Materials & Design Materials and Design 28 (2007) 8–15 www.elsevier.com/locate/matdes
A novel method for materials selection in mechanical design: Combination of non-linear normalization and a modified digital logic method B. Dehghan-Manshadi a, H. Mahmudi b, A. Abedian a, R. Mahmudi
c,*
a
Department of Aerospace Engineering, Sharif University of Technology, Tehran, Iran Department of Mechanical Engineering, University of Alberta, Edmonton, Canada Department of Metallurgy and Materials Engineering, Faculty of Engineering, University of Tehran, P.O. Box 11365-4563, Tehran, Iran b
c
Received 30 March 2005; accepted 29 June 2005 Available online 15 August 2005
Abstract In this investigation, a novel numerical method is proposed for materials selection. This method is based on the well known weighting factor approach while combining non-linear normalization with a modified digital logic method. The proposed mathematical functions and their applicability to the materials selection process is verified by examining two case studies in mechanical design and comparing the results with those obtained from the classical weighted property method. It is concluded that the new approach is capable of providing more reasonable selections as opposed to those obtained from the existing method. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Selection for material properties (H); Weighting factors (H); Performance indices (H)
1. Introduction In general, in the area of decision making trusting the intuition is more common than using any kind of numerical approach. However, in those areas such as materials selection in which there are numerous different choices and many various criteria influencing the selection, a more precise approach would be required. In the manufacture of mechanical parts, a knowledge of materials properties, cost, design concepts and their interactions is required. The large number of available materials, together with the complex relationships between the various selection parameters, often make the selection process a difficult task. When selecting materials, a large number of factors should be taken into account. These factors range from mechanical, electrical *
Corresponding author. Tel.: +98 21 8012999; fax: +98 21 8006076. E-mail address:
[email protected] (R. Mahmudi).
0261-3069/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.matdes.2005.06.023
and physical properties to corrosion resistance and economic considerations. In mechanical design, however, mechanical properties are of prime concern. The most important material properties usually encountered in materials selection processes are strength, stiffness, toughness, hardness, density and creep resistance. Perhaps the first step in the material selection sequence is to specify the performance requirements of the component and to broadly outline the main materials characteristics and processing requirements [1]. Accordingly, certain classes of materials may be eliminated and others chosen as probable candidates for making the component. The relevant material properties are then identified and ranked in order of importance. Optimization techniques may then be used to select the optimal material. Several quantitative selection procedures have been developed to analyze the large amount of data involved in the selection process so that a systematic evaluation can be made. Ashby [2]
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als performance index (c). The material with the highest performance index (c) is considered as the optimum for the application. In the cases where numerous material properties are specified and the relative importance of each property is not clear, determinations of the weighting factors (a) can be largely intuitive, which reduces the reliability of selection. This problem can be solved by adopting a systematic approach to the determination of a. Using the DL approach, evaluations are arranged such that only two properties are considered at a time. Every possible combination of properties or performance goals is compared and no shades of choice are required, only a yes or no decision for each evaluation. To determine the relative importance of each property or goal a table is constructed, the properties or goals are listed in the left hand column, and comparisons are made in the columns to the right, as shown in Table 1. In comparing two properties or performance goals, the more important goal is given numerical one (1) and the less important is given zero (0). The total number of possible decisions is N = n(n 1)/2, where n is the number of properties or goals under consideration. A relative emphasis coefficient or weighting factor, a, for each goal is obtained by dividing the number of positive decisions for each goal into P the total number of possible decisions (N). In this case a ¼ 1. For scaling candidate material properties each property is scaled so that its highest numerical value does not exceed 100. When evaluating a list of candidate materials, one property is considered at a time. The best value in the list is rated as 100 and the others scaled proportionally. For a given property, the scaled value (Y) for a given candidate material is equal to
introduced materials selection charts which allows the identification, from among the full range of available materials, the subset most likely to perform best in a given application. He has also used a multi-objective optimization method to compromise between several conflicting objectives in materials selection [3]. In this approach, trade-off allows the identification of an optimal trade-off surface on which the best choices lie. More recently, he has reviewed the selection strategies for materials and processes proposing four basic steps of translating design requirements, screening, ranking and finally searching for supporting information about proper candidate materials [4]. A number of knowledge-based and intelligent data base systems has also been developed for materials selection in mechanical design. These systems are suitable tools for searching and not for calculations. They find a solution that is already stored in the data base by fast retrieval of the stored data [5–7]. Another approach to materials selection problems is the weighted properties method (WPM) which is used when several properties should be taken into consideration. This numerical method ranks the candidate materials on the basis of their performance indices, calculated from simple mathematics [1]. In the present investigation, a new numerical method is proposed to select materials for mechanical design. This method, which is based on the WPM, uses a new digital logic (DL) together with a non-linear approach for scaling the properties. The capabilities of this method are discussed and compared with those of the WPM for two example problems in mechanical design.
