Appendix E. Exercises

Transcription

1 Appendix E Exercises Appendix conens E.1 Inroducion o he exercises 558 E.2 Devising conceps 559 E.3 Use of maerial selecion chars 559 E.4 Translaion: consrains and objecives 562 E.5 Deriving and using maerial indices 565 E.6 Selecing processes 574 E.7 Muliple consrains and objecives 579 E.8 Selecing maerial and shape 587 E.9 Hybrid maerials 594

2 558 Appendix E Exercises E.1 Inroducion o he exercises The exercises are organized ino nine secions: E.1 inroducion o he exercises, E.2 devising conceps, E.3 use of maerials selecion chars, E.4 ranslaion: consrains and objecives, E.5 deriving and using maerial indices, E.6 selecing processes, E.7 muliple consrains and objecives, E.8 selecing maerial and shape, E.9 hybrid maerials. These exercises are designed o develop faciliy in selecing maerials, processes and shape, and in devising hybrid maerials when no monolihic maerial mees compleely he design requiremens. The early examples are very easy. Those ha follow lead he reader hrough he use of propery chars, ranslaion, he derivaion of indices, screening and ranking, muli-objecive opimizaion and choice of shape. Difficuly, when i arises, is no caused by mahemaical complexiy he mahs involved is simple hroughou; i arises from he need o hink clearly abou he consrains, he objecives and he free variables. Three imporan poins. 1. Selecion problems are open-ended and, generally, under-specified; here is seldom a single, correc answer. The proper answer is sensible ranslaion of he design requiremens ino maerial consrains and objecives, applied o give a shor-lis of poenial candidaes wih commenary suggesing wha supporing informaion would be needed o narrow he choice furher. 2. The posiioning of selecion-lines on chars is a maer of judgmen. The goal is o place he lines such ha hey leave an adequaely large shor lis of candidaes (aim for 4 or so), drawn, if possible, from more han one class of maerial. 3. A reques for a selecion based on one maerial index alone (such as M E 1/2 /) is correcly answered by lising he subse of maerials ha maximize his index. Bu a reques for a selecion of maerials for a componen a wing spar, for insance (which is a ligh, siff beam, for which he index is M ¼ E 1/2 /) requires more: some maerials wih high E 1/2 / such as silicon carbide, are unsuiable for obvious reasons. I is a poor answer ha ignores common sense and experience and fails o add furher consrains o incorporae hem. Sudens should be encouraged o discuss he implicaions of heir selecion and o sugges furher selecion sages. The bes way o use he chars ha are a feaure of he book is o make clean copies (or down-load hem from hp:// on which you can draw, ry ou alernaive selecion crieria, wrie commens, and so forh. Alhough he book iself is copyrighed, he reader is auhorized o make copies of he chars and o reproduce hese, wih proper reference o heir source, as he or she wishes.

3 E.3 Use of maerial selecion chars 559 All he maerials selecion problems can be solved using he CES sofware, which is paricularly effecive when muliple crieria and unusual indices are involved. E.2 Devising conceps These wo examples illusrae he way in which conceps are generae. The lef-hand par of each diagram describes a physical principle by which he need migh be me; he righ-hand par elaboraes, suggesing how he principle migh be used. Exercise E2.1 Conceps and embodimens for dus removers. We me he need for a device o remove household dus in Chaper 1, wih examples of esablished soluions. Now i is ime for more creaive hinking. Devise as many conceps o mee his need as you can. Nohing, a he concep sage, is oo far-feched; decisions abou pracicaliy and cos come laer, a he deailed sage. So hink along he lines of Figure 2.2 and lis conceps and ouline embodimens as block diagrams like he following: Concep Embodimen C1 Enrain in air sream and filer Elecric fan pulling air sream hrough paper or cloh filer in porable uni Cenral pump and filer linked o rooms by ducing Exercise E2.2 Cooling power elecronics. Microchips, paricularly hose for power elecronics, ge ho. If hey ge oo ho hey cease o funcion. The need: a scheme for removing hea from power microchips. Devise conceps o mee he need and skech an embodimen of one of hem, laying ou your ideas in he way suggesed in Exercise E2.1. E.3 Use of maerial selecion chars The 21 exercises in his secion involve he simple use of he chars of Chaper 4 o find maerials wih given propery profiles. They are answered by placing selecion lines on he appropriae char and reading off he maerials ha lie on he appropriae side of he line. I is a good idea o presen he resuls as a able. All can be solved by using he prined chars. If he CES EDU Maerials Selecion sofware is available he same exercises can be solved by is use. This involves firs creaing he char, hen applying he appropriae box or line selecion. The resuls, a Level 1 or 2, are he same as hose read from he hard copy chars (which were made using he Level 2 daabase). The sofware offers links o processes, allows a wider search by using he Level 3 daabase, and gives access o supporing informaion via he Search Web funcion.

4 560 Appendix E Exercises Exercise E3.1 Exercise E3.2 Exercise E3.3 A componen is a presen made from a brass, a copper alloy. Use he Young s modulus densiy (E ) char of Figure 4.3 o sugges hree oher meals ha, in he same shape, would be siffer. Siffer means a higher value of Young s modulus. Use he Young s modulus densiy (E ) char of Figure 4.3 o idenify maerials wih boh a modulus E > 50 GPa and a densiy < 2 Mg/m 3. Use he Young s modulus densiy (E ) char of Figure 4.3 o find (a) meals ha are siffer and less dense han seels and (b) maerials (no jus meals) ha are boh siffer and less dense han seel. Exercise E3.4 Use he E char of Figure 4.3 o idenify meals wih boh E > 100 GPa and E/ > 23 GPa/(Mg/m 3 ) Exercise E3.5 Use he E char of Figure 4.3 o idenify maerials wih boh E > 100 GPa and E 1/3 / > 3 (GPa) 1/3 /(Mg/m 3 ). Remember ha, on aking logs, he index M ¼ E 1/3 / becomes LogðEÞ ¼ 3 LogðÞ þ 3 LogðMÞ and ha his plos as a line of slope 3 on he char, passing hrough he poin ¼ 1/3 ¼ 0.33 a E ¼ 1 in he unis on he char. Exercise E3.6 Exercise E3.7 Use he E char of Figure 4.3 o esablish wheher woods have a higher specific siffness E/ han epoxies. Do ianium alloys have a higher or lower specific srengh (srengh/densiy, f /) han ungsen alloys? This is imporan when you wan srengh a low weigh (landing gear of aircraf, mounain bikes). Use he f / char of Figure 4.4 o decide. Exercise E3.8 Use he modulus srengh E f char of Figure 4.5 o find maerials ha have E > 10 GPa and f 1000 MPa. Exercise E3.9 Exercise E3.10 Exercise E3.11 Are he fracure oughnesses, K 1C, of he common polymers polycarbonae, ABS, or polysyrene larger or smaller han he engineering ceramic alumina? Are heir oughnesses G 1C ¼ K 2 1C =E larger or smaller? The K 1C E char, Figure 4.7, will help. Use he fracure oughness modulus char (Figure 4.7) o find maerials ha have a fracure oughness K 1C greaer han 100 MPa.m 1/2 and a oughness G 1C ¼ K 2 1C E (shown as conours on Figure 4.7) greaer han 10 kj/m 3. The elasic deflecion a fracure (he resilience ) of an elasic brile solid is proporional o he failure srain, E fr ¼ fr /E, where is he sress ha will cause a crack o propagae: fr ¼ p K1C ffiffiffiffiffi c where K 1C is he fracure oughness and c is he lengh of he longes crack he maerials may conain. Thus " fr ¼ 1 K 1C pffiffiffiffiffi c E

