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An algorithm for optimal material choice for multifunctional criteria
716
JAWALKER K. SRIDHAR RAO, P. N. MURTHY, C. V. S. KAMESWARA RAO and V. K. KAPUR
fabrication processes, that are to be chosen so as to satisfy a given set of design requirements and at the same time optimize an objective function. In the second approach only the available materials with known constitutent material properties are considered, while the corresponding configurations meeting the design requirements are found. Consequently a spectrum of optimum designs are available corresponding to the spectrum of materials used as input. Finally the best suited material combination is the one that minimizes an objective function such as weight; in either of the approaches only one objective function is optimized while the rest of the requirements are considered as constraints. The work of Schmit and Kicher was one of the early applications of material-- structural synthesis in a programming formulation for minimum weight. It should be remarked that a material-structural optimization problem cannot be always cast as a mathematical programming problem by considering one of the requirements as an objective function and the rest as side or behaviour constraints. This is essentially because (1) does not have materials that possess the ideal multif~ctional properties (obtained as solution of the programming problem formulation), (2) the relative importance of various design requirements cannot be reflected in such a design process. Furthermore, in situations where the designer seeks a trade-off between various design criteria, the general problem is more complex. In an attempt to formulate and solve the optimum structural design problem in its generality as discussed earlier, Krokosky [3] devised a method that accommodates the designer’s preference of the design requirements, All the various design requirements like structural, thermal and acoustic are quantitatively specified in terms of the representative quantities called system variables. Further, in addition to the most desirable and least desirable values of system variables, the different levels of desirability of various combinations of the values of system variables are reflected in the form of what is called a ranking matrix depending on the designers preference (subjective/previous judgement). Krokosky adopts a random search technique in evaluating the material with the best performance. Also in order to accommodate various material properties in a computer, Krokosky correlates various material properties in terms of one of the a pt-iori chosen design parameters like density. The merits and demerits of approach of Krokosky are discussed in detail by the authors elsewhere 171. It is not always possible to get all the requirements (such as strength, thermal resistance, sound transmission loss) in terms of one quantity like density. Also, in terms of computation it is time-consuming, considering the subjective level of the ranking matrix which is the basis for material choice. The procedure presented herein attempts to obliviate some of the above shortcomings by using a utility theory framework.
DESIGN PROCEDURE
FOR OPTIMAL
MATERWL
CHOICE
In the present paper an altogether different design procedure is adopted as compared to any of the existing design procedures. The aim is to select an optimally suitable material out of a given lot of available materials to meet a given set of design requirements. The optimality is essentially reflected by maximum utility criterion. The utilities of various design solutions according to designer’s opinion are reflected by quantitatively specifying the utility measures with each of the associated system variables. A quantitative method of arriving at such specification is possible by employing suitable
An Algorithmfor Optimal Material Choice for MultifunctionalCriteria
717
cost effectivenessmodel lie
for example those of E&&e I$] and the generalization due to Murthy f9, IO]. Murthy’s model may be written in the gener&zed form as foifows
where U’ is the total utility of the material-structural configuration system for given design conditions and (S), is the ith system variable denoting requirements and X, is an exponent characterizing the effect of variations of the system variable on utility. In general, S, may be interrelated to S,. Trade-off between various system variables can be effected by considering their effect on the utifities_ However, in many problems it may be easier to consider the utilities with respect to each system variable independently and by giving suitable weightages (independent of material choice) to reflect the relative importance of the various design conditions. The possible candidate materials with properties (relevant to the design requirements) are listed out. The field equations relating the material properties to the system variables are sufficient to evaluate ~ema~rials~ performance withrespect to each of the requirements i.e. the ~~es~ndi~g system variabIe_ These are related to utilities with respect to each system variable using Utility Theory [l I]. In this approach instead of taking subj~ti~ty at an overal level, it is considered at elemental ievel, for effectively reducing inconsistencies. The utility value corresponding to a value of system variable of a material can be readily evaluated by linear interpolation from the tabulated values of utilities specified by the designer, initially either as a continuous function or as discrete values. Such interpolated value of utility may be denoted by Dij where i denotes the material andj denotes the system variable or requirement with reference to which the material’s utility is evaluated. The utilities of various materials ~o~espondin~ to various requirements can thus be evaluated, Now with the information, to choose the material that yidds maximum utility, weightages Wj are necessary that denote the relative importances of each of the design requirements ar attributes which are independent of the choice of a particular material. The range of values of Wj can be brought so as to make the sum CW,=l,
(2)
The total utility of any material is the sum of the weighted utilities (with respect to alf the requirements and for ith materid is given by O,dD,Wjs
(3)
Knowing the vector E7’,(i.e. is for all the materials) the best material is the one that has a maximum utifity given by if * u*= M$ax{&) *
(41
The method can be further generalized to fit into probabilistic framework in which case the uncertainty in quantifying the utility measure or the realization of a value D,,, may be specified with an associated value of probability pil, so that U, takes the form
718
JAWALKER K. SRIDHARRAO, P. N. MURTHY, C. V. S. KAMFSWARARAOand V. K. KAPI~I~
where ~ij=PijDij, i, j not summed. i.e. pij indicates the probability that the utility is Dij and Dij is the expected utility. However in the following example, the situation is assumed to be deterministic and as such values Of pij are taken to be unity. For further refinement, a sensitivity analysis by perturbing individual weightage factors i.e. different combinations of Wj satisfying equation (2) is done to examine the stability of the solution. ALGORITHM Figure 1 shows the chart of the above procedure. In this I denotes the number of materials and m the number of design requirements. The inputs are utility vectors Uj associated with each of the design requirements. The number of elements in each of the vector Ui are kk in number (for example 5 in the design of tension panel that follows). The matrix [PM] is used to denote the matrix containing the properties of various materials. To descriminate the values of the system variables obtained by evaluating each material,
I : Number of materials considered
II
q
Number of properties of each materlal problem
l-----l SubroutIne
Comparison and lnterpolotion from 6; .) Compute [ D,jl i-1 )I ,~=I,rn
FIG.
