and the weight of the vessel, detachable coolant panels, and its own coolant system. The detachable coolant panels absorb the brunt of the plasma thermal cyclic load. Thermal ratcheting is caused by the superposition of a sustained mechanical load and a cyclic thermal load which combine to

395

produce plastic strain in the same direction in each cycle and subsequent failure. The separation of coolant panels and vacuum vessel prevents thermal ratcheting of either structure since sustained mechanical loads are not transferred from the vacuum vessel to the coolant panels through the shdingsupports and cyclic thermal loading of the vacuum vessel is minimized. Sliding supports also reduce the cyclic thermal strain levels in the detachable coolant panels and thus increase their expected fatigue lifetimes. (6) The EPR vacuum vessel is composed of truncated circular shell sections supported by circular rings. The circular cross-section components are easier to fabricate and analyze than a true toroidal shell, but the beveled corners at the joints between shell sections present stress problems which should be analyzed in detail. (7) The EPR vacuum vessel and its support structure are designed to equilibrate all mechanical loads by elastic stresses to avoid any failure mechanisms invo!ving plastic strain. Some plastic deformation of the coolant panels from thermal loads will be permitted since they perform no structural function. (8) The maximum allowable operating temperature for the first wall structural material (type 316 stainless steel) is set at 500°C to avoid thermal creep problems. (9) The allowable design stress for structural components analyzed in subsequent sections is set at twothirds of the yield stress to allow a margin in the elastic range to cover stress concentrations and design features not included in the initial analyses. It is customary [3] to set allowable stress limits of o m < ~ Or and Om + ob < Oy, where ore, Oh, and oy are primary membrane stress, primary bending stress, and yield stress, respectively; this assumes plastic shakedown of secondary stresses will occur in the first few load cycles in regions of stress concentration, so that subsequent cycles are elastic. However, the EPR first wall is subjected to cyclic variations from both the burn cycle and the reactor start-up/shutdown procedure. The more conservative design assumption of Om+ Ob< oy is used here to lessen the likelilaood of low-cycle fatigue failure aggravated by the severe thermal and radiation environment. It also provides a design margin to allow for features not covered in the analyses described in this paper and to allow for some of the approximating assumptions used here. As more de-

396

CK. Youngdahl / Stress analysis for a Tokamak reactor

-700

Table 1 First wall structural parameters R O, major radius o f toroidal shell R, radius of first wall cross section E, elastic m o d u l u s v, Poisson's ratio E o = El(1 - v 2) U = [3(i - v2)] - 1 / 4 a, coefficient of thermal expansion Oy, yield stress Oa, allowable stress 7, specific weight

6.25 m 2.40 m 22.6 × 106 psi 0.3 25 × 106 psi 0.778 1.03 X 1 0 - S / ° F 17 000 psi 10 000 psi 0.28 lb/in. 3

i r , ,f,,, I

~ I

'''"l

J ' ' f'"J

-500 ~-400 i.--"

_,oo - 200 -I00 0

I III

I0 100 NUMOER OF SUPPORTS, N

Fig. 4. Thermal stress in shell as a function of n u m b e r of support locations.

tailed analyses are performed, it will be possible to reduce some of the conservatism in the specification of design stress: In the sections that follow, 0 is the angular coordinate measured around the circumference of the vessel cross-section and x is the axial coordinate measured along the plasma axis. The components O0m, O0b, Oxm and Oxb denote membrane and bending stresses in the circumferential and axial directions. Membrane stresses are uniform across the wall thickness; bending stresses vary linearly across the thickness, being equal and opposite on the inner and outer surfaces. The upper and lower signs in expressions of the form OOb = -+fdenote stress values on the outer and inner surfaces of the shell wall, respectively. Table 1 gives values of frequently used parameters. The toroidal vacuum vessel is approximated by a straight circular cylindrical shell in many of the following analyses. Implications of this approximation for various loading conditions have been discussed by Chung [4].

