

.\input texp.tex


\centerline {\large Estimation of The Mechanical Properties of  Ferritic 
 Steel Welds:}
\centerline {\large Part I: Yield and Tensile Strength}

\centerline{S. H. Lalam,  H. K. D. H. Bhadeshia and D. J. C. Mackay$^{\dag}$}\medskip
\centerline{ University of Cambridge}

\centerline{ Department of Materials Science and Metallurgy} 
 \centerline{ Pembroke Street, Cambridge CB2 3QZ, U.K.}

\medskip

\centerline{ $^{\dag}$Cavendish Laboratory}

\centerline{Madingley Road, Cambridge CB3 0HE, U.K.}

\bigskip \singlespace

\sec {ABSTRACT}

{\parindent=20pt \narrower \medskip 
  
 \x The yield strength and ultimate tensile strength of ferritic steel 
 weld metal have been expressed as functions of  chemical 
 composition, the heat input during welding, and of the heat treatment 
 given after welding is completed. The method involved a neural network 
 analysis of a vast and quite general database assembled from 
 publications on weld metal properties. The outputs of the model have 
 been assessed in a variety of ways, including specific studies of model 
 predictions for the so--called carbon--manganese and $2{1\over4}$Cr1Mo 
 systems. Where possible, comparisons have also been made against 
 corresponding methods which use simple physical metallurgical 
 principles. The models created are believed to have been trained on the 
 largest weld metal database to date, and are shown to capture vital 
 metallurgical trends. The computer programs associated with the work 
 have been made freely available on the world--wide--web. 
  
 \medskip} 
  
  
 \sec {INTRODUCTION} 
  
 It has been possible for some time  to estimate the 
 microstructure of ferritic steel weld metals from their chemical 
 composition and welding parameters [1], although the full effects of 
 heat--treatment have yet to be modelled. This has been useful 
 in the development of alloys, given a broad understanding of what 
 constitutes a good microstructure. The methodology cannot, however, be 
 used directly in engineering design because that requires specific 
 values of the mechanical properties. 
  
  
 The yield strength is one of the simpler of mechanical properties and 
 to a small extent it has been possible to apply physical metallurgy 
 principles towards its estimation for weld metals [1]. The method begins 
 with the assumption that the yield strength of 
 steel microstructures can be factorised into a number of intrinsic  
 components:  
 $$\sigma = \sigma_{Fe} + \sum_i x_i\sigma_{SS_i} + 
 x_C\sigma_C + K_L(\overline{L})^{-1} + K_D \rho^{0.5}_D   
 \numeqn $$  
 where $x_i$ is the concentration of a substitutional solute  
 which is represented here by a subscript $i$. The other terms  
 in this equation can be listed as follows:  
 \medskip 
 \settabs=10 \columns 
 \+ & $K_L$ & coefficient for strengthening due to \lq grain' 
 size, ${\rm 115\,MN\,m^{-1}}$\cr 
 \+ & $K_D$ & coefficient for strengthening due to 
 dislocations, ${7.34 \times 10^{-6}\,\rm MN\,m^{-1}}$ \cr 
 \+ & $\sigma_{Fe}$ & strength of pure, annealed iron, 
 ${219\,\rm MN\,m^{-2}}$ at 300 K \cr 
 \+ &$\sigma_{SS_i}$ & substitutional solute ($i$) 
 strengthening \cr 
 \+& $\sigma_C$ & solid solution 
 strengthening due to  carbon \cr 
 \+ & $ \rho_D$ & dislocation density, typically  
 $10^{16}\,{\rm m^{-2}}$ \cr 
 \+ & $\overline L$ & measure of the ferrite plate  
 size,  typically 0.2 ${\rm \mu m}$ \cr 
 \+ \cr 
 \x The individual strengthening contributions are 
 not described here, suffice it to say that  it is possible to obtain 
 reasonable estimates for all of the coefficients from data which are 
 independent of welding. Equation~\nnumeqn\ covers just one 
 microstructure; similar equations are needed for each of the phases 
 present. The contributions from each phase then has to be 
 appropriately summed to obtain the overall strength [1]. 
  
 The application of such work is, obviously, limited since it fails 
 to address the complexity that is inherent in real welds, \ie the 
 number of variables influencing the strength is in fact far greater 
 than implied above. For example, many welds are heat--treated after 
 fabrication and there is no satisfactory theory available to deal with 
 this. Many other complexities exist, as will become evident during the 
 course of this paper. 
  
 Linear regression analysis is frequently used to overcome these 
 difficulties [2]. In this work, we apply a much more general form of 
 regression, \ie neural network analysis, to enable the estimation of 
 the yield and tensile strength of ferritic weld metals. These  
 represent by far the largest group of welding materials. The present 
 work builds on an earlier study [3] which was  more limited in scope. In a future paper we shall attempt to deal with the elongation and 
 Charpy impact properties of ferritic steel welds. 
  
 \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 [4]. 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. 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 specified 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 input.  
  
 \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 [5], 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. 
  
 In this context, a very useful measure   is the log predictive error 
 because the penalty for making a wild prediction is reduced if that wild 
 prediction is accompanied by appropriately large error bars [6]: 
 $$LPE= 
 \sum_n 
 \bigg [ {1\over 2} { 
 \bigg ( t^{(n)}-y^{(n)}  
 \bigg ) } ^2 \bigg / \sigma_y^{{(n)}^2} + log \big (  \root \of {2 \pi 
 \sigma_y^{(n)} } \big ) \bigg ]  $$ 
 Note that a larger value of the log predictive error implies a better 
 model. 
  