m = Number of system variables
kk = 2vfnw
of utility
1. Flow chart for optimal material choice.
An Algorithmfor OptimalMaterialChoicefor M~tif~~o~l
Criteria
719
from these specified a priori by the designer, the term SV is used to denote the former, and the utility levels of these are found by linear interpolation using the vector U,, S,. The rest of the steps follow closely the development given in the preceeding part of the paper. APPLICATION
OF THE DESIGN METHOD
The optimal material choice for the design of a Tension Panel, limited by minimum thickness specification of fabrication. To illustrate the method, the multifunctional design of a structural panel 1 ft square and O-1 in. thick may be considered. The thickness is specified in the present example based on fabrication limitation which at the same time satisfies the stress restriction, could be a design variable in a more general problem. The panel is required to withstand a total tension load of 20 x lo3 lb in one direction while the different design requirements for structural, thermal and accoustical are expressed Table 1 with their associated utilities. In total, ten candidate materials (f=lO) are considered and their properties are given in Table 2 fofor =7 attributes. The designer gives a weightage of 0*3 for weight, O-2 for cost, O*l each r mthe remaining five attributes. The field equations relating the material properties to system variables are as follows (4, 5, 6 12) P 1 Panel elongation = (7) (0.1 x 12) xz Panel weightlin2 = tp.
(8)
Thermal resistance = t/k.
(9)
Volumetric specific heat = KpC . Critical frequency (fJ = f Sound tr~smission
T z J
WI
01)
loss=10 log[l’~~~].
where P is the applied load=20 x IO3 lb; t is the thickness=O* in. ; p is the unit weight of the material of the panel, K is the thermal conductivity; C is the specific heat; c is velocity of sound in air 1090 ft/sec, E is Young’s modulus; Z is the cross sectional-moment of inertia of the panel,& is the critical frequency, p. is the density of air (1.29 kg/m3) and M is the mass of the panel. The materials are designated by a number that co~esponds to the serial number in the Table 2. Using the field equations (7-12) relating the material properties to system variables the value of the system variable corresponding to each material is evaluated and the results for this problem are given in Table 3. It should be noted that this need not be required to be printed out or stored in memory in a general problem while using the computer. Taking each element of the Table 3 (say weight) the corresponding utility measure is obtained by linear interpolation from Table 1. All such values of utilities are given in the form of a matrix D, (see Table 4). The weightages W, reflecting the relative importance of the various r~~~rnents are indicated in Table 4 and the total utility of each material evaluated as explained earlier is given in the last column of Table 4. It can at once be noticed that material 1 i.e. Aluminium alloy is the optimal material. Table 4 is the decision table for the given problem.
-2
-3
-4
o-02
0.03
0.05
0.06
-2
-3
-4
-5 -5 0.03
0.02
-4
1.0
5.0
O*Of
-3
0.4
0 om5
-1
U%ty
-2
o-2
0
(;S)
Elongation (in.)
s3
2
-4
0
4
-3
-5
6
10
-2
-1
Thermal resistance8 TZlity (l/10-4)
s4
-5
-4
-3
-2
-1
US Utility
0
1000
2ooo
6ooo
1OooO
5s Volumetric specific heat:
1. Utility measures with respect to system variablest
-5
-4
-3
-2
-i
u6 Utility
0
0
100
200
300
$6 Critical frequency CPS
U7
-5
-4
-3
-2
-1
Utility
0
60
80
100
120
S? Transmission loss
(Decebels)
t A utility of value --I corresponds to the most desirable design situation while the value -5 corresponds to barely acceptable situation Number of utility levels . . . 5 Number of system variables . . .7 $British units, %l/hr/ft*/of IBritish units, Btu/hr[sq’/of/ft/lb/ft3/Btu/lb/of.