3. Effect of constraints on first wall thermal stresses Stresses in the vacuum vessel resulting from the temperature difference between it and its support structure will be discussed here, and those resulting from the weight of the vessel and atmospheric pressure will be included in later sections. Let the toroidal vacuum vessel be approximated by a straight circular cylindrical shell having the same cross section and having a uniform wall thickness. Assume the shell is connected at N equally spaced lo-

cations around its circumference to supporting structure which is at a different average temperature. Then, making use of results in ref. [5], the stresses in the shell at the supports are given by _ E o oth2 OOm - -- ~ A T f (J3) COS/3,

_ E0ah OOb =+ R3 A T f ( 3 ) ( s i n 3 - 3 c o s 3 ) ,

(1)

whe re = ,IN, f (3) = 3 sin /3

[(:) 1+~-~

X 3(3 + sin 3 cos/3) - 2 sin23] ,

(2)

../

and AT is the average temperature of the vessel minus the average temperature of its supports. As the supports become continuous around the circumference of the shell, i.e. as N -->~o, OOm "+ EoaA

T,

OOb -+

O,

(3)

independently of the wall thickness. Fig. 4 shows Oom/AT, Oob/AT, and their sum as functions of number of support locations, N, for a wall thickness, h, of 2 cm. Continuous curves are shown, but only integer values of N are physically realistic. As the number of supports become large, oo/AT becomes approximately 250 psi/°F, i.e. a 100°F temperature difference between the vacuum

397

C.K. Youngdahl / Stress analysis for a Tokamak reactor

vessel and its support structure would produce a stress of about 25 000 psi, not including the stress due to vessel weight and atmospheric pressure. Therefore, supporting the first wall continuously by the blanket would present significant design problems in controlling the temperatures of first wall, blanket, and vacuum and beam ducts to avoid excessive thermal stresses in the thin-walled vacuum vessel. For simple hanger supports, i.e. N = 1, the thermal stresses due to support constraints vanish. Consequently, the support system for the first wall should constrain the vacuum vessel as little as possible to avoid excessive thermal stresses during reactor start-up and operation. The temperature drop f T through the first wall produces a bending stress given by

-3000

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Ea

o0b

=

+

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(4)

so that varying the wall thickness affects this stress component only insofar as it affects the temperature drop.

4. Stress in uniform-thickness wall, The feasibility of using a uniform-thickness vacuum vessel to sustain the atmospheric, gravitational, and thermal loadings is investigated in this section. 4.1. S o l i d wall

For uniform external pressure p, the circumferential and axial membrane stresses in a toroidal shell with circular cross sections are [6] p R ( 2 R o + R sin 0)

OOm-

2h ( R o + R s i n 0 )

(5)

'

pR Oxm -

2h "

The corresponding stresses in a straight circular cylindrical shell with the same cross-sectional dimensions and capped ends are Oom = - p R / h ,

Oxm = - p R / 2 h

.

(6)

Fig. 5 shows the stresses produced by atmospheric pressure as a function of angular position for h = 2 cm.

Approximating the toroidal shell by a straight shell is n0n-conservative at 0 = - 9 0 ° and conservative at 0 = +90°; the solutions agree at the top and bottom of the shells. Since the maximum stresses caused by the weight of the shell occur at the top and bottom and are considerably larger than those produced by atmospheric pressure, the use of cylindrical shell solutions for the vacuum vessel analysis appears reasonable. However, the reliability of this procedure requires further investigation [4]. Consider a shell hung from a rigid beam at the top. The circumferential bending stress distribution resulting from the weight of the shell is given by [5] O0b =+

63,R 2 h

[1 +½cos 0 -- (Tr-- 0) sin 0]

(7)

and the sag at the bottom is a O = 30r ~ - 8 ) ~

.

(8)

The circumferential membrane stress resulting from the weight is negligible compared to the bending stress. In general, the stress distribution in the shell resuits from support constraint against thermal expansion, atmospheric pressure, and the shell weight. The support constraint is zero for this case (N-- 1 in rigA). The maximum stress from the other two loadings occurs at the inside surface at the top of the shell and is found by adding gore from eq. (6) to aOb from eq.(7)

C.K. Youngdahl / Stress analysis for a Tokamak reactor

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I for 0 = 0. The result is shown in fig. 6 as a function of wall thickness, along with the sag at the bottom. A wall thickness of about 6 cm would be required to satisfy the design stress value of 10 000 psi. A solid wall of this thickness would be difficult to cool evenly. Supporting the shell at several points around its circumference was studied also. A significant reduction in wall thickness is then theoretically possible. However, the analysis is quite sensitive to the proper distribution of the shell weight among the supports, so that design and fabrication difficulties make this support configuration impractical. 4.2. Sandwich wall