  
 \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 [6]. There are other important features in the control of complexity which are discussed elsewhere [6].
  
 Finally, it should be noted that the analysis uses normalised values 
 of the variables in 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 is a 
 convenient way of comparing the effect of different variables on the 
 output. 
 \fagg\ shows the general structure of the simple three layer neural 
 network. 
  
    
  
 \sec{THE DATABASE} 
  
 All of the data collected are from multirun weld deposits designed for 
 low--dilution to enable specifically the measurement of all--weld 
 metal properties. Furthermore, they all represent electric arc 
 welds made using one of the following processes: manual metal arc (MMAW), 
 submerged arc (SAW) and tungsten inert gas arc (TIG). The welding process 
 itself was represented only by the level of heat input. This is 
 because a large number of published papers did not specify welding 
 parameters in sufficient detail to enable the creation of a dataset 
 without missing values. Missing values cannot be tolerated in the 
 method used here. If the effect of a welding process is not properly 
 represented by the heat input and chemical composition, then neglect 
 of any other parameters would make the 
 predictions more noisy. As will be seen later, the noise in the 
 output was found to be acceptable; the greater uncertainty arises 
 from the lack of a uniform coverage of the input space. The sources of 
 all data are listed in references [7--74].  
  
  The aim of the neural network analysis was to predict 
 the yield and tensile strength as a function of a large number of 
 variables, including the chemical composition, the welding heat input 
 and any heat treatment. The databases for the yield and ultimate 
 tensile strength (UTS) are different because the UTS database also 
 included the oxygen concentration since tensile failure should depend 
 on inclusions which nucleate voids. As a consequence, the yield 
 strength database consists of 2002 separate experiments whereas the UTS 
 database  is slightly smaller at 1972 experiments since the oxygen 
 concentration was not always reported \footnote \dag {This compares with previous work which was based on 770 yield strength and 520 UTS experiments [3]. }. Our method cannot cope with 
 missing values of any of the variables. In 14 cases the sulphur and phosphorus concentrations were not available. Since these impurities might be important, it would not be satisfactory to set them to zero. Missing values of sulphur and phosphorus were  therefore set at the average of the database.  
  
  
  
  
 \ssec{The Yield Strength Database} 
  
 \tablaa\ shows the range, mean and standard deviation of each variable 
 including the output (yield strength). The purpose here is simply to 
 list the variables and provide an idea of the range covered.  It is 
 emphasised however, that unlike linear regression analysis, the 
 information in \tablee\ cannot be used to define the range of 
 applicability of the neural network model. This is because the inputs 
 are in general expected to interact. We shall see later that it is the 
 Bayesian framework of our neural network analysis which allows the 
 calculation of error bars which define the range of useful 
 applicability of the trained network. A visual impression of the 
 spread of data is shown in \fagg. It can be concluded from {\figg} that the effect on yield strength of carbon, manganese, silicon, nickel, molybdenum and heat input have been systematically studied. Hence, future experiments could focus on examining the effect of chromium in the range 3--8 wt\%, vanadium (0.1--0.2 wt\%), cobalt at all concentrations but in greater variety of alloy systems, tungsten at low and high concentrations, titanium and boron in high strength weld. The effect of tempering temperature in the range 250--500\deg also needs to be studied. 
  
 \topinsert  
  \thicksize=0.7pt \thinsize=0.5pt 
 \tablewidth=6.4truein {\smalltype  
 \begintable 
 {\bf Input element  } | {\bf Minimum  } |{\bf  Maximum  } | {\bf Mean  } | {\bf Standard Deviation } \cr  
  Carbon     (wt\%) \hfill  |  0.01 \hfill  | 0.22 \hfill  | 0.072 \hfill  | 0.025 \nr 
  Silicon    (wt\%) \hfill  |  0.01 \hfill  | 1.63 \hfill  | 0.344 \hfill  | 0.138\nr 
  Manganese  (wt\%) \hfill  | 0.27 \hfill  | 2.31 \hfill  | 1.192 \hfill  | 0.41 \nr 
  Sulphur    (wt\%) \hfill  | 0.001 \hfill  | 0.14 \hfill  | 0.009 \hfill  | 0.006 \nr 
  Phosphorus (wt\%) \hfill  | 0.001 \hfill  | 0.25 \hfill  | 0.012 \hfill  | 0.009 \nr 
  Nickel     (wt\%) \hfill  | 0.0 \hfill  | 4.79 \hfill  | 0.43 \hfill  | 0.888 \nr 
  Chromium   (wt\%) \hfill  | 0.0 \hfill  |   12.1 \hfill  | 0.808 \hfill  | 1.952 \nr 
  Molybdenum (wt\%) \hfill  | 0.0 \hfill  | 2.4 \hfill  | 0.221 \hfill  | 0.368 \nr 
  Vanadium   (wt\%) \hfill  | 0.0 \hfill  |  0.32 \hfill  | 0.026 \hfill  | 0.06 \nr 
  Copper     (wt\%) \hfill  | 0.0 \hfill  | 2.18 \hfill  | 0.063 \hfill  | 0.185 \nr 
  Cobalt     (wt\%) \hfill  | 0.0 \hfill  |  2.8 \hfill  | 0.007 \hfill  | 0.115 \nr 
  Tungsten   (wt\%) \hfill  | 0.0 \hfill  |  3.86 \hfill  | 0.091 \hfill  | 0.427\nr 
  Titanium   (ppm wt.) \hfill  | 0.0 \hfill  |  900 \hfill  |  64.9 \hfill  | 112.14 \nr 
  Boron      (ppm wt.) \hfill  | 0.0 \hfill  |   195 \hfill  | 5.8 \hfill  |  19.08\nr 
  Niobium    (ppm wt.) \hfill  | 0.0 \hfill  |  1770 \hfill  |  69.6 \hfill  | 168.13 \nr 
  Heat input (kJ mm$^-$$^1$) \hfill  |  0.55 \hfill  | 7.9 \hfill  | 1.6 \hfill  | 1.234\nr 
  Interpass temperature ($^\circ$C) \hfill  | 20 \hfill  | 375 \hfill  | 207.8 \hfill  |  52.67 \nr 
 Tempering temperature ($^\circ$C) \hfill  | 20 \hfill  | 780 \hfill  | 358.3 \hfill  | 249.29 \nr 
 Tempering time  (h) \hfill  | 0.0 \hfill  |  50 \hfill  | 6.5 \hfill  | 6.45 \nr 
 Yield Strength (MPa) \hfill  |  288 \hfill  | 1003 \hfill  | 533.9 \hfill  | 113.64 
 \endtable }
 {\tabtit{\tablee : } {The inputs for the yield strength model. }} 
 \endinsert 
  