-5
-1
0
-1
uzit,
Qb;zm$
cost
Weight
ut Utility
s2
Sl
TABLE
Titanium alloy zirconiuulalloy Tinahoy Ultra high strength steel Low expansion nickel alloy Nickel ahoy Tin bronze Chromium copper Cobalt ahoy
Ahlminiumalloy
Material 2.82 4.73 664 7.30 7-83 8.19 8.4 8.86 8.86 9.13 zz 355 130 280 360
:Iz 80
106
Young’s modulus E Psi 105
Cost per in* of panel units/in*
o-0510 O-684 2.88 O-l064 O-0283 O-296 o-315 0535 O-256 0.495
Weight, lbs per sq. in. of panel
0*0102 0*0171 0-024 0.0264 0.0283 O-0296 o-0315 0.032 O-032 o-033 l-89 1.05 l-43 25 O-66 0.83 0.56 1.54 0,715 0555
Elongation in. x10*
35:; 0.45 4.69
O-62 856 8-75 2.27 5.05 2.33
Thermal resistance t/k British units x 104
The various quantities in this table constitute the matrk SV in the flow chart.
Material
1;; 18
135 9.8 39.6 37 16.6 36.0 15
Thermal conductivity K Btu/hr/sql/oF/ft
5450 375 277 840 975 1830 1390 1390 10000 1225
Volumetric specific heat British units
1;
: 120 16 1 10 10 16.7
Unit cost unit per lb
TABW!3. Evaluation of materials fsvstem variables)
Number of materials considered 1=9 Number of material properties relevant to problem 11=6. The matrix PM denotes the material property matrix in the flow chart.
2 3 4 5 6 7 8 9 10
1
Sk No,
Unit weighty lb/f0 OXi-
TABOO 2, Materials considered and their properties [161
147 142 196 272 145 156 139 236 160 143
Critical frequency cps
88 170
O-23 0.13 o-07 O-05 o-12 0.123 o-11 0.09 0.10 o-12
1: 106 101 102 102 106 102 102
97
Transmission loss Decebels
: 284 40 205 26 18 165
ksi
Max. stress allowable
C
Specislc heat Btu/lb/d?
-1.25 -3.45 -4.45 -3.06 -1.14 -2.47 -2.51 -3.56 -2.23 -3.50
0.2
0.3
-1.5 -1.85 -2.4 -264 -2.83 -2.95 -3.03 -3.2 -3.2 -3.3
cost
Weight
-4.69
-1.30 -1.34 -3.89 -2.48 -3-84 -2.2 -3.5 -4.18 -3.68
-3.05 -3.43 -4.3 -2.3 -2.6 -2.16 -3.54 -2.42 -2.15
0.1
-3.89
0.1
Elongation
Thermal resistance
-2.12 -4.63 -4.7 -4.16 -4.08 -2.95 -4.24 -3.6 -1.0 -3.71
0.1
Volumetric specific heat
Materials utility matrix (decision table)
t Optimal material (No. 1 in Table 2 i.e. Aluminium alloy).
!Y 10
Material 1 2 3 4 5 6 7
System variable Design weightages Wf
TABLE 4
-2.53 -2.58 -2+4 -1.28 -2.55 -244 -2.16 -1.74 -2.4 -257
0.1
Critical frequency
-2.4 -2.2 -2-o -1.7 -1.95 -1.9 -1.9 -1.7 -1.9 -1.9
0.1
Transmission loss
-2.627 -2.961 -2.955 -2.393 -2.752 -2.722 -3.080 - 2.658 -3.097
-2.2631
Total material utility
An Algorithm for Optima1 Material Choice for Multifunctional Criteria
723
In the above problem, the minimum thickness governed the design. However, the thickness could be increased for each material so that the maximum expected utility for all the material-thickness combinations can be found thereby getting the optimal material -thickness combination for the multifunctional performance. In a general problem of this kind, for usual loading conditions, the minimum thickness is governed by prescribing maximum allowable stresses. So each material would be designed for the full stress condition. Since minimum thickness of a particular material need not ensure optimality for multifunctional criteria, each material is to be worked for various levels of stress within maximum permissible stresses, say, 95, 90, 85, 80, 75 per cent etc. (closer intervals may be taken if necessary). By this procedure, material-thickness combinations are examined. This ensures consideration of trade-offs between structural, thermal, acoustic, and economic performance. The procedure is repeated. By means of maximum expected utility criteria with requisite sensitivity analysis (for combinations of Wj, and several combinations for p,& the optimum material-desi~ parameters ( in this case thickness) is found. A slightly allied problem is that of selecting different materials where each material may have a band of performances depending upon composition, manufacturing process which can be varied. These may be composite materials which can be “designed” for different strength, thermal, sound, performances by adopting suitable technologies. For this, a cost-effectiveness models with studies of trade-offs between various parameters that aflect utilities (equation I) is important to obtain an optimum composite (muItiphase) material. These belong to the category of tailor-made materials. Fi~r-reinfor~d plastics for aerospace industry is an example of such potentialities in material design.
