DECLARATION 2 PUBLICATIONS
6.2 Fractional factorial design
Fractional factorial design is used in situations where it is impractical or too costly to perform full factorial research (i.e. full population of experiments). The idea of the fractional factorial design is to run only a subset of experiments but still to obtain representative results. There are k number of factors and L number of levels in a factorial design matrix (i.e. Lk-1 designation). The results obtained and conclusions drawn from a reduced experimental study are valid for the whole experimental region covered by the control factors.
Model significance is tested using ANOVA. In the ANOVA table, F-values compare the model variance (i.e. term mean square) with residual (error) variance (i.e. residual mean square). Values close to unity indicate that a parameter has an insignificant effect on the response. Probability values lower than 0.05 signify that the model is adequate within the selected confidence interval.
Precision adequacy, which is a ratio of the range of predicted values at points of the design to the average predicted error, indicates the model’s ability to discriminate. A precision adequacy value of 4 or higher indicates adequate discrimination. The elimination of insignificant model terms is called model reduction and is achieved through regression analysis. The achievement of the highest possible performance (which is the desired outcome of the objective function) occurs when the optimum design factors have been determined. Objective function may minimise, maximise or target the assigned nominal value. The steps involved in the DoE method used in this study are shown in Figure 6.1.
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Figure 6-1: The DoE Process
6.2.1 SAW process parameters
The main parameters of the SAW process include welding current, arc voltage, travel speed, electrode stick-out, wire-feed rate, type of flux, depth of flux and polarity. It is clear from the summary of previous studies on parametric optimisation (Chapter 2) that four SAW parameters have consistentl displayed significant effects on the outputs – the outputs being weld geometry,
Identify desired quality characteristics
Select process parameters and responses Step 1: Identification of Influential Conditions
Narrow parameters down to critical ones
Define operational range and factor levels
Determine parameter degrees of freedom Step 2: Selection of Critical Parameters
Select appropriate DoE matrix
Assign parameters to matrix Step 3: Developing DoE Matrix
Perform experiments according matrix parameters
Record results
Step 4: Conducting Experiments
Analyse model significance through ANOVA
Eliminate insignificant model terms
Present objective functions Step 5: Analysis of Results
110 | P a g e HAZ size, mechanical properties and residual stress. These four variables are welding current (I), arc voltage (V), travel speed (S) and wire-feed rate (FR). These parameters are therefore considered in this study.
An additional parameter was used in the experiments for this research, namely welding mode.
The welding mode of the SAW machine used can be set either to CA (constant amperage) or to CW (constant wire). Each of the four parameters, but not welding mode, has upper (+) and lower (-) limits that were chosen according to safe operational ranges. The output variables included bead width (BW), bead height (BH), bead penetration (BP), HAZ size (HAZ), hardness (VHN) and residual stress (RS). Figure 6.2 illustrates the bead geometry of the envisaged multi-pass full penetration weld.
Figure 6-2: Weld Bead Geometry
Table 6.1 shows the two levels of parameters applied in the welding experiments. The range of values was chosen according to guidelines from the SAW machine operating manual and the researcher’s experience from similar studies. When choosing the operational range for SAW parameters, care should be taken to include only parametric combinations that will not result in burn-through (i.e. too much heat) or lack of penetration (i.e. too little heat).
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Table 6-1: Welding Parameters
Parameter Units Lower L (-) Upper L (+)
Current (I) Amps 360 380
Voltage (V) Volts 25 30
Speed (S) mm/s 8 10
Wire-feed Rate (FR) mm/s 16.7 18.3
The response variables were chosen according to their effect on the resultant residual stress fields, and hence fatigue life, of a welded pressure vessel structure. The integrity of weld bead geometry indicates the quality of the weld, and it influences weld strength and life expectancy of the structure. The residual stress affects the fatigue life of a welded structure either by assisting crack initiation and growth (tensile) or by retarding crack growth and increasing the toughness (compressive). Hardness influences hydrogen-induced cracking, which shortens the fatigue life of the welded structure. The DoE matrix chosen in this study had five factors, four of which had two levels, namely an L5-1 design.
Table 6.2 shows the resultant matrix with five columns and two levels. Experiments were conducted in a random order to lessen noise sources. Response variables were captured for each specimen after the third pass. (The procedure for conducting the experiments was discussed in Chapter 4.)
Table 6-2: DoE Matrix for this study S.No. Input Parameters
I V S FR Mode
1 + + + + CA
2 + - - + CA
3 - - - + CA
4 - - + + CA
5 + - + + CA
6 - + - - CW
7 - + - + CW
8 - - - + CW
9 - - - - CW
6.2.2 Conducting the experiments
Experiments were conducted according to the DoE matrix in Table 6.2. Nine experimental iterations, comprising three runs each, were performed using the mechanised SAW machine with
112 | P a g e settings as shown in Table 6.1. The experimental responses were determined as described in Chapter 4, and the experimental results were recorded according to the matrix discussed in Chapter 7. The objective of this study was to minimise RS, BW, BH, VHN and HAZ, and to maximise BP. As mentioned earlier, the specimen numbers do not necessarily follow a sequence because some of them were spoiled during the experiment [e.g. through burn-through or lack of penetration (LOP)] and were not used further.
6.2.3 Statistical analysis and optimisation
ANOVA was used to analyse the statistical differences between the mathematical means of multiple samples, through subdividing the total sum of squares (R2). ANOVA calculates the level of significance associated with the effect of a specific process parameter on the response variables. It is also used to test the adequacy of a developed model. The mathematical model was designed through DesignExpert® Ver.10 program, using data from experiments. The software automatically assigns letters to the columns containing factor effects (i.e. factor-effect columns are labelled A, B, C, D and E). To understand which letter was assigned to which factor effect, the following notational replacements were made:
I A, V B, S C, FR D, MODE E,
The objective function that represents each of the response variables can be expressed according to the following general function:
𝑦 = 𝑓(𝐼, 𝑉, 𝑆, 𝐹𝑅) (6.1)
Multiple regression analysis (MRA) was performed to determine the coefficients of each parameter. MRA is a statistical multivariate data analysis tool that maps relationships between input process parameters and responses. The resultant polynomial equation for each response variable becomes:
𝑓(𝑥) = 𝛽 + 𝛽 𝐼 + 𝛽 𝑉 + 𝛽 𝑆 + 𝛽 𝐹𝑅 (6.2)
where f(x) is the response variable, β is the coefficient and β0 is the model constant.
In DesignExpert, the factor “MODE” was entered as a categorical rather than a numeric value.
This meant that coded factors of MODE could have coefficients, but the actual factors could not.
Each objective function therefore ended with two equations, one in each mode. The equation in terms of coded factors can be used to make predictions about the response for given levels of each factor. By default, the higher level of each factor was coded as +1, whereas the lower level
113 | P a g e factors were coded as -1. The coded equation is useful for identifying the relative impact of the factors by comparing the factor coefficients.
It was mentioned earlier that an effective strategy for solving a multi-objective problem is through a multi-objective genetic algorithm (MOGA). Like most engineering problems, the welding process parametric optimisation problem (WPPOP) is a complex problem with multiple objectives that are usually in conflict. The MOGA used to solve the WPPOP in this study followed the generalisation outlined in Figure 2.7 in Chapter 2. The algorithm followed a Pareto strategy, in which all objectives were assigned the same level of priority (Fonceca, 1995). The optimisation was performed using the “multi-objective optimisation” toolbox in MATLAB®
software. The 14 objective functions, which were aligned to the response variables, were all loaded and optimised at the same time to determine the Pareto set of non-dominated solutions.