MME445: Lectue 20 Mateials selection The mateials index A. K. M. B. Rashid Pofesso, Depatment of MME BUET, Dhaka Leaning Objectives Knowledge & Undestanding Elementay knowledge of how to expess design objectives as Mateial Indices Skills & Abilities Ability to use Mateial Indices to ank mateial choices Values & Attitudes Develop confidence in using Mateial Indices as an engineeing tool Resouces M F Ashby, Mateials Selection in Mechanical Design, 4 th Ed., Ch. 05 1
Outline of this lectue The mateial indices Stuctual index A Remainde: The Selection Stategy of Mateial Design equiements: expessed as Constaints and Objectives Data: Mateial attibutes Pocess attibutes Documentation Able to be moulded Wate and UV esistant Stiff enough Stong enough As cheap as possible (As light as possible) Compaison engine Sceening Ranking Documentation Final selection Density Pice Modulus Stength Duability Pocess compatibility Moe. 2
Common constaints and objectives Common constaints Meet a taget value of Stiffness Stength Factue toughness Themal conductivity Electical esistivity Magnetic emanence Optical tanspaency Cost Mass Common objectives Minimise Cost Mass Volume Impact on envionment Heat loss Maximise Enegy stoage Heat flow 3
Examples of tanslations Tanslation fo the cokscew leve Function Constaints Objective Fee vaiables Leve (beam loaded in bending) Stiff enough Stong enough Some toughness Resist coosion in wine and wate Length L specified Minimise cost Choice of mateial Choice of coss-sectional aea functional constaints geometic constaint Tanslation fo a polystyene (PS) CD case Function Constaints Objective Fee vaiables Contain and potect a CD (panel) Optically clea Ability to be injection moulded High factue toughness High toughness Dimension to hold a CD Minimise cost Choice of mateial Thickness of the case functional constaints geometic constaint 4
Mateial Indices The mateial index is a combination of mateial popeties which chaacteizes the pefomance of a mateial in a given application when mateial index maximises, pefomance is also maximised Stuctual elements pefom physical function. Aspects of the pefomance of the component can be descibed by its functional equiements (F), geomety (G) and mateial popeties (M). Thus, the pefomance metic (P) of a stuctual membe can be witten mathematically as P = f (F, G, M) eq.(1) In many situations, the function F, geomety G, and mateial paametes M, ae independent of each othe and ae sepaable. In these cases, eq.(1) can be e-witten as P = f 1 (F). f 2 (G). f 3 (M) eq.(2) stuctual index mateial index Optimum choice of mateial is often independent of geomety (G) and functional equiements (F). So, we can select optimum mateials independent of the details of F and G. The pefomance index can then be optimised by maximising o minimising the mateial popeties alone. Thus, the equation fo pefomance index P becomes P = f (M) = Mateial Index, M eq.(3) 5
Simple one-popety indices Design equiement Potective viso fo motocyclists Constaints Tanspaent - of optical quality Able to be moulded The mateial index: Choose mateial with lagest K 1c Objective As tough as possible maximize factue toughness K 1c The mateial index: Choose mateial with smallest C m Altenative objective As cheap as possible minimize mateial cost C m Multiple-popety indices E = Young s modulus = Density s y = Yield stength C m = mateial cost/kg 6
Example 1 Mateial index fo light, stong tie Maximize the pefomance of a tie od that must cay a tensile foce F* without failue and be as light as possible. The length L is specified but the coss-section aea A is not. maximizing pefomance minimizing the mass while still caying the load F* safely L stong stuctual membe loaded in tension 7
Objective function: minimise mass m = (A L) eq.(1) Constaints: length L is specified must not fail unde load F* constaint on aea: A must be sufficient to cay F* F* / A s f eq.(2) Fee vaiables: coss-sectional aea, A mateial choice m = (A L) F* / A s f eq.(1) eq.(2) Eliminate the fee vaiable A fom the objective equation, eq.(1) using the constaint equation, eq.(2) : m F*. L. s f mateial popeties functional constaint geometic constaint The pefomance P will be maximized (i.e., the mass m will be minimized) when the mateial popeties will be minimized, i.e., the lightest tie that will cay F* safely is that made of the mateial with the smallest value of ρ/σ f. It is moe usual when dealing with specific popeties to expess them in a fom fo which a maximum is sought. So the mateial index is: M = s f 8
Example 2 Mateial index fo light, stiff panel A panel is a flat slab, like a table top. Its length L and width b ae specified but its thickness h is fee. It is loaded in bending by a cental load F. The stiffness constaint S* equies that it must not deflect moe than δ. The objective is to achieve this with minimum mass, m. stong light panel loaded in bending Objective function: m = (A L) = (b h L) eq.(1) Constaints: Bending stiffness S must be at least S* C 1 E I C 1 E b h 3 S = = S* 12 L 3 L 3 Eliminating the fee vaiable h fom the objective function eq.(1) using eq.(2) eq.(2) C 1 = constant, depends only on the distibution of the loads (we don t need its value) I = second moment of aea fo a ectangula section = bh 3 /12 12 S* m = ( b 2/3 L 2 ) C 1 1/3 E 1/3 Mateial index: M = E 1/3 9
Poblem Design equiements fo the light, stiff beam Function Constaints Objective Fee vaiable Beam Stiffness S* specified Length L Section shape squae Minimise mass Choice of mateial Aea A of coss-section Second moment of aea, I, fo a squae beam b I = 4 = 12 A 2 Beam bending stiffness, S 12 12 S* m = (L 5/2 ) C 1 1/2 E 1/2 C 1 E I S* = L 3 M = E 1/2 What will be the mateial index fo a light, stong beam? 10
Minimising mateial cost: Cheap tie ods The object is to minimise cost athe than mass Mateial pice = C m Tk / kg Cost of mateial = m C m Mateial index fo cheap, stong tie od (stength and cost specified) s f M = (hee is eplaced by C m ) C m The stuctual index The efficiency of mateial usage in mechanically loaded components depends on the poduct of thee factos: the mateial index, a facto descibing section shape, and a stuctual index, which contains elements of the G and F Examination of the stuctual index often becomes useful when stuctues ae scaled in size In design fo minimum mass, a measue of the efficiency of the design is given by the quantity (m / L 3 ), which has the dimension of density the lowe this pseudodensity, the lighte the stuctue fo a given scale and, thus, the geate the stuctual efficiency 11
Fo the light stong tie od m F*. L. s f m L 3 F* L 2 s f Fo the light stiff panel stuctual index m L 3 12 C 1 1/3 b 2 S* L 3 1/3 E 1/3 mateial index Stuctual index has the dimensions of stess it is a measue of the intensity of loading Design popotions that ae optimal (by minimizing mateial usage) ae optimal fo stuctues of any size, povided they all have the same stuctual index. Next Class MME445: Lectue 21 Mateials selection: Case studies 12