Commission of the European Communities

The computation of shakedown limits for structural components subjected to variable

thermal loading — Brussels diagrams

Report

EUR 12686 EN

Commission of the European Communities

B,

The computation of shakedown limits for structural components subjected to variable

thermal loading — Brussels diagrams

A.R.S. Ponter, S. Karadeniz, K.F. Carter University of Leicester

Department of Engineering University Road

Leicester LE1 7RH United Kingdom

Contract No RAP-054-UK

Final report

This work was performed under the Commission of the European Communities

for the Working Group 'Codes and standards' Activity Group 2: 'Structural analysis'

within the Fast Reactor Coordinating Committee

Directorate-General Science, Research and Development

PARI. FÜ^P.

N.C./EUR

L* , « l U l i *.

1990 EUR 12686 EN ÈUR

Published by the COMMISSION OF THE EUROPEAN COMMUNITIES

Directorate-General Telecommunications, Information Industries and Innovation

L-2920 Luxembourg

LEGAL NOTICE Neither the Commission of the European Communities nor any person acting on behalf of the Commission is responsible for the use which might be made of

the following information

Cataloguing data can be found at the end of this publication

Luxembourg: Office for Official Publications of the European Communities, 1990

ISBN 92-826-1340-2 Catalogue number: CD-NA-12686-EN-C

© ECSC-EEC-EAEC, Brussels • Luxembourg, 1990

Printed in Belgium

C O N T E N T S

Page

Notations and some definitions V

Foreword and Executive Summary VIII

PART I - SUMMARY AND CONCLUSIONS 1

1. Introduction 3 2. The general problem and associated Brussels diagram 5 3. Typical Brussels diagrams 8 4. Example 1 - Pure type A. The Bree problem (figure 4) 11 5. Example 2 - Type A : General case. Torispherical shell

with through thickness temperature gradient (Figure 5) 12

6. Example 3 - Transitional A/B. Bree probelm with thermal transients (Figure 6) 12

7. Examples 4 and 5 - Type B : Thermal gradients along a shell surface. Cylindrical tube with axial temperature gradient (Figure 7), and circular plate with radial temperature gradients (Figure 8) 14

8. Example 6 - Type B : Moving temperature gradients. Cylindrical tube subjected to axial load and a traversing temperature discontinuity (Figure 9) 18

9. Structure of the-report 20 10. Conclusions 20

References 22

PART II - THE INFLUENCE OF TRANSIENT THERMAL LOADING ON THE BREE PLATE 25

1. Introduction 27 2. An upper-bound approach to calculations of ratchet boundaries 29 3. The transient Bree problem 34 4. Solutions to the Bree plate 38 5. Conclusions 53

Appendix 56 Tables 58 References 61

PART III - THE PLASTIC RATCHETTING OF THIN CYLINDRICAL SHELLS SUBJECTED TO AXISYMMETRIC THERMAL AND MECHANICAL LOADING 63

1. Introduction 65 2. Finite element technique 68 3. Variation of temperature along the length of a tube 71

III

4. The effects of strain hardening upon the ratchet boundaries 76 5. Experiments on thin cylinders subject to axially moving

temperature front (7) 82 6. Other types of thermal loading of cylinders 85 7. Tube subjected to a band of pressure and axially moving

temperature fronts 90 8. Conclusions 95

References 98

PART IV - INTERACTION DIAGRAMS FOR AXISYMMETRIC GEOMETRIES 99 1. Introduction 101 2. EECS-3 102 3. Cylindrical shells 107

'4. Baylac tests 107 5. Case 2 - The Bree problem 115 6. Case 7 - Cylindrical tube with variable thickness and

variable through-thickness temperature gradient 119 7. Conical tubes 125 8. Spheroidal and composite shapes 128 9. Case 4 - Cylindrical tube with spherical cap of same

thickness 133 10. Case 5 - Cylindrical tube with spherical cap of half thickness 133 11. ASME standard torispherical head 137 12. Case 6 - Cylinder to cone to cylinder (continuous angle) 139 13. Case 9 - Cylinder to cylinder by spheroidal sections

(continuous angle) 141 14. Conclusions 143

Tables 144 References 155

APPENDIX - EXTENDED UPPER-BOUND SHAKEDOWN THEORY AND FINITE ELEMENT METHOD FOR AXISYMMETRIC THIN SHELLS 157

1. Introduction 159 2. Upper-bound theory 159 3. Upper-bound method for axisymmetric shell elements 162 4. Extended upper-bound method 168 5. Computational method 170 6. Thermo-elastic stress due to an arbitrary axial temperature

distribution along a cylindrical tube 173 References 176 Figures 177

IV

Notation and Some Definitions

x = (x, y, z), t Space and time

O, e Uniaxial stress and strain

°ij' ei j » ij Cartesian components of stress, strain and

strain rate P P e

ij • ij Plastic component of strain and strain rate

X­L Plastic strain components (see Fig. 2 of Appendix)

c c aij ♦ ij Plastic component of strain rate in upper

bound theorem and corresponding stress C f*T C

Ae^j = J êijdt Accumulated strain over cycle, compatible ° with displacement increment Au

c^

P, P Primary load

X Scalar load parameter

XL» PL Limit load value of X and P , corresponding *

Oy

to yield stress oy

Pij» Pij Residual stress field; satisfies equilibrium equations within body and zero surface tractions on surface Sp

Ä P Ojj Elastic stress field corresponding to

primary load P ^ 0 o±j Elastic stress field corresponding to

temperature distribution 0 ap Primary stress, uniform stress corresponding

to load XP

op = Qp/Qy Non­dimensional primary stress

Yield stress

Oy Mean yield stress defined by equation (7) of Part IV

0, A0 Temperature, temperature difference

y Scalar temperature parameter

0max Maximum temperature during cycle

ØR Reference temperature

0O Uniform initial temperature of structure

Øc, 6C¿, Øcf Lower temperature; constant, initial and final in temperature transients of Part II

ØJJ, Öfl-L, 8jjf Higher temperature; constant, initial and final in temperature transients of Part II

at Maximum effective thermo-elastic stress in cycle

Defined in Fig. (8) of Part III

Non-dimensional thermal stress, equals crt/Oy for Bree problem

k Knock-down factor, a = k EaA0/2, i.e. k = 1 for Bree problem

E Elastic modulus

K Slope of plastic portion of stress-strain

curve = E/K

V Poisson's ratio

or Co-efficient of thermal expansion

*

-e o

* A6

EaA6 2ay

E S P R

Regions of the Brussels Diagrams, Elastic (E), Shakedown (S), Reverse Plasticity (P) and Ratchetting (R)

Ax Movement length along cylinder of temperature front

Ax = Ax//Rh Non-dimensional form of Ax

R Radius of cylindrical shell

h Shell thickness

r0, rlf v2 Radii of curvature of axisymmetric shell,

defined in Fig. (1) of the Appendix

S Surface of body

Sp Surface area where primary load P is

applied

S u Surface area where displacements prescribed

V Volume of body

V s Volume where shakedown conditions apply

Vp Volume where reverse plasticity conditions apply

VI

hth B = — — Biot number measures the relative resistances to heat transfer of the surface compared with the shell thickness, where ht = surface heat transfer coefficient, h = shell thickness and K = thermal conductivity

KT F = Fourier number, non-dimensional transient Pch time, where T = transient time in thermal

shock, p = material density and c = thermal capacity

3 = [3(1 - v )/R hz] '* Characteristic parameter

- VII -

FOREHORD AND EXECUTIVE SUMMARY

The Commission of the European Communities is assisted in its actions regarding fast breeder reactors by the Fast Reactor Coordina­ting Committee which has set up the Safety Working Group and the Working Group Codes and Standards (WGCS). The latter's mandate is to harmonise the codes, standards and regulations used in the EC member states for the design, material selection, construction and inspection of LMFBR components.

The present report is the final report of CEC study contract N° RAP-054-UK performed under WGCS/Activity Group 2 : Structural Analysis. It corresponds to one of the priority themes of WGCS/AG2, namely the development of simplified methods for the design. The final report issued in December 1987 was updated and revised for this publication.

LMFBR structures are characterised by low primary stresses (due to dead weight and internal pressure) and variable secondary (thermal) stresses of high amplitude. For certain combinations of geometry, material properties and loading, ratchetting may occur whereby the strains undergo an increment at each cycle of the applied thermal loading until either failure occurs, or the strain becomes limited by material hardening after the accumulation of unacceptably large strains. Ruling out this phenomenon at the design stage is an important task of the designer. This task is made very easy by the Brussels diagrams presented in this report. The Brussels diagram for a given geometry and a given type of loading shows four regions in the plane amplitude of the cyclic thermal stress versus magnitude of the primary stress :

- Elastic behaviour from the beginning.

- Shakedown : after the first cycles in which plastic strains occur a residual stress field is built up and only elastic strains appear thereafter.

- Reverse plasticity : in a limited volume plastic strains occur but they do not grow cyclically due to the constraint offered by the remaining shaked-down region.

- Ratchetting.

- VIII

The report presents the theory on which Brussels diagrams are based. It is the upper bound shakedown theory, specialised for axisymmetric shell elements and in which the upper bound is minimised by linear programming techniques. This theory is extended to the reverse plasticity region and has been implemented in two finite element axisymmetric shell programs which calculate a sequence of points on the ratchetting boundary. Three classes of problems are discussed :

- The uniaxial transient Bree problem. - The cylindrical tube subjected to axial load and stationary or

moving temperature discontinuity. - A range of Brussels diagrams for axisymmetric geometries and

thermal loadings typical of LMFBRs.

The discussion includes comparisons with some experiments and considerations on the sensitivity of the diagrams to the material assumptions.

Using this methodology it is possible to construct an Atlas of Brussels diagrams covering the whole range of structural geometries and temperature histories that may be encountered in LMFBR design problems. The present work, does not cover the creep range which will be treated in another report.

L.H. Larsson CEC/DGXII-D1

IX

Part I Summary and conclusions

A.R.S. Ponter

1. INTRODUCTION

The design of liquid metal cooled fast reactors poses a range of new

problems for the structural designer. Although the level of stress due

to dead weight and liquid pressure is low, usually less than 0.25 of the

yield stress, the occurrance of periodic thermal transients can induce

substantial thermal stresses which, in the most severe conditions, can

exceed twice the yield stress. There are two broad classes of phenomena

involved. Turbulent mixing of hot and cooler liquid sodium above the

reactor core can cause rapid temperature fluctuations of a moderate mag­

nitude (less than 70°C) over a short time scale. Major peaks and troughs

at a fixed material point in the above core structure are separated by time

intervals of the order of seconds. The main concern in this case is the

possibility of thermal fatigue in the form of thermal stripping, but the

thermal stresses are not sufficiently large to induce structural distor­

tions, provided the temperature remains below the creep range. The second

class of phenomena are associated with the thermal transients which occur

when the reactor trips. It is expected that the number of such trips will

be relatively small, perhaps as many as 2000 during the lifetime of the

structure. The concern here is not so much low cycle fatigue but the

possibility that components will suffer increments of plastic strain and

displacement, which will accumulate to an unacceptable level of distortion.

The broad features of this phenomenon, which is referred to as "shakedown"

when it does not occur and "ratchetting" when it does occur, has been known

and understood for some time through the work of Miller [1], Parkes [2],

Bree [3] and Gokfeld and Cherniavsky [4]. But the solutions to specific

problems which these authors discuss have proved to be an incomplete pic­

ture of the range of circumstances which can occur in fast reactors. In

the late 1970's it was appreciated that a more systematic approach to the

problem was required which could place in the hands of the designer

sufficient information to allow him to quickly assess whether a particular

circumstance was likely to cause ratchetting. At the same time, it had

become clear that the generation of step by step finite element solutions

to specific problems failed to provide any general insight into the

behaviour of thermally loaded structures. One particular approach to the

problem was described by the author (Ponter [5]) where it was suggested

that the application of classical shakedown theory, the upper bound

theorem with an extension, could be used to construct generalised "Bree"

diagrams for a range of structural components and temperature histories.

This suggested that an "Atlas" of such diagrams, which later became known

as "Brussels diagrams" could then be used as a reference to demonstrate

the way differing types of thermal loading effected a range of structural

geometry, and thereby assist designers at the initial stages of design.

The realisation of this concept has proved more difficult than was

initially envisaged for reasons which, with the aid of hindsight, are

fairly obvious. If information of this type is to be used by designers

there must be a fair degree of confidence in the relevance and accuracy of

the diagrams. Traditionally, such confidence is built up over a period of

time through comparison with experimental results. For thermal loading

problems the range of data available is limited, and the process of forming

a comparison is quite a task in its own right. The second problem is the

reliability of the diagrams themselves. The simple examples discussed by

Ponter [5] were mainly problems where the mode of ratchetting, which forms

the input into the upper bound shakedown theorem, was either known or could

be sensibly guessed. In this case the exact, or near exact, solution to

the ratchet limit could be found with relative ease. For more complex

problems the optimal mechanism needed to be found. The development of the

finite element technique for axisymmetric shells which was capable of doing

this and the writing of the associated computer software has been no mean

task, but has now been achieved by Dr Carter with the assistance of a re­

search grant from the Science and Engineering Research Council of Great

Britain. Lastly the range of possible diagrams seemed to expand with the

length of the computer listing (the main programme has in excess of 5000

lines of code) and it has taken some time before they could be condensed

down into a smaller number of significant cases.

This introduction sets out, in fairly simple terms, what the Brussels

diagrams mean, how they were generated, and what the main classes of dia­

grams look like. The main body of the report is a more detailed discus­

sion of classes of problems with some comparisons with experimental results.

2. THE GENERAL PROBLEM AND ASSOCIATED BRUSSELS DIAGRAM

The general problem consists of a structural component which is sub­

jected to two separate loading systems. A constant load, which may be a

pressure loading or a localised loading is given by XP where X is a

scalar load parameter. In addition, the structure is subjected to a

cyclic history of temperature y0 (x,t), where y is a second scalar

parameter and 0 is a distribution of temperature which varies in both

space and time. The behaviour of such a structure for differing values

of X and y is quite complex, but we can summarise the behaviour in the

form of the general Brussels diagram shown in Fig.l. For ease of inter­

pretation the scalars X and y are not used as axes, but two equivalent

non-dimensional quantities; P/PL where PL is the limit load parameter

for yield stress ay at some reference temperature 0r and; Oţ/Oy where at is the maximum effective thermo-elastic stress due to y0 . For cases

where XP produces a uniform stress ap , then o-p/Oy is substituted for

P/PL . These are the quantities used by Bree [3],

The diagram has four separate regions, referred to as E, S, P and R.

- 5

The position of the boundaries and mode of behaviour in each region depends

upon the material behaviour assumed. There are, however, two basic models,

perfect plasticity and linear hardening, Fig. (2), which are sufficient to

encompass the range of real material behaviour. The behaviour within the

regions can be summarised as follows:

E: Purely elastic behaviour occurs. The elastic stresses nowhere exceed the initial yield surface.

S: During the first few cycles some plastic strains occur but they are limited in magnitude to the order of magnitude of elastic strain i.e. 0.1%. A residual stress field is built up which pulls the elastic stress into the yield surface. The boundary to the region ABD is the elastic shakedown limit, and no plastic deformation occurs after the first few cycles.

P: In some limited volume of the structure Vp , for shells a proportion of the thickness of the shell, the stresses cause plastic strains at the extreme of the thermal loading cycle, but the kinematic constraint of the remaining material in volume Vs prevents continued cyclic growth of displacement. After a few cycles, cyclic growth of dis­placement ceases and the accumulated strain remains small, of the order of elastic strains.

R: For a perfectly plastic material cyclic strain growth of the structures occur which become a constant increment per cycle after the first few cycles. The rate of growth can be significant for small excursions into this region. For a strain hardening material the rate of strain growth is initially close to the value for a perfectly plastic material with the same yield stress, but the rate then decreases until a limit­ing strain value is obtained. This process usually takes a signifi­cant number of cycles, in excess of 40-50 cycles.

The boundary between the S and R and the S and P region can be

predicted by classical shakedown theory and the cyclic hardening properties

of the material is not significant. The boundary between the P and R

region, however, is more sensitive to the cyclic material properties. Two

extremes can be calculated. We may assume that the material suffers no

cyclic hardening i.e. perfect plasticity or kinematic hardening, or, that the

material cyclically hardens to elastic behaviour, i.e. isotropic hardening.

The extremes are illustrated in Fig. (3). The behaviour of 316 stainless

steel lies midway between these two extremes. By looking at simple examples

°V°~y 1

A P / P L or CTp/o-y

EJ9_1 The general Brussels diagram

°-|

Fig, 2 Perfect plasticity and linear hardening

°"å

ACT—.

Isotropic hardening

316 S.S.

he Kinematic hardening

Perfect plasticity Ae

Fig._3 Cyclic stress-strain curves

- 7

(Ponter and Karadeniz [6] ) it becomes clear that the assumption of complete

cyclic hardening within the volume of material where reverse plasicity occurs

provides the more conservative boundary between the P and R region.

With this assumption it is then possible to define the ratchet boundary ABC

by using an extension of classical shakedown theory. The theory is des­

cribed by Ponter and Karadeniz [6] and all the diagrams in this Atlas are

produced using this theory. The Tresca yield condition is assumed and a

simple class of displacement field involving discrete hinge circles at nodal

points and uniform membrane deformation within elements. For limit load

calculations the assumptions are equivalent to the classic non-interactive

prismatic yield surface of Drucker and Shield [11]. The thermo-elastic

stresses are generated either analytically or by a finite element method

(using the code CONIDA kindly supplied by the UKAEA) and the optimal mech­

anism is found by converting the upper bound into a linear programming

problem, which is solved using a sparse matrix simplex method. A full

description of the theory and computational techniques are given in Part 2

for uniaxial problems and in the appendix to Part 4 for axisymmetric shells.

3. TYPICAL BRUSSELS DIAGRAMS

In the previous E.E.C, report [5] two classes of diagram were distin­

guished, termed type A and Type B . The two types are distinguished by

the following properties. In the general Brussels diagram, Fig. (1), when

the applied load P is zero and the value of 6 exceeds the line DB then

there exists a volume of the structure, Vp , where reverse plasticity occurs.

The volume can be found from the thermo-elastic solution as the regions of

the structure where the thermo-elastic stress history cannot be contained

within the yield surface by translating it by a rigid body translation in

stress space. We then imagine the structure with this volume VF removed.

If the reduced structure can now carry some applied load, then the region P

O p - H

♦J-—*T-t

2R -»■ x

*e° e,

e. a. ♦ t At

Fig._£ Example 1, Type A, the Bree problem

h=.0025

s

e 0 - ö o +

1m / y

y , ' Ito /

/ ^

1.16m

AØ

y

X

.12m ^ f \ .' \ \ Into y

y

.88m

i \ \

| V\ \ \

i \ \ i \\

h:

\ \ i \ \

! Il J .233m

Internai pressure P

h=.0025

Total length 1.5m

.10(

-± =2.08 -2- =0.00

■ i i—i—i i i — i — i — » -

Bree

A 1— \ 1 1 1 1 1 • 1 I l > I I 1-P. P,

cry(6R) GR=20°C

Fiq. 5 Example 2. Type A, Torispherical shell with through-thickness temperature gradient.

10

exists (the proof is given in [6]). A pure A type thermal loading problem

is one where this remains true however large the value of at/Oy , and it has

the property that there always exists some value of applied load P which can

be carried by the structure without ratchetting. On the other hand, if the

reduced structure is not capable of carrying any applied load, no P region

exists and the thermal loading problem is of type B , and shows a much

greater susceptability to thermal stress than type A .

Since that time is has become clear that a greater range of diagrams

exist across a complete spectrum with distinctive types of behaviour occur­

ring within each category. To indicate the range of behaviour currently

understood we describe five categories with examples, two each of type A

and B and a transitional type A/B , arranged in order of increasing

susceptability to ratchetting for low levels of mechanical load.

A. EXAMPLE 1. PURE TYPE A. THE BREE PROBLEM (FIGURE 4)

A thin walled tube is subjected to internal pressure and/or axial load.

A temperature difference A0 is induced with no thermal transients through

the thickness of the tube and then removed with or without a change in mean

temperature. The elastic stresses produced by the pressure are uniformly

distributed with value ap and the thermo-elastic stresses vary linearly

through the wall thickness with a zero value at the mid-thickness surface.

The ratchet boundary has been computed by Bree [3]. The volume Vp con­

sists of two surface layers and the remaining volume Vg forms a tube of

reduced thickness. As a result, a P region exists and the ratchet

boundary assymptotes to the O^/Oy axis as 0"p reduces to zero. For low

o"p a large value of at can be tolerated before ratchetting occurs.

11

5. EXAMPLE 2, TYPE A; GENERAL CASE TORISPHERICAL SHELL WITH THROUGH THICKNESS TEMPERATURE GRADIENT (FIGURE 5)

If the temperature gradient remains predominantly through the thickness

of the shell but the shell itself has a more complex geometry, including

changes in thickness and, perhaps, a spherical or torispherical end cap,

then the stresses induced by the applied load are no longer uniform and

the thermal stresses will be effected by the geometry. A typical example

is a torispherical cap subjected to internal pressure P with a uniform

temperature gradient A0 through the shell wall. The mechanism of plastic

collapse for at = 0 involves hinge circles which allow outward movement

of the shell cap as shown in Fig. 5. We find with increasing thermal stress

that the ratchet mechanism is very similar to this collapse mechanism and the

Brussels diagram is very similar to the classic Bree problem with the

horizontal axis given by P/PL . In Part 4 of this report a whole set of

such examples are analysed. We conclude that the classic Bree diagram gives

a conservative boundary for such problems provided that crp/ay is replaced by

P/ÍL , where PL is the limit load using the yield stress corresponding to the

maximum mean temperature during the cycle.

6. EXAMPLE 3, TRANSITIONAL A/B. BREE PROBLEM WITH THERMAL TRANSIENTS (FIGURE 6)

If the rate of surface heating in the Bree problem is sufficiently

great, the through-thickness temperature distribution has a transient phase.

The nature of the transients vary with the details of the surface temperature

history, but there are certain phenomena which always occur. The stress at

the mid-section surface does not remain at zero, as was the case in the Bree

problem, but can show a significant fluctuation. As a result, the entire

thickness of the shell experiences a fluctuating thermo-elastic stress dis­

tribution, and the volume VF can penetrate the full thickness of the shell.

- 12 -

k< y 6H¡

6£ = Constant

a. Rate element ond initial condition for thermal downshock

ÖHJ

ÕHi

©HC

— V ii

VJ . t „

A6

.

o^

IU

b. Temperature history of medium H ( Qr is constant) time

©Hi

Power on Shutdown trar.sient c. Temperature distributions

Compressive R

Pò»eroff

Fig. 6 Example 3, Transitional A/B, Bree problem with thermal transients

13

There is, in addition, the influence of the variation of the yield stress with

temperature. Both these effects cause the ratchet boundaries for both

positive and negative ratchetting to meet at a cusp which is marked as C in

Fig. 6. For zero applied load the compressive ratchetting will occur at point

D at a finite value of a^ . The exact geometry of the Brussels diagrams

depends, however, on a number of factors. These include whether the transient

is associated with an upshock or a downshock or a double-sided shock. In

addition, the effects of surface heat transfer, given by the Biot number, and

the rapidity of the surface temperature change, given by the Fourier number,

are quite significant. In Part 2 of this report a range of such cases are

discussed in detail.