  
  
 \ssec{The UTS Database} 
  
 \tablaa\ shows the range, mean and standard deviation of each variable 
 including the output (ultimate tensile strength). The corresponding 
 visual impression of the  UTS database is similar to that of the yield strength. The UTS contains an extra input variable oxygen \figg, the effect of which at higher concentrations (above 900 ppm wt.) needs to be studied.
  
 \thicksize=0.7pt \thinsize=0.5pt 
 \tablewidth=6.4truein {\smalltype \begintable 
 {\bf Input element  }\hfill | {\bf Minimum  }\hfill |{\bf  Maximum  }\hfill | {\bf Mean  }\hfill | {\bf Standard Deviation } \cr  
 Carbon     (wt\%)  \hfill | 0.01 \hfill | 0.22 \hfill | 0.072 \hfill | 0.024 \nr 
 Silicon    (wt\%) \hfill |  0.01 \hfill | 1.63 \hfill | 0.345 \hfill | 0.142 \nr 
 Manganese  (wt\%) \hfill | 0.27 \hfill | 2.31 \hfill | 1.191 \hfill | 0.410 \nr 
 Sulphur    (wt\%) \hfill | 0.001 \hfill | 0.14 \hfill | 0.009 \hfill | 0.006 \nr 
 Phosphorus (wt\%) \hfill | 0.001 \hfill | 0.25 \hfill | 0.012 \hfill | 0.009 \nr 
 Nickel     (wt\%) \hfill | 0.0 \hfill | 4.79 \hfill | 0.426 \hfill | 0.900 \nr 
 Chromium   (wt\%) \hfill |  0.0 \hfill | 12.1 \hfill | 0.748 \hfill | 1.810 \nr 
 Molybdenum (wt\%) \hfill | 0.0 \hfill | 2.4 \hfill | 0.219 \hfill | 0.370 \nr 
 Vanadium   (wt\%) \hfill | 0.0 \hfill | 0.32 \hfill | 0.0252\hfill | 0.060 \nr 
 Copper     (wt\%) \hfill | 0.0 \hfill | 2.18 \hfill | 0.053 \hfill | 0.160 \nr 
 Cobalt     (wt\%) \hfill | 0.0 \hfill | 2.8 \hfill | 0.008 \hfill | 0.110 \nr 
 Tungsten   (wt\%) \hfill | 0.0 \hfill | 3.86 \hfill | 0.093 \hfill | 0.500 \nr 
 Oxygen     (ppm wt.) \hfill | 0.0 \hfill | 1650 \hfill |  362 \hfill |  200.8 \nr 
 Titanium   (ppm wt.) \hfill |  0.0 \hfill |  900 \hfill |   67 \hfill |  116.5 \nr 
 Boron      (ppm wt.) \hfill | 0.0 \hfill |  195 \hfill | 6 \hfill |   19.3 \nr 
 Niobium    (ppm wt.) \hfill | 0.0 \hfill | 1770 \hfill |   66 \hfill |  163.6 \nr 
 Heat input (kJ mm$^-$$^1$) \hfill | 0.55 \hfill | 7.9 \hfill | 1.56 \hfill | 1.17 \nr 
 Interpass temperature ($^\circ$C) \hfill | 20 \hfill |  375 \hfill |  209 \hfill |   51.8 \nr 
 Tempering temperature ($^\circ$C) \hfill | 20 \hfill |  770 \hfill |  368 \hfill |  241.8 \nr 
 Tempering time  (h) \hfill | 0.0 \hfill |   50 \hfill | 6.9 \hfill | 6.5 \nr 
 Ultimate Tensile Strength (MPa)\hfill |  440 \hfill | 1151 \hfill |  624 \hfill |  117.5 
 \endtable }
 {\tabtit{\tablee : } {The inputs for the ultimate tensile strength 
 model.}} 
   
