\input TEXP.TEX



\title{Modelling Creep Rupture Strength of Ferritic Steel Welds}
\centerline{D. Cole, C. Martin-Moran$^{\dag}$, A.G. Sheard$^{\ddag}$, H. K. D. H. Bhadeshia and D. J. C. MacKay$^*$}
\medskip
\centerline{ University of Cambridge}
\centerline{ Department of Materials Science and Metallurgy}
\centerline{ Pembroke Street, Cambridge. CB2 3QZ U.K.}
\medskip
\centerline{ $^{\dag}$Allen Steam Turbines}
\centerline{ Queens Engineering Works, Ford End Road}
\centerline{ Bedford. MK40 4JB U.K.}
\medskip
\centerline{ $^{\ddag}$Rolls-Royce Industrial and Marine Gas Turbines Limited}
\centerline{ Ansty, Coventry. CV7 9JR U.K.}
\medskip
\centerline{ University of Cambridge}
\centerline{ $^*$Cavendish Laboratory}
\centerline{ Madingley Road, Cambridge. CB3 0HE U.K.}

\bigskip \singlespace

\sec{ABSTRACT}

{\parindent=20 pt  \narrower \medskip
Recent changes in the design of steam turbine power plant have
necessitated the replacement of bolted flanges with welded joints. The
design process therefore requires a knowledge of the creep rupture
strength of the weld metal consumed in the welding process. This study
presents a method which can be used to estimate the creep rupture
strength of ferritic steel weld metals, from a knowledge of the creep
strength of wrought plates. The method is validated using published data.

\medskip}

\sec{INTRODUCTION}


\x The creep rupture life is frequently a   fundamental limit  on
the design of steam turbines for
power plant.  During steady operation, the steam turbine
casing achieves the live steam temperature; the steam  pressure is
contained within a casing; this gives rise to a steady
state stress which at elevated temperatures can cause the casing to
fail by a creep mechanism.


The present paper is concerned with small steam turbines ($< 40\,$MW),
of the kind described by Sheard and Raine [1]. This utilises a single
casing made in two halves which fit around the turbine rotor; the
two halves are connected at a horizontal joint using bolts. This bolting
limits the pressure differential\footnote\dag{\ie the difference in
pressure across the wall of the casing.} which can be tolerated to  about
70~bar. It is possible to overcome this limitation by using multiple
casings but there is then a dramatic increase in the
manufacturing cost.

The conditions currently desired of small steam turbines, 90~bar at
540\degg\ [2]  are beyond the capability of even the
most aggressive horizontal joint bolt design, making it necessary to use
an expensive double casing. Fortunately, an alternative to this design was
implemented recently by Mason and Sheard [3] for a 30.4~MW steam
turbine. The details of this single--casing design (\fagg), which permits
the higher pressure differential, are beyond the scope of this paper, but
there are two points which need to be emphasized in the context of the
present work:

{\parindent=20pt \narrower \medskip
\zz{(i) } As for the usual design, the steam is fed into the appropriate
part of the casing via a steam pipe. This pipe is conventionally joined
to the casing with a vertical flange joint which is bolted together. The
flange has had to be upgraded to tolerate the higher steam pressures
involved.

\zz{(ii) } The large flange of the steam pipe and its associated bolting
causes considerable steric hindrance, making it difficult to assemble
the turbine unit. 

\medskip}


\midinsert\vskip 240pt  {\special{psfile=fig1.eps hoffset=0
voffset=0 hscale=100 vscale=100}}
\vskip 20pt
\tabtit{\figg}{\ A cross section of
the advanced high--pressure inlet of the Sheard and Mason design of steam
turbine.}
\endinsert



To overcome these difficulties, it has been decided to replace the
bolted vertical joint by a welded joint. For reasons which are
proprietary, the heat affected zone of the weld is expected to behave
identically in creep to the region of the steel far away from the weld.
Therefore, it is the creep properties of the weld metal itself which
become an issue, and hence the present investigation, whose aim is to be
able to estimate the creep rupture life of the weld metal used in
fabricating the joint.