7. EXAMPLES 4 AND 5, TYPE B: THERMAL GRADIENTS ALONG A SHELL SURFACE CYLINDRICAL TUBE WITH AXIAL TEMPERATURE GRADIENT (FIGURE 7), AND CIRCULAR PLATE WITH RADIAL TEMPERATURE GRADIENTS (FIGURE 8).

An important class of problems involves temperature gradients which

are predominantly along the shell surface. In fact, in reactor design the

tubes are relatively thin and through-thickness gradients are often small.

It may be expected, therefore, that many problems fall into this category.

We consider two examples which are typical of this type. Example 4

consists of a uniform cylindrical tube, subjected to an axial load and an

axial history of temperature which fluctuates between a uniform temperature

and a maximum temperature. The detailed temperature history corresponds

to those of an experiment carried out at EDF-SEPTENin Lyon, France. The

axial temperature gradients induce through-thickness hoop stresses which

can exceed twice the yield value of the material. In addition, axial

bending moments with maximum stresses of a similar order of magnitude as the

hoop stresses are also induced. The resulting Brussels diagram shows two

distinct branches AB and BC as~shown in Fig. (7). Along AB the

mechanism is a local ratchetting mechanism due to the applied load and the

14

axial bending moments and is similar to that of the classic Bree Problem.

Along BC a reverse plasticity mechanism occurs where the large hoop stress

CJA , together with the axial load ap , cause plastic strains at two

instants during the cycle. Most of the plastic strain is in the hoop

direction, but there is an increment of axial strain each cycle.

Example 5 is similar in nature but rather different in geometry and

serves to demonstrate that the characteristics of example 4 are shared by

other problems which have the same type of thermal loading. In this case a

circular plate is simply supported at its edge and subjected to uniform

lateral pressure P and a linear radial temperature gradient with a maximum

temperature at the centre of the plate as shown in Fig. (8). Despite the

considerable differences between examples 4 and 5 their Brussels diagrams

are virtually identical in form. Along AB the ratchet mechanism is the

same as the plastic collapse mechanism; the plate deforms as a cone. Along

BC a local ratchet mechanism occurs around the edge of the plate induced by

the large hoop thermo-elastic stress variation and the shear stresses in­

duced by the transmission of the pressure P to the edge support. Again

this mechanism is a reverse plasticity mechanism through the thickness of

the plate. (A lower bound solution has been given by Cocks [8]).

The rate at which ratchetting will occur for load points in excess of BC

in both examples depends upon the details of the material behaviour. The

best way of describing the severity of ratchetting is to say that it can be

significant (see solutions by Ponter and Cocks [9]) but it might well be

small. Some detailed finite element calculations by Webster et al [10] for

example 5 shows that ratchetting occurs but at a lower rate than along AB.

It is tempting to refer to the boundaries BC as weak ratchetting boundaries

and AB as strong ratchetting boundaries. There is a possibility that

loading in excess of BC could be allowed, but there is a total lack of

experimental data with which comparisons can be made.

15

O 0.1 0.2 0.1 0.4 0.; 0.6 0.7 o.e 0.9

u

« 1-8

*y IWC) 16

1.4

12

to

0.8

0.6

0.4

0.2

0.0

1 1 ■ ■ i 1 1 r 1 1 1 1 1 1— >>

^>^

^ ^ ^ ^ \

B

\ . Expt - ¿70KN \ ■+-

\ R

S \ >>» \

^ \ >» \

^ \ x \

V \ \ \

X \ N. \

^ \ E ^ \

^ A 1 1 1 1 1 i • i i ^ i ■ ■

0.0 0.1 0.2 0.3 0.4 0.5 06 .7 A .9 1 1.1 1.2

<rr IU°C)

emax(°C)

616

549

482

415

348

282

215

148

81

14

F\g._2 Example U. Thermal gradient along a cylindrical tube,- the Lyon lest. Type B

- 16

e

a.

Local shear mechanism

Global mechanism

Temperature independent yield stress o~y

Fig._8 Example 5, Type B, Circular plate, simply supported. uniform lateral pressure P and radial linear temperature gradient.

t i 17

•S * ,

This type of problem is discussed, together with some experimental data,

in Part 3 and also in Part 4. It may well be the most important class of

thermal loading for fast reactor design and it must be emphasised that the

Brussels diagram is very different to that of the Bree problem. Ratchetting

can ocur at zero applied load.

8. EXAMPLE 6; TYPE B; MOVING TEMPERATURE GRADIENTS CYLINDRICAL TUBE SUBJECTED TO AXIAL LOAD AND A TRAVERSING TEMPERATURE DISCONTINUITY (FIGURE 9~K The last example concerns the most severe thermal loading problems of

all where an axial temperature gradient along a tube traverses a length of

the tube so that the thermo-elastic stresses are swept backwards and forwards

over a significant volume of material. In examples 4 and 5 the volume VF

penetrates the thickness of the shell but remains relatively small compared

with Vs • In this example V-p can be large and consists of the volume of

material through which the high thermal stress is swept. In detail the

example consists of an axially loaded tube with a steep discontinuity of

temperature which repeatedly traverses a length Ax of the tube. The

position of the ratchet boundary depends upon the value of Ax = Ax/ /Rh

where R is the radius and h the thickness of the tube. A whole set of

ratchet boundaries are shown in Fig. (9) for a range of values Ax. For

small Ax the boundary is similar to that of examples 4 and 5 with two

parts, the upper part involving a reverse plasticity mechanism, mode II.

However, as Ax increases the severe ratchetting Mode III, which consists

of inward displacement over Ax , occurs at decreasing applied load until,

for Ax > 3, corresponding to Ax > 0.3R for R/h = 100, the ratchet

boundary reduces to near the elastic limit. In addition, the point C ,

where reverse plasticity begins when ap = 0 , occurs when a = ay , i.e.

at one half of the thermal stress of all the other cases. When the effect

of the variation of yield stress with temperature are included then severe

18

2R

L

**? m. X

to

— Cn^^r-s

e.-ae

Temperature r AX—1

Cold front—\ L

-Hot front

Axial distance X

»* cr„

! Generat yield

Mode I

=| 'Weak' i reverse ! plasticity

Model

= = ^ ^ J = ^ 'Strong* ^ ^ I local

mechanism

Fig._9 Example 6, Type B, moving temperature discontinuity traversina length Ax of cylindrical tube. AS = Ax / /Rh

- 19 -

ratchetting can occur at zero applied load. This problem, together with

some experimental data, is discussed in Part 2. This circumstance is

clearly very severe and may well be of considerable significance in fast

reactor design.

9. STRUCTURE OF THE REPORT

The body of the report is sub-divided into three sections which discuss

distinct classes of problems, which also correspond to three differing

methods of applying the upper bound theorem.

Part 2 discusses the transient Bree problem, posed as a uniaxial prob­

lem, where the mode of deformation is known and calculations for the ratchet

limit become a simple integration procedure. The shakedown theory and its

application are explained in full and a number of cases are discussed in­

volving both single sided upshocks and downshocks and double-sided shocks

in terms of the Biot and Fourier numbers. This class of problems form the

transition from type A to type B.

In Part 3 a detailed description is given of the behaviour of a cylind­

rical tube subjected to axial load and either stationary or moving temperature

discontinuity. The emphasis is placed upon the sensitivity of the diagrams

to the material assumptions and correlation with the limited available

experimental data.

The final part 4 contains a wide range of diagrams involving axisymmetric

geometries using a general purpose computer code. These include a class of

problems originally suggested by Working Group 2 of the EEC Fast Reactor Co­

ordinating Committee and are known as the Bergamo Set. A description of the

computational techniques is given in the appendix.

10. CONCLUSIONS

The theoretical extremes of the ratchet boundary in the Brussels diagram

are; a vertical line, i.e. thermal stresses have no effect whatsoever and; the

20

ratchet limit coincident with elastic limit, so that no S or P regions

exist. The range of problems discussed in the report shows that both these

extremes can be achieved and that there is a gradation of cases of increasing

severity which lie between these extremes. The classic Bree diagram can now

be seen as a significant but particular case which lies within this spectrum

and that there are other forms of Brussels diagrams which are possibly more

significant for fast reactor design.

This work represents a first systematic attempt to understand the effect

of thermal cycling. It constitutes, in essence, a theoretical conjecture,

as the amount of experimental data currently available for low mechanical

loads is very limited, although what data there is tends to support the

theory. It is hoped that this work might stimulate further experimental

work, particularly into the behaviour of the less severe type B problems

such as examples 4 and 5 where "weak" ratchetting occurs.

All the calculations have been carried out in terms of a yield stress

Oy , usually taken as the 0.1% offset yield stress when comparisons are

given with experimental data. With such a definition the plastic strain

within the shakedown limit cannot be precisely known as it depends, amongst

other things, upon the initial state of residual stress in the shell. How­

ever, shakedown theory and experimental evidence indicates that the plastic

strain will remain of the same order as elastic strains, i.e. in the range

0.1 - 0.3% strain within the ratchet limit, and will rapidly grow once the

ratchet limit is exceeded, except in the "weak" ratchetting cases where the

strain growth is dependent upon the particular material and the details of

the loading history.

In conclusion the authors would like to emphasise that the Brussels

diagrams characterise a particular aspect of the behaviour of shells. The

extension to creep deformation and rupture is discussed in a companion

report [12], where comparisons, based upon the Brussels diagram, is made

21

with the solutions of O'Donnell and Porowski [13,14], and the CEA efficiency

diagram [15]. Work is currently underway within the UK to use these

results to established improved design code rules which will allow a more

accurate and flexible set of restrictions on thermal stresses than those

currently provided by either ASME Section III or RCC-MR.

The authors hope that this report will encourage an improved under­

standing of structural behaviour for these complex loading problems.

Suggestions of particular cases of interest would be welcomed.

References

[1] MILLER, P.R.

"Thermal stress ratchet mechanism in pressure vessels", J Basic Engineering, Trans ASME, Series D, 1959: 81, ppl90-196.

[2] PARKES, E.W.

"Structural effects of repeated thermal loading" in Thermal Stress, Benham et al. (eds), Pitman and Son Ltd., London 1964.

[3] BREE, J.

"Elasto-plastic behaviour of thin tubes subjected to internal pressure and intermittent high-heat fuses with application to fast nuclear reactor fuel elements", J Strain Analysis, 1967: 2, No.3, PP226-238.

[4] GOKFELD, D.A. and CHERNIAVSKY, O.F.

"Limit analysis of structures at thermal cycling", Sijthoff and Noordhoff, Alpen aan den Rijm, The Netherlands, 1980.

[5] PONTER, A.R.S.

"Shakedown and ratchetting below the creep range", Report EUR 8702 EN, Commission of the European Communities Directorate-General for Science, Research and Development, Office for Official Publications of the European Communities, L2985, Luxembourg.

[6] PONTER, A.R.S. and KARADENIZ, S.

"An extended shakedown theory for structures that suffer cyclic thermal loading" Parts I and II, J Appi. Mechanics, Trans ASME, 52, PP877-882 and pp883-889.

22

[7] CARTER, K.F. and PONTER, A.R.S.

"A finite element and linear programming method for the extended shakedown of axisymmetric shells subjected to cyclic thermal loading", Department of Engineering, University of Leicester, Report no. 86-XX, 1986.

[8] COCKS, A.CF.

"Lower-bound shakedown analysis of a simply supported plate carrying a uniformity distributed load and subjected to cyclic thermal loading", Int. J. Mech. Sci. 1984: 26, pp471-475.

[9] PONTER, A.R.S. and COCKS, A.CF.

"The incremental strain growth of an elastic-plastic body loaded in excess of the shakedown limit", Jn. Applied Mechanics, Trans ASME, 1984: Paper 84 - WA/APM-10, and "The incremental strain growth of elastic-plastic bodies subjected to high loads of cyclic thermal loading", Op. Sit. 1984: Paper 84 -WA/ADM-11.

[10] WEBSTER et al.

Private Communication.

[11] DRUCKER, D.C. and SHIELD, D.

"Limit analysis of symmetrically loaded thin shells of devolution", Trans ASME, Jn. Applied Mechanics, 1959: 26, p61.

[12] PONTER, A.R.S. and COCKS, A.CF.

"Computation of shakedown limits for structural components (Brussels Diagram) Part II - The Creep Range. Final Report RAP-066-UK (AD), EEC Fast Reactor Co-ordinating Committee, 1986.

[13] O'DONNELL, W.J. and POROWSKI, J.S.

Trans ASME, Jn. of Pressure Vessel Technology, Vol. 96, 1974, pl26.

[14] POROWSKI, J.S., O'DONNEL, W.J. and BADLANI M.

Welding Research Council Bulletin 273, 1982.

[15] CLEMENTS, G. and ROCHE, R.

General review of available results of progressive tests of structures and structural components. In: Ratchetting in the Creep Range by Ponter, A.R.S., Cocks, A.CF., Clement, C , Roche, R., Corradi, L. and Franchi, A., Report EUR 9876 EN. Directorate-General, Science Research and Development Commission of the European Community, Brussels, 1985.

23

Part II The influence of transient thermal loading

on the Bree plate S. Karadeniz, A.R.S. Ponter

1. INTRODUCTION

Many components of power producing plants are subjected to thermal

transients during start-up and shut-down conditions, but generally the time

scale of the temperature changes means that near quasi-static temperature

gradients are maintained. There are, however, some exceptional circum­

stances when extremely rapid changes induce transient thermal and thermo-

elastic fields. For example, the particular thermal properties of liquid

sodium and the rapid response of the core of a fast breeder reactor result

in rates of changes of temperature on the surface of components as high as

40 Ks_1[l].

In the context of the fast reactor it is- necessary to ensure that

structural components do not exhibit progressive distortion during the

reactors lifetime. The ASME [2] design codes treat stresses due to thermal

transients as F stresses, i.e. local stresses which can cause localised

plastic strain but are not a source of general deformation of the structural

component. As a result, they are only taken into account when assessing the

possibility of fatigue failure. It seems advisable to test this hypothesis

by the solution of some relevant problems involving only plastic deformation

(i.e. no creep) and this forms the main objective of this section. In fact,

we discover that transient thermal fields can have a significant effect upon

the potential for strain growth of components and, as a result, it appears

that the ASME code does not fully take into account the effect of F

stresses.

Some particular solutions to such problems have been published by

Goodman [3] who extended the classic Bree solution for quasi-static thermal

fields [4] to include through thickness thermal transients for an elasto-

perfectly plastic material, assuming a temperature independent yield stress

and computed the ratchet boundary where an increase in loading would cause a

rapid progressive plastic strain growth. From a computer study Goodman

found that for the single-sided thermal downshock there is a reduction in

27 -

Œ/O-y

I.HA PP \ \ \

Reversed \ \

plasticity \ P

K.H.

PP Perfect Plasticity

K.H. Kinematic hardening

I. H. Isotropic hardening (Complete cyclic hardening)

Ratchetting

R

10 X / X L

CTf' Maximum thermo-elastic effective stress XL

: Plastic limit load parameter

Fjg. 1_ Schematic representation of general problem

- 28

allowable thermal loading for small mechanical loading but the effect Is

less distinct for a double-sided downshock.

In this section of the report we discuss a wide range of such problems,

using the extended upper bound theory of Ponter and Karadeniz [5]. As the

problem involves uniaxial strain growth which is constant through the wall

thickness of the tube, the application of the theory is relatively simple.

This gives an opportunity for the discussion of the general techniques for

the construction of the Brussels diagram in the simplest of contexts. A

full discussion of the numerical techniques for a wider class of problems,

axisymmetric shells, is given in the appendix to Part 4. Those readers who

wish to avoid discussions of the shakedown theory may proceed to section (3).

In section (2), the theory is briefly described and in section (3) a set

of solutions of the Bree problem with thermal transients are presented.

2. AN UPPER BOUND APPROACH TO CALCULATIONS OF RATCHET BOUNDARIES

The general problem is shown schematically in Fig. (1) where a struc­

ture with volume V and surface S is subjected to constant loads XP

over part of S, S , and zero displacements over the remaining surface Su .

Within V a non-steady cyclic temperature field 0(x,t) occurs. The

material suffers both elastic strains e ^ and plastic strain e¿.¡ and the

total strain is given by

eij = eij + eij + a,Sij (8 " 6o> ( 1 )

where a is the linear coefficient of thermal expansion and 0O some con-p

stant reference temperature. If E±j is represented by one of the clas­sical plasticity models (perfect plasticity, kinematical hardening or iso­tropic hardening) then the general features of the structural responses are similar but not identical, and are shown schematically in Fig. (1) where at is the maximum effective thermo-elastic stress during the thermal cycle. There are four regions in this (A.,at) interaction diagram:

(1) Region E: the elastic stresses do not exceed initial yield

29

(2) Region S: some plastic strain occurs during the first few cycles but shakedown subsequently occurs

(3) Region P: cyclic plastic straining occurs over a confined volume but no incremental growth of the structure occurs

(4) Region R: for a perfectly plastic material steady incremental strain growth occurs. For the two hardening models the structure initially shows substantial rate of strain growth which assymptotes to a final value.

The detailed calculation of the boundary between the R region and

the P and S regions, line ABC can only be precisely defined for per­

fect plasticity where there is a distinct load level at which incremental

strain growth occurs. For the hardening models the boundary is less clearly

defined and varies, to some degree, with the definition of tolerable plastic

strains and the initial residual stress field assumed. It is observed how­

ever, for linear hardening models, that the line AB is defined reasonably

well for both isotropic and kinematic hardening by the perfectly plastic

shakedown boundary for the same initial yield stress. The boundary BC ,

however, is influenced by the presence of cyclic hardening, a feature

included in an isotropic hardening model but excluded from both perfect

plasticity and kinematic hardening. From experiments on a two-bar struc­

ture composed of 316 SS at 400°C Ponter and Karadeniz [5] showed that the

actual load level at which plastic strain increased rapidly occurred along

a line which lay between ABC' and ABC . For the two-bar structure the

lines BC' and BC are quite far apart, but this seems to be an extreme

case. For the classic Bree problem they are identical [5] and for cases

involving transient thermal loading, where the perfectly plastic solution

has been evaluated on a computer, the difference is small (see section 3).

We find that the evaluation of the line ABC' can be done directly from the

thermo-elastic solution and, as this line is conservative, we adopt it as

the most appropriate definition of a ratchet boundary. The resulting cal­

culation requires knowledge of the elastic properties and the variation of

a proof stress with temperature, i.e. the information which is customarily

30

available to a designer.

The theory may be described in two parts for the evaluation of line

AB and line BC' . The region ABD is characterised by the existence of

a residual stress field Pjj so that the stress history

CTij = *°ij (x) + CTij (X't) + Pij (2)

satisfies the yield condition

Õ < CJy (9(t)) (3)

where a is an appropriate effective stress, and C7y a yield stress which ~p ~6 varies with temperature. Here Oj_j and O M denote the elastic stresses

due to P¿ and due to 0(x,t) respectively where, in each case, u± = 0

or Su . Combination of (2) and the maximum work principle results in an

upper bound [5], which will now be discussed. c We define a compatable strain increment field de-n with a correspon-

c ding displacement field du^ . We will be concerned with problems where

the history of thermo-elastic stress follows a near linear path in stress

space and, as a result, plastic strains will occur at most at two instants

t = tx and t = t2 during the cycle, i.e.

c i 2 de-Lj = de-jj + de±¡ ( 4 )

1 2

where neither de^j nor de^j need be compatible. Using the maximum work

principle [5] the following upper bound can be evaluated

] [Gij (9X) dEij + ajj (62) dGij] dV > X j Pi du£ dS

f [a?j (tx) deíj + a?j (t2) dejj] dV (5)

31

where Qjj ( k) (k = 1,2) is the point on the yield surface with the k

associated plastic strain de-M at time t^ when the instantaneous tem­perature is 0fc . The evaluation of the bound can be more easily under­stood if inequality (5) is rearranged as

pi dui dS < f (a£j (Bj) - al3 (tx)) delj

-e + (aij (02) - aij (t2)) de ij dV (6)

The minimum value of the right hand side which yields the exact solution

requires both the optimal mechanism du^ "and the optimal sub-division into

dGji and de^j For the problems to be discussed here the mechanism is 1 2

known a - priori and the optimal sub-division requires either de-jj or de^j

to be zero. As a result, the minimum of the right hand side merely requires

the identification of the instant tx (or t2) for which (cjjj (Øj) -~0 •> ! 0"ij (t1)J d£ji is a minimum, which can be accomplished by a simple search

procedure.

When the maximum effective thermo-elastic stress exceeds ay(0x)+ cry(02)

then the total volume V comprises two sub-volumes; Vs where the history ~0 Gji may, by a rigid body translation in stress space, be contained within the yield surface at all times and VF , where Ot > ay(01)+ ay(02) where Q a-M cannot be so translated and must, therefore, cause reverse plasticity.

For the boundary BC', the upper bound (5) for positive cr now has

the form J [Qij (0X) del] + oli ( 9 2

) d£ij]dv > X P^du^ ds

f *>fì i ~6 2 , + d i j ( t x ) d e u + a i j ( t 2 ) d e i j ] dV

32

o co

-«—

f1 ^-f h

* CT,

Fig. 2 The Bree problem and the definition of VF and V«;

j [â±j (tx) + o±J (t2)] de j dV (y) vF

the formal proof of which is given in the Appendix.

An important corollary to this result is that the shakedown condition

can only be satisfied if there exists a region of Vg capable of transmit­

ting the load AP^ through the structure. For such problems the structure

is capable of carrying some load in excess of the reverse plasticity limit

and a P region exists. Such problems have been termed type A by Ponter

([5] of Part 1) and include the classic Bree problem. However, the volume

of reverse plasticity Vp contains a mechanism which can be activated by

the load AP^ then ratchetting can occur once the reverse plasticity limit

is exceeded and no P region exists. This situation has been termed a

type B problem. The transient Bree problem discussed here has features

of both situations and therefore forms a transitional type A/B .

3. THE TRANSIENT BREE PROBLEM

Consider the problem shown in Fig. (2) where a plate of thickness h p is restrained from curvature and subjected to a constant average stress a

in the x direction and zero average stress in the z direction. A

cyclic thermal history 0(y,t) is created by cyclic variation of the sur­

face temperature 6(0,t) and 0(h,t) .

If we adopt a Tresca yield condition then the plastic strain field for p positive a has the simple form

C c e c d£x = constant, dey = 0, dez = - dex (8)

c c and dux = dex . x (9)

The bound (6) becomes, for-small A0,

toy (9j) - ôx (tj)] dy , • (10)

34

and the exact solution requires the location, at each y , of the instant

t, , when the integrand is a minimum. For negative cr , the strain field

is reversed in sign and the corresponding result is:

,h Ic^lh < [ [cry (Øj) + ax (t^)] dy . (11)

In both (10) and (11) the optimal choice of t yields equality. For large

A0 when a volume Vp exists, the corresponding results are; for positive

cA < f ' j [âjítj) + <jj(t2)]dy + | 2[ay(01 )-âx(t1)]dy + | ' jKâxí t^+â^tpjdy

and for negative a

rh.

( 1 2 )

l^lh ^ J ^[c&V+iSctpidy + J 2[ay(02)+ax(t2)]dy + | 3j[(ax(t1)+ax(t2)]dy

( 1 3 )

where h , h and h are shown schematically in Fig. (2). It can be

seen that, in all cases, the problem is reduced to a single integral.

The calculation has been carried through for three separate cases; a

single-sided downshock, a single-sided upshock and a double-sided downshock.