DISCUSSION
AND CONCLUSIONS
A gernal design problem can hardly be cast into the form of a mathematical programming form. This is essentially because of the conflicting design requirements and the varying stabilities of variable materials. No single criterion like minimum weight is adequate to judge the quality of a design. Essentially the problem is one of multifactor optimization with associated measures of trade off between various conflicting design criteria. The attempts made by other workers like Hill [13] and NASA Committee [14] essentially attempt to invoke utility theory application on a global situation, while it is known that such utility measures are better employed effectively at a subsystem level [I 1, 151. Also the use of random search techniques and fitting curves relating various design requirements to one material property are not physi~lly realistic and can lead to answers far from realistic optimal solution. The problem of optimal material choice is essentially discrete in nature, since the available materials and properties are such. It is known from mathematical programming theory that the optimality of a solution obtained by solving an originally discrete programming problem by an equivalent continuous one and rounding off the solution to the nearest integers, can be far removed from optimality. The present design methodology and algorithm is essentially to treat the problem without sacrificing its significance. The methodology is rational in the sense, that objective nature of the problem is separated from the subjective ones which are treated at the most elemental levels in a consistent manner. The evaluation of a materials, performance by its utility measure is a systematization of the check list method and the decision process, Extensive information or prior knowledge is assumed to be available in fixing the utilities
724
JAWAKERK. SRIDHARRAO, P. N. MURT~IY,C. V. S. KAMESWARARAO and V. K. KAPUR
and the weightages rationally. It should, however, be remarked that the present contribution is one of exploratory nature and the potentialities and its ramifications should be examined by applying to more complicated design situations. The development of applications of utility theory for each situations involving models of cost-effectiveness assumes priority. The development of Bayesian statistical decision theory coupled with integer (discrete) nonlinear programming is helpful for material-configuration-design synthesis for optimal structural design. REFERENCES [l] P. N. MURTHYand J. K. SRIDHARRAO, Some considerations in structural design processes. Presented at 20th Annual General Meeting, Aero. Sot. India, Bangalore (1968). [2] C. C. CHAMIS,Closing materials research design cycle. J. Engng Mech. Div., ASCE 95, 1255-1268 (1969). [3] L. A. SCHMITand T. P. KICHER, Synthesis of materials and configuration selection. J. Struct. Div., ASCE 88,79-102 (1962). [4] E. M. KROK~~KY,The ideal multifunctional constructural material. J. Struct. Div., ASCE 94,959-98 L (1968). [5] C. P. SMOLENSKIand E. W. KORKOSKY,Optimal multifactor design procedure for sandwich panels. J. Struct. Div., ASCE 96, 823-837 (1970). [6] E. M. KROKOSKY,Optimal multifunctional material systems. J. Engng Mech. Div., ASCE 97 (1971). [7] J. K. SRIDHARRAO et al., Discussion of Paper 5 above, Forthcoming. [8] R. E. BLAKE, Predicting structural reliability for design decisions. J. Spacecraft Rocket 4, 392-398 (1967). [9] P. N. MURTHY, Design for cost effectiveness. In 2. Lecture Notes, Intensive Course on Optimization in Structural Design, ZIT, Kanpur (1969). Lecture notes, Intensive Course on [IO] P. N. MURTHY, Decision analysis in engineering problems. Probabilistic Methods in Engineering, ZZT, Kanpur (1972). [ll] P. C. FISHBURN,Decision and Value Theory. Wiley, New York (1964). [12] C. 0. MACKAYand L. T. WRIGHT, Periodic heat flow in composite walls or roofs. Trans. ASHVE 52,283 (1946). [13] P. H. HILL, The Science of Engineering Design. Holt, Reinhard, 132 (1970). 1141 Weighted index method of material selection, NASA Committee on Materials Research for Supersonic Transports (referred to in [lo]). [15] N. G. NAIR, Ph.D. Thesis, ZZTKanpur, India, Chap. 3 (1969). [16] Materials in Design Engineering, Materials Selection Issue (1970). [17] Materials in Design Engineering, Materials Selector Issue. 62 (5), 121 (1965). (Received 24 February 1972)
JAWALKER K. SRIDHAR RAO, P. N. MURTHY, C. V. S. KAMESWARA RAO and V. K. KAPUR
fabrication processes, that are to be chosen so as to satisfy a given set of design requirements and at the same time optimize an objective function. In the second approach only the available materials with known constitutent material properties are considered, while the corresponding configurations meeting the design requirements are found. Consequently a spectrum of optimum designs are available corresponding to the spectrum of materials used as input. Finally the best suited material combination is the one that minimizes an objective function such as weight; in either of the approaches only one objective function is optimized while the rest of the requirements are considered as constraints. The work of Schmit and Kicher was one of the early applications of material-- structural synthesis in a programming formulation for minimum weight. It should be remarked that a material-structural optimization problem cannot be always cast as a mathematical programming problem by considering one of the requirements as an objective function and the rest as side or behaviour constraints. This is essentially because (1) does not have materials that possess the ideal multif~ctional properties (obtained as solution of the programming problem formulation), (2) the relative importance of various design requirements cannot be reflected in such a design process. Furthermore, in situations where the designer seeks a trade-off between various design criteria, the general problem is more complex. In an attempt to formulate and solve the optimum structural design problem in its generality as discussed earlier, Krokosky [3] devised a method that accommodates