Another design concept that was investigated was using sandwich wall construction for the shell. Consider a shell wall of thickness h composed of two stainless steel cover plates of thickness ho connected by a core which is k% stainless steel. Since the core provides the shear connection between the cover plates, it must be securely attached to them and strong enough to cause the sandwich wall to bend as a unit. The maximum bending stress produced by the shell weight occurs at the top and is given by (Ob)max=_+

-~'/~

sag for uniform thickness shell

97R2h [ 2(1 - k)h o + kh~ 2ho. ~-0~-~-~-0-~- ] .

(9)

3

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Fig. 7. Maximum stress for sandwich-wall shell supported from top. The sag at the bottom is 7r2_8 AD

R2

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I(Ob)maxl•

(10)

Fig. 7 shows the maximum stress from the shell weight and atmospheric pressure as a function of wall thickness for various wall constructions. For a design stress of 10 000 psi, the savings in mass to absorb radiation and require cooling is significant compared to a solid wall, as is shown in table 2. The major drawback of the sandwich wall is fabrication; fastening the core to the cover plates with suf-

Table 2 Shell mass comparison for sandwich wall * Configuration

h (cm)

Massper unit axial length (kg/m)

Solid wall h O = 1 cm, k = 25% h o = 1 cm, k = 10%

6 5 4.25

7030 3220 2610

* For maximum stress of 10 000 psi.

399

CK. Youngdahl / Stress analysis for a Tokamak reactor

ficient rigidity to prevent crushing of the cross section in bending would require a difficult and expensive welding operation. The ring supported uniform shell construction discussed in the next section is comparable to the sandwich wall construction in weight and is much easier to fabricate.

- 2 5 --

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5. Stress in ring-stiffened shell From fig. 6 it is apparent that the main contribution to the stress in a uniform-wall shell hung from the top is the distortion of the cross section caused by the shell weight. Reinforcing rings reduce this distortion and result in thinner-wall, lighter-weight construction for a given peak stress. Because of the numerous penetrations of the first wall for vacuum and injection ports, there are not many locations suitable for continuous reinforcing rings. Rings spaced about 1 m apart and straddling the vacuum ports clear the injection ports and can provide adequate support for the vacuum wall and its coolant system. 5.1. Effect o f ring weight, shell weight, and pressure differential on ring dimensions

Consider stainless steel reinforcing rings of width c, depth b, and spacing L connected by sections of thin-walled, circular cylindrical shell of radius R and thickness h (see fig. 8). Assume that the rings are hung from the top. The bending stress in the ring resulting from the weight of the shell which it supports is [5] aOb = + 67hR2(L - c) [1 + ½cos 0 - (Tr - 0) sin 0]. cb2 (11)

The bending stress in the ring from its own weight is [5] gob = + 6 7 R 2 [1 +½COS 0 -- (~r-- 0)sin 0]. b

(12)

The circumferential membrane stresses for both these loadings are negligible compared to the bending stresses. The membrane stress in the ring from atmospheric pressure p acting on the evacuated ring and shell structure is [5] Gem =

-pR

2#x/ChR + c 2phx/-h-R + cb

'

(13)

HE L L WEIGHT

_

S

TOTAL

I-

-I0

--

-5

--

RING

EXTERNAL PRESSURE

O

5

I0

15

RING DEPTH b,

20

25

cm

Fig. 8. Maximum stress in reinforcing ring resulting from various loads.

where # is defined in table 1. The total circumferential stress is the sum of the components given by eqs. (11), (12), and (13); its maximum occurs at the inside surface at the top (0 = 0). Fig. 8 shows the three components and their total as a function of ring depth b, for L = 1 m, c = 5 cm, h = 2 cm and parameters given in table 1. (This value o f h is based on the buckling analysis described in the next section.) For a design stress of 10 000 psi, fig. 8 indicates that a ring depth of about 19 cm would be required. If the ring width is increased to 10 cm, the required ring depth is reduced to about 15 cm. The average mass of these ring and shell structures per unit axial length of shell are 3340 and 3870 kg/m for the 5 and 10 cm width rings, respectively. These contrast with 7030 kg/m for the uniform wall shell. The ring dimensions determined above would not be considered unreasonable for most structural applications of the size of the first wall torus; however, for an EPR design, rings of these sizes might be difficult to cool, interfere with the blanket, or cause assembly problems. Consequently, methods were investigated for reducing the size of the rings while maintaining a design stress of 10 000 psi and the 1 m ring spacing.