  
  
 \sec { THE YIELD STRENGTH MODEL} 
  
 Some eighty yield strength neural network models were trained on a 
 training dataset which consisted of a random selection of half 
 the data (1001) from the yield strength dataset. The remaining 1001 data 
 formed the test dataset which was used to see how the model 
 generalises on unseen data. Each model contained the 19 inputs listed in 
 Table~1 but with different  numbers of hidden units or the random seeds 
 used to initiate the values of the weights. 
 \fagg\ shows the results. As expected, the perceived level of noise 
 ($\sigma_\nu$) in the normalised yield strength decreases as the model 
 becomes more complex, \ie the number of hidden units increases.  This 
 is not the case for the test error, which goes through a minimum at 
 three hidden units, and for the log predictive error which reaches a 
 maximum at 6 hidden units.  
  
  
 The error bars presented throughout this work represent a 
 combination of the perceived level of noise $\sigma_\nu$ in the output 
  and the fitting uncertainty estimated from the Bayesian framework. It 
 is evident that there are a few outliers in the plot of the predicted 
 versus measured yield strength for the test dataset. Each of these 
 outliers has been investigated and found to represent unique data 
 which are not represented in the training dataset. For example, there 
 is a weld with a sulphur concentration of 0.15 wt.\%\ and another with 
 a phosphorus concentration of 0.25 wt.\%, both extremely high and 
 unusual level of impurities in weld metals. 
  
  
  
 It is possible that a committee of models can make a more reliable 
 prediction than an individual model [75]. The best models are ranked 
 using the values of the log predictive errors (\figg). 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 \figg. 
  
 The  test error associated with the best single model is clearly greater 
 than that of any of the committees (\figg). The committee with twenty eight 
 models was found to have an optimum membership with the smallest test 
 error. The committee was therefore retrained on the entire data set without changing the complexity of any of its member models. The final comparison between the predicted and measured values of the yield strength for the 
 committee of twenty eight is shown in \fagg. The details of the twenty eight members of the optimum committee are presented in \tablaa. 
  
  
 \midinsert \thicksize=1pt \thinsize=0.8pt 
 \tablewidth=5.2truein \begintable 
 {\bf Model  }&  {\bf Hidden units  }& {$\sigma_{\nu}$ }  | {\bf Model   }& {\bf Hidden units  } & { $\sigma_{\nu}$ }\cr 
   
  {\bf 1 } &  6  &    0.044062  |  {\bf 15 }  &  5  &  0.066848 \nr  
  {\bf 2 } &  5  &    0.048167  |  {\bf 16 }  &  2  &  0.064810 \nr  
  {\bf 3 } &  6  &    0.043382  |  {\bf 17 }  &  3  &  0.059811 \nr
  {\bf 4 } &  7  &    0.040949  |  {\bf 18 }  &  9  &  0.036538 \nr  
  {\bf 5 } &  4  &    0.053474  |  {\bf 19 }  &  2  &  0.066146 \nr  
  {\bf 6 } &  4  &    0.052702  |  {\bf 20 }  &  3  &  0.060487 \nr
  {\bf 7 } &  3  &    0.058641  |  {\bf 21 }  &  2  &  0.066151 \nr  
  {\bf 8 } &  3  &    0.060561  |  {\bf 22 }  &  9  &  0.035645 \nr  
  {\bf 9 } &  5  &    0.047489  |  {\bf 23 }  &  10  &  0.033591 \nr
  {\bf 10 } &  4  &    0.047489  |  {\bf 24 }  &  5  &  0.045975 \nr  
  {\bf 11 } &  5  &    0.046503  |  {\bf 25 }  &  9  &  0.034897 \nr  
  {\bf 12 } &  7  &    0.039873  |  {\bf 26 }  &  4  &  0.051630 \nr 
  {\bf 13 } &  3  &    0.064206  |  {\bf 27 }  &  7  &  0.039677 \nr
  {\bf 14 } &  2  &    0.067036  |  {\bf 28 }  &  8  &  0.036436 
  
 \endtable \tabtit{\tablee : } {The 28 members of the optimum committee. $\sigma_{\nu}$ is the perceived level of noise in the yield strength and the complexity of the model increases with the number of hidden units.} \endinsert 
  
 

  
 \tablee\ indicates the significance ($\sigma_w$) of each of the 
 input variables, as perceived by first five neural network models in the committee. The  $\sigma_w$ value represents the extent to which a particular input explains the variation in the output, rather like a partial correlation  coefficient in linear regression analysis. 
  
 The post-weld heat  treatment temperature on the whole explains a large 
 proportion of the variation in the yield strength \fagg. All of the 
 variables considered are found to have a significant effect on the 
 output indicating a good choice of inputs. 
 
  
 \sec {THE ULTIMATE TENSILE STRENGTH MODEL} 
  
  The models were trained 
 on 1972 individual experimental measurements, of which a random half 
 of the data formed the training dataset and the other half the test 
 dataset.  The procedures are otherwise identical to those described 
 for the yield strength model, resulting in the characteristics 
 illustrated in {\fagg}  and the performance of the optimum committee 
 of best models is illustrated in \fagg . The perceived significance of first  five models are shown in  \fagg. Here the additional input varibale oxygen shows more significance along with post-weld heat treatment variables.
  