the designer’s preference of the design requirements, All the various design requirements like structural, thermal and acoustic are quantitatively specified in terms of the representative quantities called system variables. Further, in addition to the most desirable and least desirable values of system variables, the different levels of desirability of various combinations of the values of system variables are reflected in the form of what is called a ranking matrix depending on the designers preference (subjective/previous judgement). Krokosky adopts a random search technique in evaluating the material with the best performance. Also in order to accommodate various material properties in a computer, Krokosky correlates various material properties in terms of one of the a pt-iori chosen design parameters like density. The merits and demerits of approach of Krokosky are discussed in detail by the authors elsewhere 171. It is not always possible to get all the requirements (such as strength, thermal resistance, sound transmission loss) in terms of one quantity like density. Also, in terms of computation it is time-consuming, considering the subjective level of the ranking matrix which is the basis for material choice. The procedure presented herein attempts to obliviate some of the above shortcomings by using a utility theory framework.
DESIGN PROCEDURE
FOR OPTIMAL
MATERWL
CHOICE
In the present paper an altogether different design procedure is adopted as compared to any of the existing design procedures. The aim is to select an optimally suitable material out of a given lot of available materials to meet a given set of design requirements. The optimality is essentially reflected by maximum utility criterion. The utilities of various design solutions according to designer’s opinion are reflected by quantitatively specifying the utility measures with each of the associated system variables. A quantitative method of arriving at such specification is possible by employing suitable
An Algorithmfor Optimal Material Choice for MultifunctionalCriteria
717
cost effectivenessmodel lie
for example those of E&&e I$] and the generalization due to Murthy f9, IO]. Murthy’s model may be written in the gener&zed form as foifows
where U’ is the total utility of the material-structural configuration system for given design conditions and (S), is the ith system variable denoting requirements and X, is an exponent characterizing the effect of variations of the system variable on utility. In general, S, may be interrelated to S,. Trade-off between various system variables can be effected by considering their effect on the utifities_ However, in many problems it may be easier to consider the utilities with respect to each system variable independently and by giving suitable weightages (independent of material choice) to reflect the relative importance of the various design conditions. The possible candidate materials with properties (relevant to the design requirements) are listed out. The field equations relating the material properties to the system variables are sufficient to evaluate ~ema~rials~ performance withrespect to each of the requirements i.e. the ~~es~ndi~g system variabIe_ These are related to utilities with respect to each system variable using Utility Theory [l I]. In this approach instead of taking subj~ti~ty at an overal level, it is considered at elemental ievel, for effectively reducing inconsistencies. The utility value corresponding to a value of system variable of a material can be readily evaluated by linear interpolation from the tabulated values of utilities specified by the designer, initially either as a continuous function or as discrete values. Such interpolated value of utility may be denoted by Dij where i denotes the material andj denotes the system variable or requirement with reference to which the material’s utility is evaluated. The utilities of various materials ~o~espondin~ to various requirements can thus be evaluated, Now with the information, to choose the material that yidds maximum utility, weightages Wj are necessary that denote the relative importances of each of the design requirements ar attributes which are independent of the choice of a particular material. The range of values of Wj can be brought so as to make the sum CW,=l,
(2)
The total utility of any material is the sum of the weighted utilities (with respect to alf the requirements and for ith materid is given by O,dD,Wjs
(3)
Knowing the vector E7’,(i.e. is for all the materials) the best material is the one that has a maximum utifity given by if * u*= M$ax{&) *
(41
The method can be further generalized to fit into probabilistic framework in which case the uncertainty in quantifying the utility measure or the realization of a value D,,, may be specified with an associated value of probability pil, so that U, takes the form
718
JAWALKER K. SRIDHARRAO, P. N. MURTHY, C. V. S. KAMFSWARARAOand V. K. KAPI~I~
where ~ij=PijDij, i, j not summed. i.e. pij indicates the probability that the utility is Dij and Dij is the expected utility. However in the following example, the situation is assumed to be deterministic and as such values Of pij are taken to be unity. For further refinement, a sensitivity analysis by perturbing individual weightage factors i.e. different combinations of Wj satisfying equation (2) is done to examine the stability of the solution. ALGORITHM Figure 1 shows the chart of the above procedure. In this I denotes the number of materials and m the number of design requirements. The inputs are utility vectors Uj associated with each of the design requirements. The number of elements in each of the vector Ui are kk in number (for example 5 in the design of tension panel that follows). The matrix [PM] is used to denote the matrix containing the properties of various materials. To descriminate the values of the system variables obtained by evaluating each material,
I : Number of materials considered
II
q
Number of properties of each materlal problem
l-----l SubroutIne
Comparison and lnterpolotion from 6; .) Compute [ D,jl i-1 )I ,~=I,rn
FIG.