5 E.3 Use of maerial selecion chars 561 Maerials ha can deflec elasically wihou fracuring are herefore hose wih large values of K 1C /E. Use he K 1C E char of Figure 4.7 o idenify he class of maerials wih K 1C > 1 MPa.m 1/2 and high values of K 1C /E. Exercise E3.12 One crierion for design of a safe pressure vessel is ha i should leak before i breaks: he leak can be deeced and he pressure released. This is achieved by designing he vessel o olerae a crack of lengh equal o he hickness of he pressure vessel wall, wihou failing by fas fracure. The safe pressure p is hen p 4 1 K 2 1C R f where f is he elasic limi, K 1C is he fracure oughness, R is he vessel radius. The pressure is maximized by choosing he maerial wih he greaes value of M ¼ K2 1C y Use he K 1C f char of Figure 4.8 o idenify hree alloys ha have paricularly high values of M. Exercise E3.13 A maerial is required for he blade of a roary lawn-mower. Cos is a consideraion. For safey reasons, he designer specified a minimum fracure oughness for he blade: i is K 1C > 30 MPa.m 1/2. The oher mechanical requiremen is for high hardness, H, o minimize blade wear. Hardness, in applicaions like his one, is relaed o srengh: H 3 y where f is he srengh (Chaper 4 gives a fuller definiion). Use he K 1C f char of Figure 4.8 o idenify hree maerials ha have K 1C > 30 MPa.m 1/2 and he highes possible srengh. To do his, posiion a K 1C selecion line a 30 MPa.m 1/2 and hen adjus a srengh selecion line such ha i jus admis hree candidaes. Use he Cos char of Figure 4.19 o rank your selecion by maerial cos, hence making a final selecion. Exercise E3.14 Exercise E3.15 Exercise E3.16 Bells ring because hey have a low loss (or damping) coefficien, ; a high damping gives a dead sound. Use he loss coefficien modulus ( E) char of Figure 4.9 o idenify maerial ha should make good bells. Use he loss coefficien modulus ( E) char (Figure 4.9) o find meals wih he highes possible damping. The window hrough which he beam emerges from a high-powered laser mus obviously be ransparen o ligh. Even hen, some of he energy of he beam is absorbed in he window and can cause i o hea and crack. This problem is minimized by choosing a window maerial wih a high hermal conduciviy (o conduc he hea away) and a low expansion coefficien (o reduce hermal srains), ha is, by seeking a window maerial wih a high value of M ¼ = Use he char of Figure 4.12 o idenify he bes maerial for an ulra-high powered laser window.

6 562 Appendix E Exercises Exercise E3.17 Exercise E3.18 Exercise E3.19 Exercise E3.20 Exercise E3.21 Use he hermal conduciviy elecrical resisiviy ( e ) char (Figure 4.10) o find hree maerials wih high hermal conduciviy,, and high elecrical resisiviy, e. Use he srengh maximum service emperaure ( f T max ) char (Figure 4.14) o find polymers ha can be used above 200 C. (a) Use he Young s modulus relaive cos (E C R ) char (Figure 4.18) o find he cheapes maerials wih a modulus, E, greaer han 100 GPa. (b) Use he srengh relaive cos ( f C R ) char (Figure 4.19) o find he cheapes maerials wih a srengh, f, above 100 MPa. Use he fricion coefficien char (Figure 4.15) o find wo maerials wih excepionally low coefficien of fricion. Idenical casings for a power ool could be die-cas in aluminum or molded in ABS or polyeser GFRP. Use he appropriae producion energy char of Figure 16.7 o decide which choice minimizes he maerial producion energy. E.4 Translaion: consrains and objecives Translaion is he ask of re-expressing design requiremens in erms ha enable maerial and process selecion. Tackle he exercises by formulaing he answers o he quesions in his able. Do no ry o model he behavior a his poin (ha comes in laer exercises). Jus hink ou wha he componen does, and lis he consrains ha his imposes on maerial choice, including processing requiremens. Funcion Consrains Objecive Free variables Wha does componen do? Wha essenial condiions mus be me? Wha is o be maximized or minimized? Wha parameers of he problem is he designer free o change? Here i is imporan o recognize he disincion beween consrains and objecives. As he able says, a consrain is an essenial condiion ha mus be me, usually expressed as a limi on a maerial or process aribue. An objecive is an quaniy for which an exremum (a maximum or minimum) is sough, frequenly cos, mass, or volume, bu here are ohers, several of which appear in he exercises below. Take he example of a bicycle frame. I mus have a cerain siffness and srengh. If i is no siff and srong enough i will no work, bu i is never required o have infinie siffness or srengh. Siffness and srengh are herefore consrains ha become limis on modulus, elasic limi, and shape. If he bicycle is for sprin racing, i should be as ligh as possible if you could make i infiniely ligh, ha would be bes of all. Minimizing mass, here, is he objecive, perhaps wih an upper limi (a consrain) on cos. If insead i is a shopping bike o be sold hrough supermarkes i should be as cheap as possible he cheaper i is, he more will be sold. This ime minimizing cos is he objecive, possible wih an upper limi (a consrain) on mass. For mos bikes, of course, minimizing mass