The solutions are dependent upon the non-dimensional groups which govern the

transient thermal distribution. We assume a linear heat transfer relation­

ship between the temperature in the media 0H and 0C within which the

temperature changes take place and those on plate surfaces 0(y = 0) and

0(y = h);

QH = - ht (0(0,t) - 0H (t))

= + ht (0(h,t) - 0C (t))

(1A)

35

where ht is the heat transfer coefficient and Qţj and Qc are the heat

transfer per unit area through the plate surfaces in the y direction.

The plate material itself is characterised by a coefficient of thermal con-

1 0 position •£

Fig. (3): Transient temperature profiles due to thermal downshock on

one surface, B = 810, F = 0.0056.

36

ductivity K , density p and specific heat c . The transient tempera­

ture fieids [3,7], corresponding to the media temperature history of the form

shown in Fig. (4) where the temperature changes between its extreme values

at a constant rate in time Ţ , are functions of four non­dimensional groups,

the Biot and Fourier numbers B and F , non­dimensional distance and time.

e = f (B , F , I , J ) (15) h T

hth KT

where B = —r— and F = K pch2

The Biot number measures the relative resistance of the plate surface and

plate thickness to heat transfer. In the context of the sodium cooled

reactor the relevant range of values will be characterised by extreme values

B = 160 and B = 810. We find, in fact, that the solutions are insensitive

to B in this range as, effectively, the sodium/steel interface has neg­

ligible relative resistance to heat transfer.

The Fourier number F indicates the speed of heating or cooling of the

plate. Thus a large value of F implies a very slow rate of change of

temperature. As a wide range of values of F are possible we compute

solutions for 0.0014 < F < 50 which covers a practical range. The details

of the thermo­elastic solutions are given by Karadeniz [6],

In order to include the effect of temperature on the stress distribu­

tions it is assumed that in the reversed plasticity region, where the his­

tories of peak stresses cannot be contained within the yield surface, the

ratio of peak stresses under tension and compression will be the same as the

ratio of the monotonie yield stresses at the two relative temperatures, i.e.

r 9/­ x­.max

[ax(t2)piin ay[0(t2)] <16>

where t and t are the instants of time during the transient process at 1 2

37

which the stress extremes occur.

In the analysis material properties characteristic of type 316 SS are

used and these are listed in Tables (1) and (2). The numerical values

assigned to the dimensional parameters ht , K , p and t are also pres­

ented in Table (1) .

The mechanical and thermal load components are characterised by the -p -0 dimensionless measures a and O where

-P _ o9 -0 _ EaA6 ^ " ay(0R) ' ° * 2gy(eR) ( 1 7 )

and where 0 R is a convenient reference temperature.

4. SOLUTIONS TO THE BREE PLATE

The plate is assumed to have hot coolant at temperature OH adjacent

to the one surface and cold coolant at temperature QQ adjacent to the other

surface as shown in Fig. (4a). The temperature distribution at t = 0 is

linear through the thickness of the plate.

For the present problem it is possible to produce two types of single-

sided rapid thermal transients. These depend on whether the thermal shock

is applied as a change in the temperature OH of the medium in contact with

the surface H from an initial temperature 0 ^ to a final temperature ØHf

along a ramp which is linear with respect to time, as shown in Fig. (4b) or

it is applied as a change in temperature 0Q , of the medium in contact with

the surface C from an initial temperature 0 ^ to a final temperature 0Cf

as shown in Fig. (5). These cases will be called a thermal downshock (drop

in temperature) and a thermal upshock (increase in temperature), respectively.

(a) Solutions to the single-sided thermal downshocks

In order to obtain the response of the plate element to the temperature

gradient, the plate was sub-divided into 50 through-thickness integration

intervals. The transient stress distributions within each of these inter­

vals were computed from the transient temperature distribution using a

38 -

h y 6H¡

9 = Constant

Q- Plate element and initial condition for thermal dswnshock

6 H «

6H¡

e»

b. Temperature history of medium H ( 8 r is constant) time

er e.

Power on " Shutdown transient c. Temperature distributions

Power off

Fig. (4) : Plate element and the de ta i l s of temperature his tory for

s ingle sided thermal downshock.

39

'Vi, Qs Cors*an1

g Piote element ond initio! conditions for thermol upshock

6. cf

e CI

^ «cf

6H

■ *

b. Temperture history of medium C

e, Hf

time

Power on Trcnsient

c. Temperature distributions

End of transient

Fig. (5): Plate elements and the details of temperature history for

single sided thermal upshocks.

­ 40

numerical integration technique. Values of the temperature to an accuracy

of better than two significant figures were obtained from the summation of

50 terms of the series solutions to the temperature distribution problem.

To obtain the same accuracy for the thermal stresses 45 time steps were used,

time intervals starting from t = 0 to t = 140T, where T represents the

duration of the cooling ramp in seconds.

Fig. (3) shows the temperature profiles during the thermal downshock

for a Biot number of 810 and a Fourier number of 0.0056. The resulting

stress profiles together with the envelope of such profiles are shown in Fig.

(6) for various values of t/x .

In the first set of calculations the fixed temperature 0Q = 8jjf was

chosen as 21°C. In order to assess the effects of the Biot number on the

ratchet boundary, the calculations were carried out with the Biot numbers 810

and 100. The Fourier number was kept constant at 0.0056 in both calcula­

tions. The computed contours providing the limits to the non-ratchetting

area for tensile and compressive mechanical loadings are shown in Fig. (7)

together with the boundary given by Goodman [3] for a perfectly plastic

material with a temperature independent yield stress and the boundaries

corresponding to Bree's quasi-steady thermal cycle solution. It can be seen

that the ratchet boundary shows only a slight dependence on the Biot number

for this range. Nevertheless, the extreme case, when the ratchet boundary

corresponds to the smallest value of mechanical load, occurs for larger values

of the Biot number. It can also be seen that the rapid thermal downshock

reduces the non-ratchetting area. For positive o the ratchet boundary is —9 in good agreement with the boundary given by Goodman [3] for o < 4.0 when

the temperature independent yield stress is adopted. If the thermal load

exceeds this value, a compressive ratchetting begins to occur and a further

increase in the thermal load will result in compressive ratchetting for the

lower mechanical loads. Goodman did not report this phenomenon in [3] but

reported that he was unable to generate stable solutions for small mechanical

loads. - 41 -

a.

1 0

Envelope of stress profiles

­ 1­0­

Fig. (6): Thermal stress profiles for various values of t/x , single

sided downshocks, B = 810, F = 0.0056.

♦ ♦ Bret, Température independent yield stress B=100, » . . . . . .

« «—» « - B=810 » » » « B=100 Temperature dependent yield stress B=810 » » « » Bree. Average temperature dependent yield stress

• • • • Goodman's Perfect plasticity solution, B »610. (Temperature independent yield stress)

Tensile

Ratchetting

05 Mechanical load

Fig. (7): The effects of Biot number on the ratchet boundary for single

sided thermal downshocks, F = 0.0056, 6 = 6D = 21°C . C K

42

F.0-0056 F»O-07

o » O l Fa 0•112 *—« F = 1 • 1837

Bree. Temp, independent material prop. — » — ♦ Bree. Average temperature

dependent yield stress Operating points for the high temperature components in the primary circuit of the Commercial Fast Reactor 11 1

Tensile Ratchetting

% 6 R = 370*C

Fig. (8): The effects of Fourier number on the ratchet boundary for

single sided thermal downshocks B = 810, 6 = 6 = 370°C. C K

A further reduction is obtained when the temperature dependent yield

stress is adopted in the calculations. For ã > 0.2 a worse case may be

conservatively predicted by the analysis of Bree, if the yield stress is

replaced by the average value of the yield stresses at the two extreme tem­

peratures. However, as the transient thermal load increases then compressive

ratchetting in the absence of a mechanical load starts to develop at about

ã =2.4 and the tensile and compressive ratchet boundaries coincide at about

ãP = 0.20 and a = 3.5 .

The comparisons between the calculated ratchet boundaries corresponding

to tensile and compressive mechanical loadings suggests that the thermal down­

shock applied on one side of the plate has its greatest effects when the

43

mechanical load is compressive. This Is not a surprising result since the

integration of the area under the envelope of the compressive stress profiles

in Fig. (6) is larger than the tensile stress profiles. In addition to this,

the temperature dependence of yield stress will introduce additional assymetry

since the yield stress reduces with increasing temperature and highest temper­ie

atures occur when ax < 0 .

The fixed temperature 8C chosen in the above calculations was 21°C.

However, in practice 0C can be as high as 370°C [1]. In order to assess

the effects of both the thermal downshocks at such a high temperature and the

Fourier number on the ratchet boundary, a computer investigation was carried

out with the fixed temperature 0C = 370°C. In these calculations the Biot

number was kept as 810 and the changes in the yield stress with temperature

were included. The results of such an investigations are shown in Fig. (8)

for tensile and compressive mechanical loadings which show that this set of

solutions possesses similar characteristics to those obtained for the fixed

temperature 0C = 21°C. It is again found that the extreme case occurs when

loading is compressive. It is seen that for the present case the boundary

given by Bree with the average temperature dependent yield stress may give

rise to non-conservative estimates of non-ratchetting regions for the smaller

Fourier numbers.

(b) Solutions to the single-sided thermal upshocks

The computer investigation was extended to examine the effects of thermal

upshocks on the ratchet boundary. The first set of calculations were again

carried out with the fixed temperature 6ci = 21°C. The ratchet boundaries

so calculated are shown in Fig. (9) for tensile and compressive loadings. It

can be seen that the extreme case corresponds, as before, to the largest Biot

and smallest Fourier numbers.

The comparisons between the computed ratchet boundaries and the boun­

daries corresponding to the quasi-static case shows that the rapid thermal

44

upshocks cause considerable reductions in allowable combinations of the load

components, especially for compressive loadings when the temperature depen­

dence of yield stress is taken into account. However, the effect is much

greater for tensile loading when changes in the yield stress during the

transient are ignored. This suggests that the extreme case for the single-

sided upshock depends upon the relationship between yield stress and

temperature. The reason for this marked difference in behaviour between

up- and down- shocks may be explained in terms of the temperature distri­

butions and the stress histories. For upshocks, integration of the area

under the envelope of the tensile stress curves due to thermal loading alone

is larger than that for the compressive stresses. This leads to opposite

behaviour for the two loading cases; when the temperature dependence is con­

sidered, an asymmetry is introduced at the expense of tensile loading, since

the yield stress reduces sharply with increasing temperature and the hot

regions of the plate are in compression.

As in the previous case the second set of calculations was carried out

with a fixed temperature 0C = 370°C. The resulting Bree diagrams are

shown in Fig. (10) for tensile and compressive mechanical loadings. This

set of solutions shows many similar features to those described above for the

fixed temperature 6C1 = 21°C. There is, however, a significant difference;

ratchetting occurs at a reduced tensile load when & = 0 . As the thermal

load increases then tensile ratchetting occurs.

It can be seen from these calculations that, for thermal upshocks, the -P ratchet boundary for small a is very sensitive to the variation of yield

stress with temperature.

(c) Equal thermal downshocks on both surfaces

This section considers the plate problem with a uniform initial tempera­

ture profile as shown in Fig. (11). A double-sided thermal downshock occurs

when the temperature of the surrounding coolant is suddenly reduced to a

45

lower value, i.e. the plate is fully immersed in the coolant so that the same

temperature history is applied to both outer surfaces of the plate. Since

the temperature distributions are symmetrical it is only necessary to consider

the semi-thickness 0 < y < h/2 , taking the co-ordinate origin at the centre

line of the plate as shown in Fig. (11) and making surface C the adiabatic

mid-surface so that ht = 0.

The influences of the Biot and Fourier numbers on the ratchet boundaries

are examined in a manner similar to that in the case of the single-sided

thermal downshocks. The results of such calculations are presented in Fig.

(12) for F = .0014 and for B = 810 and B = 100.

The ratchet boundary shows only a slight dependence on the Biot number.

Note also that for tensile mechanical loads the onset of ratchetting is in

good agreement with the boundary proposed by Bree when the temperature indep­

endent yield stress is adopted. However, when the changes in yield stress

with temperature are considered, considerable reductions in the allowable

combinations of load components occur and the ratchet boundary corresponding

to the quasi-static case with the average temperature dependent yield stress

gives rise to a non-conservative boundary. It is again of interest to note

that as the thermal load increases, ratchetting starts to develop in the —9 absence of a mechanical load at about a = 2.30. A further increase in the

thermal load will lead to a compressive ratchetting for small mechanical

loading and the tensile and compressive ratchet boundaries coincide at about -P -6

a = 0.12 , a = 3.25. Comparison between the boundaries for tensile and

compressive mechanical loadings demonstrates that the rapid thermal downshock

has the largest effect on the ratchet boundary for the compressive loading and

the reduction in yield stress at high temperatures may have the effect of

hastening compressive ratchetting at the expense of shakedown or reversed

plasticity. This marked difference that occurs between the two types of

loading situations is due to the reduction in yield stress with increasing

- 46

♦ Bree,Temperaturi independent yield stress Bree.Averoge temperature dependent yield stress

■ » » B»100 F=0,O056 Temperature dependent yield stress B.810 F= 0.0056 B=100 F=00056 Temperature Independent yield stress B=100 F=0-6

Tensile

Ratchetting

Vţ.n-c

0-5 Mechanical load

Fig. (9): The effects of Biot and Fourier numbers on the ratchet boundary

F=0OO56 - . F = 0-07

F=0-112 * F . M 8 3 7 - Bree temperature independent

yield stress ' Bree average temperature dependent 1 yield stress

C.F Reactor operating points

Tensile Ratchetting

0-5 Mechanical load

Fig. (10) : The effects of Fourier number on the ratchet boundary for

single sided thermal upshocks, B = 810, 9D = 0 = 370°C . K C

47

y=-h

e. k' y=*h

eH

Coolant Coolant

hc=o

a Initio! conditions for thermal downşhock on both surfaces

e,

e

Power on Shut-down transient Power of*

b Temperature distributors

Fig. (11) : Details of temperature his tory for a double sided thermal

downshock.

- 48

Brtt Temperature independent yield strest Bree Avtragi temperature depndent yiald stress

• • • Goodmark solution for perfect plasticity B*1O0> Temperature ¡ndtptndtnt yield stress

— B*eio, —• B»100. Temperature dtpendtnt yitld stress — BsSIO, « » H M

Tensile Ratchetting

W 2 1 t

VBR'

Fig. (12): The effects of Biot number on the ratchet boundary for double

sided thermal downshocks, F = 0.0014, 6n = 9„„ = 370°C . K rir

Bree, temperature independent yield stress ♦ — ♦ — ♦ Bree average temperature dependent yield stress

E a;

-x F = CK5

■* » F= 1-1837

-• • F = 50 . ... Operating

points tor the C F Reactor

Tensile

Ratchetting

eHr-e„=37o"c

Mechanical load CP/cr (6R)

Fig. (13): The effects of Fourier number on the ratchet boundary for

double sided thermal downshocks, B = 810, 6„F = 9 = 370°C. ­ 49 ­

temperature and the asymmetry in the stress profiles for thermal loading

alone.

As in the previous sections, the second set of calculations was carried

out with a fixed temperature 0u^ = 0ç = 370°C. In order to examine the

effects of the Fourier number (i.e. the effects of plate thickness or the

duration of cooling ramp) different values were assigned to the Fourier

number, while the Biot number was kept constant at 810 by adjusting the heat

transfer coefficient h . The results are shown in Fig. (13). The solu­

tion for F = .0014 shows similar characteristics to that obtained for the

fixed temperature 0Q = 21°C. For tensile loading the ratchet boundary for

F = .0014 and the boundary given by Bree with the average temperature

dependent yield stress agree fairly well provided O > 0.4 . But for small

mechanical loads the boundary corresponding to the average temperature depen­

dent yield stress does not provide a conservative prediction of the onset of

ratchetting. As in the previous cases, for larger values of thermal load,

ratchetting occurs in spite of a zero mechanical load. For the present case -0 this value is given as a = 3.80 . A further increase results in a com­

pressive ratchetting for small mechanical loads and the two ratchet boundaries -n -6

coincide at about Cr = 0.1 , O = 4.75 . It is also seen in this figure

that as the Fourier number increases (i.e. an increase in the plate thickness

or a decrease in the duration of cooling ramp) then the effect of thermal

stresses will decrease, and as a consequence of this, the area in which no

ratchetting occurs will increase. This increase will be larger for tensile

loading than for compressive, since the hot regions of the plate are subjected

to larger compressive stresses.

Following Goodman's approach [3], starting with a solution for a fixed T

and h , the variation of the onset of ratchetting with the plate thickness

may be evaluated by making use of the Fourier number concept. If ã = 5.0 is

taken as a realistic limit to the thermal stresses to be encountered in

- 50

practice for double-sided thermal downshocks and assuming that the cooling

ramp duration T and quantities K, p and c are constant, one can cal­

culate a critical thickness h for which ratchetting would not occur for

given combinations of thermal and mechanical load components. Fig. (14)

shows the variation of computed critical thicknesses with mechanical loads for

the particular material properties of Tables (1 & 2), the Biot number of 810

and the cooling ramp duration of 10 seconds. The effects of cooling ramp

duration T on the ratchet boundary may similarly be examined by use of the

Fourier number concept. By keeping the Fourier number constant one can

obtain a relationship between the plate thickness and the cooling ramp time

T which, if obeyed, should result in a safe design, i.e.

h < h / ^ - , a6 < 5.0 (18)

where h is taken from Fig. (14). It should be noted, however, that this

result is dependent on the material parameters chosen and the types of tran­

sient thermal loading cycle.

(d) Consequences for fast nuclear reactor design

In order to show the importance of rapid thermal transients in Liquid

Metal Fast Reactor design the following calculation was undertaken:

Using the following values, taken from [1] and Tables (1 & 2); and

Sub-assembly maximum nominal temperature 600°C

Core mixed outlet temperature 540°C

Core inlet temperature 370°C

Rate of temperature transient 40°C/sec.

At F u l l Power

E = 1.708 105MN/m2

a = 16.71 10~61/°C

the maximum thermo-elastic stresses which may occur in the primary circuit

51

I l

ib

1

ib

-a a

1-00-1

■90

•80i

•70

•60-1

•50

-40-

■30-

•20

•10-1

-1-0

-0-9-

-0-8-

-0-7

-0-6

-0-5-I

-0-4-

-0-3-

-0-2-

-0-1-I

Goodman [3I,B = 810,remperature independent yield stress Present study, temperature dependent yield stress j f l -E (ep ) a (eR)A9 < 50

2ery(eR) ©R =370°C a

p> 0 0

{.tu i t t t ,1 I i , i i l I , l i . i . , l n I , ,i ,

a * < 5-0 e R = 370°C cr

p<0

4 5 6 7 8 9 JO 11 12 13 cms Plate thickness \

Fig. (14): Variation of allowable mechanical load with thickness for

double sided thermal downshocks, B = 810, 10 second cooling

ramp, material of Tables (1) and (2).

52

may be evaluated as

C - e ^ I EgA9J _ [ a J = 2a y (370°) - 3 - 1 6

and ,11 f - e q i , EgA9 _

lCT J 2a y (370°) - 2 ' 3 3

where

A9 = 230°C (Sub-assembly max. outlet temp. - Core inlet temp.)

AG = 170°C (Core mixed outlet temp. - Core inlet temp.)

These stress values exceed the classical shakedown limit by substantial

margins. The operating lines which correspond to these thermal loadings are

shown in Figs. (8), (10) and (13).

For the material data given in Tables (1 & 2), the 5.75 second cooling

ramp (for A9 ) and a Biot number of 810, the critical plate thickness hj_«s

below which ratchetting need not be expected, for: single-sided upshocks;

single-sided downshocks and double-sided downshocks can be calculated using

the Fourier number concept. The resulting critical thicknesses so obtained

and the corresponding allowable mechanical loads are given in Tables (3) and

(4) together with the allowable mechanical loads given by Bree for analogous

quasi-static thermal loading.

5. CONCLUSIONS

The results of a series of computations of the behaviour of the Bree

plate subjected to single-sided rapid thermal down- and up- shocks and double-

sided thermal downshocks have been presented and discussed. As a result of

this study the following contributions have been made to the understanding of

the effects of rapid thermal transients on the behaviour of a Bree plate

taking into account, in a conservative manner, cyclic hardening.

53

(1) It has been shown that, for a Bree plate subjected to various types

of rapid thermal transient loadings, the extended upper bound

shakedown technique can be particularly useful in predicting

structural behaviour.

(2) Rapid thermal transients applied to only one surface or both sur­

faces of the Bree plate, will induce ratchetting at lower combina­

tions of mechanical and thermal load components than predicted by

Bree for analogous quasi-steady loading.

(3) The extreme case, when the ratchet boundary corresponds to the

smallest value of | c^ | , is given by large Biot and small Fourier

numbers.

(4) When a rapid down-shock is applied to one surface of the plate, the

extreme case occurs for compressive loading, whereas if the plate

is subjected to a thermal upshock the extreme case strongly

depends upon the variation of yield stress with temperature. If

the variation is large, the extreme case can occur for compressive

loading. On the other hand, for small variation in yield stress

with temperature, the extreme case occurs for tensile loading. If

equal thermal downshocks are applied to both surfaces then the

extreme case occurs for compressive loading.

(5) As the thermal load increases then ratchetting becomes possible in

the absence of a mechanical load. This ratchetting will be com­

pressive if thermal downshocks are applied. For thermal up-

shocks, this ratchetting can be either tensile or compressive

depending upon the variation of yield stress with temperature.

As a result of this, compressive ratchetting can occur for small

tensile mechanical loads when the plate is subjected to thermal

downshocks applied on one surface of the plate or both surfaces.

- 54

For thermal upshocks the occurrence of compressive ratchetting in

the presence of small compressive mechanical loads, depend on the

variation of yield stress with temperature. The former case

occurs when the variation is small.

(6) These results indicate the importance of taking the variation of

yield stress with temperature into account. Any analysis ignoring

this effect may lead to erroneous predictions of structural

behaviour.

(7) For transient thermal loadings a critical plate thickness may be

evaluated as a function of mechanical load by use of the Fourier

number concept, which should result in a safe performance.

All these calculations were carried out using conservative assumptions about

the material behaviour. They display the type of behaviour which may occur

and indicate that thermal transient effects can be significant.

55 -

Appendix Proof of inequalities (5) and (7) The maximum work principle [8] requires that for any stress state a. .

which satisfies the yield condition

(a*. ­ a?.)de?. ¿ o (Al) ij xy ij

K J

c c where de.. and a. . denote the plastic strain increment and associated

ij ij * 1 2

yield point stress. For the component strains de.. and de. . we may write

(ø*k ­ a*.)de*. * 0 , (A2)

where

and

* k ­P .9 a. . = a ij

. . + a:.(tj + p. . , k = 1, 2 (A3) ij i j

v ky K

ij ' ' *■ •*

de. . = de. . + de. . , ij ij ij

summing (A2)over k, integrating over the volume V and applying the *P

principle of virtual work to the resulting term involving a.. , yields

P.du? + 1 1

SP

(ó\ .(t,)de*. + ¿e.(tOde?.)dV

ijv V 13 ij

v 2J i j J

(CT?.(01)deK + a?.(9_)de2.)dV + *■ ij *• V ij i j

v 2J ijJ p..de?.dV $ 0 (A4) ij ij

as de., are compatible and p.. is a residual stress field, the last term in (A4) is zero, yielding inequality (5) of the main text.

When a. . (t) cannot be contained within the yield surface by a rigid body translation in stress space in volume Vn, it is necessary to assume

r

that the extreme stresses are related to each other by a relationship such as equation (16), which was the form used in the calculations.