The methodology involves the creation of neural network models which
adequately capture the complexity of the creep rupture phenomenon. Such
models rely for their training on experimental data, which are available
in large quantities for wrought power plant steels, but not for the
actual weld metals. For this reason, we shall begin by justifying the
idea to use wrought steel data to estimate weld metal properties. The
idea itself is validated later in the paper.

\sec{Comparison of Weld Metal \&\ Wrought Plate}

Weld metals and steels of matching composition seem to have
similar creep rupture properties. In fact, the chemical compositions
of weld metals and corresponding steel plates are not very
different (\fagg). Of course, weld metal will, in general, have a higher oxygen
and nitrogen concentration but the former should not affect creep
resistance. Although differences in the nitrogen concentration are
important, they can easily be taken into account both in predicting
carbonitride formation and in the neural network model where
nitrogen is an input. 

The microstructure of an as--deposited weld metal is naturally
radically different from that of a wrought steel. However, even
this is unimportant because the severe tempering heat
treatments used following the welding procedure  
essentially wipe out the original microstructure and replace it
with one which is tempered and similar to that of the steel plate. It
is probably for this reason that the welding process itself is
found not to influence the creep rupture life [4]. 


\midinsert\vskip 330pt  {\special{psfile=fig2.eps hoffset=80
voffset=0 hscale=100 vscale=100}}
\vskip 10pt
\tabtit{\figg}{A comparison of the
chemical compositions of wrought plates and corresponding weld metals of
the kind used in the power plant industry [5].}
\endinsert


\sec{Creep Rupture Strength -- the Variables}

The basic principles of alloy design
for creep resistance are   well--established and well--founded on
experience. The steels must have a stable  microstructure which
contains fine alloy carbides to resist the motion of dislocations;
however, changes are inevitable over the long service time so that
there must be sufficient solid solution strengthening to ensure long
term creep resistance. There may be other requirements such as
weldability, corrosion and oxidation resistance. It is nevertheless
difficult to express the design process quantitatively given the
large number of interacting variables.  

The variables taken into account in the present work are listed in 
\tablaa. We note that the entire information about
microstructure and properties is in principle to be found in this set of
parameters since the chemical composition and heat treatment are
comprehensively included. There may, of course, be many other
independent variables that might be considered important in creep
analysis, but these are for the  moment neglected for two reasons. 
Firstly, an empirical analysis requires experimental data; an over
ambitious list would simply reduce the dataset since publications
frequently do not report all of the necessary parameters. Secondly,
the effect of any missing variables would simply be reflected in the
uncertainties of prediction. If the predictions are noisy then they
can be improved with carefully designed experiments at a future
date. Bearing this in mind, the results to be presented are based on
some 5420 sets of experiments obtained from the published
literature [6--45]. This compares to earlier work on the creep rupture stress of wrought steels by Brun \et [46], which used a total of 2066 lines. We now proceed to describe briefly the methodology.