7 E.4 Translaion: consrains and objecives 563 and cos are boh objecives, and hen rade-off mehods are needed. They come laer. For now use judgmen o choose he single mos imporan objecive and make all ohers ino consrains. Two rules-of-humb, useful in many ranslaion exercises. Many applicaions require sufficien fracure oughness for he componen can survive mishandling and accidenal impac during service; a oally brile maerial (like un-oughened glass) is unsuiable. Then a necessary consrain is ha of adequae oughness. This is achieved by requiring ha he fracure oughness K 1C > 15 MPa.m 1/2. Oher applicaions require some duciliy, sufficien o allow sress redisribuion under loading poins, and some abiliy o bend or shape he maerial plasically. This is achieved by requiring ha he (ensile) duciliy E f > 2%. (If he CES sofware is available i can be used o impose he consrains and o rank he survivors using he objecive.) Exercise E4.1 Exercise E4.2 Exercise E4.3 Exercise E4.4 A maerial is required for he windings of an elecric air-furnace capable of emperaures up o 1000 C. Think ou wha aribues a maerial mus have if i is o be made ino windings and funcion properly in a furnace. Lis he funcion and he consrains; se he objecive o minimize cos and he free variables o choice of maerial. A maerial is required o manufacure office scissors. Paper is an abrasive maerial, and scissors someimes encouner hard obsacles like saples. Lis funcion and consrains; se he objecive o minimize cos and he free variables o choice of maerial. A maerial is required for a hea exchanger o exrac hea from geo-hermally heaed, saline, waer a 120 C (and hus under pressure). Lis funcion and consrains; se he objecive o minimize cos and he free variables o choice of maerial. A C-clamp (Figure E.1) is required for processing of elecronic componens a emperaures up o 450 C. I is essenial ha he clamp have low hermal ineria so ha i reaches emperaure quickly, and i mus no charge-up when exposed o an elecron beam. The ime i akes a componen of hickness x o reach hermal equilibrium when he emperaure is suddenly changed (a ransien hea flow problem) is x2 2a where he hermal diffusiviy a ¼ /C p and is he hermal conduciviy, he densiy and C p he specific hea. X Figure E.1 C-clamp.

8 564 Appendix E Exercises Lis funcion and consrains; se he objecive o minimize cos and he free variables o choice of maerial. Exercise E4.5 Exercise E4.6 Exercise E4.7 Exercise E4.8 A furnace is required o siner powder-meal pars. I operaes coninuously a 650 C while he pars are fed hrough on a moving bel. You are asked o selec a maerial for furnace insulaion o minimize hea loss and hus o make he furnace as energy-efficien as possible. For reasons of space he insulaion is limied o a maximum hickness of x ¼ 0.2 m. Lis he funcion, consrains, objecive and free variable. Ulra-precise bearings ha allow a rocking moion make use of knife-edges or pivos (Figure E.2). As he bearing rocks, i rolls, ranslaing sideways by a disance ha depends on he radius of conac. The furher i rolls, he less precise is is posiioning, so he smaller he radius of conac R he beer. Bu he smaller he radius of conac, he greaer is he conac pressure (F/A). If his exceeds he hardness H of eiher face of he bearing, i will be damaged. Elasic deformaion is bad oo: i flaens he conac, increasing he conac area and he roll. A rocking bearing is required o operae in a micro-chip fabricaion uni using fluorine gas a 100 C, followed by e-beam processing requiring ha all srucural pars of he equipmen can be earhed o preven sray charges. Translae he requiremens ino maerial selecion crieria, lising funcion, consrains, objecive and free variable. The sandard CD ( Jewel case) cracks easily and, if broken, can scrach he CD. Jewel cases are made of injecion molded polysyrene, chosen because i is ransparen, cheap, and easy o mould. A maerial is sough o make CD cases ha do no crack so easily. The case mus sill be ransparen, able o be injecion molded, and able o compee wih polysyrene in cos. A sorage heaer capures hea over a period of ime, hen releases i, usually o an air sream, when required. Those for domesic heaing sore solar energy or energy from cheap off-peak elecriciy and release i slowly during he cold par of he day. Those for research release he hea o a supersonic air sream o es sysem behavior in supersonic fligh. Wha is a good maerial for he core of a compac sorage maerial capable of emperaures up o 120 C? Moion Moion Load P Load P R R Block Block Figure E.2 Ulra-precise bearings wih knife-edges and pivos.

9 E.5 Deriving and using maerial indices 565 E.5 Deriving and using maerial indices The exercises in his secion give pracice in deriving indices. (a) Sar each by lising funcion, consrains, objecives and free variables; wihou having hose sraigh, you will ge in a mess. Then wrie down an equaion for he objecive. Consider wheher i conains a free variable oher han maerial choice; if i does, idenify he consrain ha limis i, subsiue, and read off he maerial index. (b) If he CES EDU sofware is available, hen use i o apply he consrains and rank he survivors using he index (sar wih he Level 2 daabase). Are he resuls sensible? If no, wha consrain has been overlooked or incorrecly formulaed? Exercise E5.1 Aperure grills for cahode ray ubes (Figure E.3). Two ypes of cahode ray ube (CRT) dominae he compuer monior and elevision markeplace. In he older echnology, color separaion is achieved by using a shadow mask: a hin meal plae wih a grid of holes ha allow only he correc beam o srike a red, green or blue phosphor. A shadow mask can hea up and disor a high brighness levels ( doming ), causing he beams o miss heir arges, and giving a blochy image. To avoid his, he newes shadow masks are made of Invar, a nickel alloy wih a near-zero expansion coefficien beween room emperaure and 150 C. I is a consequence of shadow-mask echnology ha he glass screen of he CRT curves inward on all four edges, increasing he probabiliy of refleced glare. Sony s Triniron echnology overcomes his problem and allows greaer brighness by replacing he shadow mask by an aperure grill of fine verical wires, each abou 200mm in hickness, ha allows he inended beam o srike eiher he red, he green or he blue phosphor o creae he image. The glass face of he Triniron ube is curved in one plane only, reducing glare. The wires of he aperure grill are ighly sreched, so ha hey remain au even when ho i is his ension ha allows he greaer brighness. Wha Elecron guns Deflecor coils Aperure grill Screen Figure E.3 Aperure grills for cahode ray ubes.