56

The residual stress p.. is divided into two components

1 * 2

p. . = p.. + p. . CA5) 2 1 9 1

where p.. = 0 in V_ and p.. is chosen so that a., ft) + p.. satisfies Hij F

Mi] il il

A the equation (16) (or any other appropriate relationship), as a result the

1 deviatoric component of p.. is 'determinate in Vp . We assume that where XP is applied forms part of the surface V , and hence apply inequality A4 to V ,

r P.du? + i i (â?.(tjde}. + ø6.(t0)de?.)dV _ f(<,?. (ejde*. + a?. (90)de2.)dV + ^ i;p V i] i;p 2' i]J J

v ij v V ij ij *• 2J ijJ

V V

p}. de?.dV + p?. de?. dV $ 0 ii iJ 13 J il il s V

(A6)

As p . . = 0 i n V t he corresponding i n t e g r a l in A6 i s ze ro . Fur ther

p\. d£?.dV = 0 (A7) i l i l

and hence ( p i . d£

C. ] i l i l V

dV = -

Vr

p}.de?.dV i l i l

s F which is a known quantity. If we use the isothermal condition

(A8)

then a. . (tj + p. . (5?.(tJ + 9l. .) in Vn

13 2 ij F

f 1 c p7.de?.dV = ­

. il il VF V

F

' (5?.(tJ + 5?.(tJ)de?.dV 2K 13

v r ij v 2 ij

(A9)

(AIO)

Combination of (A6), (A7), (A8) and (AIO) yields inequality (7). For the case of a temperature dependent yield value,

—7^­^r­ (a..(t,) + p..) = o (91) i ]

1 1' Mi r (a,,(t0 + p..)

ay(92) *■ ij^2 *ij

replaces (A9) and the inequality (7) may easily be extended to this case.

57

Parameter

Density, p Specific Heat, c Conductivity, K Heat Transfer Coefficient, h Modulus of elasticity, E

Coefficient of thermal expansion ex

Value

7980 kg/m3 556 J/kg °C 24.7 W/m °C 2 x 105 W/m °C 195 GN/m2 at 20°C 170 GN/m2 at 370°C 16.39 x IO"6 1/°C at 20 °C 16.71 x IO"6 1/°C at 370 °C

Table (1) : Material Parameters

Temperature °C

20 50 100 150 200 250 300 350 370 400 450 500 550 600

a MN/m 2

205 179 155 142 132 121 113 106 104 101 97 95 92 90

Table (2) : Yield strength o values for type 316 SS [2 ]

58

en to

Plate Thickness [m]

0.0756

0.0213

0.0169

0.0084

p Allowable Mechanical Load \o |/a (370°C)

Thermal Downshock -P -P a > 0 a < 0 0.24

0.25

0.26

0.29

0.04

0.13

0.17

0.27

Thermal Upshock -p -P a > 0 a < 0 0.06

0.10

0.14

0.27

0.16

0.165

0.185

0.24

Bree [4 ] with

ay(eR)+ay(eR+A9) y 2

0.28

Table (3) : Variation of Allowable Mechanical load with Thickness for Single Sided

Thermal upshock and Thermal downshock (Material of Tables (1) and Û

(2) , 5.75 second Cooling Ramp, ã =3.16, B = 810)

o

Plate thickness [m]

0.302

0.1512

0.0426

0.0338

0.0238

0.0168

0.0016

Double Sided Thermal Downshocks Allowable Mechanical Load \o? \/o (370 °C)

Tension Compression

0.22

0.225

0.31

0.385

0.54

0.68

0.86

0.08

0.12

0.28

0.35

0.50

0.67

0.86

Bree [4 ] with a =

loy C e R)+a y(e R+W 2

0.28

Table (4) : Variation of Allowable Mechanical Load with Thickness for a Double sided downshock (Material of Tables (1) and (2)

a 5.75 second Cooling Ramp, õ =3.16, B = 810).

References

[1] HOLMES, J.A.G.

"High temperature problems associated with the Design of the Commercial Fast Reactor", in "Creep in Structures", Ponter, A.R.S. and Hayhurst, D.R. (eds), 3rd IUTAM Symposium, Leicester, 1980: PP279-286.

[2] ASME Boiler and Pressure Vessel Code, Section III, Nuclear Power Plant Components, Division 1, 1974.

[3] GOODMAN, A.M.

"The influence of rapid thermal transients on elastic-plastic ratchetting", CEGB, Berkeley Nuclear Laboratories, Report no. RDB/N4492, 1979.

[4] BREE, J. "Elastic-plastic behaviour of thin tubes subjected to internal pressure and intermittent high-heat fluxes with application to Fast-Nuclear-Reactor fuel elements", J. Strain Analysis, 1967: 2, No.3, pp226-238.

[5] PONTER, A.R.S. and KARADENIZ, S.

"An extended shakedown theory for structures which suffer cyclic thermal loading", Part I: Theory. Journal of Applied Mechanics, Trans. ASME, 1985: 52, pp877-882 and Part II: Applications, Journal of Applied Mechanics, Trans. ASME, 1985: 52, pp883-889.

[6] KARADENIZ, S.

"The development of upper bound and associated finite element techniques for the plastic shakedown of thermally loaded structures", Ph.D. thesis, The University of Leicester, February 1983.

[7] CARSLAW, H.S. and JAEGER, J.C.

"Conduction of Heat in Solids", 2nd Edition, Oxford University Press, 1959, Chapter 3.

[8] MARTIN, J.B.

"Plasticity", MIT Press, Boston 1975.

61

Part III The plastic ratchetting of thin cylindrical shells

subjected to axisymmetric thermal and mechanical loading S. Karadeniz, A.R.S. Ponter, K.F. Carter

1. INTRODUCTION

An entirely different class of problems occur when the temperature

gradient occurs along the surface of a shell structure rather than through

the shell thickness. The thermo­elastic stresses involve a significant

membrane component and a situation can easily occur where the reverse

plasticity region, volume Vp , can contain an entire cross section of a

cylindrical or axisymmetric shell. As a result these problems are more

likely to be of type B where no P region exists. A full understanding

of such problems, however, requires a consideration of both the ratchet

boundary and the sensitivity of the boundary to various material effects.

This part of the report provides an introduction to the class of problems

by studying a cylindrical tube subjected to a variety of simple axial tem­

perature histories. Some comparisons with experimental data is possible

for these problems and two sets [7,8] of experimental data are used for

this purpose.

The essential feature of this type of problem is shown in Figs. (1),

(2) and (3) where an axially loaded tube is subjected to a temperature

discontinuity of amount A0 which moves over a very short distance Ax ■

The temperature discontinuity generates a discontinuity in the hoop stress

of magnitude

A ØA = EaAØ

As this discontinuity traverses the length Ax each material element ex­

periences a variation of stress of the same magnitude. Hence when A8

exceeds the value

A oc = EaAØ = 2ay

the entire thickness of the tube over the length exceeds the reverse plas­

ticity limit. The volume Vp is, therefore, a hoop of material which

can deform under the action of the axial load and hence no P region

exists; ratchetting can occur once the reverse plasticity limit is exceeded

­ 65 ­

Fig. 1 Geometry of an axially loaded tube

e

e,

Temperature General condition

Axial distance

9 1 Temperature

0O+A9

8,

r AX—1 Simplified cases

a - Moving temperature front 9 1 Temperature

e(

9R+A0

Axial distance

o < t ^

T/2^t^T

b- Stationary thermal cycling Axial distance

F[g^2 Thermal loadings

66

and the problem is of type B . The mechanism of ratchetting consists of

a reverse plasticity mechanism where plastic strains occur at two points on

the yield surface and tends to form net axial strain per cycle.

The evaluation of the ratchet limit for this class of problems invol­

ves a totally different type of numerical technique to the simple method

discussed in Part II for the transient Bree problem. The optimal mech­

anism is not known a priori and must be found by some means. The method

used has been discussed by Karadeniz and Ponter [3] and consists of a

combined finite element/linear programming technique where the optimal

mechanism is formed from amongst a class of displacement fields described

by a finite element spatial description.

In the following section the finite element method and the shakedown

theory are briefly discussed. In Section 3 a sequence of diagrams are

presented for an axially loaded tube and for moving and stationary temper­

ature distributions. We find that the ratchet limit can vary markedly

depending upon the details of the loading history. In addition, inclusion

of the variation of yield stress with temperature can have an amplified

effect for small applied loads, so that it is possible for ratchetting to

occur at zero applied load. In many realistic circumstances with moving

temperature distributions the exact history of temperature is not known and

in this case it seems unwise to exceed the elastic limits.

The effect of cyclic hardening of the material is also discussed.

In some circumstances it seems likely that ratchetting is suppressed by

the development of cyclic hardness. However, there are definite ranges

of loading where ratchetting occurs without reverse plasticity so that

cyclic hardening may be expected to have no effect. In particular it is

possible for ratchetting at zero applied load to occur even for strongly

hardening materials such as annealed 316 stainless steel. Some experi­

mental evidence is present in support of this conjecture.

To further demonstrate that the degree of severity of thermal loading

67

is not easy to predict without fairly detailed calculations, solutions are

presented for tests conducted by INSA at Lyon, France, where a hot gas jet

was diverted by a system of baffles along a narrow length of an axially

loaded tube. Although the thermal loading appears to be severe, the

interaction diagram demonstrates that it is less likely to produce ratch-

etting than a less severe moving temperature field.

Finally the interaction between concentrated loading, and a thermal

field is demonstrated by the solution of a tube problem involving a moving

temperature front and a ring of loading. It is shown that quite sudden

transitions occur as the thermo-elastic stresses approach the region of the

applied load.

2. FINITE ELEMENT TECHNIQUE

The shakedown theory and finite element techniques are discussed in

detail elsewhere [3,4,5] and here we briefly summarize the essential

features of the techniques.

The upper bound shakedown theorem [2,4,5] allows the evaluation of

an upper bound to the applied load, i.e. the axial load on the tube,

corresponding to the boundary of the region S for a prescribed history of p temperature. We define a cycle of plastic strain ¿i-j(t) which gives rise

to an accumulated strain over the cycle of thermal loading, t0< t < t0 + At

,tn+ t Aeïj - [ ° êïj(t)dt (D Jt„ -o

which is compatible with a displacement field Au¿ . The finite element

method is developed for a Tresca type yield condition where the yield sur­

face is composed of a sequence of planes in stress space. The upper bound

can then be expressed in terms of. the plastic multiplyers associated with

these planes so that the formulation reduces to the minimization of a linear

cost function, a load parameter, where the variables are the values of the

68

plastic multiplyers at a sequence of nodal points. The compatibility of the

strain field (1) and the relationship between the plastic multiplyer and

assumed displacement field is assured provided a number of linear constraint

equations are satisfied. The upper bound technique then reduces to a

linear programming problem where we seek the mechanism amongst a class

defined by the finite element-approximation which minimizes the applied load

parameter. The material is assumed to obey a Tresca yield condition and

the displacement field is chosen so that axial bending occurs at a discrete

set of nodal points [3]. If the class of displacement field includes the

exact shakedown mechanism then we find the exact value of the load parameter

at shakedown. In practice the solutions are first produced for a fairly

crude distribution of elements which is subsequently sub-divided until no

change in the load parameter occurs. As a result the computed values may

be expected to be close to the exact solution provided it is within the

general range of displacement fields adopted.

A computer programme has been written for axisymmetric loading of thin

walled tubes which takes as input an axial temperature distribution at a

sequence of times during the cycle at a sequence of points along the tube.

The thermo-elastic stress history is then computed using linear interpola­

tion spatially and a convolution integral formed from the analytic solution

for a step discontinuity in temperature. In addition the variation of

yield stress with temperature is provided in the form of a table of values.

As a result the programme is capable of providing interaction diagrams for

any history of thermal loading by scaling of the temperature history. As

output the programme produces a sequence of diagrams which gives, in

graphical form, the extremes of the thermo-elastic stresses, the interac­

tion diagram and the optimal mechanism corresponding to a sequence of

points along the shakedown boundary.

69

a/EaA9/2

o

• — • " * - » ■

x=n/2ß X=IT/ß

Fig. 3 Elastic thermal stress distribution for a tube subjected to a step change in temperature A8

3. VARIATION OF TEMPERATURE ALONG THE LENGTH OF A TUBE

A simple but not uncommon problem Involves a tube which is periodically

subjected to an increase in temperature along part of its length, so that it

is subjected to a history of temperature of a type shown schematically in

Fig. (2), where an increase of temperature A9 occurs in a fairly uncon­

trolled manner over part of the tube so that the temperature front may

fluctuate axially as well as occasionally reducing to a uniform temperature.

We study this problem by looking at two simplified cases, the first shown in

Fig. (2a) where a sharp discontinuity in temperature A0 moves cyclically

over a distance Ax (a moving temperature front) and the second, shown in

Fig. (2b) where the discontinuity is imposed and then removed (stationary

thermal cycling). The thermo-elastic stresses due to a temperature discon­

tinuity A0 at x = 0 is shown in Fig. (3), where it can be seen that the

hoop stress component QQ has a maximum value of (EœA0)/2 where E and Œ

are Young's modulus and the linear coefficient of thermal expansion respec­

tively.

The interaction diagrams for the two problems, assuming a constant

value of the yield stress are shown in Figs. (4a) and (4b). In these

diagrams and all subsequent diagrams the axes are given by

P = P/pL

where PL is the plastic limit load value of the axial load P , computed

from a yield stress value ay at a reference temperature 0R ,

and a t 2 ay(0R)

where at is the maximum thermo-elastic shear stress.

In figure (4a) a sequence of shakedown boundaries are shown corres­

ponding to a range of values of the Ax in the form of the non-dimensional

variable.

- 71

3 e Ö

itr

?

E

Elastic region (E)

0.5 Mechanical Load P= P R

2hay(0R)

Fia. U a Bree type diagram for a tube subjected to a steady axial mechanical load and moving temperature fronts with temperature independent yield stress cr=o~(8R)

- 72

Temperature independent yield stress ay=Œy(9R) Temperature dependent yield stress

Hardening models

2.25

2.00

1.75

1.50

C

1.00

at=o corresponds to 8R=150°C

Plastic Shakedown (F)

\ \

\ \

V \

N \

V \

\ w K

Ratchetting (R)

Elastic

Behaviour (E)

-l 1 1 1 1 i r-

0.5 1.0 P

Fig. ¿b Modes of behaviour for a tube subjected to constant axial load and stationär y thermal cycling

73

Ax = Ax.| (2)

where V3 is a characteristic decay length of the tube and given by

3 = [3<l-v2)/R2h2]*

In both cases the boundary between the shakedown region and the rat-

chetting region can be divided into segments, along which the mechanism of

deformation remains of constant type. These mechanisms represent the mode

in which the structure would begin to ratchet if the load were increased

above the shakedown limit. In Fig. (4a) the segments are given by regions

of the diagram labelled as Mode I, II and III. The corresponding mech­

anisms are shown in Fig. (5) together with a schematic representation of the

regions of the yield surface where the plastic strains occur. The thermo-

elastic stress history at a point within the mechanism is shown as trans­

lated, by the development of residual stresses and by the applied loads, so

that the stresses at a certain instant touch the yield surface.

If we compare the boundary for small Ax in Fig. (4a) with stationary

cycling in Fig. (4b) we see that the principal difference is that the

boundaries cross the P = 0 lines at ãt = 1 and 2 respectively. The

difference can be understood from Fig. (3). When the stress distribution

moves, the variation of stress at a material point becomes 2(EocA0/2) where­

as the variation of stationary cycling is (E<xA0/2) i.e. the range of

stress at each point. In reality, of course, the temperature would not be

discontinuous and a more gradual change would take place. In this case a

moving front would correspond to the movement of the temperature profile over

a distance which is greater than the length of the temperature change. As a

result a problem can only be regarded as stationary cycling if the temperature

is maintained sufficiently stationary for this condition not to occur. In

most applications it seems unlikely that such a high degree of control can

- 74 •

MODE I øs Region LO ' ^

localized thinning due to Ae« occurring on o"x = o~y

MODE n Line CB o*

localized thinning due to Aex occurring as a resultant of plastic strains on CT = CTW and e * °\-°"Í e = au

MODE m a 0 ( n Region GL ,J-

J8

hinge - cone mechanism with axial strains

Fig. 5 Schematic representation of shakedown states and corresponding mechanisms of deformation for a tube subjected to a steady axial mechanical load and axially moving temperature fronts for regions of Fig. 4a.

75 -

be maintained. This argument indicates that it is possible to seriously

underestimate the effect of temperature variation by using an inappropriate

simplified form of the temperature history, and it seems unlikely that the

stationary cycling approximation would have much relevance to practical

circumstances.

If we now include the effect of temperature on the yield stress, the

general feature of the diagram remains unchanged, but the boundaries

corresponding to the various mechanisms are moved by differing amounts,

depending upon the temperature at which the plastic yielding occurs. In

Fig. (6) the boundaries for the moving temperature fronts are shown, using

a variation of yield stress with temperatures which correspond to the 0.2%

proof stress of Type 316 Stainless Steel. Plastic strains in Mode III

occurs in a material element when the temperature is at maximum whereas in

Mode II plastic strains occur at both the maximum and minimum temperature.

As a result the shakedown boundary corresponding to Mode III occurs at a

reduced level of A9 compared with Mode II. For sufficiently large values

of Ax Mode III boundaries cross the P = 0 axis, i.e. the tube would

ratchet in Mode III at zero applied load. These calculations are for a

perfectly plastic model which exclude the effects of strain hardening. We

now look into the effect upon this diagram of its inclusion.

4. THE EFFECTS OF STRAIN HARDENING UPON THE RATCHET BOUNDARIES

Referring again to Fig. (5) we see that for Mode I and Mode III all the

plastic strain occurs on a single part of the yield surface. If the

applied load was raised above yield, strain hardening would occur in a mono-

tonic fashion as the mechanism deformed. As a result the shakedown

boundary gives the load level at which the tube begins to exhibit signifi­

cant plastic yielding in the form of a mechanism and may be regarded as an

estimate of the yield point of the structure in the presence of thermal

76

1.00.,

075.

« AØ

0.501

0.25.

0.00

a, so corresponds to 6R=150°C

Elastic Behaviour (E)

0.5 p* 1.0 P

Fig. 6 Bree diagram for a tube subjected to a steady axial mechanical load and axially moving temperature fronts with the tempera­ture dependent yield stress

77

loading. For Mode II, however, the situation is rather different as plastic

strain occurs, within each cycle, on two sectors of the yield surface, i.e. p reverse plasticity takes place. The hoop component of plastic strain ¿o

P cancel over the cycle but a net increase Aex occurs due to yielding under

compression. In many alloys cyclic hardening would occur in these circum­

stances which would tend to suppress this type of mechanism. We can estimate

an extreme mode of behaviour by assuming that the yield surface increases

in size to accommodate the variation in thermo-elastic stress thereby com­

pletely suppressing reverse plasticity mechanism. This was done by adopting

two assumptions, isotropic and non-istropic hardening as shown in Fig. (7).

With this adaptation the structure can only ratchet in mechanisms of the type

of Mode I and III. In practice, Mode III mechanisms always occurred. The

resulting new ratchet boundary for the two models are shown in Fig. (8) for

the moving temperature front with a temperature independent yield stress.

The difference between the two solutions is not great and it implies that the

real ratchet boundary lies somewhere between two extreme assumptions i.e. for

Ax = 0.6, between RS and RT . This argument indicates that for a given

Ax there exists a particular load p* with a corresponding value of A6*

which divides the behaviour of the structure into two distinct regions. For

A8 < A6* and P > P* then the ratchet boundary can be expected to give a

good indication of the load level at which substantial plastic strains begin

to occur. For A9> A0* and P < P* the behaviour is very sensitive to

the detail of the material behaviour and the structure may or may not ratchet.

Certainly below the perfectly plastic line there is no danger of ratchetting.

Despite this uncertainty we can, however, show that under some circumstances

ratchetting will certainly occur at zero applied load even for a cyclically

strongly strain hardening material such as 316 SS. In Fig. (9) we have

plotted from Fig. (4a) and Fig. (6) the variation of p* with Ax . For a

78

- Partial isotropic hardening model

Fig. 7-Non-isotropic hardening model

79 -

¿'Partial isotropic hardening model

¿'Non isotropic hardening model

R Rate netting (R)

1:

2:

3:

Ex 0.60

1.20

2.10

Fjg._8 Effects of material hardening on the shakedown limits for a tube subjected to an axial mechanical load and axially moving temperature fronts with temperature independent yield stress o~y=cry(9R)

80

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

\ :

.C L)

O

C o

r Z

0.1

0.0

x □

Temperature independent yield stress Temperature dependent yield stress (eR = 150° C)

Ratchetting (Global mechanism)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 Distance of travel AX

Fig._9 Effects of AX on P *

81 -

temperature independent yield stress P* goes to zero as Ax increases to

infinity, i.e. ratchetting will certainly occur at P* = 0 only for very

large Ax . However, for a temperature dependent yield stress P* = 0 when

Ax - 1.8 for material data which is typical of type 316 Stainless Steel.

With Poissons ratio V = 0.3 and — = 400 then this value corresponds to n

Ax = 0.17R. Therefore, any movement of the temperature front over a

length of the order of the radius of the tube well certainly give rise to

radial plastic displacement.

These calculations indicate that quite small movements of a temperature

front will cause ratchetting of a tube even at zero applied load. In a

practical circumstance, the temperature front would involve a temperature

gradient over a length of the tube. The effect of this would be to reduce

the maximum thermo-elastic stress to kE<xA0/2 where k < 1 . For example,

in experiments described in the next section 0.14 < k < 0.48. This has

two effects, it increases the range of A0 required to cause ratchetting

and, at the same time, increases the difference between the yield values at

the two temperature levels 0R and 0R + A0. The latter effect increases

the tendancy for ratchetting to occur at zero applied load and as a result

the estimate of Ax =1.8 will be reduced in most practical circumstances.

In the following section we describe some experimental results due to

Bell [19] and compare them with shakedown prediction for a moving temperature

front. The experimental results confirm that ratchetting will occur even

when the thermal loading exceeds the shakedown limit by only a small amount.

5. EXPERIMENTS ON THIN CYLINDERS SUBJECT TO AXIALLY MOVING TEMPERATURE FRONTS [7]

The experiments consisted of two types:

a) Cold Front Experiment

A cylinder of 316 Stainless Steel of outer diameter 140mm and wall

82

thickness 0.381mm was heated over a length of 165mm using RF heating. The

tube was then lowered into water at room temperature at a rate which allowed

the formation of a steep temperature front along the length of the tube but

almost uniform temperature through the thickness.

We discuss two of their tests, Experiments 1A and 2, which were con­

ducted under near identical conditions with A9 of 530°C and 515°C except

that in experiment 2 the tube was subjected to an axial load of 20MN/m whereas in experiment 1A the tube was load free. The value of k , the

reduction in maximum thermo-elastic stress compared with a temperature dis­

continuity, lay within the range

0.31 < k < 0.46

depending upon the instant during the cooling of the cylinder. The shake­

down boundary was evaluated using the more severe temperature front through­

out the movement of the cylinder and the result is shown in Figure (10)

assuming both temperature independent yield stress (taken as the 0.2% proof

stress of the lower temperature) and a linear variation of yield stress with

temperature. As A0 was relatively larger than the cases discussed in the

previous solution the difference between the solutions is much larger and the

ratio of the values of ãt for zero applied load for the two calculations is

given approximately by the ratio of the yield stresses at the two extreme

temperatures. The operating points of the experiment are also shown and can

be seen to be far in excess of the shakedown limit. The tubes showed

excessive ratchetting showing a hoop plastic strain of 0.47% strain in the

first cycle and a mean constant rate 0.16% strain/cycle from the 5th to the

25th cycle when the experiment ended with a noticably misshapen cylinder.

