2.3 - Materials Selection for best performance. Outline. Deriving performance indices. Performance maximizing criteria

Transcription

1 2 - Ashb Method Materials Selection for best performance Outline Deriving performance indices Performance maximizing criteria Selection with multiple constraints Resources: M.. Ashb, Materials Selection in Mechanical Design Butterworth Heinemann, 1999 Chapters 5 and 6 The Cambridge Material Selector (CS) software -- Granta Design, Cambridge (

2 Materials selection and function Design involves choosing a material, process and part shape to perform some function. unction dictates the choice of both materials and shape. In man cases materials choice is not directl dependent of shape. Performance Indices UNCTION Tie Beam ach combination of OBJCTIV Minimum cost unction Constraint ree variable CONSTRAINTS Has a characterising material index Shaft Column Mechanical, Thermal, lectrical... Minimum weight Maximum energ storage Minimum environ. impact Stiffness specified Strength specified atigue limit Geometr specified INDX M = σ Minimise this!

3 Performance Indices Performance Indices

4 Performance Indices Deriving Performance Indices: Procedure Identif the attribute to be maximized or minimized (weight, cost, energ, stiffness, strength, safet, environmental damage, etc.). Develop an equation for this attribute in terms of the functional requirements, the geometr and the material properties (the objective function). Identif the free (unspecified) variables. Identif the constraints; rank them in order of importance.

5 Deriving Performance Indices: Procedure Develop equations for the constraints (no ield; no fracture; no buckling, maximum heat capacit, cost below target, etc.). Substitute for the free variables from the constraints into the objective function. Group the variables into three groups: functional requirements,, geometr, G, and material properties, M, thus, we can write: p = f 1 () f 2 (G) f 3 (M) Read off the performance index, expressed as a quantit f 3 (M), to be minimized or maximized. Deriving Performance Indices: ight, strong tie Strong tie of length and minimum mass Area A = length = densit σ = ield strength

6 Deriving Performance Indices: ight, strong tie Minimize mass, m, of a solid clindrical tie rod of length, which carries a tensile force with safet factor S f. The mass is given b: m = A where A is the area of the cross-section and is the densit. This is called the unction The length and force are specified; the radius r is free. The section must, however, be sufficient to carr the tensile load, requiring that: / A = σ f / S f where σ f is the failure strength. Deriving Performance Indices: ight, strong tie liminating A between these two equations gives: Note the form of this result. m = ( S f ) ( ) { / σ f } or m = σ The first bracket contains the functional requirement that the specified load is safel supported. The second bracket contains the specified geometr (the length of the tie). The last bracket contains the material properties.

7 Deriving Performance Indices: ight, strong tie We want to minimize the performance, m, while meeting the functional and geometric requirements. This means we want the smallest value of { / σ f } or the largest value of M = σ f / Ashb defines this latter quantit as the performance index. Deriving Performance Indices: ight, strong tie unction Constraints Tie-rod Minimise mass m Area A ength is specified Must not fail under load Adequate fracture toughness Strong tie of length and minimum mass = length = densit σ = ield strength ree variables Material choice Section area A STP 1 Identif function, constraints, objective and free variables

8 Deriving Performance Indices: ight, strong tie unction Constraints Tie-rod Minimise mass m: m = A (1) Area A ength is specified Must not fail under load Adequate fracture toughness Strong tie of length and minimum mass = length = densit σ = ield strength ree variables Material choice Section area A STP 2 Define equation for objective -- the performance equation Deriving Performance Indices: ight, strong tie unction Constraints Tie-rod Minimise mass m: m = A (1) Area A ength is specified Must not fail under load Adequate fracture toughness quation for constraint on A: /A < σ (2) Strong tie of length and minimum mass = length = densit σ = ield strength ree variables Material choice Section area A STP 3 If the performance equation contains a free variable other than material, identif the constraint that limits it