10 566 Appendix E Exercises index guides he choice of maerial o make hem? The able summarizes he requiremens. Funcion Consrains Objecive Free variables Aperure grill for CRT Wire hickness and spacing specified Mus carry pre-ension wihou failure Elecrically conducing o preven charging Able o be drawn o wire Maximize permied emperaure rise wihou loss of ension Choice of maerial Exercise E5.2 Maerial indices for elasic beams wih differing consrains (Figure E.4). Sar each of he four pars of his problem by lising he funcion, he objecive, and he consrains. You will need he equaions for he deflecion of a canilever beam wih a square crosssecion, given in Appendix A, Secion A.3. The wo ha maer are ha for he deflecion of a beam of lengh L under an end load F: ¼ FL3 3EI and ha for he deflecion of a beam under a disribued load f per uni lengh: ¼ 1 fl 4 8 EI where I ¼ 4 /12. For a self-loaded beam f ¼ Ag where is he densiy of he maerial of he beam, A is cross-secional area and g he acceleraion due o graviy. (a) Show ha he bes maerial for a canilever beam of given lengh L and given (i.e., fixed) square cross-secion ( ) ha will deflec leas under a given end load F is ha wih he larges value of he index M ¼ E, where E is Young s modulus (neglec self-weigh) (Figure E.4(a)). (b) Show ha he bes maerial choice for a canilever beam of given lengh L and wih a given secion ( ) ha will deflec leas under is own self-weigh is ha wih he larges value of M ¼ E/, where is he densiy (Figure E.4(b)). (c) Show ha he maerial index for he lighes canilever beam of lengh L and square secion (no given, ha is, he area is a free variable) ha will no deflec by more han under is own weigh is M ¼ E/ 2 (Figure E.4(c)). (d) Show ha he lighes canilever beam of lengh L and square secion (area free) ha will no deflec by more han under an end load F is ha made of he maerial wih he larges value of M ¼ E 1/2 / (neglec self weigh) (Figure E.4(d)). Exercise E5.3 Maerial index for a ligh, srong beam (Figure E.5). In siffness-limied applicaions, i is elasic deflecion ha is he acive consrain: i limis performance.

11 E.5 Deriving and using maerial indices 567 (a) L Fixed (b) Force f per uni lengh F, δ Fixed (c) Force f per uni lengh Free (d) L Free F, δ Figure E.4 Maerial indices for elasic beams wih differing consrains. In srengh-limied applicaions, deflecion is accepable provided he componen does no fail; srengh is he acive consrain. Derive he maerial index for selecing maerials for a beam of lengh L, specified srengh and minimum weigh. For simpliciy, assume he beam o have a solid square cross-secion. You will need he equaion for he failure load of a beam (Appendix A, Secion A.4). I is F f ¼ I f y m L where y m is he disance beween he neural axis of he beam and is ouer filamen and I ¼ 4 /12 ¼ A 2 /12 is he second momen of he cross-secion. The able iemizes he design requiremens: Funcion Consrains Objecive Free variables Beam Lengh L is specified Beam mus suppor a bending load F wihou yield or fracure Minimize he mass of he beam Cross-secion area, A Choice of maerial

12 568 Appendix E Exercises L Free F Figure E.5 Maerial index for a ligh, srong beam. F 2r H Figure E.6 Maerial index for a cheap, siff column. Exercise E5.4 Maerial index for a cheap, siff column (Figure E.6). In he las wo exercises he objecive has been ha of minimizing weigh. There are many ohers. In he selecion of a maerial for a spring, he objecive is ha of maximizing he elasic energy i can sore. In seeking maerials for hermal-efficien insulaion for a furnace, he bes are hose wih he lowes hermal conduciviy and hea capaciy. And mos common of all is he wish o minimize cos. So here is an example involving cos. Columns suppor compressive loads: he legs of a able; he pillars of he Parhenon. Derive he index for selecing maerials for he cheapes cylindrical column of specified heigh, H, ha will safely suppor a load F wihou buckling elasically. You will need he equaion for he load F cri a which a slender column buckles. I is F cri ¼ n2 EI H 2 where n is a consan ha depends on he end consrains and I ¼ r 4 /4 ¼ A 2 /4 is he second momen of area of he column (see Appendix A for boh). The able liss he requiremens: Funcion Consrains Objecive Free variables Cylindrical column Lengh L is specified Column mus suppor a compressive load F wihou buckling Minimize he maerial cos of he column Cross-secion area, A Choice of maerial

13 E.5 Deriving and using maerial indices 569 W W δ δ 2a 2a Figure E.7 Exercise E5.5 Indices for siff plaes and shells. Indices for siff plaes and shells (Figure E.7). Aircraf and space srucures make use of plaes and shells. The index depends on he configuraion. Here you are asked o derive he maerial index for (a) a circular plae of radius a carrying a cenral load W wih a prescribed siffness S ¼ W/ and of minimum mass, (b) a hemispherical shell of radius a carrying a cenral load W wih a prescribed siffness S ¼ W/ and of minimum mass, as shown in he figure. Use he wo resuls lised below for he mid-poin deflecion of a plae or spherical shell under a load W applied over a small cenral, circular area. Circular plae: ¼ 3 Wa 2 4 E 3 ð1 2 Þ 3 þ 1 þ Hemispherical shell : ¼ A Wa E 2 ð1 2 Þ in which A 0.35 is a consan. Here E is Young s modulus, is he hickness of he plae or shell and v is Poisson s raio. Poisson s raio is almos he same for all srucural maerials and can be reaed as a consan. The able summarizes he requiremens: Funcion Consrains Objecive Free variables Siff circular plae, or Siff hemispherical shell Siffness S under cenral load W specified Radius a of plae or shell specified Minimize he mass of he plae or shell Plae or shell hickness, Choice of maerial Exercise E5.6 The C-clamp in more deail (Figure E.8). Exercise E4.4 inroduced he C-clamp for processing of elecronic componens. The clamp has a square cross-secion of widh x and given deph b. I is essenial ha he clamp have low hermal ineria so ha i reaches emperaure quickly. The ime i akes a componen of hickness x o reach

14 570 Appendix E Exercises L M F H X X M Figure E.8 The C-clamp in more deail. hermal equilibrium when he emperaure is suddenly changed (a ransien hea flow problem) is x2 2a where he hermal diffusiviy a ¼ =C p and is he hermal conduciviy, he densiy and C p he specific hea. The ime o reach hermal equilibrium is reduced by making he secion x hinner, bu i mus no be so hin ha i fails in service. Use his consrain o eliminae x in he equaion above, hereby deriving a maerial index for he clamp. Use he fac ha he clamping force F creaes a momen on he body of he clamp of M ¼ FL, and ha he peak sress in he body is given by ¼ x M 2 I where I ¼ bx 3 /12 is he second momen of area of he body. The able summarizes he requiremens. Funcion Consrains Objecive Free variables C-clamp of low hermal ineria Deph b specified Mus carry clamping load F wihou failure Minimize ime o reach hermal equilibrium Widh of clamp body, x Choice of maerial Exercise E5.7 Springs for rucks (Figure E.9). In vehicle suspension design i is desirable o minimize he mass of all componens. You have been asked o selec a maerial and dimensions for a ligh spring o replace he seel leaf-spring of an exising ruck suspension. The exising leaf-spring is a beam, shown schemaically in he figure. The new spring mus have he same lengh L and siffness S as he exising one, and mus deflec hrough a maximum safe displacemen max wihou failure. The widh b and hickness are free variables.