\midinsert {\fiverm  \tablewidth=4truein \begintable
Variable & Range & Mean & Standard deviation \cr
{\bf Test Conditions} & & & \nr
log(life/h) & -0.22-+5.29 & 3.00 & 0.28 \nr
Temperature (K) & 723-977 & 867 & 60.97 \nr
{\bf Composition} & & & \nr
Carbon (wt\%) & 0.004-0.23 & 0.11 & 0.044 \nr
Silicon (wt\%) & 0.01-0.86 & 0.28 & 0.17 \nr
Manganese (wt\%) & 0.01-0.92 & 0.50 & 0.12 \nr
Phosphorus (wt\%) & 0.001-0.029 & 0.013 & 0.0074 \nr
Sulphur (wt\%) & 0.001-0.02 & 0.0074 & 0.0048 \nr
Chromium (wt\%) & 2.17-12.9 & 8.20 & 3.46 \nr
Molybdenum (wt\%) & 0.04-2.99 & 0.84 & 0.53 \nr
Tungsten (wt\%) & 0.01-3.93 & 0.50 & 0.79 \nr
Nickel (wt\%) & 0.01-2.00 & 0.23 & 0.28 \nr
Copper (wt\%) & 0.01-0.87 & 0.069 & 0.099 \nr
Vanadium (wt\%) & 0.01-0.28 & 0.13 & 0.10 \nr
Niobium (wt\%) & 0.005-0.312 & 0.037 & 0.045 \nr
Nitrogen (wt\%) & 0.001-0.165 & 0.030 & 0.027 \nr
Aluminium (wt\%) & 0.001-0.057 & 0.011 & 0.012 \nr
Boron (wt\%) & 0.00-0.051 & 0.0013 & 0.0041 \nr
Cobalt (wt\%) & 0.00-3.09 & 0.13 & 0.49 \nr
Tantalum (wt\%) & 0.00-0.1 & 0.0007 & 0.0066 \nr
Oxygen (wt\%) & 0.003-0.035 & 0.0099 & 0.0031 \nr
Rhenium (wt\%) & 0.00-1.69 & 0.014 & 0.12 \nr
{\bf Normalising} & & & \nr
Temperature (K) & 1123-1453 & 1283 & 69.16 \nr
Duration (h) & 0.17-33 & 2.05 & 3.75 \nr
Cooling rate & & & \nr
in furnace & 0-1 & 0.060 & 0.24 \nr
in air & 0-1 & 0.58 & 0.49 \nr
oil quenched & 0-1 & 0.25 & 0.43 \nr
water quenched & 0-1 & 0.11 & 0.31 \nr 
{\bf Tempering} & & & \nr
Temperature (K) & 823-1133 & 979 & 74.88 \nr
Duration (h) & 0.50-32 & 3.64 & 6.17 \nr
Cooling rate & & & \nr
in furnace & 0-1 & 0.060 & 0.24 \nr
in air & 0-1 & 0.88 & 0.33 \nr
oil quenched & 0-1 & 0.030 & 0.17 \nr
water quenched & 0-1 & 0.032 & 0.18 \nr
{\bf Annealing} & & & \nr
Temperature (K) & 300-1023 & 465.7 & 282.5 \nr
Duration (h) & 0.5-50 & 4.38 & 8.37 \nr
Cooling rate & & & \nr
in furnace & 0-1 & 0.054 & 0.226 \nr
in air & 0-1 & 0.946 & 0.226 

\endtable}
\tabtit{\tablee : }{Details  of inputs in database - composition and heat
treatments.}
\endinsert


\sec{THE METHOD}

A neural network is a general method of regression  analysis in which a
very flexible non--linear function is fitted to experimental data, the
details of which have been extensively reviewed [47]. It is,
nevertheless, useful to present some salient features in order to
place the technique in context. 

The flexibility  of the non--linear function scales
with the number of hidden nodes
$i$. Thus, the dependent variable $y$ is given in the present work by 
$$ y = \sum_i w_i^{(2)} h_i + \theta^{(2)} \numeqn $$ where
$$h_i = \tanh \bl(\sum_j w^{(1)}_{ij}x_j + \theta^{(1)}_i\br)
\numeqn$$ where $x_j$ are the $j$ variables on which the output $y$
depends, $w_i$ are the weights (coefficients) and
$\theta_i$ are the biases (equivalent to the constants in linear
regression analysis). The combination of equation~\nnumeqn\ with a set of
weights, biases, value of $i$ and the minimum and maximum values of the
input variables defines the network completely. Notice that the
complexity of the function is related to the number of hidden units. The
availability of a sufficiently complex and flexible function means that
the analysis is not as restricted as in linear regression where the form
of the equation has to be defined explicitly before the analysis. 
       
The neural network can capture interactions between the inputs because the hidden units are nonlinear. The nature of these interactions is implicit in the values of the weights, but the weights may not always be easy to interpret. For example, there may exist more than just pairwise interactions, in which case the problem becomes difficult to visualise from an examination of the weights. A better method is to actually use the network to make predictions and to see how these depend on various combinations of inputs.