    
 \midinsert{\smalltype\thicksize=1pt \thinsize=0.8pt 
 \tablewidth=5.2truein \begintable 
 {\bf Model  }& {\bf Hidden units  } & { $\sigma_{\nu}$ }  
 | {\bf Model   }& {\bf Hidden units  } & { $\sigma_{\nu}$ } \cr  
  {\bf 1 } &  8  &  0.029323 | {\bf 21 } &  7 &  0.027016 \nr  
  {\bf 2 } &  9  &  0.026503 | {\bf 22 } &  8 &  0.060074 \nr  
  {\bf 3 } &  6  &  0.027177 | {\bf 23 } &  3 &  0.028483 \nr  
  {\bf 4 } &  10 &  0.025500 | {\bf 24 } &  3 &  0.039529 \nr  
  {\bf 5 } &  9  &  0.031034 | {\bf 25 } &  2 &  0.065098 \nr  
  {\bf 6 } &  6  &  0.026604 | {\bf 26 } &  2 &  0.052619 \nr  
  {\bf 7 } &  4  &  0.027218 | {\bf 27 } &  2 &  0.029716 \nr  
  {\bf 8 } &  10 &  0.026010 | {\bf 28 } &  2 &  0.045941 \nr  
  {\bf 9 } &  6  &  0.025530 | {\bf 29 } &  4 &  0.053212 \nr  
  {\bf 10 } &  5 &  0.031240 | {\bf 30 } &  9 &  0.053929 \nr  
  {\bf 11 } &  8 &  0.026471 | {\bf 31 } &  4 &  0.033334 \nr  
  {\bf 12 } &  9 &  0.026485 | {\bf 32 } &  3 &  0.034551 
  
 \endtable}\tabtit{\tablee : } {The hidden  units and $\sigma_{\nu}$ in the 
 optimum UTS committee model.} 
 \endinsert 
  
  
  
  
 \sec {APPLICATION TO C--Mn WELDS} 
  
 Carbon--manganese weld metals refer to a popular class of   
 ferritic steels in which the substitutional solutes other than silicon 
 and manganese are generally kept to low concentration levels. They are 
 interesting because there is a great deal already known about them, 
 making it easy to interpret the physical significance of the neural 
 network model. Furthermore, there exists an alternative 
 semi--empirical model for the estimation of the yield and tensile 
 strengths of such multirun welds [76] enabling a further comparison. 
 The semi--empirical model is henceforth referred to as the \lqq 
 physical model" or PM for short. The basic values of the variables used 
 in applying the model to carbon--manganese welds are listed in 
 \tablaa. The specified low--temperature heat treatment is simply a 
 standard hydrogen removal treatment (250\deg for 14 h) applied to most welds before mechanical testing. 
  
 \midinsert 
 \vskip 0.5 in 
  \thicksize=0.7pt \thinsize=0.5pt 
 \tablewidth=3.0truein {\smalltype \begintable  
 Carbon     (wt\%)  \hfill | 0.06 \nr 
 Silicon    (wt\%) \hfill | 0.50 \nr 
 Manganese  (wt\%) \hfill | 1.50 \nr 
 Sulphur    (wt\%) \hfill | 0.006 \nr 
 Phosphorus (wt\%) \hfill | 0.008 \nr 
 Nickel     (wt\%) \hfill | 0.0 \nr 
 Chromium   (wt\%) \hfill | 0.0  \nr 
 Molybdenum (wt\%) \hfill | 0.0 \nr 
 Vanadium   (wt\%) \hfill | 0.0 \nr 
 Copper     (wt\%) \hfill | 0.0\nr 
 Cobalt     (wt\%) \hfill | 0.0\nr 
 Tungsten   (wt\%) \hfill | 0.0 \nr 
 Oxygen     (ppm wt.) \hfill | 300 \nr 
 Titanium   (ppm wt.) \hfill | 0.0 \nr 
 Boron      (ppm wt.) \hfill | 0.0 \nr 
 Niobium    (ppm wt.) \hfill | 0.0 \nr 
 Heat input (kJ mm$^-$$^1$) \hfill | 1.00 \nr 
 Interpass temperature ($^\circ$C) \hfill | 175 \nr 
 Tempering temperature ($^\circ$C) \hfill | 250 \nr 
 Tempering time  (h) \hfill |  14.0  
 \endtable} 
 \tabtit{\tablee : }{ The inputs relevant for a typical \lqq 
 carbon--manganese" weld metal made using the manual metal arc welding 
 process.} 
 \endinsert  
  
  
 The results as a function of the carbon and manganese 
 concentrations are illustrated in 
 \fagg\ for a variety of interesting cases. The calculated yield 
 strength is in all cases found to be consistent with that expected from 
 the physical model, although there are systematic differences at high 
 yield strength values for all cases other than at the highest manganese 
 concentration. However, the deviations are all within the error bounds 
 of the neural network model for yield strength. The major 
 discrepancies arise with the UTS especially at high UTS values. It is 
 believed that the physical model is poorly constructed since the UTS 
 is essentially taken arbitrarily to be linearly related to a single 
 variable, the yield strength, \fagg\ shows the comparision between the measured and strength estimation by physical model. Physical models at higher strength values behaved very poorly, it estimated the strength higher than the measured. 
  