2. Selection methods Y ¼ 2.1. The existing WPM
numerical value of property 100. maximum value in the list
ð1Þ
For properties like cost, corrosion or wear loss, weight gain in oxidation, density, etc., a lower value is more desirable. In such cases, the lowest value is rated as 100 and Y is calculated as
In the well known WPM, each material property is assigned a certain weight, depending on its importance. A weighted property value is obtained by multiplying the scaled value of the property by the weighting factor (a). The individual weighted property values of each material are then summed to give a comparative materi-
Y ¼
minimum value in the list 100. numerical value of property
ð2Þ
Table 1 Determination of relative importance of performance goals using the DL method [1] Goals
A B C D E *
Number of possible decisions [N = n(n 1)/2] 1
2
3
4
1 0
1
0
1
0 1
6
7
1 0
0
1
8
1 0
1 0
a ¼ positive Ndesicions .
5
0
9
Relative emphasis coefficient (a)*
3 2 1 2 2
0.3 0.2 0.1 0.2 0.2
10
0 1
Positive decisions
0 1
10
B. Dehghan-Manshadi et al. / Materials and Design 28 (2007) 8–15
For material properties that can be represented by numerical values, application of the above procedure is simple. However, with properties like corrosion and wear resistance, machinability and weldability, etc., numerical values are rarely given and materials are usually rated as very good, good, fair, poor, etc. In such cases, the rating can be converted to numerical values using an arbitrary scale. For example, a corrosion resistance rating of excellent, very good, good, fair, and poor can be given numerical values of 5, 4, 3, 2, and 1, respectively. Then, the material performance index is n X c¼ Y i ai ; ð3Þ
done by means of a linear function which yields scaled properties (Y) between 0 and 100. In the proposed method, however, two different non-linear functions are suggested for the normalization purpose Y ¼ a1 lnðb1 X þ c1 Þ; b2 þ c2 ; Y ¼ a2 ln X
ð4Þ ð5Þ
where Y is the scaled property, X is the numerical value of property and a1, a2, b1, b2, c1 and c2 are constants. Eq. (4) is for those properties for which a higher value is desirable while, Eq. (5) is proposed for properties for which a lower value is of prime concern. The main task in the new method is the determination of the constants a1, a2, b1, b2, c1 and c2. For Eq. (4), this should be done using boundary conditions of Y = 100 at X = 0 and Y = +100 at X = Xmax. This equation is plotted in Fig. 1. As it can be observed, another boundary condition is needed to obtain the constants in Eq. (4). This is related to the intersection of the curve with the X axis, parameter Xc, which enables the designer to assign a critical value to the property less than which, the scaled property becomes negative and thus the performance index is reduced. Therefore, the third bound-
i¼1
where i is summed over all the n relevant properties. 2.2. The proposed method 2.2.1. Modification of the digital logic approach There exist some disadvantages in both digital logic and the scaling procedure of the previous method. In the DL method, the least important goal or property is given zero (0) in all comparisons; therefore, the positive decisions for such a goal and its relevant weighting factor, a, would be zero (0). This implies that this property will be expelled from the material selection process and does not play any role in the selection process. In the method suggested here, however, we assign a value of one (1) to the less important property and three (3) to the more important one, when two properties are considered at a time (Table 2). By doing this, the least important property still remains in the selection list. Another striking advantage of the method is that two properties with equal importance can have equal numerical values of two (2). This is not possible in the original DL method, which assigns either zero (0) or one (1) to any pair of properties. 2.2.2. Non-linear normalization In any selection method which uses weighting of properties, the ranking is based on the performance indices calculated from scaling candidate material properties. For the WPM, the scaling or normalization is
Fig. 1. A schematic representation of non-linear normalization functions for those properties for which a higher value is desirable.
Table 2 Determination of relative importance of performance goals in the proposed method Goals
A B C D E *
Number of possible decisions [N = n(n 1)/2] 1
2
3
4
3 1
3
3
2
1 1
6
7
1 3
1
1
N 0 ¼ 2nðn 1Þ.