m = Number of system variables
kk = 2vfnw
of utility
1. Flow chart for optimal material choice.
An Algorithmfor OptimalMaterialChoicefor M~tif~~o~l
Criteria
719
from these specified a priori by the designer, the term SV is used to denote the former, and the utility levels of these are found by linear interpolation using the vector U,, S,. The rest of the steps follow closely the development given in the preceeding part of the paper. APPLICATION
OF THE DESIGN METHOD
The optimal material choice for the design of a Tension Panel, limited by minimum thickness specification of fabrication. To illustrate the method, the multifunctional design of a structural panel 1 ft square and O-1 in. thick may be considered. The thickness is specified in the present example based on fabrication limitation which at the same time satisfies the stress restriction, could be a design variable in a more general problem. The panel is required to withstand a total tension load of 20 x lo3 lb in one direction while the different design requirements for structural, thermal and accoustical are expressed Table 1 with their associated utilities. In total, ten candidate materials (f=lO) are considered and their properties are given in Table 2 fofor =7 attributes. The designer gives a weightage of 0*3 for weight, O-2 for cost, O*l each r mthe remaining five attributes. The field equations relating the material properties to system variables are as follows (4, 5, 6 12) P 1 Panel elongation = (7) (0.1 x 12) xz Panel weightlin2 = tp.
(8)
Thermal resistance = t/k.
(9)
Volumetric specific heat = KpC . Critical frequency (fJ = f Sound tr~smission
T z J
WI
01)
loss=10 log[l’~~~].
where P is the applied load=20 x IO3 lb; t is the thickness=O* in. ; p is the unit weight of the material of the panel, K is the thermal conductivity; C is the specific heat; c is velocity of sound in air 1090 ft/sec, E is Young’s modulus; Z is the cross sectional-moment of inertia of the panel,& is the critical frequency, p. is the density of air (1.29 kg/m3) and M is the mass of the panel. The materials are designated by a number that co~esponds to the serial number in the Table 2. Using the field equations (7-12) relating the material properties to system variables the value of the system variable corresponding to each material is evaluated and the results for this problem are given in Table 3. It should be noted that this need not be required to be printed out or stored in memory in a general problem while using the computer. Taking each element of the Table 3 (say weight) the corresponding utility measure is obtained by linear interpolation from Table 1. All such values of utilities are given in the form of a matrix D, (see Table 4). The weightages W, reflecting the relative importance of the various r~~~rnents are indicated in Table 4 and the total utility of each material evaluated as explained earlier is given in the last column of Table 4. It can at once be noticed that material 1 i.e. Aluminium alloy is the optimal material. Table 4 is the decision table for the given problem.
-2
-3
-4
o-02
0.03
0.05
0.06
-2
-3
-4
-5 -5 0.03
0.02
-4
1.0
5.0
O*Of
-3
0.4
0 om5
-1
U%ty
-2
o-2
0
(;S)
Elongation (in.)
s3
2
-4
0
4
-3
-5
6
10
-2
-1
Thermal resistance8 TZlity (l/10-4)
s4
-5
-4
-3
-2
-1
US Utility
0
1000
2ooo
6ooo
1OooO
5s Volumetric specific heat:
1. Utility measures with respect to system variablest
-5
-4
-3
-2
-i
u6 Utility
0
0
100
200
300
$6 Critical frequency CPS
U7
-5
-4
-3
-2
-1
Utility
0
60
80
100
120
S? Transmission loss
(Decebels)
t A utility of value --I corresponds to the most desirable design situation while the value -5 corresponds to barely acceptable situation Number of utility levels . . . 5 Number of system variables . . .7 $British units, %l/hr/ft*/of IBritish units, Btu/hr[sq’/of/ft/lb/ft3/Btu/lb/of.
