Materials Selection Without
Shape - Case Studies
Table legs
Flywheels
Springs
Pressure vessels
Precision Devices
Buildings
Materials for Table Legs
Aim: cylindrical legs as light and thin as possible
constraint: resistance to buckling
m   r l
2
P  Pcrit 
 2 EI
l
12
 4P 
first eliminate r and get m  

 
14
4P
for slenderness r   3 
 

2
 3 Er 4
4l 2
  
l   1 2 
E 
2
M 1  E1 2 
14
1 2 1 
l   
E
M2  E
Materials Selection for
Table Legs
Materials Selection for
Table Legs
Material
M1
M2
Comment
(GPa1/2m3/Mg) (Gpa)
Wood
5-8
4-20 Outstanding M1; poor M2.
Cheap, traditional, reliable.
CFRP
4-8
30-200 Outstanding M1 and M2,
but expensive
GFRP
3.5-5.5
20-90 Much cheaper than CFRP,
but not as good.
Ceramics
4-8
150- Outstanding M1 and M2.
1000 Eliminated by brittleness
Materials for Flywheels
Flywheel store energy.
Currently made of lead, cast iron, steel,
composites - a strange diversity!
Aim: highest stored energy per unit weight,
without failing
1

2
Stored energy U  J  R 4t 2
2
4 2
Mass of flywheel disk m   R t
Materials for Flywheels
The quantity to be maximized
U 1 2 2
 R
m 4
The maximum principle stress in a spinning
disk of uniform thickness
 max
 3   2 2

 R    f
 8 
Eliminating R gives
f
M

  f
U 
2


m  S f 3     



Materials Selection for
Flywheels
Materials Selection for
Flywheels
Material
Ceramics
Composites:
CFRP
Composites:
GFRP
Beryllium
High strength steel
High strength Al
alloys
High strength Mg
alloys
Ti alloys
Lead alloys
Cast iron
M (kJ/kg)
200-2000
(compression only)
200-500
100-400
300
100-200
100-200
100-200
100-200
3
8-10
Comment
Brittle and weak in tension - eliminate
The best performance – a good choice
Almost as good as CFRP and cheaper.
Excellent choice.
Good but expensive, difficult to work and
toxic
All about equal in performance
Steel and Al-alloys cheaper than Mg- and
Ti alloys
High density makes these a good (and
traditional) selection when performance is
velocity-limited, not strength-limited
Materials for Springs
Spring is a device for storing
energy
Springs come in many shapes
and have many purposes, e.g.
axial springs, leaf springs,
helical springs, spiral springs,
torsion bars, etc.
Materials for Springs
Maximum energy stored in axial spring
1f
W
3 E
2
For torsion bars
1f
W
2 E
2
For leaf springs (bending deformation) W 
The geometry and form of the spring is
immaterial to the objective function
 2f
For efficient small spring M 1 
E
For efficient light spring
 2f   f


M2 
 
E   
2
2

1 f
4 E
E
 

Materials Selection for
Efficient Small Spring
Materials Selection for
Efficient Small Spring
Material
M1 (MJ/m3)
Ceramics
10-100
Comment
Brittle in tension; good only in compression.
Spring steel
10
The traditional choice; easily formed and heat treated.
Ti alloys
10
Expensive, corrosion-resistant.
CFRP
8
Comparable in performance with steel; expensive
GFRP
5
Almost as good as CFRP and much cheaper.
Glass
10
Brittle in torsion, but excellent if protected against
damage; very low loss factor
Nylon
3
The least good; but cheap and easily shaped, but high
loss factor
Rubber
20
Better than spring steel; but high loss factor
Materials Selection for
Efficient Light Spring
Materials Selection for
Efficient Light Spring
Material
M2 (MJ/kg)
Ceramics
5-100
Comment
Brittle in tension; good only in compression.
Spring steel
2
Poor, because of high density.
Ti alloys
3
Better than steel; corrosion-resistant; expensive.
CFRP
4
Better than steel; expensive
GFRP
3
Better than steel; less expensive than CFRP
Glass
10
Brittle in torsion, but excellent if protected
Wood
1-2
On a weight basis, wood makes good springs.
Nylon
2
Rubber
20-50
As good as steel, but high loss factor.
Outstanding; ten times better than steel, but with high
loss factor.
Materials for Safe
Pressure Vessels
Daily examples: aerosol can, boiler, etc.
Small vessels are designed to “yield before
break”; the distortion easy to detect and the
pressure released safely.
Large vessels are designed to “leak before
crack”; i.e. critical crack length for unstable
propagation is larger than vessel wall
thickness; leak is easily detected and releases
pressure gradually
Materials for Safe Small
Pressure Vessels
 The hoop stress of a thin-wall spherical vessel of
radius R is   pR   f
2t
 If the vessel contains no cracks or flaws of diameter
greater than 2ac, then the stress required for crack
CK Ic