8
2 2
3 2
a ¼ positive Ndesicions ;
5
3
9
Relative emphasis coefficient (a)*
11 4 7 7 11
0.275 0.100 0.175 0.175 0.275
10
1 3
Positive decisions
1 3
B. Dehghan-Manshadi et al. / Materials and Design 28 (2007) 8–15
ary condition would be Y = 0 at X = Xc. Solving the three mentioned conditions, the constants are obtained as follows: a1 ¼ ln
100 ;
ð6Þ
Xc
X max X c
X max 2X c ; b1 ¼ X c ðX max X c Þ Xc c1 ¼ . X max X c
ð7Þ ð8Þ
According to the proposed equation, the scaled property function is indeterminate at X = Xmax/2. To evaluate the determinate form of the function, the LÕHopitalÕs rule is used. The obtained function would be of the following form: Y ¼
200X 100. X max
ð9Þ
The final relationships for this case are then ( Y ¼ a1 lnðb1 X þ c1 Þ for X c 6¼ X max =2; Y ¼ 200X 100 X max
for X c ¼ X max =2.
ð10Þ
A similar approach can be followed for the properties for which a lower value is desirable. Eq. (5) is schematically plotted in Fig. 2. Again, the boundary conditions are Y = +100 at X = Xmin and Y = 100 at X ! +1. The critical value for the intended property is X = Xc at Y = 0. The three needed constants are calculated as: a2 ¼ ln
100 ; X min X min X c
X 2c þ 2X min X c ; X min X c X min c2 ¼ . X min X c b2 ¼
ð11Þ
ð12Þ ð13Þ
Fig. 2. A schematic representation of non-linear normalization functions for those properties for which a lower value is desirable.
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Similar to the previous case, the same procedure should be followed to obtain the determinate form of the scaled property function at X = 2Xmin 200X min 100. X The final relationships for this case are then ( Y ¼ a2 ln bX2 þ c2 for X c 6¼ 2X min ;
Y ¼
Y ¼ 200XX min 100
for X c ¼ 2X min .
ð14Þ
ð15Þ
Substituting the corresponding constants into Eqs. (10) and (15) would enable the designer to calculate the scaled properties and consequently the performance indices for the ranking of candidate materials.
3. Verification of selection methods The existing WPM and the proposed method can be applied to some known examples in mechanical design and the results could be compared with each other. Accordingly, two case studies: (i) the cryogenic storage tank for transportation of liquid nitrogen and (ii) the spar for the wing structure of a Human-Powered Aircraft (HPA) are considered. 3.1. Cryogenic storage tank for transportation of liquid nitrogen As a first step, the performance requirements of the storage tank should be translated into material requirements. In addition to having good weldability and processability, lower density and specific heat, smaller thermal expansion coefficient and thermal conductivity, and adequate toughness at the operating temperature, the material should be sufficiently strong and stiff. With seven properties to evaluate, the total number of decisions would be 21. Based on the DL and the modified DL methods, different decisions are made, as shown in Tables 3 and 4. The weighting factors can thus be calculated using the relationships given in Tables 1 and 2. The resulting weighting factors are also given in Tables 3 and 4. As can be seen, toughness is given the highest weight followed by density. The least important properties are thermal conductivity, and specific heat; other properties are in between. The properties of a sample of the candidate materials are listed in Table 5. The next step in the weighted properties method is to scale the properties given in Table 5. For the present application, materials with higher mechanical properties are more desirable and highest values in toughness, yield strength, and YoungÕs modulus are considered as 100. Other values in Table 5 are rated in proportion. On the other hand, lower values of specific gravity, thermal conductivity, thermal expan-
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Table 3 Application of digital logic method to cryoneic tank problem [1] Goals
Number of possible decisions
Toughness Yield strength YoungÕs modulus Density Thermal expansion coefficient Thermal conductivity Specific heat
1
2
3
4
5
6
1 0
1
1
1
1
1
0
7
8
9
10
11
1 0
0
0
1
1
0
12
13
0 1
1 0
0
1 0
14
15
1
16
1 0
0
18
19
20
21
1
1
0
17
1
0
1 1 0
0
0
0
1
0
0
0 1
Positive decisions
Weighting factors
18 13 10 16 13 7 7
0.214 0.155 0.119 0.190 0.155 0.083 0.083
Table 4 Application of modified digital logic method to cryoneic tank problem Goals
Number of possible decisions 1
2
3
4
5
6
Toughness Yield strength YoungÕs modulus Density Thermal expansion coefficient Thermal conductivity Specific heat
3 1
3
3
3
3
3
1 1
7
8
9
10
11
3 1
1
2
3
3 1 3
3 1
12
2 1
1
14
3
15
16
3 1 1
1
17
18
19
20
21
3
3 1
1
13
3
3 3
1 1
3
1 1
1
2 2
Positive decisions
Weighting factors
Critical value (Xc)
6 3 1 5 4
0.28 0.14 0.05 0.24 0.19
150 400 90 5 16.5
1 1
0.05 0.05
0.10 0.13
Table 5 Properties of candidate materials for cryogenic tank [1] Materials
1 Toughness indexa
2 Yield strength (MPa)
3 YoungÕs modulus (GPa)
4 Density g/cm3
5 Thermal expansionb
6 Thermal conductivityc
7 Specific heatd
Al 2024-T6 Al 5052-O SS 301-FH SS 310-3AH Ti-6Al-4V Inconel 718 70Cu-30Zn
75.5 95 770 187 179 239 273
420 91 1365 1120 875 1190 200
74.2 70 189 210 112 217 112
2.80 2.68 7.90 7.90 4.43 8.51 8.53
21.4 22.1 16.9 14.4 9.4 11.5 19.9
0.370 0.330 0.040 0.030 0.016 0.310 0.290
0.16 0.16 0.08 0.08 0.09 0.07 0.06
a b c d
Toughness index,T, is based on UTS, yield strength YS, and ductility e, at 196 °C. T = (UTS + YS)e/2. Thermal expansion coefficient is given in 106/°C. The values are averaged between RT and 196 °C. Thermal conductivity is given in cal/cm2/cm/°C/s. Specific heat is given in cal/g/°C. The values are averaged between RT and 196°C.