15 E.5 Deriving and using maerial indices 571 Load F δ L b Figure E.9 Springs for rucks. Derive a maerial index for he selecion of a maerial for his applicaion. Noe ha his is a problem wih wo free variables: b and ; and here are wo consrains, one on safe deflecion max and he oher on siffness S. Use he wo consrains o fix free variables. The able caalogs he requiremens: Funcion Consrains Objecive Free variables Leaf spring for ruck Lengh L specified Siffness S specified Maximum displacemen max specified Minimize he mass Spring hickness Spring widh b Choice of maerial You will need he equaion for he mid-poin deflecion of an elasic beam of lengh L loaded in hree-poin bending by a cenral load F: ¼ 1 FL 3 48 EI and ha for he deflecion a which failure occurs max ¼ 1 f L 2 6 E where I is he second momen of area; for a beam of recangular secion, I ¼ b 3 /12 and E and f are he modulus and failure sress of he maerial of he beam (boh resuls can be found in Appendix A). Exercise E5.8 Disposable knives and forks (Figure E.10). Disposable knives and forks are ordered by an environmenally-conscious pizza-house. The shape of each (and hus he lengh, widh, and profile) are fixed, bu he hickness is free: i is chosen o give enough bending-siffness o cu and impale he pizza wihou excessive flexure. The pizzeria-proprieor wishes o enhance he greenness of his image by minimizing he

16 572 Appendix E Exercises b L Loading Figure E.10 Disposable knives and forks. energy-conen of his hrow-away ableware, which could be molded from polysyrene (PS) or samped from aluminum shee. Esablish an appropriae maerial index for selecing maerials for energy-economic forks. Model he eaing implemen as a beam of fixed lengh L and widh w, bu wih a hickness ha is free, loaded in bending, as in he figure. The objecive-funcion is he volume of maerial in he fork imes is energy conen, H p, per uni volume (H p is he producion energy per kg, and he densiy). The limi on flexure imposes a siffness consrain (Appendix A, Secion A.3). Use his informaion o develop he index. Flexure, in culery, is an inconvenience. Failure wheher by plasic deformaion or by fracure is more serious: i causes loss-of-funcion; i migh even cause hunger. Repea he analysis, deriving an index when a required srengh is he consrain. This is a sraighforward applicaion of he mehod illusraed by Exercise E5.2; he only difference is ha energy conen, no weigh, is minimized. The free variable is he hickness of he shaf of he fork; all oher dimensions are fixed. There are wo alernaive consrains, firs, ha he fork should no flex oo much, second, ha i should no fail. Funcion Consrains Objecive Free variables Environmenally friendly disposable forks Lengh L specified Widh b specified Siffness S specified, or Failure load F is specified Minimize he maerial energy-conen Shaf hickness, Choice of maerial The selecion can be implemened using Figures 16.8 and If he CES sofware is available, make a char wih he siffness index as one axis and he srengh index as he oher. The maerials ha bes mee boh crieria lie a he op righ. Exercise E5.9 Fin for a rocke (Figure E.11). A ube-launched rocke has sabilizing fins a is rear. During launch he fins experience ho gas a T g ¼ 1700 C for a ime 0.3 s. I is

17 E.5 Deriving and using maerial indices 573 Figure E.11 Fin for a rocke. imporan ha he fins survive launch wihou surface meling. Sugges a maerial index for selecing a maerial for he fins. The able summarizes he requiremens: Funcion Consrains Objecive Free variables High hea-ransfer rocke fins All dimensions specified Mus no suffer surface meling during exposure o gas a 1700 C for 0.3 s Minimize he surface emperaure rise during firing Maximize he meling poin of he maerial Choice of maerial This is ricky. Hea eners he surface of he fin by ransfer from he gas. If he hea ransfer coefficien is h, he hea flux per uni area is q ¼ hðt g T s Þ where T s is he surface emperaure of he fin he criical quaniy we wish o minimize. Hea diffuses ino he fin surface by hermal conducion. If he heaing ime is small compared wih he characerisic ime for hea o diffuse hrough he fin, a quasi seady-sae exiss in which he surface emperaure adjuss iself such ha he hea enering from he gas is equal o ha diffusing inwards by conducion. This second is equal o ð q ¼ T s T i Þ x where is he hermal conduciviy, T i is he emperaure of he (cold) inerior of he fin, and x is a characerisic hea-diffusion lengh. When he heaing ime is shor (as here) he hermal fron, afer a ime, has peneraed a disance x ð2aþ 1=2 where a ¼ /C p is he hermal diffusiviy. Subsiuing his value of x in he previous equaion gives q ¼ C p 1=2 ðt s T i Þ x where is he densiy and C p he specific hea of he maerial of he fin.

18 574 Appendix E Exercises Proceed by equaing he wo equaions for q, solving for he surface emperaure T s o give he objecive funcion. Read off he combinaion of properies ha minimizes T s ; i is he index for he problem. The selecion is made by seeking maerials wih large values of he index and wih a high meling poin, T m. If he CES sofware is available, make a char wih hese wo as axes and idenify maerials wih high values of he index ha also have high meling poins. E.6 Selecing processes The exercises of his secion use he process selecion chars of Chapers 7 and 8. They are useful in giving a feel for process aribues and he way in which process choice depends on maerial and he shape. Here he CES sofware offers greaer advanages: wha is cumbersome and of limied resoluion wih he chars is easy wih he sofware, which offers much greaer resoluion. Each exercise has hree pars, labeled (a) (c). The firs involves ranslaion. The second uses he selecion chars of Chaper 7 (which you are free o copy) in he way ha was illusraed in Chaper 8. The hird involves he use of he CES sofware if available. Exercise E6.1 Elevaor conrol quadran (Figure E.12). The quadran skeched here is par of he conrol sysem for he wing-elevaor of a commercial aircraf. I is o be made of a ligh alloy (aluminum or magnesium) wih he shape shown in Figure E.12. I weighs abou 5 kg. The minimum secion hickness is 5 mm, and apar from he bearing surfaces he requiremens on surface finish and precision are no sric: surface finish 10 mm and precision 0.5 mm. The bearing surfaces require a surface finish 1 mm and a precision 0.05 mm. A producion run of is planned. (a) Iemize he funcion and consrains, leave he objecive blank and ener choice of process for he free variable. (b) Use copies of he chars of Chaper 7 in succession o idenify processes o shape he quadran. (c) If he CES sofware is available, apply he consrains and idenify in more deail he viable processes. Figure E.12 Elevaor conrol quadran.

