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M o d e l for m a t e r i a l s o p t i m i z a t i o n

As a target function for optimization the material cost is considered. The material cost can include manufacturing costs in addition to the basic price of the semiproduct. The material cost is to be minimized at the same time as the material is selected and the geometry of the component is adopted to the chosen material. In Appendix A alternative target functions are analysed. Minimizing the material cost The material cost of a component made out of one material (such a component is referred to as a simple part) is given by Lk = Q Ck V(p)

(1)

where Q is the density, Ck is the material cost per unit weight (possibly including manufacturing costs) and V the material volume of the part. V is assumed to depend on a geometrical variable p; for example, the section thickness, the height or the width of the part. In this present paper only the case with one geometrical variable is considered. The generalization to several variables will be dealt with in a future paper. The purpose here is to minimize the material cost by selecting a material and simultaneously optimizing the sizing variable p. The design criteria are assumed to take the following form: goi (P) ~ D~ Pl~

i = i. . . . .

P ~ P2

no

(2) (3)

The function go1 Co) depends on the property D1 and the design case. Criteria for no properties are taken into account. In (3), limits Pl and P2 for the sizing variable are introduced. They play a central role in materials optimization. In fact, one of the most common mistakes is to ignore the so-called geometric condition in (3). To illustrate the form of (2), two design criteria which are frequently used in materials optimization are taken as an example. If an external load F is acting on a component there have to be lower limits on p to ensure that the elastic deflections are not too large and that plastic deformation or failure does not occur. The design criteria take the following form: F

&ma~ = -~ gE (P) ~ A2

(4)

om~` = F g o ~ ° c h a r

(5)

fs

The form of the design criteria implies that the maximum elastic displacement '~rn,,xdoes not exceed a value A2 in the presence of an external load, i.e. a minimum stiffness is specified and that the maximum stress O,n~xis less than the characteristic strength (~c,ar,e.g. yield strength of the material divided by a safety factor f,. gE and go are functions of p. If (4) and (5) are transformed to the form of (2) one finds that

0261-3069/92/030131-07 © 1992 Butterworth-Heinemann

Ltd

131

Merit parametersin materialsoptimization go1 (P) = "~?E (p)

D, = E;

(6) D2 = (Tctmr ;

gD2 (p ) = F f, g : ( p )

To obtain the material cost in closed form a Taylor expansion is performed for V(p) and go~ (P) around the initial value Po. log V(p) = log V(Po) + (Iogp - IogPo)V(Po) + . . . ( 7 )

(8) log gel (p) = log gD, (Po) -- (Iogp -- IOgPo) m/(Po) +-..

Merit parameters The material-dependent part of the minimum material cost Lkm~n has a special significance since it describes the influence of a change of material. The inverse of this part is referred to as the merit parameter which is used to rank the suitability of individual materials. It is important to give a precise definition of the merit parameter concept. For a design criterion for a property D, (cf. (2)) the corresponding merit parameter (2o is defined as the ratio between the optimum cost for a reference material and the material in question. Using (1) one finds that

~ o, =_

p " c." V(po7~)

(17)

pc. V(po,)

where

aiog v(p) v -

(9) o~iog p

m, = -

where the denominator refers to the reference material. PD is the minimum value of the sizing variable (m, =- 0) satisfying the design criterion (14).

po,= (A°~'~ "m'

a log gol (p)

\°,/

(10)

o~log p

(m, ~- 0)

(18)

In many cases an artificial reference material is chosen with all property values equal to unity (in this case no 'tilde' is used with the symbol for the merit parameter).

Introducing ADi = go, (p) pm

(11)

Bv = V(p) p-V

(12)

V (poi r~ ) QD =

(19)

Q c. v(po)

(1) and (2) can be rewritten as LK = PCk Bv p v

(13)

ADi P-% ~ Di

(14)

The minimum value of p is found by combining (3) and (14)

Using the first-order approximations in (7) and (8) the following merit parameters are obtained for the design criteria in (14)-(15):

EVlmE QE =

(20)

QCk vIr%

6rchar p,,,,, = max

Ir,,,>OI \

o,i

'

' ...

, Pl

(15)

Q, =

(21) OCk

1 Only terms where mi=~ 0 should be included in (15). Provided v > 0 the minimum material cost Lkmln is obtained by inserting (15) into (13). F/

Lkmin : P C k B v m a x

\v/rni

v/re,o

J {AD' I ..... ( D n D ~ ,p, v

\°oo)

(16)

The quantities Bv, v, Ao, mj are evaluated at the starting value, P0, of the sizing variable resulting in a first approximation Pm*n (1). Equation (15) can be iterated by evaluating ADj, mi for Pmin(1).Equation (15) usually converges rapidly and in many situations the first-order approximation is adequate. In fact, for many elementary cases the Taylor expansion terminates and (15) becomes exact. Each member in the square brackets in (15) and (16) correspondss to one design criterion. The most critical one of the design criteria gives the largest member and hence controls the value of Pm~nand Lkm~..These members and criteria are referred to as controlling. If the controlling criterion is of the type in (2) the corresponding property is also identified as controlling.

