2 - Ashby Method
2.5 - Case studies
Outline
• Materials for oars
• Materials for flywheel
• Materials for spark plug insulators
• Air cylinders for trucks
Resources:
• M. F. Ashby, “Materials Selection in Mechanical Design” Butterworth Heinemann, 1999
Chapter 6
• The Cambridge Material Selector (CES) software -- Granta Design (www.grantadesign.com)
Materials for oars
Materials for oars
Specification
Function
Objectives
Constraints
Oar: light, stiff beam
Oars are stiffness-limited,
minimum weight, designs
Minimum weight
• Stiffness S specified
• Cost within reason
• Toughness adequate
Free
variables
• Cross-section area
• Material
Multiple constraints problem
Materials for oars
F
Function
Beam (solid square section).
Objective
Minimise mass, m, where:
b
b
m = A L ρ = b2 L ρ
Constraint
L
Stiffness of the beam S:
m = mass
A = area
L = length
ρ = density
b = edge length
S = stiffness
I = second moment of area
E = Youngs Modulus
CEI
S=
L3
ρ
I is the second moment of area:
I=
Free variables
b4
12
• Material choice.
• Edge length b. Combining the equations gives:
1/ 2
⎛ 12 S L5 ⎞
⎟
m=⎜
⎜ C ⎟
⎝
⎠
⎛ ρ ⎞
⎜ 1/ 2 ⎟
⎝E ⎠
⎛ ρ ⎞
Chose materials with smallest ⎜ 1/ 2 ⎟
⎝E ⎠
Materials for oars
Specification
Function
Objectives
Constraints
Oars are stiffness-limited,
minimum weight, designs
Oar: light, stiff beam
Minimum weight
• Stiffness S specified
• Cost within reason
• Toughness adequate
Free
variables
• Cross-section area
• Material
Multiple constraints problem
Primary constraint
Performance maximizing criteria
Material index, set by objective
Material limits, set by constraints
Fracture toughness
1/2
K lc ≥ 10 MPa.m
Minimise M =
ρ
E1/ 2
Density
Modulus
Materials for oars -- hard copy chart
M=
3 4
ρ
E1/ 2
The selection
1 Wood: Cheap, light, but
variable
2
Search
area
2 CFRP: The best choice,
more control of design
1
3 Beryllium: cost (and
toxicity) rules it out
4 Ceramics: But fracture
toughness inadequate
Oars: real life
Wood: Sitka spruce,
quartered and laminated
CFRP: Tailored to rower’s
specification
Flywheels
Real flywheel -- filament-wound GFRP
Case
Rotor
Burst shield
Specification for a flywheel
Specification
Function
Objectives
Flywheel
Maximum energy/weight
• Must not disintegrate
Constraints
• Cost within reason
• Fr. Toughness adequate
Free
variables
• Material
Multiple constraints problem
Material index and constraints for flywheels
• Maximise energy/unit weight at maximum velocity
Angular velocity
Moment of inertia
Energy
Ιω 2
U =
2
Mass
m = π R 2 tρ
Must not fail
σ =
Maximise
(1)
(2)
Density
(3 + ν ) ρ R 2 ω 2
8
≤ σy
Strength
⎛ (3 + ν ) 1 ⎞
≈ ⎟
⎜
8
2⎠
⎝
σ
U
≈ y
m
2ρ
Energy/mass
⎛
πρ R 4 t ⎞
⎟⎟
⎜⎜ Ι =
2
⎠
⎝
M =
(3)
(4)
σy
ρ
Additional constraint: Fracture toughness > 15 MPa.m1/2
Material for flywheels
1
σ
M= y
ρ
2
The selection
1 Ceramics: But fracture
toughness inadequate
2 CFRP, GFRP: The best
choice
Spark-plug insulators
Specification
Spark-plug insulator
Function
Insulator
Objectives
Minimise material cost
Body
shell
• Good electrical insulator
Constraints
• Breakdown V > 20 MV/m
• Tolerate temp. > 600 C
• Resist thermal shock of 100 °C
Free
variables
Central
electrode
• Material
Multiple constraints problem
Analysis for spark plug insulators
Constraints: R > 1.1015 μohm.cm
Vmax > 20 MV/m
Tmax > 600 °C
Temperature change
Thermal Strain ε = α ΔT
(1)
T-expansion coefficient
Insulator
Young’s modulus
Stress:
σ=Eε
(2)
Elastic limit
Fracture when:
σ = σel in tension
(3)
Combining (1) to (3) gives allowable T-shock:
ΔTmax =
Body
shell
σ el
Eα
Impose ΔT > 100 °C as a constraint, then minimise material cost
Central
electrode
Materials for spark-plug insulators