9 Deriving Performance Indices: ight, strong tie unction Constraints Tie-rod Minimise mass m: m = A (1) Area A ength is specified Must not fail under load Adequate fracture toughness quation for constraint on A: /A < σ (2) Strong tie of length and minimum mass = length = densit σ = ield strength ree variables Material choice Section area A; eliminate in (1) using (2): m = σ STP 4 Use this constraint to eliminate the free variable in performance equation Deriving Performance Indices: ight, strong tie unction Constraints ree variables Tie-rod Minimise mass m: m = A (1) Area A ength is specified Must not fail under load Adequate fracture toughness Strong tie of length and minimum mass quation for constraint on A: /A < σ (2) STP 5 Material choice Section area A; eliminate in (1) using (2): m = σ = length = densit σ = ield strength Read off the combination of material properties that maximise performance

10 Deriving Performance Indices: ight, strong tie unction Constraints ree variables Tie-rod Minimise mass m: m = A (1) Area A ength is specified Must not fail under load Adequate fracture toughness quation for constraint on A: /A < σ (2) Material choice Section area A; eliminate in (1) using (2): m = σ Strong tie of length and minimum mass = length = densit σ = ield strength PRORMANC INDX Chose materials with smallest σ Deriving Performance Indices: ight, stiff tie unction Constraints Tie-rod Minimise mass m: m = A (1) Area A Stiffness of the tie S: A S = (2) Stiff tie of length and minimum mass = length = densit S = stiffness = Youngs Modulus ree variables Material choice Section area A; eliminate in (1) using (2): 2 m = S Chose materials with smallest

11 Performance Indices for weight: Tie Material properties -- the Phsicists view of materials, e.g. Cost, Densit, Modulus, Strength, ndurance limit, Thermal conductivit, C m σ σ e T- expansion coefficient, α λ the ngineers view of materials unction Stiffness Strength Tension (tie) Bending (beam) Bending (panel) Material indices -- : minimise mass / / 1/3 / /σ 2/3 /σ /σ Minimise these! Deriving Performance Indices: ight, stiff beam unction Beam (solid square section). b Constraint ree variables 12 S m = C Minimise mass, m, where: 2 m = A = b Stiffness of the beam S: CI S = 3 I is the second moment of area: 4 b I= 12 Material choice. dge length b. Combining the equations gives: 1/ 2 5 1/ 2 b = length = densit b = edge length S = stiffness I = second moment of area = Youngs Modulus Chose materials with smallest 1/ 2

12 Deriving Performance Indices: ight, strong beam unction Beam (solid square section). b Constraint ree variables Minimise mass, m, where: 2 m = A = b Must not fail under load σ M b/2 = I 3 b > 3 I is the second moment of area: 4 b I= 12 Material choice. dge length b. Combining the equations gives: b = length = densit b = edge length I = second moment of area σ = ield strength m 5/3 ( ) ( 3) σ 2/3 = 2/3 Chose materials with smallest 2/3 σ Performance Indices for weight: Beam Material properties -- the Phsicists view of materials, e.g. Cost, Densit, Modulus, Strength, ndurance limit, Thermal conductivit, C m σ σ e T- expansion coefficient, α λ the ngineers view of materials unction Stiffness Strength Tension (tie) Bending (beam) Bending (panel) Material indices -- : minimise mass / / 1/3 / /σ 2/3 /σ /σ Minimise these!