132

Qp, =

(22) Q Ck

Alternative target functions In the analysis in the section on minimizing the material cost it was assumed that the minimizing of the material cost was the target function of the optimization. Many other target functions can be considered. If the weight or the volume of the component is to be minimized, the previous analysis still applies except for the replacements c k --1

(weight optimization)

(23)

QCk --1

(volume optimization)

(24)

which should be made, i.e. the material cost per unit weight or unit volume should be replaced by unity. Unfortunately, these types of optimization rarely give results which are considered as meaningful because cost is entirely ignored [12]. For example, if stiffness is the controlling design criterion, (20) indicates that a material with maximum specific modulus (E/o) or elastic modulus (E) should be chosen. Many times engineering ceramics like SiC [12] are suggested in such a

MATERIALS& DESIGN Vol. 13 No. 3 1992

R Sandstr~m Table 1

Values o f the constant ~ for different l o a d cases

Load distribution

/

k

Type of support

,L

,l \ \ %

I.

\

J

1

1

1

11

6

15

8

60

1

5

1

48

76.9

384

76.9

1

1

1

1

107

164

185

210

1

1

1

1

192

382

384

382

process which, only in exceptional cases, are reasonable candidate materials. A better approach is to minimize the material cost taking the value of each kilo of saved material, c,, into account. In fact, also in this case, the previous analysis is valid provided c k is replaced by c k + c, (see Ref. 2) Ck--C k J~- Cw (optimization taking the value of weight

saving Cw into account)

(25)

In this way well-defined results are obtained. Another common case is the optimization of the performance. For the same reason as above such an optimization is rarely meaningful without cost restrictions. In Appendix A two cases are considered: namely, maximizing the load F and minimizing the elastic deflections. It turns out that the merit parameters are close to those for cost minimization. In fact, from (20), (21), (A5), (A6) and (A8) one finds that

QE

=

vlm~ --

~

~

FJ v/me ~"~.~E

Q, = Q v/mo Fa

(26) (27)

Assuming v, m E and mo to be positive, this implies that the materials are ranked in the same order whether the material cost is optimized at a given performance or the performance is optimized at a given cost. This is the main reason for primarily considering cost optimization since cost is a limiting factor in most material selection cases. MATERIALS& DESIGNVol. 13 No. 3 1992

Merit parameters for engineering beams To illustrate the theory derived in the previous section, it is applied to engineering beams. The loading of the many technical components can be represented by engineering beams. How to handle them in materials optimization is consequently of great practical value. Design criteria (3)-(5) are used. The basic equations are derived in Appendix B. In the equations, a number of design parameters are involved such as the geometry, magnitude and distribution of the external loading, type of support, safety factor and maximum permissible elastic deflection. The load distribution and type of support for the beam is taken into account through two parameters # and #, Some values of these parameters are given in Tables 1 and 2. Further values can be found in design handbooks, see e.g. [13]. Parameters which are needed in materials optimization are shown in Table 3 for beams with different cross-sections. The corresponding merit parameters are presented in Table 4. The quantity M is for the minimizing of the material cost in accordance with (20)-(22) given by M = Qc,

(28)

For other target functions (28) has to be replaced (cf. (23)(25)). From Table 4 it is evident that the merit parameters depend on the beam cross-section, sizing variable and controlling material property but their form is not influenced by the magnitude and distribution of the load, type of support, the safety factor, the maximum permissible deflection or geometrical parameters other than the sizing variable. This is characteristic of elementary load cases. Since three design criteria are involved there are three merit parameters in each 133

Medt parameters in materials optimization

Table 2

Values of the constant I~ for different load cases

Load distribution

r

Type of support

2 3

I

1

1

7.81

7.81

\ \

3

7

\

16

60

1

1

1

1

8

10

12

10

\ \ \ \ \

/i

J fb~ J

I'

Table 3(a) Beam cross-section

Moment of inertia

Materials selection parameters of beams

Sizing Given parameter parameter(s)