400

CK. Youngdahl / Stress analysis for a Tokamak reactor

5.2. Effect o f support location on ring dimensions One way by which smaller rings could be utilized is by moving the support location away from the top or bottom of the ring. Consider symmetric supports located at an angle/3 from the top of the ring. The bending stress in the ring resulting from the weight of the shell which it supports is [5] aOb = +- 6 7 h R 2 ( L - c) K(O, cb ~

/3),

(deg.) 0 22.5 45 67.5 90

0ma x (deg.)

K(0max, ~3)

0 22.5 90 67.5 66.8, 113.2

1.50 0.619 -0.308 0.175 -+0.0917

(14)

and the bending stress in the ring from its own weight is O0b

Table 3 Maximum stress location for ring-reinforced shell

= + 6"yR2IK(O,/3). - b

(15)

In the above, K is defined by g(o,/3) = cos/3 - (zr -/3) sin/3 +(21- + sin:/3) cos 0 + 0 sin0, 0~<0 -,-
thickness h of 2 cm. This maximum stress is the sum of the bending stresses given by eqs. (14) and (15), with K being given in table 3, plus the membrane stress resulting from atmospheric pressure given by eq. (13). A significant reduction in ring depth is possible for a given maximum stress if the supports are moved away from the top of the ring. This occurs because the ring tends to distort less under a distributed weight if the supports are moved out. Table 4 gives ring dimensions and vessel mass per unit axial length for various shell constructions. For the thinner ring configurations, the rings account for only about 10% of the total mass, so further mass reduction

Fig. 9 shows K as a function of 0 for several values of /3. The maximum stress occurs where [KI is a maximum; let 0max denote this location for a given support location. Table 3 gives 0max and K(0max,/3) for various/3. Fig. 10 shows the maximum circumferential stress in a ring supported at various locations as a function of ring depth b for a ring width c of 5 cm and a shell

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30

60

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120

150

0 180

Fig. 9. K(O, [3) for various values of support location t3.

I

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5

I0 15 20 RING DEPTH b, cm

1

I

, 25

Fig. 10. Maximum stress in reinforcing ring supported at various locations.

-

401

C.K. Youngdahl / Stress analysis for a Tokamak reactor Table 4 Shell mass comparison for various vessel designs * Configuration

h (cm)

/3 (deg.)

c (cm)

b (cm)

Mass per unit length, (kg/m)

Uniform wall Ring-reinforced Ring-reinforced Ring-reinforced Ring-reinforced Ring-reinforced Ring-reinforced

6 2 2 2 2 2 2

0 0 0 22.5 45 67.5 90

-

19 15 12 8 6 4.5

7030 3340 3870 2930 2690 2580 2490

5 10 5 5 5 5

* For maximum stress of 10 000 psi.

would have to come primarily from a decrease in the shell wall thickness h; this thickness is determined by buckling considerations, as discussed in section 4.

5.3. Tapered rings Another technique for reducing the dimensions of a reinforcing ring is to vary the depth of the ring around its circumference. Since the stress varies around the circumference of a uniform ring, as shown in fig. 9, a more uniform stress state would be achieved by reducing ring dimensions in lightly loaded regions. Because the maximum stress in a ring supported at the top is at the top, its depth at the sides can be reduced by about one-third. The rings would then not intrude as far into the inner blanket region where space is at a premium. A similar reduction would not be possible for a ring supported away from the top because the maximum stress location then moves toward the side of the ring. A tapered ring would be more expensive to design and fabricate than a uniform ring. On the other hand, if the ring is supported at the top it might be easier to combine its support structure with the vacuum port support structure and thereby result in less interference with blanket assembly and disassembly than a ring supported at the sides.