\ssec{Error Estimates}

The input parameters are generally assumed in the analysis  to be
precise and it is normal to calculate an overall error by comparing the
predicted values $(y_j)$ of the output against those measured $(t_j)$, for
example,
$$ E_D \propto \sum_j (t_j - y_j)^2 \numeqn $$ $E_D$ is expected to increase if important input variables have been
excluded  from the analysis.  Whereas $E_D$   gives an overall  perceived
level of noise in the output parameter, it is, on its own, an unsatisfying
description of the uncertainties of prediction. 

MacKay has developed a particularly useful treatment of neural networks in
a Bayesian framework [47], which allows the calculation of error bars
representing the uncertainty in the fitting parameters. The
method recognises that there are many functions which can be fitted or
extrapolated into uncertain regions of the input space, without unduly
compromising the fit in adjacent regions which are rich in accurate data.
Instead of calculating a unique set of weights, a probability distribution of
sets of weights is used to define the fitting uncertainty. The error bars
therefore become large when data are sparse or locally noisy.

The error bars presented throughout this work therefore represent a
combination of the perceived level of noise in the output (the creep
rupture strength) and the fitting uncertainty as decribed above.


\ssec{Overfitting}

A potential difficulty with the use of powerful non--linear   regression
methods is the possibility of overfitting  data. To avoid this,
the experimental data can be divided into two sets, a {\it
training} dataset and a {\it test} dataset. The model is produced using
only the training data. The test data are then used  to check that the
model behaves itself when presented with previously unseen data. The
training error tends to decrease continuously as the model complexity
increases. It is the minimum in the test error which enables that model
to be chosen which generalises best on unseen data [47].

The discussion of overfitting is rather brief because the problem does not simply involve the minimisation of test error. There are other parameters which control the complexity, which are adjusted automatically to try to achieve the right complexity of model [47].


\sec{THE ANALYSIS}

The aim of the neural network  in this case was to predict the creep
rupture stress as a function of the variables shown in \tablee. Those
variables include a comprehensive  description of chemical
composition and three heat treatments (each characterised by
a temperature, duration and cooling rate from the treatment temperature).
If the steel had only undergone two heat treatments, the annealing heat
treatment was set to 300 K to correspond to room temperature. The time to
rupture was expressed in logarithmic form, along with the output creep
rupture stress, to improve the accuracy of the model.

All the 37 different input variables  and the output were normalised
within the range
$\pm$0.5 as follows:
$$x_N={x-x_{min}\over{x_{max}-x_{min}}}-0.5$$
where $x$ is the original value from the database,  $x_{max}$ and
$x_{min}$ are the respective maximum and minimum of each variable in the
original data and $x_N$ is the normalised value. This step is not
essential to the running of the neural network but later allows a
convenient way to compare the results of the output. \fagg{ } shows the
general structure of the neural network.



\midinsert\vskip 165pt  {\special{psfile=pictures/hiddenunits.eps hoffset=150
voffset=-70 hscale=100 vscale=75}}
\vskip 80pt
\tabtit{\figg}{\ Structure of neural network}
\endinsert

For several runs of the neural network, \fagg\ shows the model perceived noise $\sigma_\nu$ in the creep rupture
stress. As expected, $\sigma_\nu$ decreases as the fitting function
becomes more flexible with larger numbers of hidden units.
\vskip 20pt

\midinsert\vskip 130pt {\special{psfile=pictures/model/sigmav.eps hoffset=30 voffset=-30 hscale=75 vscale=75}}
\vskip 10pt
\tabtit{\figg}{\ Variation in $\sigma_v$ as a function of the number of hidden units. Several values are presented for each set of hidden units because the training for each network started with a variety of random seeds.}
\endinsert


By contrast, \fagg{ } shows the test error at first decreases and then
levels out, with minima at 18 to 22 hidden units.

\midinsert\vskip 130pt {\special{psfile=pictures/model/testerror.eps hoffset=30 voffset=-75 hscale=75 vscale=75}}
\vskip 60pt
\tabtit{\figg}{\ The test error as a function of the number of hidden units.}
\endinsert

\fagg\ shows the comparisons made for the best model, consisting of 22 hidden units, on the split database. The comparison for the seen data with the model shows very good agreement as should be expected. The analysis of unseen data by the neural network also shows good agreement.