  
 An interesting feature of strengthening due to substitutional solutes 
 is the synergistic effect with carbon. {\fagg} shows that the 
 dependence of the strengthening effect of molybdenum on the carbon 
 concentration is particularly large, the effect of molybdenum in strengthening the weld is greaterthan that of Cr or Mn. This is consistent with published literature [76]. Elements such as molybdenum and vanadium  are associated with strong secondary hardening effects which frequently trigger a reduction in toughness. In ordinary carbon--manganese multirun welds, the secondary microstructure, \ie regions of weld metal which are tempered by subsequent weld runs, lose  most of their microstructural 
 strength. This is not necessarily the case in weld metal containing 
 strong carbide formers. For example, it is well--established that the 
 yield strength calculated using equation~1 is always underestimated 
 with molybdenum--containing welds, the degree of underestimation 
 increasing with the molybdenum concentration [1]. The behaviour 
 observed in {\figg} is not therefore surprising. 
  
 The sensitivity to carbon concentration, and the net magnitude of the 
 strengthening effect decreases for the ultimate tensile strength 
 (\figg ). This is expected since the UTS is measured at large plastic 
 strains whereas the yield strength is much more sensitive to the 
 initial microstructure. 
  
 The predicted dependence of the strengthening effect of niobium on the 
 carbon concentration is shown in \figg . The strength increment plotted on the vertical axis is based on average effect on niobium in the concentration range 0-1500 parts per million by weight, for any given carbon concentrations. The increment per weight percent of niobium is obviously very large and this may be reason why niobium is generally found to be determinantal to toughness [77]. 

   
 \fagg\ shows other predictions; although there are no surprises, it is 
 worth noting the error bars. These error bars can be used to identify 
 regions of the input space where further experiments would be useful. 
 For example, the prediction uncertainties associated with niobium, or 
 with large heat inputs, are much larger than say with changes in the 
 manganese concentration. This is where future experiments could be 
 focussed. 
  
  
  
 \sec{APPLICATION TO $2{1\over4}$Cr1Mo WELDS} 
  
 The $2{1\over4}$Cr1Mo weld metal system is designed 
 primarily for applications where the components will 
 serve at elevated temperatures (450--565\degg) for 
 long periods of time ($\simeq$30 years). This in 
 contrast to carbon--manganese weld metals which are used in 
 structural applications such as buildings and bridges which are 
 essentially at ambient temperature. Consequently, the post--weld 
 heat treatment is of vital importance to $2{1\over4}$Cr1Mo weld 
 metals, not only to relieve residual stresses but also to generate 
 a stable microstructure in which the carbides hinder creep 
 deformation.  The basic values of the variables 
 used in applying the models to $2{1\over4}$Cr1Mo welds are listed in 
 \tablaa. The specified high--temperature heat treatment is a typical 
 post--weld heat treatment (PWHT). 
  
 \topinsert 
 \vskip 0.5 in 
  \thicksize=0.7pt \thinsize=0.5pt 
 \tablewidth=3.0truein {\smalltype \begintable 
 Carbon     (wt\%)  \hfill | 0.11 \nr 
 Silicon    (wt\%) \hfill | 0.20 \nr 
 Manganese  (wt\%) \hfill | 0.80 \nr 
 Sulphur    (wt\%) \hfill | 0.002 \nr 
 Phosphorus (wt\%) \hfill | 0.005 \nr 
 Nickel     (wt\%) \hfill | 0.20 \nr 
 Chromium   (wt\%) \hfill | 2.25  \nr 
 Molybdenum (wt\%) \hfill | 1.0 \nr 
 Vanadium   (wt\%) \hfill | 0.0 \nr 
 Copper     (wt\%) \hfill | 0.0\nr 
 Cobalt     (wt\%) \hfill | 0.0\nr 
 Tungsten   (wt\%) \hfill | 0.0 \nr 
 Oxygen     (ppm wt.) \hfill | 300 \nr 
 Titanium   (ppm wt.) \hfill | 0.0 \nr 
 Boron      (ppm wt.) \hfill | 0.0 \nr 
 Niobium    (ppm wt.) \hfill | 0.0 \nr 
 Heat input (kJ mm$^-$$^1$) \hfill | 1.5 \nr 
 Interpass temperature ($^\circ$C) \hfill | 200 \nr 
 Tempering temperature ($^\circ$C) \hfill | 690 \nr 
 Tempering time  (h) \hfill |  8.0 
 \endtable} 
 \tabtit{\tablee : } {The inputs relevant for a typical  
 2$1\over4$Cr - 1Mo wt\% weld metal.} 
 \endinsert 
  
 It is notable from the predictions illustrated in \fagg\ that there are 
 greater uncertainties (larger error bars) associated with the estimation 
 of mechanical properties for the $2{1\over4}$Cr1Mo system when 
 compared with the carbon--manganese welds. This is largely because 
 there are fewer data available for $2{1\over4}$Cr1Mo welds. 
  