5.4. Effect of coolant panel weight on ring dimensions In the analyses described in subsections 5.1 and 5.2 above the weight of the coolant panels and other coolant system hardware supported by the vacuum vessel was not included in sizing the reinforcing rings. Let the weight of the coolant panels, vacuum vessel

coolant channels, and associated piping, manifolding, and supports be equivalent to an additional thickness h* of stainless steel shell. The circumferential bending stress in the ring corresponding to this additional weight is

Oob = -+

67h *R 2L K "0 ~-~ ( ,/3),

(17)

where K is again defined by eq. (16). Fig. 11 shows the maximum stress in the ring resulting from the weight of the shell [eq. (14)], the weight of the ring

-20 =4cm

;

"~

-15

-5 I0

i !ilcm 15 20 RING DEPTH b,

25

30

cm

Fig. 11. Effect of coolant panel weight on reinforcing ring depth; support at top.

402

CK, Youngdahl / Stress analysis for a Tokamak reactor

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- -

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q

-3.0 -20

------2.5

h=2cm

4¢m

p=15psi

2 3 h@:

-15

o-x~(O) R= 240 cm

/ -2o -

/

%

t J

/ / /

/3=45*

~

/

-

/

-I0

I

_

--

C =Scm

-5

/

/

- I .0

--

h=2cm

L=Im

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R= 240 crn

//

-0.5

I 0

5

I I0 RING DEPTH b, crn

I 15

20

Fig. 12. Effect of coolant panel weight on reinforcing ring

0

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L

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5

I0

15

20

depth; support at 45 ° .

25

RINGDEPTH b, cm Fig. 13. Discontinuity stresses at shell-ring intersection resuiting from vacuum load.

[eq. (15)], the pressure differential across the vacuum wall [eq. (13)], and the weight of the first wall coolant system [eq. (17)] for a 5 cm-wide ring supported from the top. Fig. 12 gives the maximum stress if the supports are moved to a 45 ° location. For the EPR design, h* is approximately 2 cm. The design stress of 10 000 psi then corresponds to ring depths of 26 cm and 11 cm for these cupport cases. Increasing the ring widths to 10 cm results in a decrease of required ring depths to 19 cm and 8 cm for/3 = 0 ° and/3 = 45 °, respectively. 5.5. Stress concentrations The geometric discontinuity in wall thickness at the junction of ring and shell results in the portion of shell near the relatively stiffer ring being more highly stressed then the remainder of the shell [7]. The stress concentrations caused by the pressure differential across the vacuum vessel wall and by the average difference between shell temperature and ring temperature are described here. The axial bending stress in the shell near the ring

resulting from the external atmospheric pressure p is -

exp

I

(18)

The circumferential membrane stress O0m in the shell at the ring is equal to the corresponding stress in the ring as given by eq. (13); it decays with the same exponential behavior as Oxb to - p R / h as x increases. Fig. 13 shows the stress components Oxb and O0m in the shell at the ring as a function of ring depth. For the ring sizes treated here, the discontinuity stresses are less than 2000 psi. If the ring is at an average temperature Tr and the shell is at an average temperature Ts, the axial bending stress and circumferential membrane stress in the shell at the ring are 3tl2 cb Ea(Tr

Oxb(0)

= -+

O0m(0)

- 2 / a h % / ~ + bc

2uhv~

Ts)

+ bc

bc Ea(Tr - Ts)

'

(19)

403

C.K, Youngdahl / Stress analysis for a Tokamak reactor

400

I

I

I

200

I

%co)/(T r -T s) 300

. . . .

O-xblOl/lTr_Ts ) I SHELL

-----

5m/%-rs

/

150-

I R~NG

200 --

c=10era_L?.___ ~

IOO90*- 8 0 ~70m o

~.

I00

C

o

m SO~o ,.i

,," 5 0 o.

(.9

z40B

-10O

J

30-

~ " ~

-200

~ 20-

R = 240 cm h=Zcm

-300

-400

9

15-

I

I

5

10

I

15 RING DEPTH b, cm

I 20

25 I0

Fig. 14. Discontinuity stresses at shell-ring intersection resuiting from temperature difference. The variation with x is similar to that in eq. (18). The corresponding circumferential membrane stress in the ring is

2I~-R'h Eot(T r - Ts) ODin =

21shv:Rh + be

(20)

The stress components resulting from the temperature difference Tr - Ts are shown in fig. 14. For rings of the size considered here, the shell stresses are approximately 200 psi/°F, so reasonably uniform temperature distributions must be provided by the structural coolant system to avoid large localized thermal stresses.