\midinsert\vskip 130pt {\special {psfile=pictures/comparison/comp.eps hoffset=30 voffset=-50 hscale=75 vscale=75}}
\vskip 60pt
\tabtit{\figg}{\ Comparisons between seen and unseen data}
\endinsert


\midinsert\vskip 200pt {\special {psfile=bar.eps hoffset=30 voffset=-50 hscale=75 vscale=75}}
\vskip 60pt
\tabtit{\fagg}{\ Comparison of values of $\sigma_w$ for the top five models}
\endinsert

The parameter $\sigma_w$ indicates the importance of an input in terms of its variation having an effect on the output of the model. \figg\ compares the value of $\sigma_w$ for a selection of inputs for the top five models. A high value of $\sigma_w$ for a specific input can be caused by the corresponding variable inducing a large variation in the output, but it can be seen from \figg{ } that different models can assign varying significance to the same input. In such cases, it is possible that a committee of models can make a more reliable
prediction than an individual model. The best models are ranked
using the values of the test errors. Committees are then formed by
combining the predictions of the best $L$ models, where
$L = 1, 2, \ldots$; the size of the committee is therefore given by the
value of $L$. A plot of the test error of the committee versus its size 
 gives a minimum which defines  the optimum size of the committee, as
shown in \fagg.


\midinsert\vskip 60pt {\special{psfile=pictures/com/committee.eps hoffset=10 voffset=-110 hscale=75 vscale=75}}
\vskip 90pt
\tabtit{\figg}{\ Comparison of test error of increasing size of committees.}
\endinsert

The  test error associated with the best single model is clearly greater
than that of any of the committees. It was determined in this case that a
committee of eleven models would be the best choice, being the committee of
the lowest test energy. The committee was then retrained on the entire
data set without changing the complexity of any of its members.

The predictions of the committee trained on the entire data set can be compared with the original dataset as shown in \fagg. It can be seen that there is excellent agreement between the values and that the error bars are extremely small.

\midinsert\vskip 80pt {\special{psfile=pictures/com/target.eps hoffset=80 voffset=-110 hscale=75 vscale=75}}
\vskip 110pt
\tabtit{\figg}{\ Comparison of predicted values and experimental values for the committee.}
\endinsert


\sec{APPLICATION OF THE MODEL}

\fagg { }shows predictions using the neural network in examining the creep
rupture strength of a standard 2.25CrMo steel and a modern 10CrMoW heat
resistant steel (\tablaa).

\midinsert {\sevenrm \tablewidth=5truein \begintable
Steel & 2.25Cr & 10Cr | Steel & 2.25Cr & 10Cr \cr
Norm temp & 1203K & 1338K | Cr & 2.4 & 10.61 \nr
Duration & 6h & 2h | Mo & 1.01 & 0.44 \nr
Cooling rate & water q. & in air | W & 0.01 & 1.87 \nr
Tempering & 908K & 1043K | Ni & 0.14 & 0.32 \nr
Duration & 6h & 4h | Cu & 0.16 & 0.86 \nr
Cooling rate & in air & in air | V & 0.01 & 0.21 \nr
Annealing & 873K & 1013K | Nb & 0.005 & 0.01 \nr
Duration & 2h & 4h | N & 0.0108 & 0.064 \nr
Cooling rate & in air & in air | Al & 0.018 & 0.022 \nr
C wt\% & 0.15 & 0.12 | B & 0.0003 & 0.0022 \nr
Si & 0.21 & 0.05 | Co & 0.05 & 0.015 \nr
Mn & 0.53 & 0.64 | Ta & 0.0003 & 0.0003 \nr
P & 0.012 & 0.016 | O & 0.01 & 0.01 \nr
S & 0.012 & 0.001 | Re & 0.0003 & 0.0003 

\endtable}
\tablee { }  Heat treatments and compositions of two common power plant
steels
\endinsert