 Another striking feature is that the sensitivity of the strength to  
 alloying elements, in the PWHT condition, is far smaller than in the 
 as--welded condition. This is not surprising given the severe nature 
 of the post--weld heat treatment at 690\degg\ for 8 hours. It is 
 emphasised that although the yield and tensile strengths are not 
 particularly sensitive to composition in the PWHT condition, this will 
 not be the case for creep properties where the tempering heat 
 treatment is essential for the generation of alloy carbides and to 
 provide a microstructure which has long term stability. 
  
  
    
 \sec{SOFTWARE} 
  
 We have examined a vast number of other trends predicted by the 
 models. It is not reasonable to present all the results in this paper, 
 suffice it to say that the trends have been found to be reasonable from 
 a metallurgical point of view. 
  
 Models like these cannot be fully studied because the number of 
 possibilities is very large indeed. In addition, although the 
 carbon--manganese and $2{1\over4}$Cr1Mo have been illustrated in the 
 present paper, the work is much more widely applicable since the 
 database from which the neural network models were created covers a 
 vast range of alloys. 
  
 The software capable of doing these calculations can be obtained 
 freely from 
 $$http://www.msm.cam.ac.uk/map/map.html$$ 
  
  
  
 \sec{SUMMARY \&\ CONCLUSIONS} 
  
 The yield strength and ultimate tensile strength of ferritic steel 
 weld metal have been analysed using a neural network method within a 
 Bayesian framework. The data used were mostly obtained from the 
 published literature and represent a wide cross--section of alloy 
 compositions and arc--welding processes.  
  
 Trends predicted by the models appear to be consistent with those 
 expected metallurgically, although it must be emphasised that only the 
 simplest of trends have been examined since the number of variables 
 involved is very large. The models can be applied widely because the 
 calculation of error bars whose magnitude depends on the local 
 position in the input space is an inherent feature of the neural 
 network used. The error bar is not simply an estimate of the perceived 
 level of noise in the output but also includes an uncertainty 
 associated with fitting the function in the local region of input 
 space. This means that the method is less dangerous in extrapolation 
 or interpolation since it effectively warns when experimental data are 
 lacking or are exceptionally noisy. The work has clearly identified regions of the input space where further experiments should be encouraged.
  
 \sec {ACKNOWLEDGEMENTS} 
  
 The  authors are thankful to the to the Cambridge Commonwealth 
 Trust for funding this work and to Kenneth Mitchell of National Power 
 for further financial support. We would like to acknowledge Professor 
 Alan Windle of the University of Cambridge for the provision of 
 laboratory facilities.  
  
  
 \sec{REFERENCES} 
  
\def\ref#1#2#3#4#5#6#7{\hoffset=6.0mm \parindent=-6.0mm 
\singlespace \rightskip=6.0mm \singlespace {#1.} #2: {\it 
#4} {\bf #5} (#3) #6\vskip 0.5truemm} 

{\parskip=0mm \parindent=10pt \narrower 
  
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\medskip}



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\smallskip
{\tabtit{Fig. 1 : }{ Schematic illustration of the input, hidden and output layers of the neural network model utilised here.}} 
\endinsert

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\vskip 1.55 in
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\vskip 1.55 in
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 {\tabtit{Fig. 2 : } {The database values of each variable versus the 
 yield strength.}}
\endinsert


\topinsert
\vskip 2.3 in
\special{psfile=YSHU_Sigma.eps hoffset=20 voffset=0 hscale=45 vscale=45}
\special{psfile=YSHU_energy.eps hoffset=220 voffset=0 hscale=45 vscale=45}

\vskip 2.5 in
\special{psfile=YSHU_Log.eps hoffset=20 voffset=-15 hscale=45 vscale=45}
\special{psfile=YScommittee.eps hoffset=210 voffset=-15 hscale=45 vscale=45}

\bigskip
\vskip 2.8 in
\special{psfile=YSOT_TRAIN.eps hoffset=0 voffset=0 hscale=45 vscale=45}
\special{psfile=YSOT_PRED.eps hoffset=213 voffset=0 hscale=45 vscale=45}

 {\tabtit{Fig. 3 :  }{Characteristics of the yield strength model.}}
\endinsert

\topinsert
\vskip 2.3 in
\special{psfile=YSCOM_RESULT.eps hoffset=80 voffset=-10 hscale=45 vscale=45}

 {\tabtit{Fig. 4 : }{Comparison of the predicted and measured values 
 of yield strength for the optimum committee model.}}

\vskip 3.5 in
\special{psfile=best_YS_sigma_w.eps hoffset=0 voffset=-50 hscale=80 vscale=80}

 {\tabtit{Fig. 5 : }{The perceived significance $\sigma_w$ values of first best five yield strength models for each of the inputs.}}

\endinsert


\midinsert
\vskip 2.3 in
\special{psfile=UTSHU_Sigma.eps hoffset=-20 voffset=0 hscale=45 vscale=45}
\special{psfile=UTSHU_energy.eps hoffset=190 voffset=0 hscale=45 vscale=45}

\vskip 2.4 in
\special{psfile=UTSHU_Log.eps hoffset=-20 voffset=-10 hscale=45 vscale=45}
\special{psfile=UTScommittee.eps hoffset=175 voffset=-10 hscale=45 vscale=45}

\vskip 2.8 in
\special{psfile=UTSOT_TRAIN.eps hoffset=4 voffset=0 hscale=45 vscale=45}
\special{psfile=UTSOT_PRED.eps hoffset=215 voffset=0 hscale=45 vscale=45}
 {\tabtit{Fig. 6 : }{Characteristics of the ultimate tensile strength model.}}
\endinsert