Experiment 2 showed similar behaviour. In both cases the rachetting was

outwards, whereas the mechanism from the shakedown calculation was inwards.

However, the loading was far in excess of shakedown and, perhaps, it is no

surprise that the mechanism has changed.

83

1.50 -

1.375

1.25

1.125 -

1.00

0.875-

0.75 .

0.625

1A (SE) ■ 2 (SE)

■+—+

■ • — • ■

Temperature independent yield stress Œy=CTy(8R) for both hot & cold fronts

Temperature dependent yield stress Experiment(1A) Temperature dependent yield stress Experiment (5) Operating points Severe extreme temperature profile Gentle extreme temperature profile

!' o-y(20°C)

Ratchetting (R)

Shakedown (S)

Fig JO Operating points and calculated shakedown limits for tests (1A). and (5).

84

b) Hot Temperature Front, Experiment 5

A front of increasing temperature was induced by moving a cylinder

initially at room temperature through a high power single turn RF coil at

a speed of about lOmm/sec. The increase of temperature was about 600°C

but the shape of the temperature profile was less severe than in the cold

front experiments so that less severe thermo-elastic stresses were induced

with a factor k = 0.14.

The shakedown boundaries are shown in Fig. (11), again for a temperature

independent and temperature dependent yield stress. In this case the

operating point lies only 12.5% in excess of the predicted shakedown limit.

The experiment was continued for 60 cycles during which a total hoop strain

of 2.5% occurred with an assymptotic rate over the final 40 cycles of .01%

per cycle. The mode of deformation was very similar in form to the shake­

down mechanism. The experiment confirms that ratchetting of a significant

magnitude occurs at zero applied load, once the shakedown limit has been

exceeded.

6. OTHER TYPES OF THERMAL LOADING OF CYLINDERS

A sequence of ratchetting experiments have been carried out by Cousin

and Julien, at INSA [8] . In the tests a cylindrical tube of ICL/67SPH

Stainless Steel (similar in composition to 316 Stainless Steel) of diameter

400mm, wall thickness 2mm and lm in length was used. An axial temperature

profile over a short length of the tube was induced by circulating combus­

tion gas from a burner past a sequence of baffles. By spraying water over

a section adjacent to the hot gases, high axial temperature gradients could

be induced. A complete cycle of temperature for a particular experiment is

shown in Figure (12). A very high temperature gradient is induced over a

short length of tube; with the temperature varying over the cycle between

room temperature and 480°C. Although this type of cycling appears to be

85 -

Temperature independent yield stress + + Temperature dependent yield stress

• Operating point for test (5) 6R = 20°C

100

075.

CD < a

" 0-50J CD

CSI

II

0-25

> ^ Shakedown \

Ratchetting

Elastic region

ao ^ F " ■ 1 « I ¡ I I I

P=PR/2ho-y(0R) Fig. 11 Operating points and computed shakedown

boundaries for test(5)

- 86

very severe we find, in fact, that it is less likely to induce ratchetting

than the temperature histories discussed in the last section.

The interaction diagram was constructed from the temperature history of

Figure (12) by linearly scaling the entire history. The yield stress

variation with temperature was given by the 0.2% proof stress. The resulting

diagram is shown in Figure'(13) which includes the operating point of the

experiment. The shakedown boundary consists of two parts, each of which

involve a particular mechanism.

Section AB: Axial strains occur over a short length of tube near the

position of the maximum temperature due to the presence of a large variation

in the thermo-elastic axial bending moment at this point. The ratchetting

is induced, therefore, by linear through-thickness thermo-elastic stress in

the same manner that ratchetting occurs in the Bree plates problem [1],

although in that case the stresses arise from a through-thickness temperature

difference. The boundary AB can be seen to be very close to the Bree

solution.

P + \ õt = 1 (3)

and the deviation from the formula arises from the decrease in yield stress

with increasing maximum temperature.

Section BC: Axial strains occur over a short length of tube adjacent

to the step temperature gradient due to a large value of hoop stress inducing

reverse plasticity.

As a result the behaviour has features of both the classic Bree problem

(axial ratchetting induced by linear through-thickness stresses) and

stationary cycling (line BC of Fig. (4b)). But for a moving temperature

front Fig. (4a) the value of ãt at the ratchet limit is reduced by a

factor of 2.

87

9 ra 500

450

400

350

300

250

200

150

100

50

1

165s 150s - " " "~"

135s-—

-

120s ■

9 0 s —

60s

30s —

i i

l 1 1

IÆ 1 8 0 s

ţ \ \ _ _ _ _ 1 9 5 s

Aff l i 210s

\/V\VOK

i i i

i

-285s

^ %

i

i i

_1258s

i i

-

-

-

-

-

-

■

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 METERS

Fig. 12 Temperature distributions for Lyon test

88

Temperature dependent yield stress T 1 r

an i(U°C) 0.1

Fig. 13 Bree diagram for Lyon test

89 -

It can be seen that the operating point of the experiment was in the

ratchetting region R . An estimate of the assymptotic plastic strain over

many cycles can be obtained for the mechanism which occurs on section AB by

using the approximation [6] .

AeP = KAcTp

S s where ACTD = (aD

- aD ) where <j is the value of the mean applied stress d a

P/2irRh at the shakedown limit and K = — , the slope of the uniaxial stress-deP

strain curve above yield. The material data allows the calculation of K

which is reasonably independent of temperature. Contours of constant

Ae = 0.5% and 1% are shown indicating that about 1% accumulated strain

might be expected after a number of cycles. Full inelastic analysis yields

a similar value [9] . The experiments showed a larger amount of plastic

strain which may be attributable to the fact that the cycle time was quite

large (two cycles per 24 hours). As a result some logarithmic creep will

have been induced as this material shows such creep in the temperature range

0OC < e < 350°C .

This example further demonstrates that the precise behaviour depends

upon the details of the thermal loading history.

7. TUBE SUBJECTED TO A BAND OF PRESSURE AND AXIALLY MOVING TEMPERATURE FRONTS

The limit load analysis of an infinitely long isotropic thin tube of

perfectly pastic material obeying the Tresca yield criterion has been given

by Drucker [10]. Insofar as the authors are aware no attempt has been made

to investigate the behaviour of cylindrical tubes under a band of pressure

or a ring of force in the presence of thermal loads, although such loading

situations are encountered in .structures operating at elevated temperatures.

In this section we discuss the behaviour of a thin cylindrical tube under

a band of pressure P and a temperature front of magnitude A6 which travels

- 90

repeatedly in alternating directions over a length of tube as shown in Fig.

(14). The problem is an interesting one and yet is sufficiently simple to

serve as an example of problems involving thermal loading interacting with a

localised mechanical load.

The computer programme used is a modified form of that used in the

analysis of the problem of a tube under axial load. The assumptions made

about material constitutive relations are identical to those described in the

previous sections but the effects of temperature on the material properties

are ignored here for simplicity.

The calculations are carried out in two phases. First we assume that

a band of pressure was being applied at a distance from the temperature discon­

tinuity (Case I). Then we considered that the band of pressure was being

applied within the sweep of the moving temperature front (Case II). The

Cases I and II are illustrated in Fig. (14).

For Case I the computations were performed for differing values of the

length of travel Ax with the aim of providing some information to assess

the effects of the variation of distance between the traversed region and the

section of the tube which is subjected to the band of pressure loading. To

compare the predictions of the limit load given by Drucker [10] and that

predicted by the present technique, the calculations were carried out with

õt = 0 . It was found that the difference between the predictions was less

than one per cent. The interaction diagram for a range of values of Ax is

shown in Fig. (15). As it is seen, if the temperature front moves a small

amount, i.e. the traversed region is far away from the region where a band of

pressure is applied, the thermal loads have little effect on the load carrying

capacity of the tube. As a result the non-ratchetting lines are insensitive

to the length of the front and the magnitude of thermo-elastic stress at .

For intermediate values of Ax the sensitivity of the boundaries to the length

of travel Ax and the magnitude of thermo-elastic sress at increases with

91

co

0

9 R

Temperature

Cold Front

AX

I—

_L

9R+A9

Hot Front

a- Axially Moving Temperature Fronts

t

A

,, ,

Ab

Casei

1=0-3 j

,~T Case II

■f— -4

A l = 1 - 3 5 f ' i i

:iQ LU iïAf.

Uf

2R

^

b-Bandof Pressure Loadings ß = [3( 1 - V2

) / R V ] ° 25

Fiq. 14 Geometry and loading programme

Fig. 15

1.0 2.0 Mechanical Load P R = p

2höy The interaction diagram for a tube under a band of pressure and QXiallv moving temperature fronterQseD

93 -

increasing Ax up to a certain value, i.e. the boundary of the traversed

region coincides with a boundary of the region on which the band of pressure

is applied. For values of Ax larger than this value the sensitivity to the

length of travel reduces with increasing Ax . It can also be seen that for

a range of values of Ax , the non-ratchetting lines become horizontal to the

mechanical load axis. The reason for this sharp reduction in the allowable

mechanical load component can be explained in terms of the location of the

hinge circles. For ãt > 1.0 the shakedown condition demands that the

material within the traversed region must satisfy the reversed plasticity

condition. However, if one of the side hinge circles forms within this

volume, or in a region beyond the traversed region, the contributions to

the load carrying capacity of the tube from such a hinge will be zero since

no mechanical load can be transmitted through this volume without causing

ratchetting. Similarly if all the hinge circles form within the traversed

region or in a region where the thermo-elastic stress history cannot be

accommodated within the yield surface, which contains a mechanism of

deformation, then there exists no reversed plasticity region since the

behaviour above the shakedown limits is determined by whether there exists a

region capable of transmitting the applied loads through the structure.

If ãt < 3.0 is considered to be a realistic limit to the thermal loads,

the reversed plasticity region may be divided into three sub-regions from

consideration of the locations of the hinge circles. If the operating points

fall within the region marked I in Fig. (15) then all the hinge circles form

on the same side of the traversed region and contribute to the load carrying

capacity. If an operating point lies within the region marked II then one

of two side hinge circles forms within the traversed region, whereas if the

operating point falls within the region marked III then one of the side hinge

circles and the central hinge circle form within or beyond the traversed region.

As a consequence of this, the ratchetting lines which cross the line dividing

94

region II from region III will have a horizontal section corresponding to the

reduction in load carrying capacity of the tube due to the formation of the

central hinge circle within the traversed region, as shown in Fig. 15.

The computations were repeated for the Case II. The resulting interaction

diagram is shown in Fig. (16) for a range of values of Ax . It can be

readily seen that for this case there exists no reversed plasticity region.

In this case, unlike in the previous cases, the length of the region in which

a global mechanism forms increases with increasing magnitude of the thermal

load ãt • There exists only one type of mechanism of deformation, that is

a global mechanism, i.e. three hinge circles separating cone like regions of

radial deformation.

In order to assess the effects of hardening above the shakedown limits

the calculations were carried out assuming that the material behaved in a

manner similar to that described in section 4 and Fig. (7) in the reversed

plasticity region and obeyed the Tresca yield condition elsewhere. The

results of such calculations for various values of Ax are also shown

schematically in Fig. (16). As is seen, this set of solutions show a

similar characteristic to those obtained in the previous tube problems and

requires no further comment. The only difference that occurs is in the

length of mechanism of deformation which increases with increasing at.

8. CONCLUSIONS

There has been increasing reliance upon full inelastic analysis in

nuclear industry for the validation of structural designs using available

non-linear finite element codes. However, such solutions do not directly

help the designer to understand the nature of complex loading systems such

as severe thermal loading, as the answers are specific to a particular

circumstance and give no general picture of structural response. In this

section we have described the use of a simplified shakedown technique to

95

1.25.

1.00

0.7S CD <

Ib*

TD Ö O

Ö E c _

0-50

0.25

0.00

\ | \ \

, - . . \ \ \ ((hardening mfcdels) \\

\í \

f Plastic Shakedown \

Elastic Behaviour

Fig. 16

1 :

2 :

3 :

4:

5:

6:

1-

8:

AX 0.30

0.60

0.90

1.50

2.10

2.40

2.70

6.00

1.0 Mechanical Load

Rate h et ting

Bree diagram for a tube subjprtpfi t° n hand of pressure and axially moving temperature fronts

96

compute interactive diagrams for certain important types of loading. Such

information provides, in a simple and graphic form, the entire range of

ratchetting response of the structure for varying severity of themal loading.

By combining a number of particular cases it is then possible to draw some

general conclusions^about the influence of thermal loading of thin circular

cylinders.

The most significant conclusion to these calculations was that the

variation of yield stress with temperature and cyclic hardening can signi­

ficantly effect structural response. For tubes subjected to moving temp­

erature fronts over very short lengths of tube structural ratchetting can

occur in the absence of applied loads even when the material is strongly

cyclically hardening. Some experiments conducted by Bell [19] give re­

sults which are consistent with our calculations. On the other hand, a

history of temperature which involves the near proportional increase and

decrease of a temperature distribution is far less likely to produce struc­

tural ratchetting. Comparison between our calculation and tests carried

out by Cousin et al [8] give support to the conclusion. As a result, we

conclude that in validating experimental work on thermal loading, care must

be taken that the history of temperature is of the same type, in some detail,

as that in the industrial application. Otherwise, significant errors can

occur. In addition, it seems that the simplified forms of analysis des­

cribed here can give a better insight into the nature of the problem than

full inelastic analysis.

Further solutions have been presented which show the interaction bet­

ween localised forces, in our example a ring of load on a tube and localised

thermal loading. The two forms of loading begin to strongly interact when

the high thermal stresses occur within the plastic collapse mechanism of

the localised load. In the process, the mechanism itself changes to

include the high thermal stresses within its volume. Shakedown analysis

demonstrates these interactive effects in a very clear and simple manner.

97 -

References

[1] BREE, J.

"Elastic-plastic behaviour of thin tubes subjected to internal pressure and intermittent high-heat fluxes with applications to Fast Nuclear Reactor Fuel Elements", J. Strain Analysis, 1967: 2, No. 3, pp226-238.

[2] KOITER, W.T.

"General theorems for elastic-plastic solids", Progress in Solid Mechanics, Hill, R., and Sneddon, I., (eds), North Holland Press, Amsterdam, 1960: 2, ppl67-219.

[3] KARADENIZ, S. and PONTER, A.R.S.

"A linear programming upper bound approach to the shakedown limit of thin shells subjected to variable thermal loading", J. Strain Analysis for Engineering Design, 1984: 19, pp221-230.

[4] PONTER, A.R.S. and KARADENIZ, S.

"An extended shakedown theory for structures that suffer cyclic thermal loading, Part I: Theory", J. of Applied Mechanics, Trans. ASME, 1985: 52, pp877-882.

[5] PONTER, A.R.S. and KARADENIZ, S.

"An extended shakedown theory for structures that suffer cyclic thermal loading, Part II: Applications", J. of Applied Mechanics, Trans. ASME, 1985: 52, pp883-889.

[6] COCKS, A.CF. and PONTER, A.R.S.

"Accumulation of plastic strain in thermal loading problems for a linear hardening material", to appear.

[7] BELL, R.T.

"Ratchetting experiments on thin cylinders subjected to axially moving temperature fronts", UKAEA, Risley Nuclear Power Development Establishment, Report ND-R-835(R), Risley, October, 1980.

[8] COUSIN, M. and JULIEN, J.F.

"Specifications de l'essai pour step II benchmark calculations", Institut National des Sciences Applique de Lyon, France, May, 1983.

[9] CORSI, F.

Private communication.

[10] DRUCKER, D.C.

"Limit analysis of cylindrical shells under axially symmetric loading", Proc. 1st Midwest Conf. Solid Mechanics, Urbana, II, 1953: PP158-163.

98

Part IV Interaction diagrams for axisymmetric geometries

K.F. Carter, A.R.S. Ponter

1. INTRODUCTION

Previous sections of this report have been concerned with the applica­

tion of the upper bound techniques to particular types of structures and

thermal loading by using special features of the geometry with the results

presented in the form of Brussels diagrams. A computer code, EECS3, has now

been generated at the University of Leicester which is capable of producing

such diagrams for a wide range of axisymmetric thin shells subject, in

principle, to an arbitrary history of thermal loading. The method used is

based upon the technique described by Karadeniz and Ponter [1] and a full

description is given in the appendix to this section. In this section we

give results for a range of cases, typical of fast reactor design, which

demonstrate the effects of variations in shell thickness, axial and through-

thickness temperature gradients and geometries composed of cylindrical,

spherical and conical sections. In all cases continuity of the tangent angle

to the shell mid-section is maintained. The cases chosen are based upon a

set suggested by WG2 of the EEC Fast Reactor Co-ordinating Committee under the

chairmanship of Dr Tonnorelli to which have been added further cases,

including a problem suggested by Guy Baylac of EDF (the Baylac test). Their

assistance in this matter is gratefully acknowledged.

The cases in this section demonstrate the type and range of information

which may be gained through the application of these new numerical techniques.

The information is broader in scope and more easily understood in terms of

design restrictions than conventional finite element methods. A typical

design question, such as the amount the pressure or temperature distribution

needs to be changed to avoid excessive deformation, can be more easily

answered through a technique which concentrates on the problem of finding the

load levels at which significant deformation begins to occur. As far as the

authors are aware, this is the first time classical shakedown theory has been

successfully employed in this way and it seems likely that there will be many

101

other applications in the future of this type of technique.

In the following section the general form of the component parts of EECS3

are described. This is followed by a description of Brussels diagrams and

associated mechanisms for a set of cases. We then conclude that, despite

many differences in detailed behaviour, there are general trends which

suggest that certain master diagrams may cover ranges of useful cases.

2. EECS-3

The program takes as input the basic physical dimensions of the axisym-

metric shape, the material data (including yield stress as a function of

temperature) and the temperature distribution. The thermo-elastic stress for

the temperature distribution, which can vary axially and through the thickness

of the shell anywhere within the material volume, is calculated by a finite

element elastic stress program CONIDA [2], supplied by the United Kingdom

Atomic Energy Authority. The program then calculates the Brussels diagram

for the shell subject to a proportional temperature history and constant

mechanical load, which can be axial loading (tension or compression), internal

or external pressure, or a band of internal or external pressure. The

solution is subject to boundary constraints such as zero displacements normal

or tangential to the mid-surface or axisymmetric axis (as required by loading

type) at the ends of the body.

The program initially establishes a finite element structure based on a

minimum number of elements in each geometrical section, and then increases the

density of elements, by bisection, at positions where the thermo-elastic

stress is largest. For the solutions discussed here, a maximum of 40 elements

were used. The entire Brussels diagram is obtained by linear scaling of the

temperature history 0(x,t) = g(x.»t)(0max_0o^ ^y a factor ^ t o produce a

sequence of distributions 9 (x,,t) differing only in magnitude;

eX(x,t) = e0 + x g(x , t ) (6 m a x -e 0 ) (1)

- 102 -

e ° c

600-

500-

¿oo-

300-

200-

100 -

Fig. la - Yield surface showing plastic multiplier directions Solid line - Tresca yield condition Dashed line - 12 A yield condition

DCWG [¿] Recommended values 316 Stainless steel

0 0

F i g . lb

50 100 150 200 a-y(MPa)

Yield stress values vs temperature for Type 316 Stainless Steel from DCWG recommended data (4) - See Table 1

103

where 0O is the initial temperature and 6max (or 0min for downshocks) is

the temperature having the largest difference from 0O . The function

g(x,t) is the normalized shape function of the temperature distribution. For

experimentally obtained temperature distributions the correct solution will

correspond to the value of at/üy(0o) = ãt obtained when X = 1 where at is the maximum shear stress in the thermo-elastic distribution. ay(0o) is

the plastic yield stress at 0O . The temperature distribution can then be

characterized by its knockdown factor k , which is defined as the maximum

thermo-elastic stress of the temperature distribution divided by the maximum

thermo-elastic stress for a step discontinuity having the same maximum temper­

ature difference A0 = (0max~0o)

k = at(0max)/(EaA0/2) (2)

The value for k lie in the range 0 < k < 2 .

The program incorporates the same extension to the upper bound shakedown

theorem as discussed in Section 2 which allows calculation of the shakedown/

ratchetting boundary in the P region where the thermo-elastic stress cannot

be contained within the yield surface within a volume VF . In this case

The mean value of the thermo-elastic stress history is set to zero within the

volume Vp , and then the calculation for the shakedown boundary is carried

out using this assumption in the region Vp . The method is discussed in

detail in the appendix to this section.

Throughout these calculations the yield criteria used is based on the

Tresca yield surface (illustrated in Fig. 1) in terms of the meridional and

circumferential stress components. The yield stress values are calculated by

linear interpolation within a table of data values of yield stress against

temperature. The program is also capable of using a 12 X yield surface which

has only a 3% error in comparison with the Von Mises yield surface, however the

increased accuracy results in a significant increase in computer time and

104

storage required.

Curvature in the meridional direction is concentrated at plastic hinges

at the nodal points between elements and linear variation in A^ is assumed

between nodes. As the structure and loadings are both completely axisymmetric,

there is no curvature in the circumferential direction. Consistancy between

the displacement components and plastic strains expressed in terms of the

plastic multiplyers \± is assured by integration of the strain-displacement

relationships. As a result the meridional curvature, when derived from the

displacement fields, is small but non-zero. However, the energy dissipated

due to curvature within elements is assumed zero as, in terms of the A's

curvature is calculated in the plastic hinges. In certain exceptional cases,

usually only found at high values of at or where the geometry is very rapidly

varying this effect can be significant. However, as the program does not

account for this mode of energy dissipation the resultant mechanical load will

always be less than the true value. Thus this method is always conservative

in these conditions.

The cases studied in this report are based on a set of typical thermal

loading problems in reactors, known as the Bergamo set, proposed by Working

Group 2 of the Fast Reactor Co-ordinating Committee. Throughout this report

the Case Numbers correspond to those of the Bergamo set. Case 1, not dis­

cussed in detail in this report, is à sphere with a through thickness tempera­

ture gradient. The Brussels diagram for this case is exactly the same as for

the Bree problem and a representative mechanism is shown by Case 5 which

involves a spherical end section. Case 3 of the Bergamo set is a cylinder

with an axial temperature gradient. Examples of this type have been

discussed in great detail in Section 3 of this report.

105

1

.35m

,

jh=.oii

1

.2275m

^

i

.045

.5m

jh=.020

1 }

.2275m

Geometry and temperature distributions for Baylac tests See Table 1

UPSHOCK max

—J .063

Fig._2

106

3. CYLINDRICAL SHELLS

The program EECS3 is capable of calculating Brussels diagrams for all the

cylindrical cases discussed in Section 3, with the exception of cases

containing multiple temperature distributions, although a small modification

to include such cases can easily be incorporated. However the program can

also handle cases for any geometry which involves changes in shell thickness

and also temperature distributions including linear changes in temperature

across the shell wall. This is accomplished by specifying the temperatures

^in(z±*t) and 6our(zi,t) on the inner and outer surfaces respectively at a

number of specific points z± along the axisymmetric axis, and using linear

interpolation to deduce the temperature between these points and through the

thickness. When changes in thickness are incorporated, the thickness is

assumed to vary linearly along the length of an element. The thermo-elastic

stress for the temperature distribution and specific geometry is calculated

by CONIDA, and is then used in the calculation of the Brussels diagram.