sion coefficient, and specific heat are more desirable for this application. Accordingly, the lowest values in the table were considered as 100 and other values rated in proportion according to Eq. (2). The performance indices, which are calculated according to Eq. (3) and ranking of candidate materials, are given in Table 6. A similar procedure can be adopted for the proposed method using Eqs. (10), (15), and the Xc values specified by the designer. The calculated scaled properties together with the weighting factors given in Table 4 are used in Eq. (3) to obtain the performance indices, as
shown in Table 6. It is observed that the WPM predicts that stainless steel 301 would have the best performance followed by titanium alloy, Inconel 718, stainless steel 310, aluminium alloys and brass. The same ranking is predicted by the new method for the first five choices. However, the performance indices for the last three choices are negative according to the new method. This implies that the proposed approach expels the possibility of selecting both aluminium alloys and brass for this application while, the old method ranks them as possible choices at the end of the list.
B. Dehghan-Manshadi et al. / Materials and Design 28 (2007) 8–15
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Table 6 Performance index and ranking of candidate materials according to two methods Materials
WPM
New method
Performance index (c)
Rank
Performance index (c)
Rank
Al 2024–T6 Al 5052–O SS 301–FH SS 310–3AH Ti–6Al–4V Inconel 718 70 Cu–30 Zn
42.2 40.1 70.9 50.0 59.8 53.3 35.9
5 6 1 4 2 3 7
1.17 8.75 47.40 31.88 43.52 33.44 3.07
5 7 1 4 2 3 6
3.2. Spar for the wing structure of a Human-Powered Aircraft (HPA)
epoxy in the fifth place, Ti alloys and S-glass–70% epoxy continuous fibers in between. The main difference between two selection methods is that in the old approach Balsa wood stands in the sixth place, being above Al 7075–T6 and Al 2024–T4 which are well known aerospace materials, while the new method suggests that Balsa with a negative performance index stands one from the bottom of the list and cannot be considered as a possible choice for the spar of an aircraft. This is a reasonable and obvious result that is not observed by the old WPM. The cause of such a misleading prediction is that the WPM which is based on a linear scaling method, overestimates the contribution of the low density of Balsa wood to its overall performance. The proposed method, however, uses a non-linear function which shows saturation in scaled property for all conditions. This reduces the effect of a specifically low or high property value in the total performance evaluation. Furthermore, the possibility of having negative scaled properties such as young modulus and strength of Balsa levels off the pronounced effect of its low density.