\vskip 10pt
\midinsert\vskip 50pt {\special{psfile=pictures/plate/7.eps hoffset=40 voffset=-350 hscale=75 vscale=75}}
\vskip 350pt
\tabtit{\figg}{\ Neural network results for well known power plant steels}
\endinsert


The error bounds represent  the uncertainty in fitting the non-linear
function to the training data, as 65\% confidence limits with the $\sigma_\nu$ added quadratically. The calculated lifetime of each
steel was examined at temperatures of 550, 600 and 650 $^{\circ}$C. The
network correctly predicted that the creep strength of the two steels
would reduce with increasing lifetime and temperature and also that the
10CrMoW steel is more creep resistant than 2.25CrMo. The error bars,
represented by dashed lines, are smaller when the neural network is
confident of the predictions that it is making. Increases in error bar
size, such as at long life for the 2.25CrMo alloy at 650$^{\circ}$C,
indicate that a lack of experimental data is restricting the accuracy of
the predictions.

The behaviour of the model can be assessed further by examining creep
rupture strength of a variety of steels and determining if the predictions made by the neural network agree with practical results or microstructural theory. In one case the creep rupture stress of an uncommon 3Cr steel was examined using the neural network and then compared to the results for the 2.25Cr steel previously examined. It could be expected that the 3Cr steel should have a consistently higher creep rupture stress than the 2.25Cr steel due to the increase in chromium content, but it can be seen from \fagg{ } that they have comparable creep rupture lives. \figg{ } also shows that the uncertainty for the 3Cr steel is much greater than that of the 2Cr, this being due to the amount of information in the database on 3Cr steel being much less than the amount of data on 2Cr.

\midinsert\vskip 30pt {\special{psfile=pictures/3cr/3.eps hoffset=30 voffset=-110 hscale=75 vscale=75}}
\vskip 110pt
\tabtit{\figg}{\ Comparison of a 3Cr steel and a 2.25Cr steel at 600$^{\circ}$C}
\endinsert

This phenomenon can be explained by microstructural theory and is an excellent result for the neural network. The microstructural trends were examined by Robson and Bhadeshia [48] and the results of this research are represented in \fagg, showing plots of carbide volume fraction versus time for 3Cr steel and for 2.25Cr steel.

\midinsert\vskip 200pt  {\special{psfile=fig12.eps hoffset=0
voffset=0 hscale=100 vscale=100}}
\vskip 20pt
\tabtit{\figg}{\ Plot of calculated carbide volume fraction versus time at 600$^{\circ}$C for the 2.25Cr alloy and the 3Cr alloy (Robson and Bhadeshia [47]).}
\endinsert



It is apparent that the rate at which $M_{23}C_{6}$ precipitates in 3Cr steel is very much greater than that of the 2.25Cr steel. It can also be seen that the maximum volume fraction of $M_{2}X$ obtained by the 3Cr alloy is much less. These plots show that, in the 2.25Cr steel, $M_{2}X$ starts to precipitate well before $M_{23}C_{6}$, forming a large volume fraction which then suppresses $M_{23}C_{6}$ formation. However, in the higher chromium steel both carbides tend to precipitate at similar times, and both phases are then competing for solute. The $M_{23}C_{6}$ phase dominates and $M_{2}X$ dissolves. The $M_{2}X$ precipitate is considered to be a factor in improving creep rupture strength, whereas $M_{23}C_{6}$ is considered less effective in resisting creep deformation due to its large size. This then explains why the 3Cr steel is not as creep resistant as expected.

The neural network model may be used to examine the influence of any of the input variables on the output. Hence, the effect of varying composition in a particular alloy at certain conditions may be determined. \fagg\ shows trends with increasing tungsten content, with a maximum at approximately 3wt\% for the 10Cr alloy examined previously. Although the uncertainty of the predictions is large, the decline at greater than 3wt\% is expected due to the formation of $\delta$-ferrite which is generally considered detrimental to creep properties. The National Physical Laboratory's Metallurgical and Thermochemical data Bank (MTDATA [49]) was used to determine at what weight per cent of tungsten that $\delta$-ferrite started to form. The result was that at 2.7wt\%, $\delta$-ferrite formation commenced, agreeing well with the neural network results.