resistance,
shock
Thermal
shock
resistance (C)ΔT (C)
Thermal
10000
1000
The selection:
alumina
Thermal shock - cost
Search
region
Silica
Silicon Nitrides
Glass Ceramics
Aluminas
Aluminium Nitrides
100
Additional constraints:
Electrical resistivity > 1.1015 μohm.cm
Breakdown potential > 20 MV/m
Maximum service temperature > 873 K
10
1
10
100
Price (typical) (GBP/kg)
Approximate material cost ($/kg)
Case study: Air cylinders for trucks
Design goal: lighter, cheap air cylinders for trucks
Compressed air tank
Design requirements for the air cylinder
t
Specification
Function
Pressure vessel
Objectives
• Minimise mass
• Minimise cost
•
•
•
•
•
Constraints
Free
variables
Pressure p
Dimensions L, R, pressure p, given
Must not corrode in water or oil
Working temperature -50 to +1000C
Safety: must not fail by yielding
Adequate toughness: K1c > 15 MPa.m1/2
2R
L
R = radius
L = length
ρ = density
p = pressure
t = wall thickness
• Wall thickness, t;
• Choice of material
Performance metrics for the air cylinder
• Thin-walled pressure vessels are treated as membranes. The
approximation is reasonable when t < b/4
• The stresses in the wall do not vary significantly with radial distance, r
σr
σz
σθ
σθ =
p ⋅ 2bL p b
=
2tL
t
σr = −
σz =
pe + pi
p
=−
2
2
p ⋅ πb2 pb
=
2πbt 2t
b
⎛
⎜t <
4
⎝
⇒
b
⎞
> 4⎟
t
⎠
Performance metrics for the air cylinder
t
Volume of material in cylinder wall
Objective 1
(
)
Pressure p
m = 2πR L t + 4πR2t ρ
2R ⎞
⎛
= 2πR L t⎜1 +
⎟
L ⎠
⎝
Aspect ratio
Constraint
L
Q
R = radius
L = length
ρ = density
p = pressure
t = wall thickness
σy= yield strength
Sf = safety factor
Q = aspect ratio 2R/L
σ
pR
σ=
< y
t
Sf
Eliminate t to give:
Metric 1
Objective 2
Metric 2
2R
⎡ρ⎤
m = 2 πR 2 L (1 + Q) p Sf ⎢ ⎥
⎢⎣ σ y ⎥⎦
C = Cm m
⎡C ρ ⎤
C = 2 πR 2 L (1 + Q) p Sf ⎢ m ⎥
⎢⎣ σ y ⎥⎦
Substitution: Relative performance metrics
• This is a problem of substitution. The tank is currently made of a plain
carbon steel.
• The mass m and cost C of a tank made from an alternative material M,
differs (for the same strength) from one made of Mo by the factors
C ⎛⎜ Cm ρ ⎞⎟ ⎛⎜ σ y,o ⎞⎟
.
=
Co ⎜⎝ σ y ⎟⎠ ⎜⎝ Cm,o ρo ⎟⎠
m ⎛⎜ ρ ⎞⎟ ⎛ σ y,o ⎞
⎟
.⎜
=
mo ⎜⎝ σ y ⎟⎠ ⎜⎝ ρo ⎟⎠
For plain carbon steel
ρo /σ y,o = 0.03
• Explore the trade-off between
and
m
and
mo
C
Co
Cm,o ρo /σ y,o = 0.02
Trade-off plot
Trade-off
surface
10
Additional
constraints:
K1c >15 MPa.m1/2
Cu-alloys T
max > 373 K
Zn-alloys
Tmin < 223 K
Water: good +
Organics: good +
1
Mild steel
Ni-alloys
High-C steel
Low alloy steel
Al-alloys
0.1
Mg-alloys
Al-SiC Composite
0.1
GFRP
1
CFRP
Ti-alloys
10
100
Price * Density / Elastic limit
Cost relative to plain carbon steel, C/Co
Trade-off plot
The four sectors of a trade-off plot for substitution
Lead alloys
Mass relative
to plain
carbon
Density
/ Elastic
limit steel, m/mo
Mass relative
to plain
carbon
Density
/ Elastic
limit steel, m/mo
Lead alloys
10
D. Worse by
both metrics
B. Cheaper
but heavier
Cu-alloys
Zn-alloys
1
Mild steel
Ni-alloys
High-C steel
Low alloy steel
C. Lighter
but more
expensive
Al-alloys
0.1
M g-alloys
A. Better by
both metrics
0.1
Al-SiC Composite
GFRP
1
CFRP
10
T i-alloys
1 00
Price * Density / Elastic limit
Cost relative to plain carbon steel, C/Co
High-C steel, low alloy steel and Al-alloys all offer reduction in mass and cost
Problem solution
• Selected material M (or materials) to substitute carbon steel
ρ , σy , Cm
• Mass m and cost C of a tank made from the alternative material M
m ⎛⎜ ρ ⎞⎟ ⎛ σ y,o ⎞
⎟
.⎜
=
mo ⎜⎝ σ y ⎟⎠ ⎜⎝ ρo ⎟⎠
C ⎛⎜ Cm ρ ⎞⎟ ⎛⎜ σ y,o ⎞⎟
=
.
Co ⎜⎝ σ y ⎟⎠ ⎜⎝ Cm,o ρo ⎟⎠
m
ρo /σ y,o = 0.03
Cm,o ρo /σ y,o = 0.02
• Free variable (thickness)
σ=
σ
pR
< y
t
Sf
t
C