Spar is the major element of the wing in any aircraft. This element that acts as a beam is extended across the wing of aircraft and is joined to fuselage. This part tolerates all of the aerodynamic and static loads applied directly and indirectly on the wing. Due to the limited allowed weight of the structures in ultra light aircrafts, particularly Human-Powered Aircrafts (HPA) material selection and design of their parts are of considerable importance [8]. The main properties for materials used in spar structure are YoungÕs modulus, tensile strength, compressive yield strength and especially the specific gravity. Here, there are six properties and thus, 15 possible decisions for which all weighting factors are calculated according to both methods, as shown in Tables 7 and 8. In this case study, specific gravity is the main property with the highest weight followed by young modulus and tensile strength. The next three properties have the same importance and weighting factors of 0.070 and 0.116 in the old and new methods, respectively. The properties of a group of the candidate materials are listed in Table 9. After scaling the properties and considering the weighting factors, the performance indices are calculated for both methods and shown in Table 10. It can be inferred from this table that the ranking of the first five choices are the same in both selection methods. Carbon-63% epoxy stands first and E-glass–73%
4. Summary and conclusions The new method proposed in this paper has some significant advantages over the previous WPM. The MDL, in comparison with the existing DL method, does not have the elimination problem of the least important criterion. Using non-linear functions for normalizing the
Table 7 Application of digital logic method to an HPA spar problem Goals
Price Tensile strength YoungÕs modulus Density Compressive strength Creep resistance
Number of possible decisions 1
2
3
4
5
0 1
0
0
1
0
1 1
6
7
8
9
0 1
0
1
1 0 1
1 0
0 1
10
11
1
12
14
1 0 0
1 0
Positive decisions
Weighting factors
1 3 4 5 1 1
0.07 0.20 0.27 0.33 0.07 0.07
15
1
0 0
13
1 0
14
B. Dehghan-Manshadi et al. / Materials and Design 28 (2007) 8–15
Table 8 Application of modified digital logic method to an HPA spar problem Goals
Number of possible decisions 1
2
3
4
5
Price Tensile strength YoungÕs modulus Density Compressive strength Creep resistance
1 3
1
1
2
2
3 3
6
7
8
9
1 3
1
3
3 1 3
3 2
10
1 2
11
3
12
14
3 1 1
3 1
Weighting factors
Critical value (Xc)
7 11 13 15 7 7
0.116 0.183 0.216 0.250 0.116 0.116
20 1000 60 3 40 2
15
3
1 1
13
Positive decisions
2 2
Table 9 Properties of candidate materials for an HPA spar Materials
1 Price
2 Tensile strength (MPa)
3 YoungÕs modulus (GPa)
4 Density gr/cm3
5 Compressive strength (MPa)
6 Creep Resistance (25 °C)
Al 7075-T6 Al 2024-T4 Ti-6Al-4V Ti-2Fe-3Al-10V E-glass 73%-Epoxy E-glass 56%-Epoxy E-glass 65%-Polyester S-glass 70%-Epoxy Continuous Fibers S-glass 70%-Epoxy Fabric Carbon 63%-Epoxy Aramid 62%-Epoxy Balsa
3.5 3.5 21 22 2.6 2.5 2.5 9 8 45 20 6
581 425 1008 1295 1642 1028 340 2100 680 1725 1311 28.5
70.0 72.5 112.0 120.0 55.9 42.8 19.6 62.3 22.0 158.7 82.7 7.0
2.6 2.6 4.4 4.5 2.17 1.97 1.8 2.11 2.11 1.61 1.38 0.22
581 425 1008 1295 410 290 90 550 180 900 300 17.5
Good Good Excellent Excellent Average Weak Weak Average Average Average Average Average
Table 10 Performance index and ranking of candidate materials according to two methods Materials
WPM Performance index (c)
Rank
Performance index (c)
Rank
Al 7075-T6 Al 2024-T4 Al-4V-Ti Ti-2Fa-3Al-10V E-glass 73%-Epoxy E-glass 56%-Epoxy E-glass 65%-Polyester S-glass 70%-Epoxy Continuous fibers S-glass 70%-Epoxy Fabric Carbon 63%-Epoxy Aramid 62%-Epoxy Balsa
33.62 31.73 43.45 49.13 40.19 31.08 19.82 42.43 20.32 70.56 38.10 40.53
8 9 3 2 5 10 12 4 11 1 7 6
30.48 27.28 38.81 44.49 34.07 17.42 3.73 35.32 2.76 46.72 27.57 0.85
6 8 3 2 5 9 12 4 10 1 7 11
data, helps to achieve more reasonable results by not emphasizing on any of the high and low extremes. The use of a critical value in the proposed method plays a very vital role since it enables the designer to influence the decision making process. In fact, this change is toward making the procedure an intelligent one by significantly enhancing the human role in the selection process. The possibility of obtaining a negative performance index in the new method implies that the material
New method
is eliminated from the list since it has not been a suitable choice for the specific application. Finally, the applicability of the new method in mechanical design has been verified by two case studies. For the cryogenic tank, the new results are matching the acceptable answers provided by the existing method while, for the wing spar of the HPA, the new method offers much more reasonable solutions as compared to the questionable results obtained from the old method.
B. Dehghan-Manshadi et al. / Materials and Design 28 (2007) 8–15
Acknowledgment The authors thank Mr. Hesam Mahmudi for his helpful suggestions.
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