-5
-1
0
-1
uzit,
Qb;zm$
cost
Weight
ut Utility
s2
Sl
TABLE
Titanium alloy zirconiuulalloy Tinahoy Ultra high strength steel Low expansion nickel alloy Nickel ahoy Tin bronze Chromium copper Cobalt ahoy
Ahlminiumalloy
Material 2.82 4.73 664 7.30 7-83 8.19 8.4 8.86 8.86 9.13 zz 355 130 280 360
:Iz 80
106
Young’s modulus E Psi 105
Cost per in* of panel units/in*
o-0510 O-684 2.88 O-l064 O-0283 O-296 o-315 0535 O-256 0.495
Weight, lbs per sq. in. of panel
0*0102 0*0171 0-024 0.0264 0.0283 O-0296 o-0315 0.032 O-032 o-033 l-89 1.05 l-43 25 O-66 0.83 0.56 1.54 0,715 0555
Elongation in. x10*
35:; 0.45 4.69
O-62 856 8-75 2.27 5.05 2.33
Thermal resistance t/k British units x 104
The various quantities in this table constitute the matrk SV in the flow chart.
Material
1;; 18
135 9.8 39.6 37 16.6 36.0 15
Thermal conductivity K Btu/hr/sql/oF/ft
5450 375 277 840 975 1830 1390 1390 10000 1225
Volumetric specific heat British units
1;
: 120 16 1 10 10 16.7
Unit cost unit per lb
TABW!3. Evaluation of materials fsvstem variables)
Number of materials considered 1=9 Number of material properties relevant to problem 11=6. The matrix PM denotes the material property matrix in the flow chart.
2 3 4 5 6 7 8 9 10
1
Sk No,
Unit weighty lb/f0 OXi-
TABOO 2, Materials considered and their properties [161
147 142 196 272 145 156 139 236 160 143
Critical frequency cps
88 170
O-23 0.13 o-07 O-05 o-12 0.123 o-11 0.09 0.10 o-12
1: 106 101 102 102 106 102 102
97
Transmission loss Decebels
: 284 40 205 26 18 165
ksi
Max. stress allowable
C
Specislc heat Btu/lb/d?
-1.25 -3.45 -4.45 -3.06 -1.14 -2.47 -2.51 -3.56 -2.23 -3.50
0.2
0.3
-1.5 -1.85 -2.4 -264 -2.83 -2.95 -3.03 -3.2 -3.2 -3.3
cost
Weight
-4.69
-1.30 -1.34 -3.89 -2.48 -3-84 -2.2 -3.5 -4.18 -3.68
-3.05 -3.43 -4.3 -2.3 -2.6 -2.16 -3.54 -2.42 -2.15
0.1
-3.89
0.1
Elongation
Thermal resistance
-2.12 -4.63 -4.7 -4.16 -4.08 -2.95 -4.24 -3.6 -1.0 -3.71
0.1
Volumetric specific heat
Materials utility matrix (decision table)
t Optimal material (No. 1 in Table 2 i.e. Aluminium alloy).
!Y 10
Material 1 2 3 4 5 6 7
System variable Design weightages Wf
TABLE 4
-2.53 -2.58 -2+4 -1.28 -2.55 -244 -2.16 -1.74 -2.4 -257
0.1
Critical frequency
-2.4 -2.2 -2-o -1.7 -1.95 -1.9 -1.9 -1.7 -1.9 -1.9
0.1
Transmission loss
-2.627 -2.961 -2.955 -2.393 -2.752 -2.722 -3.080 - 2.658 -3.097
-2.2631
Total material utility
An Algorithm for Optima1 Material Choice for Multifunctional Criteria
723
In the above problem, the minimum thickness governed the design. However, the thickness could be increased for each material so that the maximum expected utility for all the material-thickness combinations can be found thereby getting the optimal material -thickness combination for the multifunctional performance. In a general problem of this kind, for usual loading conditions, the minimum thickness is governed by prescribing maximum allowable stresses. So each material would be designed for the full stress condition. Since minimum thickness of a particular material need not ensure optimality for multifunctional criteria, each material is to be worked for various levels of stress within maximum permissible stresses, say, 95, 90, 85, 80, 75 per cent etc. (closer intervals may be taken if necessary). By this procedure, material-thickness combinations are examined. This ensures consideration of trade-offs between structural, thermal, acoustic, and economic performance. The procedure is repeated. By means of maximum expected utility criteria with requisite sensitivity analysis (for combinations of Wj, and several combinations for p,& the optimum material-desi~ parameters ( in this case thickness) is found. A slightly allied problem is that of selecting different materials where each material may have a band of performances depending upon composition, manufacturing process which can be varied. These may be composite materials which can be “designed” for different strength, thermal, sound, performances by adopting suitable technologies. For this, a cost-effectiveness models with studies of trade-offs between various parameters that aflect utilities (equation I) is important to obtain an optimum composite (muItiphase) material. These belong to the category of tailor-made materials. Fi~r-reinfor~d plastics for aerospace industry is an example of such potentialities in material design.