4. BAYLAC TESTS

A simple example of a cylindrical tube with a change in thickness subject

to an axially varying temperature distribution (Type B) is provided by the

'Baylac' tests, the geometry of which is illustrated in Fig. (2). This con­

figuration is particularly suitable as it shows competition between mechanisms

involving the thin cold part of the tube and the thicker hot part, with the

thermo-elastic stresses determining the locality of the mechanism. The

Brussels diagrams for four cases of this type have been calculated involving

two separate temperature gradients, shown in Fig. 2, with either an axial load

or internal pressure. The geometric and material data and the temperature

distributions being given in Table 1.

For axial loading the mechanism is localized in the thin section of the

tube for ãj- < 1.2 , where the Brussels diagram is almost exactly that of a

107 -

1.0..

ay(fì0)

-1.0

1 h ks 0.915 k= 0.680

l9«ox-9o) = 2o-yieo)/E*

-\ h H 1 1 h 0.0 0.50

Maximum and mimimum thermo-elastic stresses for the Baylac tests in the meridional (axial) direction a. for temperature difference

1.0-A cr0 a y(8 0)

-LO-

H h -Ì h

k* 0.915 -^ k=0.680 •9««-ÖJ»20L(S0)/E«

H h H h 0.0 0.25

Fig. 2 b 0.50

Maximum and mimimum thermo-elastic stresses for the Baylac tests in the circumferential (hoop) direction aQ for temperature

108

2.0-

o-y(e0)

io -

o.o.

Axial loading k=.915 Axial loading k=.68 \ \ \ ' Internal pressure k=.915 V \ \

— — Internal pressure k=.68

0.0

Fig. 3 Master diagram for Baylac tests

- 109

Wlz) tt<ÍT TRI ni m n^

L <j t=0 9 max = 20 P/PL=1.0

U(z) - H 1 1 1 1 IllIllllllHHWl H—I 1 H-H

Fiq. 3a Deformation mechanism for the Baylac Test - See Table 2 Internal Pressure - k=0.680 U(z) - Axial displacement W(z) - Radial displacement Tick marks denote the axisymmetric element structure

w(z) ^1 1 'L CTt=0

9max = 20 P/PL = 0.577

U(z) - M 1 1 1 1 l l t l l l l l l l B I W I — i — i — i — m i

Fig. 3b Deformation mechanism for the Baylac Test - See Table 2 Internal Pressure - k=0.680 U(z) - Axial displacement W(z) - Radial displacement Tick marks denote the axisymmetric element structure

110

W(z)[ 1

cjt = 1.69 S max = 338 P / P L / 0.273

U(z) 11 i i i imiuiiHmwr^^ I i l ^ -

Fiq. 3c Deformation mechanism for the Baylac Test - See Table 2 Internal Pressure - k=0.680 U(z) - Axial displacement W(z) - Radial displacement Tick marks denote the axisymmetric element structure

W(z)l

L CT,= 0.11 e = ¿1

max M ' f PL= 0.979

U(z) - M 1- /miniiimmm i i i i 11 Í

Fig. 3d Deformation mechanism for the Baylac Test - See Table 2 Axial Loading - k=0.680 U(z) - Axial displacement W(z) - Radial displacement Tick marks denote the axisymmetric element structure

111 -

W(z)

L (7t = 1.38 9mox = 2.13 P/PL=0.785

U(z)

l i i i i i HttW—I—I—I—H+

Fiq. 3e Deformation mechanism for the Baylac Test ­ See Table 2 Axial Loading ­ k=0.915 U(z) ­ Axial displacement W(z) ­ Radial displacement Tick marks denote the axisymmetric element structure

W(z) ■ «-i-rriT.

U crt = 1.69 ömox = 338 P/PL = 0.135

U(z) ■H—I—I—I—Mil Mimmi I I I MM

Fig._3f Deformation mechanism for the Baylac Test ­ See Table 2 Axial Loading ­ k=0.680 U(z) ­ Axial displacement W(z) ­ Radial displacement Tick marks denote the axisymmetric element structure

­ 112

W(z)[

| r

■ ui IP^ o-, = 124 e _ = 254

max P/PL= 0.777

U(z) f I inniii

1

Deformation mechanism for the Baylac Test - See Table 2 Axial Loading - k=0.680 U(z) - Axial displacement W(z) - Radial displacement Tick marks denote the axisymmetric element structure

113 -

tube with temperature independent yield stress, subjected to a step temperature

distribution whose thermo-elastic stress has the same knockdown factor, k

For internal pressure the behaviour at low at is progressively compromised by

incursion of the hinge/cone mechanism into the hot part of the tube, and

localization of the mechanism towards the position of the change in thickness.

The mechanisms from these Baylac tests are summarized in Table 2. At the top

of the Brussels diagram two distinct types of behaviour are seen. For the

sharper temperature distribution (k = 0.915) the usual reverse plasticity

mechanism (Xi = X2//2) is encountered. However, the lower knockdown factor

cases (k = 0.680), result in a four hinge mechanism involving a Xi hinge/

cone mechanism in the thin section of the tube changing to a X3 hinge/cone

mechanism with associated axial stretching, located across the change in

thickness and into the thick part of the tube. The relative ratios of this

composite mechanism changes between the axial loading and the internal pressure

cases. This composite mechanism occurs below the reverse plasticity line.

The master diagram for the Baylac tests is shown in Fig. (3), together

with illustrative examples of typical mechanisms reported in Table 2. All

these cases can be be shown to be conservatively bounded by the following

equations.

P/P!+ at/4CTy = 1 Axial Load i

1 P/2PJ + at/2av = 1 (3)

P/P! + at/4CTy = 1 Internal Pressure i

1 PMPi + at/2Gy = 1

where ay is taken to be the yield stress at the maximum temperature, P is

the mechanical load at the ratchetting boundary and Pi is the corresponding

limit load. In addition, Brussels diagrams for this problem is conservatively

approximated by the Brussels diagram for a Type B step temperature distribution

of the same knockdown factor in a tube whose thickness is given by the thin

section of the Baylac test geometry, again assuming a constant yield stress

- 114

corresponding to the value of the maximum temperature.

CASE 2 - THE BREE PROBLEM

The ratchetting boundary for cylindrical tubes with through thickness

temperature distributions has been calculated by Bree [3] for the simplified case

of a thin cylinder subjected to a temperature distribution given by

e = e 0 + ( e m a x - e 0 ) ( i / 2 - h ) (4)

where h is the normalized distance across the thickness of the cylinder

-1/2 < h < 1/2. Thus the inner surface cycles between 0O and 6max while

the outer surface remains at 0O. The thermo-elastic stress in this case is

given by

°<b = aQ = hEa(0max-eo)/(l-v) (5)

The solution diagram found by Bree is shown in Fig. (4), for a temperature

independent yield stress. The Bree solution for a cylinder under internal

pressure predicts that the ratchet boundary varies as

P / P j + a t / 4 a y = 1 a t < 2 a y (6)

(P/P1).(at/ay) = 1 at > 2ay

The Bree problem for a tube whose geometrical and material properties are

given in Table 3 has been solved using EECS3. The solution for internal

pressure loading is a A, 3 hinge mechanism shown in Fig. (5) for all values

of at . This mechanism is caused by the boundary condition of zero radial

displacement at the ends of the tube. The resultant Brussels diagram for a

temperature independent yield stress using uniaxial (QQ only) or biaxial

thermo-elastic stress is within 1% of the analytic solution of Bree. The

difference is due to the contribution of the hinges to the deformation

mechanism, caused by the boundary conditions. The mechanism for axial loading

cases with biaxial thermo-elastic stress is a single node X2 axial stretch

115

í-a.1

1 P/PL Fig. 4 ­ Analytic ratchetting boundary calculated by Bree (3) for a

tube with temperature independent yield stress under internal pressure

W(z) —■^­rrrrTtTmTTTTT­r­».—

In» 1

L. (Jt=0 9mox=20 P/PL=1.0

M|z j Ţ z Axisymmetric axis

n i n n i ni min n i munit

Fig. 5 ­ Deformation mechanism for Bree problem (Case 2) ­ See Table 3 Internal pressure with end U(z) ­ Axial displacement W(z) ­ Radial displacement Tick marks denote the axisymmetric element structure

­ 116

ay(90)

5.0-

¿.0--

3.0-

2.0--

L O ­

CO

Analytic bree line [3] Case 2 cr(8)=316SS Case 2 cry (8) = cry(T0)

0.0

_Fig. 6a Master diagram for Bree problems (Case 2) - o./olQ) vs P/P-, Brussels diagrams for axial loading or internal pressure, with a linear through thickness temperature distribution, using temperature independent and temperature dependent yield stress Coincident lines for a = 316 Stainless Steel

Internal Pressure - Uniaxial Thermo-elastic stress Internal Pressure - Biaxial Thermo-elastic stress Axial Loading - Biaxial Thermo-elastic stress

Same lines coincident for a = a (8 )

117

8--

4--

2--

Analytic Bree line [3] Case 2 cr (9) = 316SS Case 2 cry(e) = o-y(80)

0.0

Fig. 6b

P x o-y(80)

P Lxã y

Master diagram for Bree problems (Case 2) renormalized with respect to mean yield stress - o./~a vs P.CT (0 )/(p.,.ã ) Key as for Figure 6a

118

at an arbitary node, all nodes being equivalent, and gives the exact solution.

A basic master diagram composed of temperature independent and temper­

ature dependent yield stress cases is shown in Fig. (6a). The same calcula­

tions but with both axes renormalized with respect to the mean yield stress is

shown in Fig. (6b). The mean yield stress is defined as

- = 1 y [0max~0oJ

°max ay(0) dø (7) e0

It can be seen from the basic master diagram that all six cases lie on two lines

corresponding to the temperature independent and dependent solutions. The

analytic Bree solution, given by Equation (4), is also drawn and is coincident

with the temperature independent line. The renormalized master diagram shows

that the temperature dependent lines are shifted outside the analytic solution

at all temperatures, and thus this renormalization constitutes a conservative

rule for Type A (through thickness only) temperature distribution within tubes,

where the thickness is constant and the temperature varies linearly throughout

the thickness.

6. CASE 7 - CYLINDRICAL TUBE WITH VARIABLE THICKNESS AND VARIABLE THROUGH-THICKNESS TEMPERATURE GRADIENT

Another example of a tube with varying thickness is Case 7 of the Bergamo

set, the geometry for which is illustrated in Fig. (7) and tabulated with the

temperature distribution and the material properties in Table 4. Case 7 has

been solved as a thermal upshock under internal pressure loading, with one

end of the tube acting as an enclosing plate giving an axial component to the

internal pressure. The temperature distribution along the tube has been

estimated using a simple formula which relates the temperature at the inner

and outer surfaces. This can be expressed as

9out - O m + B 6ex)/(l + B) (8)

119

•35m

LL

h=0.01

.2275m

h=0.02

SI E

.045 .135m 0¿5

h=0.01

I T

.2275m .68m

8 max

8,

Inner surface

Outer surface " " ■ — _ ^ " " ^

2/3 max

1/2 e

max

2/3 9max UPSHOCK

Geometry and temperature distribution for Case 7 ­ See Table 4

120

where Q±n and 0Out a r e t^ie i-nner and outer surface temperatures respec­

tively and 0ex is a fluid temperature adjacent to the outer surface of the

tube. The Biot number B is a measure of the relative resistance to heat flow

of the tube metal to the adjacent fluid, and is defined as

B = W.H/K (9)

where W is the heat transfer coefficient of the surface interface, K is

the conductivity of the tube metal and H is the thickness of the tube. A

large Biot number implies that the resistance to heat flow is principally

within the tube metal whereas a small Biot number indicates that the greater

resistance to heat flow is in the tube/fluid interface. Thus B = 1 implies

that the surface and the tube transfer the same amount of heat per unit area

for identical temperature differences. The values chosen in this particular

case are B = 1 for the thicker section and B = 1/2 for the ends of the tube

where the thickness is half that of the middle section. The essential assump­

tions are that the tube is filled with liquid sodium, which being a very good

heat conductor, means that the inner surface of the tube can be regarded as

being at the same temperature as the liquid sodium. The Biot numbers chosen

approximately correspond to the outer surface being in contact with air, and

for simplicity of calculation of the thermo-elastic stress 0ex = 0 is chosen.

This results in the upshock temperature distribution shown in Table 4, for

which the axial and hoop components of the thermo-elastic stress envelopes,

calculated by CONIDA, are shown in Figs. (8a) and (8b) respectively.

The resultant Brussels diagram is very similar to that of the Bree prob­

lem. The master diagram for Case 7, renormalized with respect to the mean

yield stress is illustrated in Fig. (9) and the associated mechanisms tabu­

lated in Table 5. It can be seen from the master diagram that the renorm­

alized ratchetting boundary for Case 7 again lies outside the analytic Bree

line for all values of at , and that the changes in mechanism have little

121

1.0 -

CT«,

0.0

-1.0

J I I L I , 1 L J I L

iem,x-eol=2cry(eo)/Ect T 1 1 1 i 1 1 1 1 1 r -0.20 0.40 0.60 z

Fig. 8a Maximum and mimimum thermo-elastic stresses for Case 7 in meridional (axial) direction a, for temperature differenc

the :e given

- Fig. 8b Maximum and mimimum thermo-elastic stresses for Case 7 in the circumferential (hoop) direction cr. for temperature difference given by Û9 = 2 a ( 0 ) / Ea J y o

122

'y 8--

6 "

H \

t*-

2--

0.0

\ h

Analytic bree line [3] Case 2 Case 7

P x <ry(80) PLx cry

Fig^9

Master diagram for Case 7 renormalized with respect to mean yield stress - <*t/õ vs P. a ( 0Q)/( P^ .ã ) Internal Pressure with End Plate Included for comparison

Analytic Bree line given by equation (6) Case 2 - a = 316 Stainless Steel

- 123

W(z) A A

U(z)

k Πt=0

Ømax =20 P/PL=1.0

L Axisymmetric axis

l l l l l I I I I I III I I I HIM I I 1 H-H4

Fia 9a

W(z)

er,=2.11 9max=322 P/PL = (U24

L U(z) Axisymmetric gxis

H-H 1 I I I MUH Ml I MII - H I H4+4

Fiq. 9b

W(z) s

Á. o-, = 3.21 9max^81 P/PL=0.2^1 L

U(z) Axisymmetric axis

+f-H 1 I III l l l l l l IUI I Mill I I 1 H+++-

Fiq. 9c Deformation meclianisms for Case 7 Description of individual figures given in Table 5 Internal Pressure with End Plate U(z) - Axial displacement W(z) - Radial displacement Tick marks denote the axisymmetric element structure

- 124

effect on the shape of the ratchetting boundary.

CONICAL TUBES

The program is able to calculate the ratchetting boundaries for all types

of conical sections. As a comparison with other upper-bound and lower-bound

techniques, the limit load has been calculated for the geometry and material

properties given in Table 6a. The resultant mechanism is shown in Fig. (10),

compared with that calculated by Morelle [5] and Franco [6]. All three \ l

hinge/cone mechanisms in the wider part of the cone are nearly identical.

Comparative values for the limit load are given in Table 6b. It can be seen

that the present technique gives the lowest upper bound consistant with being

above the highest lower bound. The result by Morelle, while being a lower

upper bound is below both lower bound loads.

The calculation of Brussels diagrams has been carried out for two repres­

entative conical shapes, the geometry of which is given in Table 6c and the

material properties are as given in Table 6a. The temperature distribution

and thermo-elastic stress are the same as those used in the Bree problem.

The mechanisms along the ratchetting boundaries are the same as the limit load,

except at the very top of the Brussels diagram where very high curvature

collapse mechanisms occur, which is because EECS3 does not account for the

energy dissipated by changes of curvature within elements. Thus these mechan­

isms are not valid and show the limits at high cr of the technique. The

master diagram renormalized with respect to mean yield stress is shown in Fig.

(11). Again the diagrams for both Cones A and B are both outside the

analytic Bree line in this renormalized plot. For comparison the results of

the Bree problem for a tube (Case 2) are shown. It would appear that as the

cone angle changes from a tube towards a plate geometry, the mechanical load

for a given thermal load decreases slightly, as the Case 2, Cone B and Cone A

rachetting boundaries indicate.

125

resent technique [5]

Fig. 10 Comparison of the mechanism of deformation of the present technique with those obtained by Morelle[5] and Franco [6]

- 126 -

PLXCTy

Master diagram for Cones (See Table 6c) renormalized with respect to mean yield stress - <**./<* vs P.ff (9 )/(P,.a ) Pure Internal Pressure Included for comparison

Analytic Bree line given by equation (6) Case 2 - a » 316 Stainless Steel

127

8. SPHEROIDAL AND COMPOSITE SHAPES

To illustrate the ability of the program to calculate Brussels diagrams

for a wide variety of composite axisymmetric cases, a few particular examples

have been chosen, based on the Bergano set. For these spheroidal shapes

three characteristic distances, shown in Fig. (12), are defined as follows

r - Radius perpendicular to the axisymmetric axis

r - Radius of curvature of element mid-surface

r - Distance from the axisymmetric axis perpendicular to the mid-surface alomg a radius of curvature

r - Distance of the centre of curvature of r perpendicularly to the axisymmetric axis

Where a spherical cap meets the axes of symmetry special boundary conditions

can be analytically derived and included in the present upper bound formu­

lation. The essential requirement is that the strain increments in the

meridional and circumferential directions, EA and CQ respectively, become

equal as a point approaches the axis and zero on the axis itself.

It is'worth noting that under certain circumstances it is not possible

to recover known analytic solution mechanisms, even though the mechanical load

at the ratchetting boundary is computed accurately. This is due to a number

of mechanisms having the same or near identical mechanical loads. Approxima­

tions in the finite element method, including the use of axisymmetric assump­

tions for a spherically symmetric geometry provide sufficient perturbations to

the problem to change the optimal mechanisms. This is illustrated by the

solution for a perfect sphere (Case 1) where the technique used does not have

sufficient plastic multipliers to give the analytic solution. Thus the

solution shows the correct limit load and ratchetting boundary, but the

mechanisms are not those of the analytic solution.

128 -

Fig. 12

Characteristic radii for Spheroidal and Composite Shapes

W(z) ,-T-TTTrr

U(z) " L (T t=0

9„„ =20°C max

P/PL = 1.0

■+-H 1 I I I I I I 1 I I I I I I I I I I I I 1—H

Rg.J3 Deformation mechanism for Case 4 ­ See Table 7 Pure Internal Pressure U(z) ­ Axial displacement W(z) ­ Radial displacement Tick marks denote the axisymmetric element structure

129

5.0--

cry(60) ¿.0-1-

3.0..

Master diagram for Spheroidal and Composite Shapes - a./a (6 ) vs P/P, t y o Brussels diagrams for linear through thickness temperature distribution, using temperature dependent yield stress a - 316SS Case 4 - Pure Internal Pressure * Case 5 - Pure Internal Pressure - See Table 8 ASME Torispherical Head - Pure Internal Pressure - See Table 9 Case 6 - Internal Pressure with End Plate - See Table 10 Case 9 - Internal Pressure with End Plate - See Table 11 Included for comparison

Analytic Bree line given by equation (6) Case 2 - a = 316 Stainless Steel

130

9. max

600--

¿00 ■-

200--

-i 1 f- H 1 1 H

CASES 2 and L ASME TORISPHERICAL HEAD CASE 5 IUPSHOCK) CASE 6

— CASE 9

ff,«3BSS

0.0 p/p,

max

FigJŞ Diagram of maximum temperature in temperature distribution 0, plotted against normalized mechanical load P/P, for Spheroidal and Composite Shapes Key as for Figure 14 Analytic Bree line not shown

131

Fig 16

Pxo-y(80) PLXCTy

Master diagram for Spheroidal and Composite Shapes renormalized

with respect to mean yield stress - <*t/«J vs P. a {Q0)/(?i • % )

Key as for Figure 14

- 132

9. CASE 4 - CYLINDRICAL TUBE WITH SPHERICAL CAP OF SAME THICKNESS

The first spheriodal shape is a simple tube with a spherical cap end under

internal pressure (Case 4), the geometry of which is given in Table 7 together

with the boundary conditions and the material properties. Again the tempera­

ture distribution and the thermo-elastic stress are those used in the Bree

problem (Type A) and given in equations (4) and (5) respectively.

The solution mechanism, a \ 1 hinge/cone in the tube part, is shown in

Fig. (13), and is exactly the same as that obtained for Case 2 (Bree problem)

for all values of ãt . The Brussels diagram normalized with respect to the

limit load is shown in Fig. (14) with those for other spheroidal and composite

shapes discussed below. The corresponding diagram of maximum temperature 0 m a x

against mechanical load normalized by the limit load is shown in Fig. (15).

Finally the master diagram renormalized with respect to the mean yield stress

is shown in Fig. (16) for all these cases. In all three of these figures the

ratchetting boundary line for Case 4 is exactly coincident with that for Case

2 which is also shown for comparison.

10. CASE 5 - CYLINDRICAL TUBE WITH SPHERICAL CAP OF HALF THICKNESS

This example from the Bergamo set combines a tube of one thickness with a

spherical cap of half the thickness, illustrated in Fig. (17) and given with

boundary conditions and material properties in Table 8. This has been solved

under internal pressure for the temperature distribution which is also given in

Table 8. These values have been calculated using the same assumptions for the

Biot numbers as in Case 7. The thermo-elastic stress envelopes for this

temperature distribution, as calculated by CONIDA, are shown in Figs. (18a) and

(18b). To a good approximation the thermo-elastic stress envelope is the

same as that given by equation 5 in terms of the local thickness and through-

thickness temperature gradients.

The resultant Brussels diagram is again included in Figs. (14), (15) a

- 133 -

h=.0050

h=.0025

Geometry and temperature distribution for Case 5

e max

9

1/2 e max

Inner surface

Outer surface 2/3 e max UPSHOCK

Fig. 17

134

1.0-

'0 S^o)

Maximum and mimimum thermo-elastic stresses for Case 5 in the meridional (axial) direction a. for temperature difference given by AT = 2 ff (0O) / Ea 9

H 1 1 1 1 1 1 1 1 — 1.0-

0" e ^ 9 o ) +

0.0

-1.0-

J~ le^-ej^iej/E« 4-

T r 0.0 — I — 1.0 T r

Fig. 18b 2.0 z

Maximum and mimimum thermo-elastic stresses for Case 5 in the circumferential (hoop) direction aQ for temperature difference given by ÛT = 2 t T

v( 6o ) ^ E a

- 135

axis Axisymetnc

Fig. 19a

Deformation mechanisms for Case 5 Pure Internal Pressure U(z) ­ Axial displacement W(z) ­ Radial displacement Tick marks denote the axisymmetric element structure

W(z)f ■.-r-T-TTr»...

Axisymetric axis

U(z)

Fig. 19b

136 -

(16). There are two mechanisms along the ratchetting boundary. At gt less

than 0.4 the mechanism is a spherical cap deformation shown in Fig. (19a) for

which the mechanical load is exactly that of a sphere (Case 1). At ^jt greater

than or equal to 0.4 the mechanism changes to the Xi hinge/cone mechanism in

the tube part, shown in Fig. (19b), as in Case 2 and Case 4. However the

thickness is increased in the tube part, and thus the ratchetting boundary lies

outside the line of Case 2 for the hinge/cone mechanism at at > 0.4 . Thus

the ratchetting boundary for this case also lies outside the analytic Bree line

when renormalized with respect to the mean yield stress.