19 E.6 Selecing processes 575 Figure E.13 Casing for an elecric plug. Exercise E6.2 Casing for an elecric plug (Figure E.13). The elecric plug is perhaps he commones of elecrical producs. I has a number of componens, each performing one or more funcions. The mos obvious are he casing and he pins, hough here are many more (connecors, a cable clamp, faseners, and, in some plugs, a fuse). The ask is o invesigae processes for shaping he wo-par insulaing casing, he hinnes par of which is 2 mm hick. Each par weighs abou 30 grams and is o be made in a single sep from a hermoplasic or hermoseing polymer wih a planned bach size of The required olerance of 0.3 mm and surface roughness of 1mm mus be achieved wihou using secondary operaions. (a) Iemize he funcion and consrains, leave he objecive blank and ener choice of process for he free variable. (b) Use he chars of Chaper 7 successively o idenify possible processes o make he casing. (c) Use he CES sofware o selec maerials for he casing. Exercise E6.3 Ceramic valves for aps (Figure E.14). Few hings are more irriaing han a dripping ap. Taps drip because he rubber washer is worn or he brass sea is pied by corrosion, or boh. Ceramics wear well, and hey have excellen corrosion resisance in boh pure and sal waer. Many household aps now use ceramic valves. The skech shows how hey work. A ceramic valve consiss of wo disks mouned one above he oher, spring-loaded so ha heir faces are in conac. Each disk has a diameer of 20 mm, a hickness of 3 mm and weighs abou 10 grams. In order o seal well, he maing surfaces of he wo disks mus be fla and smooh, requiring high levels of precision and surface finish; ypically olerance <0.02 mm and surface roughness <0.1mm. The ouer face of each has a slo ha regisers i, and allows he upper disk o be roaed hrough 90 (1/4 urn). In he off posiion he holes in he upper disk are blanked off by he solid par of he lower one; in he on posiion he holes are aligned. A producion run of is envisaged. (a) Lis he funcion and consrains, leave he objecive blank and ener choice of process for he free variable. (b) Use he chars of Chaper 7 o idenify possible processes o make he casing. (c) Use he CES sofware o selec maerials for he casing.

20 576 Appendix E Exercises Top, upper disk Spring-loaded ceramic disks Slo Hole Boom, upper disk Figure E.14 Ceramic valves for aps. Figure E.15 Shaping plasic boles. Exercise E6.4 Shaping plasic boles (Figure E.15). Plasic boles are used o conain fluids as various as milk and engine oil. A ypical polyehylene bole weighs abou 30 grams and has a wall hickness of abou 0.8 mm. The shape is 3-D hollow. The bach size is large (1,000,000 boles). Wha process could be used o make hem? (a) Lis he funcion and consrains, leave he objecive blank and ener choice of process for he free variable. (b) Use he chars of Chaper 7 o idenify possible processes o make he casing. (c) Use he CES sofware o selec maerials for he casing. Exercise E6.5 Car hood (bonne) (Figure E.16). As weigh-saving assumes greaer imporance in auomobile design, he replacemen of seel pars wih polymer-composie subsiues

21 E.6 Selecing processes 577 Hood Figure E.16 Car hood (bonne). Figure E.17 Complex srucural channels. becomes increasingly aracive. Weigh can be saved by replacing a seel hood wih one made from a hermoseing composies. The weigh of he hood depends on he car model: a ypical composie hood weighs is 8 10 kg. The shape is a dished-shee and he requiremens on olerance and roughness are 0.2 mm and 2 mm, respecively. A producion run of 100,000 is envisaged. (a) Lis he funcion and consrains, leave he objecive blank and ener choice of process for he free variable. (b) Use he chars of Chaper 7 o idenify possible processes o make he casing. (c) Use he CES sofware o selec maerials for he casing. Exercise E6.6 Complex srucural channels (Figure E.17). Channel secions for window frames, for slide-ogeher secions for versaile assembly and for ducing for elecrical wiring can be complex in shape. The figure shows an example. The order is for 10,000 such secions, each 1 m in lengh and weighing 1.2 kg, wih a minimum secion of 4 mm. A olerance of 0.2 mm and a surface roughness of less han 1mm mus be achieved wihou any addiional finishing operaion. (a) Lis he funcion and consrains, leave he objecive blank and ener choice of process for he free variable. (b) Use he chars of Chaper 7 o idenify possible processes o make he casing. (c) Use he CES sofware o selec maerials for he casing.

22 578 Appendix E Exercises Exercise E6.7 Selecing joining processes. This exercise and he nex require he use of he CES sofware. (a) Use CES o selec a joining process o mee he following requiremens: Funcion Creae a permanen bu join beween seel plaes Consrains Maerial class: carbon seel Join geomery: bu join Secion hickness: 8 mm Permanen Waerigh Objecive Free variables Choice of process (b) Use CES o selec a joining process o mee he following requiremens: Funcion Creae a waerigh, demounable lap join beween glass and polymer Consrains Maerial class: glass and polymers Join geomery: lap join Secion hickness: 4 mm Demounable Waerigh Objecive Free variables Choice of process Exercise E6.8 Selecing surface reamen processes. This exercise, like he las, requires he use of he CES sofware. (a) Use CES o selec a surface reamen process o mee he following requiremens: Funcion Increase he surface hardness and wear resisance of a high carbon seel componen Consrains Maerial class: carbon seel Purpose of reamen: increase surface hardness and wear resisance Objecive Free variables Choice of process (b) Use CES o selec a joining process o mee he following requiremens: Funcion Apply color and paern o he curved surface of a polymer molding Consrains Maerial class: hermoplasic Purpose of reamen: aesheics, color Curved surface coverage: good or very good Objecive Free variables Choice of process