6. Buckling A detailed buckling analysis of the EPR vacuum vessel composed of a ring and stringer reinforced polygonal shell of an approximate toroidal shape would be beyond the scope of the conceptual design effort. Buckling loads for some simpler idealized structural configurations were determined to provide a basis for sizing the vacuum wall thickness.

/

r

,

1.5

2

2.5

i

i

3

4

WALL THICKNESS

i

,

I

5 6 h,cm

i

7

8910

I

Fig. 15. Buckling pressures for various shell structures.

Consider first a uniform-wall shell without reinforcing rings or stringers. Sobel and Flfigge [8] give buckling pressures for thin-walled toroidal shells as a function of the ratios of major to minor radius, Ro/R, and minor radius to wall thickness, R/h. Their results are shown in fig. 15 as a function of wall thickness for R0 = 6.25 m and R = 2.4 m. For comparison, the buckling pressure of a long cylindrical shell of radius R is shown also; it is given by [5] (P)buck = Eoh 3/4R 3.

(21)

The toroidal shell can withstand a considerably higher pressure because of the double curvature of its wall. The polygonal EPR vessel is composed of truncated sections of cylindrical shells. Although it lacks the double curvature of the toroidal shell, its beveled joints should provide a stiffening effect. Consequently, the results for a polygonal vessel should lie between those for toroidal and cylindrical vessels. The buckling mode cohfiguration for a toroidal shell are quite complex; moreover, for EPR dimensions, Sobel and Fldgge show that the buckling pressures for the three lowest modes almost coincide. In commenting on these re-

404

C K. Youngdahl / Stress analysis for a Tokamak reactor

sults, Baker et al. [9] state: 'For values of [Ro/R] less than 6.3, the buckling pressure should be verified by test'. The ratio Ro/R for the EPR vessel is 2.6, and the buckling modes corresponding to its polygonal shape are probably more complex than those for the ideal toroidal shell. Fig. 15 also shows buckling pressures for reinforced cylindrical shells. For short sections of cylinder between rigid reinforcing rings spaced a distance L apart, the buckling pressure is [5] 2Eh (P)buck

X

-

R(2

(

+q2)

q4

\n2(1 +q2)2

+ n2h2(1 + q2)2 ~ 12(1 - v2)R 2 ] '

(22)

=

(23)

and the mode number n is the integer which gives the lowest value for p for each combination of R, h, and L. The results shown in fig. 15 are for L = 1 m and show that the buckling pressure for a ring-reinforced cylindrical shell is much higher than that for an unreinforced cylinder. The EPR rings are too thin to be considered rigid, but the stringers betwec, the rings also provide some stiffening for the shell. The last curve on fig. 15 shows buckling pressure for a cylindrical shell panel between rigid stringers, given by [5]

(P)buck kEoha/R3, =

7. Strain cycling of coolant panels The coolant panels are subjected to two types of thermal cycling, the burn cycle of approximately 1 min duration and the reactor start-up/shutdown cycle of widely variable duration. Fig. 16 shows idealized temperature variations in the hottest EPR coolant panel during a typical burn cycle of duration At [I]; Ti(t) is at the interface with the coating, To(t) is at the outer surface exposed to the coolant, and A ~ t ) = ri(t) - t o ( t ) .

where

q lrR/nL

should be performed to verify this and to provide a reliable basis for reducing the wall thickness.