11. ASME STANDARD TORISPHERICAL HEAD

The ratchetting boundary has been calculated for the ASME Standard Toris­

pherical Head under internal pressure, for which independent calculations of

the limit load by Drucker and Shield [9] are available. The geometry, boundary

conditions and material properties for a particular torispherical head are

given in Table 9. The limit load collapse mechanism is shown in Fig. (20a)

which is very similar to the mechanism given by Drucker and Shield. The only

substantial difference is that the mechanism of Drucker and Shield extends

slightly into the tube section, whereas in the present calculation the mech­

anism starts at the boundary between the tube section and the spheroidal

knuckle section. This seems a more likely mechanism as a tube requires more

energy to distort than the weaker knuckle section. The limit load, divided

by the yield stress at 0O , for the present calculation is .626 x 10~3

whereas the Drucker and Shield result gives .675 x 10~3.

The Brussels diagram for the Type A temperature distribution of equation

(4) and the thermo­elastic stress given in equation (5), is shown in Figs.

(14),(15) and (16). The associated ratchet boundary mechanisms from 0 to

2.5 o'ţ. are virtually identical to the limit load mechanism. At higher

the hinge/cone mechanism at the knuckle becomes progressively sharper as seen

137

Wlz)

•L Œ, = 0 Ømax =

P/PL = 1.0 Ømax = 20

U(z) -<—I—I—I—I—H 1—I—I—H-#

Fig. 20a

W(z)

Ulzl

■L

L

ãt=2.63 emax=257 P/P, = 0.249

- I—l—I—H 1 1 1—K-W

Fig. 20b

W(z)

L (Tt= 3.40 Ømax=326 P/PL= 0.075

U(z) < 1 1—I 1 1 M 1 I I

Fig. 20c Deformation mechanisms for ASME Tor i sphe r i ca l Head Pure I n t e r n a l Pressure U(z) - Axial displacement W(z) - Radial displacement Tick marks denote the axisymmetric element s t r u c t u r e

138

in Figs. (20b) and (20c) which are typical examples. The work done by these

mechanisms is principally in the movement of the spherical cap in the direction

of the internal pressure, enabled by the knuckle deforming, being the weakest

part of the structure.

The master diagram renormalized with respect to the mean yield stress

Fig. (16) shows that the ratchetting boundary lies outside the analytic Bree

line except for at > 4.0 which corresponds to an upshock of 270°C. The

mechanisms in the high at region are substantially sharper and the changes

in curvature within the knuckle elements are an order of magnitude larger

than those at low at •

12. CASE 6 - CYLINDER TO CONE TO CYLINDER (CONTINUOUS ANGLE)

This example from the Bergamo set shows the ability of EECS3 to cope with

composite axisymmetric shapes where the angle defining the geometry is con­

tinuous. This case consists of a tube connected by a short spheroidal

section to a cone, which leads to a tube of twice the diameter of the previous

tubular section via another short spheroidal section. The geometry, boundary

conditions and material properties are given in Table 10. The ratchet

boundary has been solved for Type A thermal loading, the temperature distri­

bution being given by equation (4) and the thermo-elastic stress by equation

(5) as in the Bree problem. The mechanical loading is internal pressure with

an end plate at the end of the larger tube section giving an additional axial

component to the mechanical loading.

The resultant Brussels diagram is again illustrated in Figs. (14),(15)

and (16). The limit load mechanism is shown in Fig. (21a) and shows remark­

able similarity to the ASME Standard Torispherical case, the work done by the

mechanism being in the movement of the end plate in the direction of the

internal pressure. Again the knuckle with the largest radius deforms as it

is the weakest part of the structure. As the thermal load increases the

139

W(z) emnv=20 'max P/PL=1.0

U(z) ++* 1 1 Hm 1 1 1 <illllll I I 1 HHH y^m

Fiq. 21a

W(z)

Qmax =89

U(z)

P/PL= 0.853 i

< + m — i — i — \ m — i — i — i num i i—i—mw+-

Fiq. 21b

W(z)

U(z)

<rt = 3.95 * 9max=376

P/PL=0.U h -Må

m—\—i—i f mim i i — i — m m -

Fiq.21c Deformation mechanisms for Case 6 Internal Pressure with End Plate U(z) - Axial displacement W(z) - Radial displacement Tick marks denote the axisymmetric element structure

140

mechanism remains that of the limit load to approximately 0.75at» at which

point the mechanism starts to become progressively sharper as shown in Figs.

(21b) and (21c). At the highest value of at (4.28) the change of curvature

within the knuckle elements is an order of magnitude larger than for the limit

load.

The master diagram, shown in Fig. (16), which has been renormalized with

respect to the mean yield stress shows that the ratchetting boundary lies out­

side the analytic Bree line except at very high at > 4.0 (which corresponds

to an upshock of approximately 350°C) where the mechanisms involve large

changes of curvature within the elements.

13. CASE 9 - CYLINDER TO CYLINDER BY SPHEROIDAL SECTIONS (CONTINUOUS ANGLE)

In this case the geometry is similar to that of Case 6, consisting of a

tube connected to a tube of twice the radius by two spheroidal sections of the

same curvature. The geometry, boundary conditions and material properties are

given in Table 11. The thermal loading is again Type A, and is given by

equations (4) and (5). The mechanical loading is the same as Case 6, i.e.

internal pressure with an end plate.

Here the structure does not contain a weak knuckle section, thus the limit

load mechanism illustrated in Fig. (22a) is a Xi hinge/cone entirely local­

ized in the section of tube with the larger radius. This mechanism persists

to approximately 0.7at . Above this value an interesting composite mechanism

takes place, shown in Fig. (22b), where there is an axial stretching mechanism

at the boundary between the small radius tube and the first spheroidal section,

which results in work done by end plate extension, and a much smaller Xi

hinge/cone mechanism in the larger radius tube section. This composite

mechanism continues from 0.7 ot to approximately 1.5 at , during which the

Xi hinge/cone part becomes smaller and smaller, so that at 1.5 at the

mechanism only involves the axial stretch at the boundary between the smaller

141

• - Z

Fig. 22a

e max = 89 P/PL= 0.753

U(z) Hill I I t l * T I 1—l-H-H—I 1 1 1 I I I 1 I mi»

Fig. 22b

U(z)

cjt=1.5A Ömox=158 P/PL=CU98

Hill U M ' 1—I I MM I—I M I M I I

Fig. 22c. Deformation mechanisms for Case 9 Internal Pressure with End Plate U(z) - Axial displacement W(z) - Radial displacement Tick marks denote the axisymmetric element structure

142

radius tube and the first spheroidal section. This mechanism is shown in Fig.

(22c), and is the mechanism of the ratchetting boundary up to 3.4 at where

mechanisms involving large changes in curvature within elements start to occur.

The Brussels diagrams for Case 9 are again shown in Figs. (14),(15) and

(16). The associated master diagram renormalized with respect to the mean

yield stress (Fig. (16)) shows that the renormalized ratchet boundary lies

outside the analytic Bree line except at very high at (corresponding to an

upshock of approximately 300°C) where the mechanisms become unreliable as' they

involve large changes of curvature within elements.

14. CONCLUSION

In the final section of the report we have given Brussels diagrams for a

range of geometries, from solutions generated by a finite element method

incorporated in a computer code EECS3. We observe that the mechanisms assoc­

iated with the ratchet boundary take a wide variety of forms, but some general

conclusions can be drawn. For all these problems where the thermal gradient

is through the shell thickness, the classical Bree solution for a uniform

cylindrical tube yields a safe bound when the uniform yield stress in that

solution is chosen as the mean yield value, defined by equation (7). The

comparison for a range of cases is shown in Fig. (16). For cases where the

temperature gradient is entirely axial along a cylindrical tube, there is some

evidence from the Baylac tests and from other tests not reported here, that

the Brussels diagrams can be approximated by the diagram for a linear axial

gradient in a uniform term, where the gradient is chosen so that the knockdown

factor coincides.

These solutions begin to yield an insight into the range of Brussels

diagrams which may be of use in design. At the present time the results of

these and many other calculations are being used to define a set of master

diagrams which encompass a range of practically useful cases as the basis for

design code rules in Fast Reactor design. - 143 -

Table 1 Baylac Tests

Geometry

Axial Position (m)

0.0 0.2275 0.2725 0.5

Temperature Distributions

Temperature

e max

Radius

0.35 0.35 0.35 0.35

(m)

e

Axial Position (m) k = 0.680

0.0

0.2185

0.2815

0.5 max Boundary Conditions 1) Radial Displacement set to zero at extremes of tube. 2) Axial Displacement set to zero at end of tube (z = 0),

Thickness (m)

0.011 0.011 0.020 0.020

Axial Position (m) k = 0.915

0.0

0.2365

0.2635

0.5

Material Properties

0O

E

a

V

av(0o)

20°C

.195 x 10+12 N/m2

.1639 x IO"1* /C

.3

.205 x 10+9 N/m2

Yield values vs. temperature given by DCWG[4] - see Fig. la

6 °C

20 50 100 150 200 250 300

Limit Load

0-y(6) MPa

205 179 155 142 132 121 113 ""

8 °C

350 370 400 450 500 550 600

Oy(0) i 106 104 101 97 95 92 90

Axial Loading Internal Pressure

.242 x IO"1 ay(60)

.407 x IO"1 ay(80)

144

Table 2 Summary of Mechanisms for Baylac Tests

k = 0.680 Axial Loading

0 - 1.2 at X2 single node axial stretch in thin part of tube Figure 3d

1.2 - 1.6 ot A3 hinge/cone mechanism (axial stretch) across change in thickness Figure 3g

1.6 at -4 hinge/cone mechanism Ai hinge/cone in thin part (small) A3 hinge/cone across change in thickness into thick part (large) - Figure 3f

01

k = 0.680 Internal Pressure

0 - 1.2 at Ai hinge/cone in thin part - Figure 3a

1.2 - 1.6 ot Ai hinge/cone around change in thickness Figure 3b

1.6 at -4 hinge/cone mechanism Ai hinge/cone in thin part (large) A3 hinge/cone across change in thickness into thick Part (small) - Figure 3c

k = 0.915 Axial Loading

0 - 1.3 at A2 single node axial stretch in thin part

1.3 - 1.8 ot single node (A1/A2/2) reverse plasticity at start of change in thickness (thin end) Figure 3e

1.8 at -single node (A1/A2/2) reverse plasticity at end of change in thickness (thick end)

k = 0.915 Internal Pressure

0 - 1.3 at Ai hinge/cone in thin part

1.3 - 1.9 at Ai hinge/cone around change in thickness

1.9 at -single node (A1/A2/2) reverse plasticity at end of change in thickness (thick end)

Table 3 Bree Problem (Case 2)

Geometry

Axial position (m)

0.0 1.0

Radius (m)

1.0 1.0

Thickness (m)

0.0025 0.0025

Temperature Distribution

As given by equation (4) in text.

Boundary Conditions

1) Radial Displacement set to zero at extremes of tube. 2) Axial Displacement set to zero at end of tube (z = 0)

Material

e0

E

a

V

ay(90)

P roperties

20°C

.195 x 10+12 N/m2

.1639 x 10-lt /C

.3

.205 x 10+9 N/m2

Yield values vs. temperature given by DCWG[4]

Limit Load

Axial Loading .157 x 10-1 ay(80) Internal Pressure .253 x 10~2 ay(60)

146

Table 4 Tube with Variable Thickness and Through Thickness Temperature Gradient (Case 7)

Geometry

Radius (m) Thickness (m) Axial Position (m)

0.0 0.2275 0.2725 0.4075 0.4525 0.68

Temperature Distributions

Axial Position (m)

0.0

0.2275

0.2725

0.4075

0.4525

0.68

Boundary Conditions

0.35 0.35 0.35 0.35 0.35 0.35

0.01 0.01 0.02 0.02 0.01 0.01

Initial Inner

e0

e0

e0

e0

e0

e0

Surface

Outer

e0

e0

6o

e0

e0

e0

6n

Temperatures

= 20°C

Upshock Inner

"max

"max

"max

^max

°max

"max

- Case 7 Outer

26max/3

20max/3

°iax'2

°iax'2

2emax/3

26max/3

1) Radial Displacement set to zero at extremes of tube. 2) Axial Displacement set to zero at end of tube (z = 0)

Material Properties

E

a

V

ay(0o)

N/m2

/C

N/m2

Upshock - Case 7

.195 x 10+12

.1639 x IO"1*

.3

.205 x 10+9

Yield values vs. temperature given by DCWG[4]

Limit Load

Internal Pressure .363 x 10"1 ay(80)

147

Table 5 Ratchetting Boundary Mechanisms for Case 7

0 - 2.1 at Xi hinge/cone in thin part of tube - Figure 9a

2.1 - 3.2 Oţ X2 single node axial stretch in thin section of tube, just after change in thickness - Figure 9b

3.2 - 4.5 at Xi hinge/cone in thin part of tube localized towards the change in thickness - Figure 9c

Table 6a Cone Test

Geometry

Axial Position (m) Radius Perpendicular to Thickness (m) Axisymmetric axis (m)

0.0 0.0 0.05 2.5 1.0 0.05

Boundary Conditions

1) Displacement normal to the mid-surface set to zero at the extremes of the tube

2) Displacement tangential to the mid-surface set to zero at the end of the cone (z = 0)

Material Properties

E N/m2

a /C

V

ay(60) N/m2

Yield values vs.

.195 x 10+12

.1639 x 10-1*

.3

.205 x 10+9

temperature given by DCWG[4],

148

Table 6b Limit Load for Cone

Reference

Present

Morelle [5]

Franco [6]

Biron et al [7]

Nguyen et al [8]

Type of Formulation

Upper Bound

Upper Bound

Upper Bound

Upper Bound Lower Bound

Upper Bound Lower Bound

Pi/cry(eo)

0.0518

0.0A82

0.0521

0.0532 0.0504

0.0541 0.0496

Table 6c

Geometry Cone

Axial Position

0.0 1.0

Geometry Cone

A

(m)

B

Axial Position (m)

Radius Perpendicular to Axisymmetric Axis (m)

1.25 0.125

Radius Perpendicular to Axisymmetric Axis (m)

1.25 0.125

Thickness (m)

0.05 0.05

Thickness (m)

0.05 0.05

0.0 2.25

Boundary Conditions

1) Displacement normal to the mid-surface set to zero at the extremes of the tube

2) Displacement tangential to the mid-surface set to zero at the end of the cone (z = 0)

Limit Load

Pure Internal Pressure

Cone A -l

Cone B

.506 x IO-1 ay(0o) .518 x 10 -i

149

Table 7 Parameters for Case 4

1.0 1.0 0.0

Radius (m) r0 rx r2

1.0 1.0

0.0 1.0 1.0

Thickness (m)

0.0025 0.0025

0.9925

Geometry

Axial Position (m)

0.0 1.0

2.0

Boundary Conditions

1) Displacement normal to the mid-surface set to zero at the end of the tube (z = 0)

2) Displacement tangential to the mid-surface set to zero at the end of the tube (z = 0)

3) Spherical Cap a) Displacement tangential to the mid-surface set to zero at the

centre of the cap (z = L) b) c)

Material

e0

E

a V

ay(90)

£e - e* deQ/dsq>= de^/ds

Properties

N/m2

/c

N/m2

= 0.0

20°C

.195 x 10+12

.1639 x 10-1*

.3

.205 x 10+9

Yield values vs. temperature given by DCWG[4]

Limit Load -,-2 Pure Internal Pressure .253 x 10"z ay(0o)

150

Table 8 Parameters for Case 5

Geometry

Axial Position (m)

0.0 1.0

1.174

2.0

Temperature Distributions

Axial Position (m)

0.0

1.0

1.174

2.0

Boundary Conditions

r 1.0 1.0

0.985

0.0

Radius ro

0.0

0.0

(m) r i

1.0

1.0

r2 1.0 1.0

1.0

1.0

Initial Inner Outer

Thickness (m)

0.005 0.005

0.0025

0.0025

Surface Temperatures Upshock - Case 5 Inner Outer

e0

e0

e0

e0

e, e. max -'max Jmax

-"max

emax/2

"max'2

20max/2

20max/2

0n = 20°C

1) Displacement normal to the mid-surface set to zero at the end of the tube (z = 0)

2) Displacement tangential to the mid-surface set to zero at the end of the tibe (z = 0)

3) Spherical Cap a) Displacement tangential to the mid-surface set to zero at the

centre of the cap (z = L)

c) deø/dsT= de<j)/ds = 0.0

Material Properties

20°C 6c E

a v

N/m2

/C

ay(60) N/m2

.195 x 10 + 1 2

.1639 x 10_,f

.3

.205 x 10+9

Yield values vs. temperature given by DCWG[4]

Limit Load

Pure Internal Pressure .500 x 10 _ z ay(90)

151

Table 9 Parameters for ASME Standard Torispherical Head

Geometry

Axial Position (m) Radius (m) r0 *i

0.88

0.0

0.12

2.0

r2 1.0 1.0

2.0

2.0

Thickness (m)

0.0025 0.0025

0.0025

0.0025

0.0 1.0 1.161 1.0

1.267 0.936

1.5 0.0

Boundary Conditions 1) Displacement normal to the mid-surface set to zero at the end of the

tube (z = 0) 2) Displacement tangential to the mid-surface set to zero at the end of

the tube (z = 0) 3) Spherical Cap

a) Displacement tangential to the mid-surface set to zero at the centre of the cap (z = L)

b) CQ = £fy c) deø/ds = de^/ds =0.0

Material

e0

E

a V

ay(e0)

Properties

N/m2

/c

N/m2

20°C

.195 x 10+12

.1639 x 10_lt

.3

.205 x 10+9

Yield values vs. temperature given by DCWG[4]

Limit Load 1-3 Pure Internal Pressure .626 x IO-' Cfy(60)

152

Table 10 Parameters for Case 6

Geometry

Axial Position

0.0 0.450

0.535 0.965

1.050 1.5

(m)

Boundary Conditions

r

0.5 0.5

0.535 0.965

1.0 1.0

radius (m) Thickness (m) r0 ri r2

0.62 0.12

0.88 0.12

0.5 0.0025 0.5 0.0025

0.757 0.0025 1.365 0.0025

1.0 0.0025 1.0 0.0025

1) Displacement normal to the axisymmetric axis set to zero at the ends of the structure

2) Displacement tangential to the axisymmetric axis set to zero at the end of the tube (z = 0)

Material

e0

E

a V

av(90)

Properties

N/m2

/c

N/m2

20°C

.195 x 10 + 1 2

.1639 x IO-1*

.3

.205 x 10+9

Yield values vs. temperature given by DCWG[4]

Limit Load

Internal Pressure .110 x 10~2 ay(90)

153'

Table 11 Parameters for Case 9

Geometry

Axial Position

0.0 0.317

0.75

1.183 1.5

(m)

Boundary Conditions

r

0.5 0.5

0.75

1.0 1.0

Radius (m) r0 r!

1.0 0.5

0.5 0.5

r2

0.5 0.5

1.5

1.0 1.0

Thickness

0.0025 0.0025

0.0025

0.0025 0.0025

1) Displacement normal to the axisymmetric axis set to zero at the ends of the structure

2) Displacement tangential to the axisymmetric axis set to zero at the end of the tube (z = 0)

Material

6o E

a

V

ay(60)

Properties

N/m2

/c

N/m2

20°C

.195 x 10 + 1 2

.1639 x IO-4

.3

.205 x 10+9

Yield values vs. temperature given by DCWG[4]

Limit Load -2 Internal Pressure .275 x 10 0"y(6o)

154 -

References

[1] S. KARADENIZ & A.R.S. PONTER

An extended shakedown theory for structures that suffer cyclic thermal loading, Parts 1 and 2, J. Appi. Mechanics, Trans. ASME, 1985: 52, 877. Ibid. 1985: 52, 883.

[2] P.W. CLARKE

CONIDA: A finite element program for the stress analysis of axisymmetric thin shells, United Kingdom Atomic Energy Authority, 1974: HMSO Report 2382(R).

[3] J. BREE

Elastic-plastic behaviour of thin tubes subjected to internal pressure and intermittent high heat fluxes with application to fast nuclear reactor fuel elements, J. Strain Analysis, 1967: 2, 226.

[4] Interim D.C.W.G. recommendation note on allowable design limits for type 316 stainless steel in the treated solution condition, UKAEA, Risley Nuclear Development Estalishment, Report no. CFR/DCWE/P(80) 269.

[5] P. MORELLE

Numerical shakedown analysis of axisymmetric sandwich shells, To be published.

[6] J.R.Q. FRANCO

Ph.D thesis, Leicester University 1987.

[7] A. BIRON & U.S. CHAWLA

Numerical method for limit analysis of rotationally symmetric shells, Bulletin de l'Académie Polonaise des Sciences, 1970: 18, 109.

[8] D.H. NGUYEN, M. TRAPELETTI & D. RANSART

Bornes quasi-inferieures et bornes superieures de la pression de ruine des coques de revolution par la methode des elements finis et par la programmation non limeaire, Int. J. Non-linear Mechanics, 1978: 13, 79.

[9] R.T. SHIELD & D.C. DRUCKER

Limit strength of thin walled pressure vessels with an ASME standard torispherical head, 3rd Congr. of Appi. Mech., 1958: 665.

[10] Limit analysis of symetrically loaded thin shells of revolution, ASME J. Appi. Mech., 1959: 26, 61.

- 155

Appendix Extended upper bound shakedown theory and the finite

element method for axisymmetric thin shells

1. INTRODUCTION

The method developed to establish the position of the shakedown/rat-

chetting boundary is based upon the upper-bound shakedown theorem [6].

This theorem bounds the magnitude of the mechanical load for a prescribed

temperature cycle by equating the internal rate of plastic energy dissipa­

tion to the rate at which the applied mechanical loads do work on mechanisms

of deformation of the body. The optimal mechanism of deformation is chosen

by linear programming methods to be that which has the lowest energy

dissipation for a given load within the constraints of the structure.

Thus the upper-bound estimate of the shakedown/ratchetting boundary will be

either correct or high by the least possible amount within a class of

mechanisms.

2. UPPER-BOUND THEORY

The upper-bound theorem [6] relates the energy dissipated by a mech­

anism of deformation to the work done by the thermal and mechanical loads,

where the material is assumed to be perfectly plastic. The energy dissi­

pated by the mechanism is the total plastic energy dissipated over the

loading cycle which is given by

, c where V denotes the volume of the body and the superscript c denotes

'T

( [ aïj(x,t) êij(x.t) dt dV (D

that the stress O M a n d strain rates ¿£j are related through the

associated flow law. The work done by the mechanical load \|>p on dis­

placement can be written as

* P.AUC ds (2) JS

where S denotes the surface area of the body and AU the plastic dis-

159

placements of the deformation mechanism. The mechanical load is assumed

to be constant throughout the thermal loading cycle (t = 0 to t = T) . » c «c

The displacement AU is related to the history of Btrain £±¡ through the condition that the accumulated strain

Ae ij e. . dt ij (3)

is compatible with AU . The work done by the thermal loads is an

integral over the thermo-elastic stresses O.. (x,t) in the body,

V á±.(x,t) ¿ijíx.t) dt dV (4)

In terms of these quantities the upper bound theorem is expressed in the

inequality

,T

V aJjU.t) ¿J.(x,t) dt dV > ip 'ij P.AU ds

(5)

V .c a, ,(x,t) è (x,t) dt dV

1J 1J

i.e. the energy dissipated must be greater than or equal to the work done

by the system. The equation (5) may now be rearranged to give a form

suitable for minimization

.c [a±.(x,t) - a±,(x,t)] | (x,t) dt dv > \|> p.AU ds (6)

If we now require that

P.AU ds = 1, (7a)

inequality (6) simplifies to

[a°(x,t) - â (x,t)] ljj(x,t) dt dV > \|t (7b)

Thus the problem can be reduced to a minimization over the volume of the

body throughout the loading cycle.