23 E.7 Muliple consrains and objecives 579 E.7 Muliple consrains and objecives Over-consrained problems are normal in maerials selecion. Ofen i is jus a case of applying each consrain in urn, reaining only hose soluions ha mee hem all. Bu when consrains are used o eliminae free variables in an objecive funcion (as discussed in Secion 9.2) he acive consrain mehod mus be used. The firs hree exercises in his secion illusrae problems wih muliple consrains. The remaining wo concern muliple objecives and rade-off mehods. When a problem has wo objecives minimizing boh mass m and cos C of a componen, for insance a conflic arises: he cheapes soluion is no he lighes and vice versa. The bes combinaion is sough by consrucing a rade-off plo using mass as one axis, and cos as he oher. The lower envelope of he poins on his plo defines he rade-off surface. The soluions ha offer he bes compromise lie on his surface. To ge furher we need a penaly funcion. Define he penaly funcion Z ¼ C þ m where is an exchange consan describing he penaly associaed wih uni increase in mass, or, equivalenly, he value associaed wih a uni decrease. The bes soluions are found where he line defined by his equaion is angenial o he rade-off surface. (Remember ha objecives mus be expressed in a form such ha a minimum is sough; hen a low value of Z is desirable, a high one is no.) When a subsiue is sough for an exising maerial i is beer o work wih raios. Then he penaly funcion becomes Z ¼ C C o þ m m o in which he subscrip o means properies of he exising maerial and he aserisk on Z and is a reminder ha boh are now dimensionless. The relaive exchange consan measures he fracional gain in value for a given fracional gain in performance. Exercise E7.1 Muliple consrains: a ligh, siff, srong ie (Figure E.18). A ie, of lengh L loaded in ension, is o suppor a load F, a minimum weigh wihou failing (implying a consrain on srengh) or exending elasically by more han (implying a consrain on siffness, F/). The able summarizes he requiremens: Funcion Consrains Objecive Free variables Tie rod Mus no fail by yielding under force F Mus have specified siffness, F/ Lengh L and axial load F specified Minimize mass m Secion area A Choice of maerial

24 580 Appendix E Exercises Cross-secion area A Force F L Figure E.18 Muliple consrains: a ligh siff, srong ie. M 2 = Densiy/Elasic limi (Mg m 3 )/MPa e-3 Tungsen carbide Silicon Aluminum niride Silicon carbide Boron carbide Brick Borosilicae glass CFRP Sone PMMA Nylons Low alloy seels Tianium alloys Magnesium alloys Aluminum alloys Polyehylene 1e-4 1e-6 1e-5 1e-4 101e M 1 = Densiy/Modulus (Mg m 3 )/MPa Figure E.19 Maerial char. (a) Follow he mehod of Chaper 9 o esablish wo performance equaions for he mass, one for each consrain, from which wo maerial indices and one coupling equaion linking hem are derived. Show ha he wo indices are M 1 ¼ E and M 2 ¼ y and ha a minimum is sough for boh. (b) Use hese and he maerial char of Figure E.19, which has he indices as axes, o idenify candidae maerials for he ie (i) when L/ ¼ 100 and (ii) when L/ ¼ Exercise E7.2 Muliple consrains: a ligh, safe, pressure vessel (Figure E.20). When a pressure vessel has o be mobile; is weigh becomes imporan. Aircraf bodies, rocke casings and liquid-naural gas conainers are examples; hey mus be ligh, and a he same ime hey mus be safe, and ha means ha hey mus no fail by yielding or by fas

25 E.7 Muliple consrains and objecives 581 R Pressure difference p Figure E.20 Muliple consrains: a ligh, safe, pressure vessel. fracure. Wha are he bes maerials for heir consrucion? The able summarizes he requiremens: Funcion Consrains Objecive Free variables Pressure vessel Mus no fail by yielding Mus no fail by fas fracure. Diameer 2R and pressure difference p specified Minimize mass m Wall hickness, Choice of maerial (a) Wrie, firs, a performance equaion for he mass m of he pressure vessel. Assume, for simpliciy, ha i is spherical, of specified radius R, and ha he wall hickness, (he free variable) is small compared wih R. Then he ensile sress in he wall is ¼ pr 2 where p, he pressure difference across his wall, is fixed by he design. The firs consrain is ha he vessel should no yield, ha is, ha he ensile sress in he wall should no exceed y. The second is ha i should p ffiffiffiffiffi no fail by fas fracure; his requires ha he wall-sress be less han K 1C c, where K1C is he fracure oughness of he maerial of which he pressure vessel is made and c is he lengh of he longes crack ha he wall migh conain. Use each of hese in urn o eliminae in he equaion for m; use he resuls o idenify wo maerial indices M 1 ¼ y and M 2 ¼ K 1C and a coupling relaion beween hem. I conains he crack lengh, c. (b) Figure E.21 shows he char you will need wih he wo maerial indices as axes. Plo he coupling equaion ono his figure for wo values of c: one of 5 mm, he oher 5 mm. Idenify he lighes candidae maerials for he vessel for each case.

26 582 Appendix E Exercises M 2 = Densiy/Fracure oughness (kg m 3 /MPa) Alumina Silicon carbide Boron carbide Boron niride CFRP Silicon Glass Elasomers Polymer foams Sainless seel Low alloy seels Tianium alloys PTFE Polyehylene (PE) Nickel alloys Aluminum alloys M 1 = Densiy/Yield srengh (kg m 3 /MPa) Polypropylene (PP) Lead alloys Copper alloys 1000 Figure E.21 Densiy/yield srengh and densiy/racure oughness char. Force F Cross-secion area πd 2 /4 Heigh H Figure E.22 A cheap column ha mus no buckle or crush. Exercise E7.3 A cheap column ha mus no buckle or crush (Figure E.22). The bes choice of maerial for a ligh srong column depends on is aspec raio: he raio of is heigh H o is diameer D. This is because shor, fa columns fail by crushing; all slender columns buckle insead. Derive wo performance equaions for he maerial cos of a column of solid circular secion and specified heigh H, designed o