propagation is
ac
 For “yield before crack”
CK Ic
 f
ac
2


2  K Ic 
ac  C 


f



 M1 
K Ic
f
Materials for Safe Large
Pressure Vessels
 Number and sizes of cracks in large pressure vessels
changes with time due to corrosion and cyclic
loading but NDT cannot be done very frequently,
therefore the “leak before crack” strategy.
CK Ic
 Through-thickness crack must still be stable  
t 2
 The wall thick enough for the pressure without
yielding t  pR
2
2 f
4  K Ic 
 Eliminate t, gives p  2 
C R   f 
2
K
 Maximize p means
M 2  Ic
M3   f
f
Materials for Safe
Pressure Vessels
Materials for Safe
Pressure Vessels
M1 = KIc/f
(m1/2)
M3 = f
(MPa)
Tough steels
>0.6
300
Tough copper
alloys
>0.6
120
Tough Al
alloys
>0.6
80
Ti-alloys
0.2
700
High strength
Al-alloys
0.1
500
GFRP/CFRP
0.1
Material
Comment
These are the pressure-vessel steels,
standard in this application.
OFHC hard drawn copper
1000 and 3000 series Al-alloys
High yield but low safety margin.
Good for light pressure vessels
500
Materials to Minimize Thermal
Distortion in Precision Devices
 The precision of measuring device
is limited by its stiffness and the
dimensional change or distortion
caused by temperature gradients.
 Elastic deflection of the Force Loop
is allowed, provided natural
vibration frequencies are high!
 Expansion is permissible of the
Force Loop, provided distortion
does not occur!
Materials to Minimize Thermal
Distortion in Precision Devices
The temperature can be equalized by heat
q  
dT
dx
conduction
The strain related to temperature is    T0  T 
d
dT   



  q
The strain gradient (distortion) dx
dx   
Distortion is minimized by maximizing M 1  

To reduce sensitivity to external vibration,
12
E
the natural frequencies must be high M 2 

Materials to Minimize Thermal
Distortion in Precision Devices
Materials to Minimize Thermal
Distortion in Precision Devices
M1 = /
M3 = E1/2/
(W/m)
(GPa1/2/(Mg/m3))
Diamond
5108
8.6
Outstanding M1 and M2; expensive.
Silicon
4107
6.0
Excellent M1 and M2; cheap.
Silicon carbide
2107
6.2
Excellent M1 and M2; potentially cheap.
Beryllium
107
9
Aluminium
107
3.1
Silver
2107
1.0
Copper
2107
1.3
Gold
2107
0.6
Tungsten
3107
1.1
Molybdenum
2107
1.3
Invar
3107
1.4
Silicon nitride
6106
6.0
Material
Comment
Less good than silicon or SiC.
Poor M1, but very cheap.
High density gives poor value of M2.
Better than copper, silver or gold, but less good than
silicon, SiC, diamond.
M1 less good than silicon
Structural Materials for
Buildings
 Roughly half the cost of a house is the cost of materials
of which it is made (HK$300/sq. ft in Hong Kong)
 The quantities is very large, e.g. around 20000 tonnes
for a large apartment block
 Structural materials must be stiff, strong and cheap
 The critical components in building are loaded either in
bending or as columns
E1 2
M1 
CR
 2f 3
M2 
C R
Structural Materials for
Buildings
Structural Materials for
Buildings
Structural Materials for
Buildings
M1 = E1/2/CR
M3 = f2/3/CR
(GPa1/2/(Mg/m3))
(GPa2/3/(Mg/m3))
Concrete
21
25
Brick
12
25
Stone
9
25
Soft woods
3
10
2.7
7
Material
Cast iron
Steel
Reinforced
concrete
3
9
15
Comment
Use in compression only
Tension and compression, with
freedom of section shape
Summary
 The case studies illustrate how the choice of
materials is narrowed
 Most designs make certain non-negotiable
demands on the material, e.g. operation
temperature, corrosion resistance, etc.
 The choice is further narrowed by maximizing
performance.
 Final choice also depends on more detailed
information of properties, manufacturing
processes, joining, finishing, etc.