(24)

where k = 6.1 for stringers at 30 ° intervals. The reinforcement again adds significant buckling resistance to a cylindrical shell. The stiffening effect of the rings has been neglected here; on the other hand, the stringers are too thin to be treated as rigid. Because of the flexibility of the reinforcing members, the buckling mode for the EPR polygonal shell probably involves combined deformations of the rings, stringers, and shell wall. A choice o f h = 2 cm gives a large factor of safety over atmospheric pressure for the buckling pressures shown in fig. 15 for the two types of shell reinforcement. Even though these cases both involve elements of non-conservatism, the factor of safety should be large enough to assure a safe design. However, a more detailed investigation

(25)

The strain range for the burn cycle depends on the difference between the maximum and minimum values of AT, and the strain range for the reactor startup/shutdown cycle (hereafter referred to as the ON-OFF cycle) is a function of the average value of AT during the burn cycle. Two support configurations will be considered here: (a) the panel is constrained against bending and expansion by supports at a con-

800~

2I

~: 7 0 0

~e o o ~ 5°Oto

At i

-

I

=

i

0

15

30 45 TIME , sec

15

30 45 TIME, sec

i

60

,°°t °;F, 0

,

,

, 60

Fig. 16. Idealized temperature variation in hottest coolant panel during burn cycle.

405

C.K. Youngdahl / Stress analysis for a Tokamak reactor

stant temperature Ts during the burn cycle (fixed supports); (b) the panel is constrained against bending but not expansion (sliding supports). The latter support configuration is the one chosen for the EPR design since it corresponds to a significantly longer fatigue lifetime for the coolant panels. 7.1. F i x e d supports

It is assumed here that no sustained mechanical loads from the weight of the structure or external pressure are transferred through the supports to the coolant panels. Such loads could lead to thermal ratcheting, where the mechanical loads provide a bias so that thermal cycling results in plastic strain accumulation in one direction and failure. Since the coolant panels are nearly as stiff as the vacuum wall, it is difficult to design fixed supports which do not transfer a portion of the mechanical loads to the panels; consequently, the sliding supports treated later would be preferable. For fixed supports and approximately linear temperature variation through the panel thickness, the strains ei and Co at the coating interface and cooled surface, respectively, are t2

ei(t) -

-1-u

Take the support temperature Ts = 450°F, which is about the average of the coldest an d hottest panels. Using temperature Values from fig. 16,

c* = - 0 . 0 0 3 2 ,

(ci),~g = 0.0021.

(30)

For the coldest panel, c* would have about the same magnitude but the opposite sign. A mean strain alters the number of fatigue cycles to failure for a given strain range, with a mean tensile strain decreasing the lifetime and a mean compressive strain increasing it. However, the mean stain during a burn cycle may not equal the average value c* computed above because of stress relief during the operating cycle of the reactor, resulting from self-annealing of the average thermal stresses at reactor temperatures [3]. Fig. 17 shows an idealized reactor operating history. The ON and OFF phases are assumed to be approximately equal in duration and to be very long compared to the burn cycle duration At. The average interface strain in the hottest panel as it varies during the operating cycle is shown in fig. 17; i.e. during the ON phase (ei)avg is the average value over the burn cycle At allowing for stress relief during the operating

[ T i ( t ) - Ts],

°°. F I I I I I I OFF

Co(t)-

[To(t) - Ts].

--

1-v

From fig. 16 the variation of the strain ci at the interface is more severe than the variation of %. Let e* be the average value of ci during the burn cycle assuming no stress relief has occurred previously. Then,

c~-

l--t)

[(ri)a~ - rs],

TI ME.---~-

OPERATING CYCLE

(26)

,l I

(27)

Y

I

I

I

CASE A,NO STRESS RELIEF

I I I I V

CASE B, STRESS RELIEF DURING ON PHASE

where

f

(Tilavg = Alt

ri(t) dt ~

½(r2 + r~)

(28)

At The strain range during the burn cycle is (Ci)rng = I(ei)max

-

I';II

(Ci)minln t

k

k

y

k

TIME~

CASE C~ STRESS RELIEF DURING BOTH PHASES

Fig. 17. Effect of stress relief during operating cycle on -

~ 1--v

(r2,-rl).

(29)

(ei)avg.