DISCUSSION
AND CONCLUSIONS
A gernal design problem can hardly be cast into the form of a mathematical programming form. This is essentially because of the conflicting design requirements and the varying stabilities of variable materials. No single criterion like minimum weight is adequate to judge the quality of a design. Essentially the problem is one of multifactor optimization with associated measures of trade off between various conflicting design criteria. The attempts made by other workers like Hill [13] and NASA Committee [14] essentially attempt to invoke utility theory application on a global situation, while it is known that such utility measures are better employed effectively at a subsystem level [I 1, 151. Also the use of random search techniques and fitting curves relating various design requirements to one material property are not physi~lly realistic and can lead to answers far from realistic optimal solution. The problem of optimal material choice is essentially discrete in nature, since the available materials and properties are such. It is known from mathematical programming theory that the optimality of a solution obtained by solving an originally discrete programming problem by an equivalent continuous one and rounding off the solution to the nearest integers, can be far removed from optimality. The present design methodology and algorithm is essentially to treat the problem without sacrificing its significance. The methodology is rational in the sense, that objective nature of the problem is separated from the subjective ones which are treated at the most elemental levels in a consistent manner. The evaluation of a materials, performance by its utility measure is a systematization of the check list method and the decision process, Extensive information or prior knowledge is assumed to be available in fixing the utilities
724
JAWAKERK. SRIDHARRAO, P. N. MURT~IY,C. V. S. KAMESWARARAO and V. K. KAPUR
and the weightages rationally. It should, however, be remarked that the present contribution is one of exploratory nature and the potentialities and its ramifications should be examined by applying to more complicated design situations. The development of applications of utility theory for each situations involving models of cost-effectiveness assumes priority. The development of Bayesian statistical decision theory coupled with integer (discrete) nonlinear programming is helpful for material-configuration-design synthesis for optimal structural design. REFERENCES [l] P. N. MURTHYand J. K. SRIDHARRAO, Some considerations in structural design processes. Presented at 20th Annual General Meeting, Aero. Sot. India, Bangalore (1968). [2] C. C. CHAMIS,Closing materials research design cycle. J. Engng Mech. Div., ASCE 95, 1255-1268 (1969). [3] L. A. SCHMITand T. P. KICHER, Synthesis of materials and configuration selection. J. Struct. Div., ASCE 88,79-102 (1962). [4] E. M. KROK~~KY,The ideal multifunctional constructural material. J. Struct. Div., ASCE 94,959-98 L (1968). [5] C. P. SMOLENSKIand E. W. KORKOSKY,Optimal multifactor design procedure for sandwich panels. J. Struct. Div., ASCE 96, 823-837 (1970). [6] E. M. KROKOSKY,Optimal multifunctional material systems. J. Engng Mech. Div., ASCE 97 (1971). [7] J. K. SRIDHARRAO et al., Discussion of Paper 5 above, Forthcoming. [8] R. E. BLAKE, Predicting structural reliability for design decisions. J. Spacecraft Rocket 4, 392-398 (1967). [9] P. N. MURTHY, Design for cost effectiveness. In 2. Lecture Notes, Intensive Course on Optimization in Structural Design, ZIT, Kanpur (1969). Lecture notes, Intensive Course on [IO] P. N. MURTHY, Decision analysis in engineering problems. Probabilistic Methods in Engineering, ZZT, Kanpur (1972). [ll] P. C. FISHBURN,Decision and Value Theory. Wiley, New York (1964). [12] C. 0. MACKAYand L. T. WRIGHT, Periodic heat flow in composite walls or roofs. Trans. ASHVE 52,283 (1946). [13] P. H. HILL, The Science of Engineering Design. Holt, Reinhard, 132 (1970). 1141 Weighted index method of material selection, NASA Committee on Materials Research for Supersonic Transports (referred to in [lo]). [15] N. G. NAIR, Ph.D. Thesis, ZZTKanpur, India, Chap. 3 (1969). [16] Materials in Design Engineering, Materials Selection Issue (1970). [17] Materials in Design Engineering, Materials Selector Issue. 62 (5), 121 (1965). (Received 24 February 1972)