13 Deriving Performance Indices: ight, stiff panel unction Constraint ree variables Panel with given width w and length Stiffness of the panel S: CI S = 3 I is the second moment of area: 3 w t I= 12 1/ S w = 2 m C Minimise mass, m, where m = A = w t Material choice. Panel thickness t. Combining the equations gives: 1/3 t w w = width = length = densit t = thickness S = stiffness I = second moment of area = Youngs Modulus Chose materials with smallest 1/3 Deriving Performance Indices: ight, strong panel unction Constraint Panel with given width w and length Minimise mass, m, where m = A = w t Must not fail under load σ M t/2 = I I is the second moment of area: 3 wt I= 12 3 wt > 2 t w w = width = length = densit t = thickness I = second moment of area σ = ield strength ree variables Material choice. Panel thickness t. Combining the equations gives: m ( 3w) ( ) σ 3/2 = Chose materials with smallest σ

14 Performance Indices for weight: Panel Material properties -- the Phsicists view of materials, e.g. Cost, Densit, Modulus, Strength, ndurance limit, Thermal conductivit, C m σ σ e T- expansion coefficient, α λ the ngineers view of materials unction Stiffness Strength Tension (tie) Bending (beam) Bending (panel) Material indices -- : minimise mass / / 1/3 / /σ 2/3 /σ /σ Minimise these! Optimised selection using charts b UNCTION Tie Beam OBJCTIV Minimum cost b CONSTRAINTS Shaft Column Mechanical, Thermal, lectrical... Minimum weight Maximum energ storage Minimum environ. impact Stiffness specified Strength specified atigue limit Geometr specified INDX M = 1/ 2 Minimise this!

15 Optimised selection using charts og Index M = = 2 / M 2 ( ) = 2og( ) 2og( M) Contours of constant M are lines of slope 2 on an - chart Young s modulus, (GPa) Composites Woods 2 Ceramics 1 / 2 = Polmers oams lastomers Densit (Mg/m 3 ) C Metals Optimised selection using charts og Index M = = 2 / M 2 ( ) = 2og( ) 2og( M) = a x + b Young s modulus, (GPa) = og () Composites Woods Ceramics Polmers Contours of constant M are lines of slope 2 oams lastomers on an - chart = og() = Metals x = og () Densit (Mg/m 3 )

16 Optimised selection using charts og Index M = = 2 / M 2 ( ) = 2og( ) 2og( M) = a x + b Young s modulus, (GPa) = og () Composites Woods Ceramics Polmers 0.1 b = 1-2og(M) =1 oams lastomers M = 10 - = 0.31 = og() = Metals x = og () Densit (Mg/m 3 ) Optimised selection using charts og Index M = = 2 / M 2 ( ) = 2og( ) 2og( M) = a x + b b = 2-2og(M) =2 M = 10-1 = 0.1 Young s modulus, (GPa) = og () Ceramics Composites Woods 2 Polmers Metals x = og () Minimising M oams lastomers = og() = Densit (Mg/m 3 )

17 Performance Indices for weight: Stiffness UNCTION ach combination of unction Constraint ree variable Has a characterising material index OBJCTIV Minimum cost Minimum weight Maximum energ storage Minimum environ. impact CONSTRAINTS Stiffness specified Strength specified atigue limit Geometr specified INDX [ f ( ) ] M =, Minimise this! Performance Indices for weight: Stiffness Material properties -- the Phsicists view of materials, e.g. Cost, Densit, Modulus, Strength, ndurance limit, Thermal conductivit, C m σ σ e T- expansion coefficient, α λ the ngineers view of materials unction Stiffness Strength Tension (tie) Bending (beam) Bending (panel) Material indices -- : minimise mass / / 1/3 / /σ 2/3 /σ /σ Minimise these!

18 og Index Performance Indices for weight: Stiffness M = = 2 / M 2 ( ) = 2og( ) 2og( M) Contours of constant M are lines of slope 2 on an - chart Young s modulus, (GPa) Woods 1 Composites 2 3 C 1 / 3 = Ceramics 1 / 2 = Polmers oams lastomers Densit (Mg/m 3 ) C Metals = C Performance Indices for weight

19 Deriving Performance Indices for Cost and nerg To minimize Cost use the indices for minimum weight, replacing densit b C, where C is the cost per kg. M = / σ f M = C / σ f To minimize nerg use the indices for minimum weight, replacing densit b q, where q is the energ content per kg. M = / σ f M = q / σ f