The problem is reduced to a linear programming problem by first ex-

160

pressing both ¿ ­j an

d Ae ­ţ in terms of the magnitude of the plastic

multipliers corresponding to the surfaces of the Tresca or 12X surface

of Fig. 2,

k • k ê­Lj = Nj j Xk and Ae­ j = N ^ Xk , k = 1 to 6 (8)

where the yield surfaces are given by

(9)

Substituting in (7b) yields

Ţ f f faij - °ij)

Ni j *k dt dV > | {(a±5(tk) - a lj(tk)N1^} Xk dV > $ (10)

'v o

where tv is chosen so that tk

{(cr±j(tk) ­ â±j(tk)) N^} < {(a±j(t) ­ Sij(t)) N±J} for 0 < t < T (ID

A finite element approximation to the total Xk ma

y n o w De introduced,

reducing (10) to a linear form in terms of nodal values of the Xk • The

linear constraint equations are then provided by equation (7a). This is

achieved by integrating the strain displacement relationships. Displace­

ment boundary conditions and continuity conditions between elements of

differing geometric types provide further linear constraints which form

part of the linear programming formulation.

For axisymmetric shell problems discussed here the following assump­

tions were made;

(a) The Xk varied linearly between nodal points with no through­

thickness variation,

(b) mid­section values of Xk were continuous between nodes,

161

(c) at nodal points a concentrated curvature could occur in the form

of a plastic hinge.

When the plastic behaviour of a single element is investigated in

terms of generalised stresses and strains, the behaviour is identical to

the non-interactive yield surface of Drucker and Shield [2], as membrane

action and curvature in the meridional direction are uncoupled by concen­

trating the curvature at plastic hinges. Improvements on this approxima­

tion are currently under development.

3. UPPER-BOUND METHOD FOR AXISYMMETRIC SHELL ELEMENTS

The structure is divided into a series of finite elements. Four

types of elements can be so defined for axisymmetric shapes; cylindrical,

conical, toroidal and inverse toroidal. The strain/displacement relation­

ships within such elements are given by

_ dq(s) B(s) £<t> " ds " rx (12)

= a(s)Cot(|> - ß(s) (13) 6 r2

Similarly, the curvatures are given by

d rq(s) A dß(s)i H = d^ [~7^ + "ïïs- ' d1»)

= Çotj rojll + dß(s)| ( 1 5 ) o r l r, ds J

2 1

where <f denotes the meridional direction and 6 the circumferential

direction and s is the mid-surface coordinate. Oí(s) is the displace­

ment tangential to the mid-surface and ß(s) is the displacement normal

to the surface towards the axisymmetric axis. rx defines the radius of

curvature of the element mid-surface, r2 the distance along the radius

to the axis of rotation, and r the distance of the centre of curvature

of rx to the axis of rotation as illustrated in Fig. 1.

162

For a Tresca yield condition the plastic strains determining the

mechanism are a linear combination of the plastic multipliers, which are

assumed to vary linearly throughout an element, hence the strains in this

approach also vary linearly. The displacements must also be continuous

throughout the structure. It is further desirable that the two curvature

terms, particularly <, should be small or zero.

Work in bending in the meridional direction are accomodated by the use

of plastic hinges at the nodal points between elements. The energy dissi­

pated by changes in curvature within elements is thus transferred to the

energy dissipated by changes of angle in the plastic hinges. In certain

exceptional cases, usually only found at high values of the thermal load

or where the geometry is very rapidly varying, it is necessary to specif­

ically account for the energy dissipated caused by changes in curvature

within elements. In these circumstances the mechanisms found by this

technique would no longer be the lowest upper bound if the curvatures

within the elements were found to be very large. As the present method

does not account for this mode of energy dissipated the resultant mechanical

load will always be less than the true value. Thus this method is always

conservative in these conditions.

The plastic hinge angle can be expressed as

= Lim + dß(s) _ Lim _ dß(s) ( l 6 ) s+s^ ds s- Sj ds

This gives a constraint at each node relating positive hinge angle to the

normal displacements at the node. It is possible to increase the accuracy

of the yield surface by increasing the number of basic plastic multipliers

from 6 (Tresca) to 12 which decreases the error compared with the Von Mises

ellipse from 15? to 3%.

The constant mechanical load can be separated into an axial load com­

ponent and a pressure component acting normal to the surface. This allows

163

a variety of differing loading situations to be studied, namely; axial

loading (tension and compression); pure internal/external pressure;

internal/external pressure (with end plates); bands of internal/external

pressure.

Cylindrical Elements;

The expressions for the strains and curvatures reduce to

e e = ­ B(s)/r2 (17)

(18) = da(s) :<t> " ds

KQ = d2ß(s)/ds

2 : Kø = 0 (19)

which implies that a(s) varies quadratically and 3(s) varies linearly

within the element.

Conical Elements:

Here the expressions for the strains and curvatures become

_ a(B)Cos(ţ) + g(s)Sin(ţ) £9 " r2Sinc|>

= da(s) e$ ds

K<j, = d2ß(s)/ds2

Cosd) dg(s) 9 r Sin<j) ds

As the strains are linear functions of s , then the displacements must

both vary quadratically along the element.

Toroidal Elements:

(20)

(21)

(22)

(23)

For toroidal elements the equations for the strains (12) and (13) can

be separated giving a first order differential equation for the displace­

ments, which can be solved to give

­ 164 ­

a (s) = C Sin<|) + Sint|) ds [e^ - eøi^/rJD/SirKj) (24)

ß(s) = C Cos<)) - eg r2 + Cos(J) ds [e^ - eg(r2/r1 ) ]/Sin<|> (25)

where C is a constant determined by the boundary conditions. The

strains and plastic multipliers remain linear determining the analytic form

of the displacements.

Inverse Toroidal Elements:

In this case the equations for the strains and curvatures are slightly

different due to redefining the displacement directions to be consistant

with the previous element types. Again the strain/displacement relations

can be separated giving a first order differential equation for the dis­

placements , which can be solved to give

a(s) = C Sin<t> + Sin(J> I ds [e^ + eø (r2/r1 ) ]/Sin<t> (26)

3(s) = C Co8(|> - Gø r2 + Cost ds [E<J, + e0(r2/r1 ) ]/Sln<|» (27)

the constant C being determined by the boundary conditions.

Minimisation of the Upper Bound by Linear Programming Techniques

The upper bound method is then the straightforward translation of the

strain/displacement relations above into the energy dissipation and work

terms, giving in linear programming terms, a cost function and a general

constraint respectively. The minimisation takes place to find the

mechanism of smallest cost (plastic energy dissipation less thermal work

done) for a given amount of mechanical work done, subject to the boundary

constraints, mechanical work done constraint and hinge angle constraints.

It can be shown that there is no need for matching constraints between

different element types as the displacements are continuous, if and only

if the tangent angle determining the geometry is continuous.

165

The cost function is evaluated by calculating the left hand side of

equation (10). The upper bound solution determines the closest solution

to the correct solution within the class of solutions defined by the

various approximations made. In addition a consequence of the upper

bound theory is that the elastic modulus E and the coefficient of thermal

expansion a are assumed to be constant at values corresponding to the

initial temperature.

The procedure for evaluating the cost function may be summarised as

follows :

1) For each of the plastic multipliers, corresponding to a face of the

yield surface, find the minimum of the stress difference through the entire

loading cycle, for a given point in space as expressed in inequaltity (11).

This gives a set of times in the loading cycle at which the minima occur.

These times may not coincide for adjacent material points in the structure.

2) Integrate through the volume of each element using the stress differ­

ence minima determined above.

It should be remembered that both the thermo-elastic stress and the yield

stress vary throughout the loading cycle as the temperature of the material

changes.

The procedure for calculating the cost function for the plastic hinges

is slightly different. In order to facilitate linear programming in which

all the variables must be positive, the hinge angle at each node is ex­

pressed as the sum of two positive contributions

6i = 6i - dl (28)

at each hinge the plastic strain rate can be expressed as the sum of

meridional and circumferential contributions

. c, „ r .c ¿ijix.t) = (êjjíx.t) , ¿e(x,t)) (29)

166

The work dona by the circumferential term can be shown to be zero, and the

strain in the surface meridional direction can be deduced from the elon­

gation of different fibres, initially undistorted, through the thickness.

These elongations vary linearly from the mid-point h = 0 so that the

layers above are in tension and those below in compression. This gives

áeç = % h/rj (-1/2 < h < 1/2) (30)

where r^ is the radius of curvature corresponding to the angle 0^ and

H^ is the thickness at the i node. The hinge point on the yield

surface is along the dgx axis. Thus for the Tresca yield condition the

plastic hinge is on the junction between X and X and between \ and

X . However, in the limit of small hinge angles, it can be shown that 6

the hinge involves the plastic multipliers X and X only (see Figure

2). The hinges are such that de^ is positive on the outer surface, zero

on the mid-surface and negative on the inner surface for positive hinge

angles øt. Thus the active plastic multiplier above the mid-surface

(h > 0) is X , and below (h < 0) is X for øt. This is the other 2 5 -1-

way round for negative hinge angles ©Ï . The concept of the active

plastic multiplier associated with the plastic hinge is required in order

to apply the extended upper bound method. The hinge angle is assumed to

be small, which gives the cost function for plastic hinges in the 0i case to be

2ïïHi0iSint|>i V. 2

dh [ r j + H±h] h Aox ( x ± , t )

/•O

- 2TTHi0iSin<J)i (3D

dh [r* + H±h] h Aax5(xi,t) -Vi

The procedure for calculating the cost function for the plastic hinges

follows the same method as for the elements, namely;

167 -

1) Search through the loading cycle for the minimum of the stress

difference at a given node and point through the thickness.

2) Integrate through the thickness changing the sign of the plastic

strain, corresponding to the hinge angle at the mid­point.

The mechanical work done can be be expressed as the sum of contribu­

tions from the plastic multipliers within each element, taking account of

the different geometries of the element type. This forms a general con­

straint to the linear programming method determining the size of the mech­

anism of deformation.

The upper­bound mechanism is thus the solution to the linear program­

ming problem above, giving the mechanical load required to reach the

shakedown/ratchetting boundary for a given thermal loading cycle.

4. EXTENDED UPPER­BOUND METHOD

The upper­bound shakedown theorem only applies when there exists a

residual stress field p^* so that the sum with the thermo­elastic history ai1

+ Pii ll e s within the yield surface. If, at some point, there exists

no local value of p^j which satisfies this criterion then the shakedown

limit has been exceeded and localized reverse plasticity will occur.

The extension to the shakedown theory employed in EECS3 allows an estimate

of the primary load \|;p which will cause general ratchetting by allowing

localized reverse plasticity to occur. The theory is described by Ponter

and Karadeniz [6]. We sub­divide the total volume of the structure V

into a region VF where the thermo­elastic solution cannot be translated

by the addition of p ^ so that it lies entirely within yield; and its

complement Vo . Within Vp we define a residual stress field p^* so

that cjji + p;M i s contained within the Tresca yield condition for in­

creased values of o ^ , k = 1 .. 6. The particular p^ţ chosen is the

one which requires the smallest increase in a k , assuming that the

­ 168

yield value for each pair of yield surfaces of opposite sign have equal

yield values. This means that the sum o^* + piî has equal and opposite

values of shear stress corresponding to a pair of yield surfaces. In

this way we define an assymptotic (in terms of cyclic behaviour) stress

history which assumes complete cyclic hardening of the material. In [6]

Ponter and Karadeniz argue that the assumption is conservative. The rat­

chet limit is then defined as the load level \|jp corresponding to the

shakedown limit in Vg , given by the upper-bound.

i> P Au ds < JVC

[cr-Lj (x,t) - Gij (x,t) - PiJ] ¿ ± J dt dV (32)

where é^j is defined in the usual way for the entire volume V and hence

also for Vc As p.. i is a residual stress field in V then

Pij êij dt dV = Vs

Jo

,T Pij éij d t d v + Pij éij d t d v

Ae< A dV = 0 Pij ûeij

Hence the inequality (32) may be written as

(33)

* P Au ds < [Oij(x,t) - â±j(x,t)] i±j dt dV + Vs

Jo Pij AsijdV (34)

As p^* is defined in terms of o^* within Vp its value may be calculated

without difficulties.

There are two underlying problems in the argument. The method of

calculating p^î within V p assuming that there exists a distribution

within Vg which ensures that p ^ is a residual stress field. For thin

shells this does not present a problem as VF consists of regions adjacent

to the shell surfaces, and a proof of the existance of p^î can be con­

structed. Perhaps a more significant point occurs when the loads \|jp are

169 -

applied on the surface of Vp itself. In these circumstances we need to

construct, conceptually, a hydrostatic stress field within Vp which

translates the surface traction from the surface of Vp to the surface of

Vg . Again formal proof can be constructed which demonstrates that (32)

remains unchanged when this occurs.

5. COMPUTATIONAL METHOD

The upper-bound method is implemented by the programs EEC-SHAKEDOWN 1

(EECS1) and EEC-Shakedown 3 (EECS3), both of which take as input the basic

physical dimensions of the axisymmetric shape, the material data (including

yield stress as a function of temperature) and the temperature distribution.

EECS1 calculates the Brussels diagrams for cylindrical tubes subject

to single or multiple stationary axial temperature distributions or a

single moving axial temperature distribution. The thin shell thermo-

elastic stress in this case is calculated assuming uniform temperature

through the thickness and linear temperature variation between specified

points.

EECS3 finds the Brussels diagrams for axisymmetric geometries in which

the tangent angle defining the geometry is continuous, subject to a single

upshock or downshock temperature distribution, which can vary axially or

through the thickness of the shell anywhere within the material volume.

The thermo-elastic stress is calculated by a finite element elastic stress

program CONIDA [3] supplied by the UKAEA.

Both programs choose a suitable finite element structure of axisym­

metric elements. This is accomplished by first Inserting nodes at the

ends of the geometrical sections, then giving a minimum number of nodes to

each section. This is supplemented by adding nodes at the edges and

170

centre of any band of pressure and/or moving temperature front, together

with additional nodes at, and bisecting, the elements either side of any

particular point of stress maxima or minima. Finally, further nodes are

added up to the required total in areas where the nodes are most sparse.

The energy dissipation or cost function contribution for each node is

then calculated using linear variations of the strains and plastic multi­

pliers within the elements by the method discussed above. The coeffic­

ients of the constraint equations are also calculated, there being one

constrain equation for each plastic hinge (i.e. each node) and one con­

straint equation for each boundary condition. There is also one general

constraint equation governing the size of the mechanism, obtained by

setting the work done by the plastic multipliers to a constraint as in

equation (7a). The boundary constraints are usually:

a) Displacement normal to the axisymmetric axis (or shell mid-surface

depending on mechanical loading type) at one or both ends of the structure

- set to zero

b) Displacement tangential to the axisymmetric axis (or shell mid-surface

depending on mechanical loading types) at one or both ends of the structure

- set to zero

c) Spherical Cap Elements

Special analytic boundary conditions can be shown to apply at the

centre of the cap. Displacement tangential to the shell mid-surface is set

to zero and, e e - e(|>

deg/ds = de^/ds = 0

The minimum cost (energy dissipation) is found within the system of

constraints by a sparse matrix linear programming package known as XMP [4],

kindly supplied by Professor Roy Marston of the University of Arizona.

The resultant minimum cost is then the mechanical load at the ratchetting

- 171 -

boundary and the plastic multipliers active in the solution give the mech­

anism of deformation. The maximum number of plastic multipliers active

in the solution is equal to the total number of distinct constrain equa­

tions. In one or two special cases this can produce an incorrect result

as there are insufficient plastic multipliers to give the true mechanism.

However, the value of the mechanical load at the ratchetting boundary re­

mains accurate.

Having solved the linear programming problem the finite element mesh

can be refined by bisecting about the positions of hinges or distinct

plastic multipliers active in the solution. The problem can then be

resolved to give increased accuracy of solution. This may be repeated as

often as required, consistant with the cost function and constraint matrix

not becoming ill-conditioned; twice is usually sufficient for the accuracy

wanted for most problems (greater than 2 significant figures).

Finally, from the plastic multipliers and hinges active in the solu­

tion, the deformation of the structure is calculated as well as any changes

or curvature within elements.

The solution process for the mechanical load at the ratchetting boun­

dary is repeated for varying values of the thermal load, until a Brussels

diagram is calculated containing sufficient points. This is achieved by

linear scaling of the temperature history 0(x,t) = g(x,t) (8max- 60) by a

factor X to produce a sequence of distributions 6^(x,t) differing only

in magnitude

e x ( x , t ) = e 0 + x g ( x , t ) ( 6 m a x - e 0 ) (35)

where 60 is the initial temperature and 8 m a x (or 0mln for downshocks)

is the temperature having the largest difference from 0O . The function

g(x,t) is the normalized shape function of the temperature distribution.

172

For experimentally obtained temperature distributions the correct solution

will correspond to the value of at/cry(0o) obtained when \ = 1 . at is

the maximum shear stress in the thermo-elastic distribution, used with the

quantity at/ay(0o)» denoted by ät > to characterise the thermal load.

Oy(Q0) is the plastic yield stress at 0 . The temperature distribution

can then be characterised by its knockdown factor k , which is defined as

the maximum thermo-elastic stress of the temperature distribution divided

by the maximum thermo-elastic stress for a step discontinuity having the

same maximum temperature difference A0 = (Ømax ~ 0)

k = at(0max)AEaA0 /2) (36)

The values of k lie in the range 0 < k < 2 . The concept of the

characteristic length or gradient of the temperature distribution

x = AX//(RH) is not very useful in cases where either the radius R or

the thickness H vary when the thermo-elastic stress is of a significant

magnitude. During the calculation of the Brussels diagram, at the onset

of the region Vp , where the stress first exceeds twice yield, the pro­

gram bisects the thermal load points for increased accuracy.

Both programs thus accurately calculate a sequence of points on the

shakedown/ratchetting boundary, together with their associated deformation

mechanisms, to form a complete Brussels diagram.

6. THERMO-ELASTIC STRESS DUE TO AN ARBITRARY AXIAL TEMPERATURE DISTRIBUTION ALONG A CYLINDRICAL TUBE

Assuming uniform temperature through the thickness, any axial temp­

erature distribution may be represented as the sum of a number of discrete

temperature increments (steps) of height A0 over length Ax • Boundary

effects are not included as the tube is assumed to be continuous within the

range of the thermo-elastic stresses. The thermo-elastic stress due to a

discrete temperature increment (step) of A0 at a point x0 is given by

173

[5] as

a<j)(x) = ap + hßa t exp(e(x-x0)/o)) Sin( (x-x0)/œ) (37)

aø(x) = (e /2)a t exp(e(x-x0)/o)) Cos ((x-x0) /u)

+ iivßat exp(e (x-x0 )/(D) Sin ((x-x0)/w)

(38)

where a is the constant axial load, v is Poissons Ratio, h is the

normalized distance across the thickness, - 0.5 < h < 0.5, and

ß = /(3/(i-v2)) : at = EaAØ

e = 1 for (x-x0) < 0

e = -1 for (x-x0) > 0

0) = /(RH)/V(3(1-V2))

E is the modulus of elasticity and a is the coefficient of thermal ex­

pansion. R is the radius of the cylinder and H is the thickness.

The stresses for the temperature distribution are now given by

CtyU) = L± CF^U-XQ) Aôi/Axi

crø(x) = Z± QQ(X-X0) A6i/Ax±

(39)

(40)

This approximation becomes exact as the interval AXJ becomes infinitely

small and the summation becomes an integration over the length of the

temperature distribution W .

<Vx) = w d8 <VMo> ãT- dxo (41)

Qø(x) = W

(42)

For computational purposes the temperature distribution is given by the

174

temperature at a number of discrete points (0j at Xj, j = 1 ■*■ N).

Between these points the temperature is assumed to vary linearly. Thus

the integration may be split into a sum of integrations over segments Xj

t° x

j+l • I n each segment d8/dx0 is then a simple constant given by

(0j+l ­ 0j) / (*j+i ­ Xj). It should be noted that the term e in

equations (37) and (38) changes sign at the point x , giving an extra sub­

division within that particular segment, using equations (37) and (38) ,

results in simple analytic integral forms. These expressions may then be

summed over the segments specifying the temperature distribution to produce

the thermo­elastic stress at that point in the structure.

175

References

[1] S. KARADENIZ and A.R.S. PONTER

A linear programming upper bound approach to the shakedown limit of thin shells subjected to variable thermal loading. J. Strain Anal. 1984, 19, 221.

[2] D.C. DRUCKER and R.T. SHIELD

Limit analysis of symetrically loaded thin shells of revolution. ASME J. Appi. Mech., 1959, 26, 6lb.

[3] P.w. CLARKE

CONIDA: A finite element program for the stress analysis of axisymmjetric thin shells. United Kingdom Atomic Energy Authority, 197^, HMSO Report 2382(R).

[4] R.E. MARSTEN

The design of the XMP linear programming library. Asse. Comp. Mach. - Trans. Math. Soft. (ACM TOMS), 1981, 7, 481.

[5] F. ARNAUDEAU, J. ZARKA and J. GERIJ

Thin circular cylinder under axisymmetric thermal and mechanical loading. Proc. 4th Int. Conf. Struct. Mech. in React. Tech., San Francisco, USA, 1977, Vol. L, Paper L6/5.

[6] A.R.S. PONTER and S. KARADENIZ

An extended shakedown theory for structures that suffer cyclic thermal loading, Part I and II. J. Appi. Mechanics, Trans ASME, 1984, 52, pp877-882 and pp883-889.

176

Fig. 1 Characteristic radii for Spheroidal and Composite Shapes

Tresca yield surface

Fig. 2 MIM W ^MMM

177

European Communities — Commission

EUR 12686 — The computation of shakedown limits for structural components subjected to variable thermal loading — Brussels diagrams

A.R.S. Ponter, S. Karadeniz, K.F. Carter

Luxembourg: Office for Official Publications of the European Communities

1990 — X, 177 pp., tab., fig. — 21.0 x 29.7 cm

Nuclear science and technology series

EN

ISBN 92-826-1340-2

Catalogue number: CD-NA-12686-EN-C

Price (excluding VAT) in Luxembourg: ECU 15

Structures submitted to a constant primary load and a cyclic (thermal) secondary load may for certain combinations of load ratio, geometry and material properties undergo ratchetting, i.e. a situation where the strains increase at each cycle of the applied thermal load until failure or prohibitively large accumulated deformations occur. This report resulting from CEC Study Contract RAP-054-UK having mainly fast breeder reactor applications in mind, discusses the so-called Brussels diagrams which are a practical tool for the designer for assessing a particular design situation with respect to ratchetting. Brussels diagrams show four regions: elastic, shakedown, reverse plasticity and ratchetting.

The theory of Brussels diagrams is presented. It is the upper bound shakedown theory, specialized for axisymmetric shell elements and in which the upper bound is minimized by linear programming techniques. This theory is extended to the reverse plasticity region and has been implemented in two finite element axisymmetric shell programs which calculate a sequence of points on the ratchetting boundary. Three classes of problems are discussed: (i) The uniaxial transient Bree problem. (ii) The cylindrical tube subjected to axial load and stationary or moving

temperature discontinuity, (iii) A range of Brussels diagrams for axisymmetric geometries and thermal

loadings typical of LMFBRs.

The discussion includes comparisons with some experiments and con­siderations on the sensitivity of the diagrams to the material assumptions.

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