AE.,~ 2

FE 3 Solid rectangular

b h3112

b

Aa

av

~Ff~l

12

6

h3

h2

f

m

h

1

h

b

121b

61b

b

3

2

1

h

3,=blh

1213,

613,

3,

4

3

2

2bc~2

1

h

h2

Thin walled I -~ (btf+ or box shaped i h twl6)

tf

B= tJtf

2

1

h,b

b h 2 Q8

b h o~8

2 b 11c~6

b I /ct6

,f 2~2b

h tf boL6

b I Ict2

2h2'~4/ ~6

h % I ~6

t~ 2b o~zhV°~

h2tf c~6 b

h tfcleb

tf, tw, h

tw

h

h

134

h2tf b

~6

tf, tw, h

.y= blh

2

tf, t w

tf 3, (xs

1

lhx 8

1/o~2

2e./o~6

~3 /~6

1-1/ot2

3

2

h

2

trc~ 2

tf 3' ~6

MATERIALS & DESIGN VoI. 13 No. 3 1992

R SandstrSm Table 3(b)

Beam cross-section

Solid circular

@

Thin-walled circular

Moment of inertia .~-d4

I

Sizing variable

d

Materials selection parameters of beams (continued) A E" A 2

A

F/3

~Ff=l

64

32

64

BV

o

Given parameter

m

E

?r

~r d3t

8

4

8

7rt

7rt

d

d

8

4

~- d 3

7rd 2

8

4

7rt

1

7rd

1

7r 3,

~/ = tld

7r,y

Footnotes i = 0, 1, 2, (3) ..... number of webs for thin walled rectangularcross section; i = 0 (web neglected), i = 1 (I-shapedbeam), i =2 (box section) = 1 + i h tw/j b tf where j is a number. ength of beam: ~ and/~ constants (see Tables 1 and 2)

load case in Table 4. Although the form of each merit parameter is not influenced by many parameters the choice between the three merit parameters is strongly influenced by all design variables [2]. As pointed out already in the introduction the merit parameter for the geometric condition plays a central role in material optimization. For example, if the external load F is sufficiently low the geometric condition is controlling. When the lower limit values of the geometric parameters (marked with a subscript 1) are independent of the material the geometric parameter can be simplified to

Qpl = 1 / M

(29)

Qp~ is in this case the same for all load cases. Special cases of the non-geometrical merit parameters can be found in many texts, see e.g. [5]. In most contexts the merit parameters are assumed to increase when an improved material is considered. However, their inverse can also appear, in which case the merit parameter should be made as small as possible.

Conclusions (i)

A theory for materials optimization has been derived. The purpose of the theory is that it should be used in connection with quantitative materials selection. According to the model target function, the design case (e.g. load case) and geometry, including the parameters that can be varied (sizing variable), should be identified.

(ii) For the ranking of materials merit parameters are used. A precise definition of them is given. The mathematical form of the merit parameters depends on the choice of target function, design case and sizing variable but not on other design parameters such as geometry and load magnitude and distributions. MATERIALS& DESIGNVol. 13 No. 3 1992

(iii) Each design criterion is associated with one merit parameter. The central importance of the merit parameter for the geometric condition is emphasized.

(iv)

Merit parameters are derived for elementary load cases for engineering beams with different cross-sections.

Acknowledgements Financial support from Skan Aluminium, (Oslo) is gratefully acknowledged.

References 1 Moore, M A, Selection and use of materials for light design. IV Scand. Symp. Materials Science, Trondheim (1986) 237-253 2 Sandstrem, R. Systematicselection of materials in light weight design. IV Scand. Symp. Materials Science, Trondheim (1986) 255-268 3 Farag, M M; Materials and process selection in engineering, Appl. Sci. London (1979) 4 Budinski, K G. Engineering Materials; Properties and Selection. Prentice Hall, New Jersey (1989) 5 Charles, J A, Crane, F A A. Selection and Use of Engineering Materials. Butterworths, London (1989) 6 Sandstrem, R. Introductionto materialsselection.Kompendium. Dept. Materials Technology, Royal Institute of Technology, Stockholm (1989) (in Swedish) 7 Sandstrem,R. Pre-selection of aluminium alloys, Scand.AIConf., GSteborg (1987) (in Swedish) 8 Sandatrem, R. Material optimization. Kompendium. Dept. MaterialsTechnology, Royal institute of Technology, Stockholm (1984) (in Swedish) 9 Sandstrem, R. An approach to systematic materials selection. Materials and Design 6 (1985) 328-338 10 Plastics and Material Selection Handbook, MekanfSrbundet, Stockholm (1989) 11 Sandatrem, R. Control area diagram in materials optimization (1991) (to be published) 12 Sandstrem, R. New principles for systematicmaterial selection using merit parameters. Conf. Modern Design Principles, Tapir, Trondheim (1988) 55-74 13 Strength of Aluminium, Design Handbook, Alcan, Montreal (1973) 135

Merit parameters in materials optimization Table 4 Merit parameters for beams exposed to bending