\midinsert\vskip 10pt {\special{psfile=pictures/w/wat5h.eps hoffset=80 voffset=-150 hscale=75 vscale=75}}
\vskip 150pt
\tabtit{\figg}{\ Rupture stress for 10Cr alloy with varying tungsten}
\endinsert



\sec{APPLICATION TO WELD METALS}

We have already argued that weld metals and  steels of matching
composition may have similar creep rupture properties. 
 
The hypothesis can be  proved by examining the relatively limited data on
all--weld metal tests in published literature [4,18]. \fagg { } shows the encouraging agreement between the calculated creep rupture stress and the measured values [4] of 2.25CrMo welds (\tablee). The predictions were made without any
adjustment to the model, which did not interrogate any weld data during
its creation.

\midinsert \vskip 30pt {\special{psfile=pictures/2crweld/8.eps hoffset=30 voffset=-250 hscale=75 vscale=75}}
\vskip 250pt
\tabtit{\figg}{\ Calculated (filled circles) and measured (open squares) stress rupture data for 2.25CrMo weld metal versus wt\% carbon.}
\endinsert

The heat treatments were changed to represent the welding process and the post weld heat treatment (PWHT). It was determined, by trial and error, that setting the normalising process to 1200 K for 6 hours was a fair representation of the welding process itself. The tempering heat treatment was then set to the conditions of the PWHT and the annealing heat treatment was set to 300 K to show that no further heat treatments had been carried out.

\figg{ } confirms that it is reasonable to assume that the creep rupture life of weld metals can be modelled on the basis of wrought steels. The results are plotted with varying carbon since the literature from which these data were taken was, in part, examining the effect of varying carbon wt\% on weld metals.

The model may also be used to determine creep rupture stress of modern modified 9CrMoW weld metals. There is much less open literature available on this type of weld metal in terms of creep data and so there is less opportunity of testing the model. Naoi \et [18] have carried out creep tests on a 9CrMoW gas tungsten arc weld metal, the composition of which is shown in \tablaa.

\midinsert {\sevenrm \begintable
Composition (wt\%) & C & Si & Mn & P & S &Ni & Cr & Mo & W & V & Nb & N \cr
Weld Metal | 0.07 & 0.20 & 1.01 & 0.006 & 0.004 & 0.36 & 8.94 & 0.48 & 1.62 & 0.09 & 0.04 & 0.032
\endtable}
\tablee { } Chemical composition of gas tungsten arc weld metal, wt\%
vskip 20pt
\endinsert

\midinsert \vskip 20pt {\special{psfile=pictures/ninecr/9.eps hoffset=30 voffset=-250 hscale=75 vscale=75}}
\vskip 250pt
\tabtit{\fagg}{\ Calculated (filled circles) and measured (open squares) stress rupture data for 9CrMoW weld metal versus creep rupture life.}
\endinsert

\figg{ } shows that the model is also capable of predicting the creep rupture life of modified 9CrMoW weld metals. The ability to predict the creep properties of these welds has direct industrial implications, as discussed previously.

\sec{CONCLUSIONS}

A neural network model has been created to study the influence of the
chemical composition and heat treatment on the creep rupture strength of
ferritic steels. The model is based on a very large dataset accumulated
from the literature.

The model has been applied successfully to weld metal data reported in
the literature, even though these were not used in the creation of the
models. This work is helping in the choice of weld metal for the
manufacture of small power plant based on steam turbine.

\sec{Acknowledgments}

\x The authors are grateful to Rolls--Royce Allen Steam Turbines for
financing this work, and to Professor Alan Windle of the University
of Cambridge for the provision of laboratory facilities. We would like to
acknowledge the generous provision of experimental data by Dr. Eng. F.
Abe of the National Research Institute for Metals in Japan.

\vfill\eject

\sec{References}

{\parindent=10pt \narrower \parskip=0mm \medskip
\def\ref#1#2#3#4#5#6{\item{#1. } #2: {\it #3} {\bf #4}  
{#6}{(#5)}} 

\singlespace \medtype

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