406

C.K. Youngdahl / Stress analysis for a Tokamak reactor

cycle, and during the OFF phase (ei)avg is the strain allowing for stress relief. It is assumed that the supports and panels come to a common temperature during the OFF phase, which may be Ts, room temperature, or somewhere in between. If no stress relief occurs (Case A) then (ei)avg = e* [eq. (27)] during the ON phase and (ei)avg = ei = 0 during the OFF phase. If only the temperature in the ON phase is high enough to cause stress relief (Case B), then stress relief will occur during the first ON phase when the panel is permanently shortened but not in later ON phases when the strain is zero; since the panel is shortened, a residual tensile strain occurs in the OFF phases. If the OFF phase temperature is also high enough to cause stress relief (,Case C), then the average strain decays during each ON and OFF phase. All these phenomena depend on temperature, material history, radiation history, and operating cycle variations. Both strain range and mean strain are important in determining the number of cycles to failure. Table 5 lists these quantities for the hottest panel for both the burn cycle and operating cycle and for each of the stress relief cases A, B, and C shown in fig. 17. For Case C, the expected lifetime [1] is 104 burn cycles or 300 operating cycles, whichever comes sooner, for the hottest panel with fixed supports. The coldest panel would have a similar lifetime, and intermediate panels would survive longer.

surface are ot -2(1- - v) AT(t),

ei(t)-

ot - - - - -- - ~v) A t ( t (31) ). eo(t) = +- 2(1

The strains ei and eo have the same magnitude and opioosite signs. Again letting e* be the average value of ei during the burn cycle for no previous stress relief, then e*-

2(1 - v )

(AT)avg '

(32)

where 1 P (AT) a,~ - At J AT(t) dt. At The strain range during the burn cycle is

(33)

(ei)rng = [(~i)max - (ei)minlAt ot - _ _ [(AT)max -- (AT)mini . 2(1 -- v)

(34)

For the hottest panel (fig. 16) (AT)avg ~ 100°F, (AT)min

~

(AT)max ~ 180°F (at t2),

60°F (at t l ) ;

(35)

for these values e~' = - 0 . 0 0 0 7 1 ,

(ei)rng = 0.00085.

(36)

f

Table 5 still pertains, but the values of e* and (ei)rng are much lower than those for the fixed support configuration. Again assuming that stress relief Case C is appropriate, fig. 18 shows the expected number of burn cycles to failure as a function of (AT)max - ( A T ) r a i n

7.2. SlMing supports

The supports here are assumed to keep the panel fiat but not to prevent expansion or transmit any structural load to the panel. The strain variations at the interface and cooled

Table 5 Strain combinations in hottest panel. Stress relief case

Strain

range

Mean strain

Operating cycle

Burn cycle A

B

C

A

B

C

(ei)rng

(ei)rng 0 (after first ON phase)

(ei)rng between 0 and e~

leVI 1 -~-Ie i, I

[e~[ 1 ~-Ie i, I

2leVI

e~

0

CK. Youngdahl / Stress analysis for a Tokamak reactor 600

\

I

I I I IIII I

,

500 -

-

~

,u. 400 ~ .g

-

-

I

I IIPlll

I

I

I IIIIII

300

-- 250 OPERATINGCYCLES

ioo

URN CYCLES

-

~so g

_

,00

-

50

I

0

I 103

i lililJ]

I 10 4

i lilllll

./. 105

temperature differences given in eq. (35), the expected lifetime for coolant panels on sliding supports is more than 106 burn cycles or l 0 s operating cycles, whichever comes sooner.

-- 2OO

_~ 300 -

-

407

I IIIJil

0 106

CYCLES TO FAILURE

Fig. 18. Burn cycle and operating cycle lifetimes.for coolant panels on sliding supports.

and expected number of operating cycles to failure as a function o f (AT)avg for sliding supports [1 ]. F o r the

References [1] W.M. Stacey, Jr. et al., ANL/CTR-76-3, Argonne National Laboratory (1976). [2] W.M. Stacey, Jr. et al., Tokamak experimental power reactor, ASME Paper No. 76-WA/NE-11 (1976). [3] D. Burgreen, Design Methods for Power Plant Structures (C.P. Press, 1975). [4] H. Chung, ANL/CTR/TM-69 (1976). [5] R.J. Roark and W.C. Young, Formulas for Stress and Strain, 5th edn (McGraw-Hill, 1975). [6] W. F1Fagge,Stresses in Shells, 2nd edn (Springer-Verlag, 1973). [7] S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, 2nd edn (McGraw-Hill, 1959). [8] L.H. Sobel and W. Fl~lgge,AIAA J. 5 (1967) 425. [9] E.H. Baker, L. Kovalevsky and F.L. Rish, Structural Analysis of Shells (McGraw-Hill, 1972).