27 E.7 Muliple consrains and objecives 583 suppor a load F large compared o is self-load, one using he consrains ha he column mus no crush, he oher ha i mus no buckle. The able summarizes he needs: Funcion Consrains Objecive Free variables Column Mus no fail by compressive crushing Mus no buckle Heigh H and compressive load F specified Minimize maerial cos C Diameer D Choice of maerial (a) Proceed as follows: (1) Wrie down an expression for he maerial cos of he column is mass imes is cos per uni mass. (2) Express he wo consrains as equaions, and use hem o subsiue for he free variable, D, o find he cos of he column ha will jus suppor he load wihou failing by eiher mechanism (3) Idenify he maerial indices M 1 and M 2 ha ener he wo equaions for he mass, showing ha hey are M 1 ¼ C m c and M 2 ¼ C m E 1=2 where C m is he maerial cos per kg, he maerial densiy, c is crushing srengh and E is modulus. (b) Daa for six possible candidaes for he column are lised in he able. Use hese o idenify candidae maerials when F ¼ 10 5 N and H ¼ 3 m. Ceramics are admissible here, because hey have high srengh in compression. Daa for candidae maerials for he column Maerial Densiy (kg/m 3 ) Cos/kg C m ($/kg) Modulus E (MPa) Compression srengh, c (MPa) Wood (spruce) , Brick , Granie , Poured concree , Cas iron , Srucural seel , Al-alloy , (c) Figure E.23 show a maerial char wih he wo indices as axes. Idenify and plo coupling lines for selecing maerials for a column wih F ¼ 10 5 N and H ¼ 3 m (he same condiions as above), and for a second column wih F ¼ 10 3 N and H ¼ 20 m.

28 584 Appendix E Exercises M 2 = (Densiy Cos/kg)/Sqr(Modulus) ($ m 3 /(MPa)) Aluminum alloys Low alloy seel Carbon seel CFRP GFRP Sainless seel Silicon Carbide Concree Wood Cas irons Sone Copper alloys M 1 = (Densiy Cos/kg)/Compression srengh ($ m 3 /(MPa) Figure E.23 Maerial char. Exercise E7.4 An air cylinder for a ruck (Figure E.24). Trucks rely on compressed air for braking and oher power-acuaed sysems. The air is sored in one or a cluser of cylindrical pressure anks like ha shown here (lengh L, diameer 2R, hemispherical ends). Mos are made of low-carbon seel, and hey are heavy. The ask: o explore he poenial of alernaive maerials for ligher air anks, recognizing he here mus be a rade-off beween mass and cos if i is oo expensive, he ruck owner will no wan i even if i is ligher. The able summarizes he design requiremens. Funcion Consrains Objecive Free variables Air cylinder for ruck Mus no fail by yielding Diameer 2R and lengh L specified Minimize mass, m Minimize maerial cos, C Wall hickness, Choice of maerial (a) Show ha he mass and maerial cos of he ank relaive o one made of low-carbon seel are given by m ¼ y;o C and ¼ C m y;o m o y o C o y C m;o o

29 E.7 Muliple consrains and objecives 585 where is he densiy, y he yield srengh and C m he cos per kg of he maerial, and he subscrip o indicaes values for mild seel. (b) Explore he rade-off beween relaive cos and relaive mass using Figure E.25, which has hese quaniies as axes. Mild seel lies a he co-ordinaes (1, 1). Skech a rade-off surface. Define a relaive penaly funcion Z ¼ m m o þ C C o where is a relaive exchange consan, and plo a conour ha is approximaely angen o he rade-off surface for ¼ 1 and for ¼ 100. Wha selecions do hese sugges? Pressure p 2R Aspec raio Q = 2R/L L Figure E.24 An air cylinder for a ruck. Lead alloys Copper alloys Mos relaive o low carbon seel Low carbon seel High carbon seel Low alloy seel Zinc alloys Aluminum alloys GFRP PEEK Tianium alloys Magnesium alloys CFRP Cos relaive o low carbon seel 100 Figure E.25 Relaive cos and relaive mass.

30 586 Appendix E Exercises Figure E.26 Insulaion walls for freezers. Exercise E Aluminum-SiC (1.0) Thermal conduciviy (W m 1 k) Alumina (1.2) Alumina (0.8) Alumina (0.6) PC (0.85) PVC (0.5) PVC (0.2) Aluminum-SiC (0.5) Aluminum-SiC (0.16) Aluminum-SiC (0.1) PVC (0.05) Phenolic (0.03) PU (0.08) 1e-3 1e-4 1e / Young's modulus (1/GPa) Figure E.27 Thermal conduciviy and elasic compliance.

31 E.8 Selecing maerial and shape 587 Insulaing walls for freezers (Figure E.26). Freezers and refrigeraed rucks have panel-walls ha provide hermal insulaion, and a he same ime are siff, srong and ligh (siffness o suppress vibraion, srengh o olerae rough usage). To achieve his he panels are usually of sandwich consrucion, wih wo skins of seel, aluminum or GFRP (providing he srengh) separaed by, and bonded o, a low densiy insulaing core. In choosing he core we seek o minimize hermal conduciviy,, and a he same ime o maximize siffness, because his allows hinner seel faces, and hus a ligher panel, while sill mainaining he overall panel siffness. The able summarizes he design requiremens: Funcion Consrains Objecive Free variables Foam for panel-wall insulaion Panel wall hickness specified. Minimize foam hermal conduciviy, Maximize foam siffness, meaning Young s modulus, E Choice of maerial Figure E.27 shows he hermal conduciviy of foams ploed agains heir elasic compliance I/E (he reciprocal of heir Young s moduli E, since we mus express he objecives in a form ha requires minimizaion). The numbers in brackes are he densiies of he foams in Mg/m 3. The foams wih he lowes hermal conduciviy are he leas siff; he siffes have he highes conduciviy. Explain he reasoning you would use o selec a foam for he ruck panel using a penaly funcion. E.8 Selecing maerial and shape The examples in his secion relae o he analysis of maerial and shape of Chapers 11 and 12. They cover he derivaion of shape facors, of indices ha combine maerial and shape, and he use of he 4-quadran char arrays o explore maerial and shape combinaions. For his las purpose i is useful o have clean copies of he char arrays of Figures and Like he maerial propery chars, hey can be copied from he ex wihou resricion of copyrigh. Exercise E8.1 Shape facors for ubes (Figure E.28) (a) Evaluae he shape facor e B for siffness-limied design in bending of a square box secion of ouer edge-lengh h ¼ 100 mm and wall hickness ¼ 3 mm. Is his shape more efficien han one made of he same maerial in he form of a ube of diameer h r y h Figure E.28 Shape facors for ubes.