\topinsert
\vskip 2.5 in
\special{psfile=UTSCOM_RESULT.eps hoffset=60 voffset=-10 hscale=45 vscale=45}

 {\tabtit{Fig. 7 : }{Comparison of the predicted and measured values 
 of ultimate tensile strength for the optimum committee model.}}

\vskip 3.5 in
\special{psfile=best_UTS_sigma_w.eps hoffset=0 voffset=-55 hscale=80 vscale=80}

 {\tabtit{Fig. 8 : }{The perceived significance $\sigma_w$ values of first best five UTS models for each of the inputs.}}

\endinsert

\midinsert
\vskip 2.3 in
\special{psfile=0.1Mn_YUC_pwht.eps hoffset=-60 voffset=0 hscale=40 vscale=40}
\special{psfile=0.3Mn_YUC_pwht.eps hoffset=108 voffset=0 hscale=40 vscale=40}
\special{psfile=0.5Mn_YUC_pwht.eps hoffset=280 voffset=0 hscale=40 vscale=40}

\vskip 2.3 in
\special{psfile=1.1Mn_YUC_pwht.eps hoffset=-60 voffset=-10 hscale=40 vscale=40}
\special{psfile=1.5Mn_YUC_pwht.eps hoffset=108 voffset=-10 hscale=40 vscale=40}
\special{psfile=1.9Mn_YUC_pwht.eps hoffset=280 voffset=-10 hscale=40 vscale=40}

 \tabtit{Fig. 9 : }{A series of calculations for carbon--manganese 
 welds, using both the neural network committee models and an 
 alternative published model [76].} 

\vskip 2.3 in
\special{psfile=EXP_PHY.eps hoffset=108 voffset=0 hscale=40 vscale=40}
\tabtit{Fig. 10 : }{Comparision between measured and published physical model calculations.}
 \endinsert 

\topinsert
\vskip 2.6 in
\special{psfile=Stg_coeff_YS.eps hoffset=-20 voffset=-10 hscale=45 vscale=45}
\special{psfile=Stg_coeff_UTS.eps hoffset=200 voffset=-10 hscale=45 vscale=45}

\bigskip
\vskip 2.6 in
\special{psfile=Stg_coeff_V.eps hoffset=-20 voffset=-10 hscale=45 vscale=45}
\special{psfile=Stg_coeff_Nb.eps hoffset=200 voffset=-10 hscale=45 vscale=45}
\bigskip

 \tabtit{Fig. 11 : }{Change in strength and the YS/UTS ratio as 
 a function of a wt\% of substitutional solute content in carbon--manganese steel 
 welds. The error bars are not included for clarity, but the 
 maximum values are 60.}
\endinsert 

\midinsert
\vskip 2.3 in
\special{psfile=C_MnYUMn_pwht.eps hoffset=-60 voffset=0 hscale=40 vscale=40}
\special{psfile=C_MnYUCr_pwht.eps hoffset=108 voffset=0 hscale=40 vscale=40}
\special{psfile=C_MnYUMo_pwht.eps hoffset=280 voffset=0 hscale=40 vscale=40}

\bigskip
\vskip 2.3 in
\special{psfile=C_MnYUNb_pwht.eps hoffset=-60 voffset=-10 hscale=40 vscale=40}
\special{psfile=C_MnYUNi_pwht.eps hoffset=108 voffset=-10 hscale=40 vscale=40}
\special{psfile=C_MnYUSi_pwht.eps hoffset=280 voffset=-10 hscale=40 vscale=40}

\bigskip
\vskip 2.3 in
\special{psfile=C_MnYUHI_pwht.eps hoffset=20 voffset=-15 hscale=40 vscale=40}
\special{psfile=C_MnYUPWHT_t_600.eps hoffset=200 voffset=-20 hscale=40 vscale=40}

\bigskip

\tabtit{Fig. 12 : } {Variations in the yield and ultimate 
 tensile strengths of carbon--manganese weld metal as a 
 function of alloying elements and heat treatment.}
\endinsert 

\topinsert
\vskip 2.35 in
\special{psfile=Cr_YUC.eps hoffset=-20 voffset=-10 hscale=45 vscale=45}
\special{psfile=Cr_YUC_pwht.eps hoffset=200 voffset=-10 hscale=45 vscale=45}

\bigskip
\vskip 2.35 in
\special{psfile=Cr_YUMo.eps hoffset=-20 voffset=-10 hscale=45 vscale=45}
\special{psfile=Cr_YUMo_pwht.eps hoffset=200 voffset=-10 hscale=45 vscale=45}

\bigskip 
\vskip 2.35 in
\special{psfile=Cr_YUCr.eps hoffset=-20 voffset=-10 hscale=45 vscale=45}
\special{psfile=Cr_YUCr_pwht.eps hoffset=200 voffset=-10 hscale=45 vscale=45}

\bigskip
 
\tabtit{Fig. 13 : } {The effect of carbon, molybdenum and chromium 
 concentrations on the strength of $2{1\over4}$Cr1Mo welds in 
 the as--welded and PWHT (690$\,^\circ$C, 8~h) conditions.}
\endinsert 

 
\vfil \eject
 \end 
  
 