Beam cross-section

Given

Design criteria Merit parameters

Sizing

parameter(s) variable Max stress ox

Rectangular

F,E, f,,

O"X =

#, $, /~2 (all cases)

Thin walled I or

box shaped

6#F[

~

°cha,

b h~

h

b

b

h

"y= blh

h

F,E, f,#,

Geometric

FE _ _3 ~12 A

h~h~

~=$

e6r

b~hl Qb~ = llblM

QE ElM

G 112/AA

EI~IM

char/~V,

0 2131AA

Q b = llh~M

Oblhl =llh M

Elra l M

chars Ivn

# Ff

ac"a'

h(bt, + ihtwl6 )

(all cases)

2

E b h3

f~

O'char/M

0"x

~, /t 2

Max elastic deflection A

2~ F f 3

A=

/max [7 h~2, b~l~] ~ 42

t~t.1; t~_~tf1 b~bl

E h2(btf + iht./6)

f,

h~hl b

',T"-[!

6 = t~t~

tf

7= bib

F, E, fs, #,

ox =

Qbihi = I l M

32# F /

IIV!

Ell31M 0"char

4 - -

A-

Ox =

d d

QE = e 12 I M

4# f f

Ochar

X d2

fs

A

~=F E 3 8 xd3tE

Qdl = 1 Id~ M

d >__d~

Z12

t ~t

1

Qt1 = lltlM

QE

Qo Och.rlM

EI M

d

112,.. o char1 nn

E l131M

d

213/ira O'char / / V l

E 1/3/M

,

, Y=t/d

136

d~d~

d~E

Qo = Ochar2/3 IM

Thin walled circular

/max (b. -(hi)

~ F f 3 64 ~ ~2

f,

lr d 3

(all cases)

©

/max(try, t. 1 16)

112~=# Gchar

t,,tw Solid circular

Qtf= 1 IM

QE E/M

u QG charlM

~h

Qdl = 11dl M Qdltl = IlM max

(t 12/% d 2~)

MATERIALS & DESIGN Vol. 13 No. 3 1992

R SandstrOm

APPENDIX A

The material dependent parts of this expression yield the merit parameters

Maximizing the performance QpE =

A.1. Load maximizing Consider a situation where design criteria (3)-(5) apply. The external load F should be maximized at a given maximum cost L 2. Using (16) this gives the following condition

I oCkBvmax

v,o]

VlmE (FA-E~

,

\ E% /

, Pl

~L2

(A1)

\ °c.,, /

AE = g E (P) P mE

(A2)

Ao = g ~ (P) pm~

(A3)

From (A1) the maximum value of F can be obtained

AE-

Fm~x = min

,

'

Gc.=

(A6)

QFa =

(OC,) m,lv A.2. Minimizing the elastic deflection

,~nin

L2

(A5)

Consider again a situation where design criteria (3)-(5) are used and a maximum cost is specified, i.e. (A1) applies. From the first member of (A1) the minimum value of the maximum allowed deflection is obtained

where

E~2

E (O,Ck) mEIv

Och"r

fsA~

L2

( QCkB~

=

mEIV

F "~E

(A7)

The corresponding merit parameter is (A4)

QAE -

E

(A8)

(Qc~ mEIV

APPENDIX B Bending of beams The beam is assumed to be exposed to a distributed bending load q(x) along its length perpendicular to its axis. The total load F is

/

F =

/q(x)

dx

(B1)

For a beam with a rectangular cross-section

bxh

I = bh3112

(B6)

c = hi2

(B7)

V = fbh

(B8)

O

where is the length of the beam. Using the engineering theory of beams the maximum elastic deflection A = , and stress ~x can be expressed as ,~,,~x = ~

FE 3

(B2)

El (B3)

where c is the maximum distance from the neutral line and I the moment of inertia of the beam cross-section. The constants $ and/~ depend on the distribution of the load and on how the beam is supported. Values for a few cases are given in Tables 1 and 2. Comparing (2), (4) and (5) to (B2) and (B3) one finds that

gE(p) = ~ I__~_l I ~ l

go(p) = /~

I~_t

MATERIALS & DESIGN Vol. 13 No. 3 1992

Ff,

(B4)

(B5)

Taking the height h as the sizing variable p, using (B6) to (B8) in (B4) and (B5) and inserting the resulting expressions into (A9)-(A14) yields

v = 1

(B9)

mE = 3; m~ = 2

(B10)

e v = fb

(Bll)

AE =

F &f3 121bA2

A~ = i~ f 6 F fslb

(B12) (B13)

In this case the constants v, mE, m a Bv, AE and A o are independent of the initial value of the sizing variable h. Values of these constants for other cross-sections and sizing variables are presented in Table 3. 137