ces edupack 2 2008

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© Granta Design, January 2008 1

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Material and process charts Mike Ashby, Engineering Department

Cambridge CB2 1PZ, UKVersion 1

1. Introduction

3. Process attribute charts

Chart P1 Material – Process compatibility matrix

Chart P2 Process – Shape compatibility matrixChart P3 Process/MassChart P4 Process/Section thicknessChart P5 Process/Dimensional toleranceChart P6 Process/Surface roughnessChart P7 Process/Economic batch size

Appendix: material indices

Table 1 Stiffness-limited design at minimum mass (cost …)Table 2 Strength-limited design at minimum mass (cost …)Table 3 Strength-limited design for maximum performanceTable 4 Vibration-limited designTable 5 Damage tolerant designTable 6 Thermal and thermo-mechanical design

2. Materials property charts

Chart 1 Young's modulus/Density

Chart 2 Strength/DensityChart 3 Young's modulus/StrengthChart 4 Specific modulus/Specific strengthChart 5 Fracture toughness/ModulusChart 6 Fracture toughness/StrengthChart 7 Loss coefficient/Young's modulusChart 8 Thermal conductivity/Electrical resistivityChart 9 Thermal conductivity/Thermal diffusivityChart 10 Thermal expansion/Thermal conductivityChart 11 Thermal expansion/Young's modulusChart 12 Strength/Maximum service temperatureChart 13 Coefficient of frictionChart 14 Normalised wear rate/HardnessChart 15a,b Approximate material pricesChart 16 Young's modulus/Relative costChart 17 Strength/Relative costChart 18a,b Approximate material energy contentChart 19 Young's modulus/Energy contentChart 20 Strength/Energy content


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Chart 1: Young's modulus, E and Density, ρ

This chart guides selection of materials for light, stiff,components. The moduli of engineering materials span arange of 107; the densities span a range of 3000. Thecontours show the longitudinal wave speed in m/s; naturalvibration frequencies are proportional to this quantity.The guide lines show the loci of points for which• E/ ρ = C (minimum weight design of stiff ties;minimum deflection in centrifugal loading, etc)

• E 1/2

/ ρ = C (minimum weight design of stiff beams,shafts and columns)

• E 1/3 / ρ = C (minimum weight design of stiff plates)

The value of the constantC increases as the lines aredisplaced upwards and to the left; materials offering thegreatest stiffness-to-weight ratio lie towards the upper lefthand corner. Other moduli are obtained approximatelyfrom E using• ν = 1/3 ; G = 3/8E; (metals, ceramics,glasses and glassy polymers)

E K ≈

• or 5.0≈ν ; 3 / E G ≈ ; (elastomers,rubbery polymers)

E 10 K ≈

where ν is Poisson's ratio,G the shear modulus and K the bulk modulus.


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Chart 2: Strength, σ f , against Density, ρ

This is the chart for designing light, strongstructures. The "strength" for metals is the 0.2% offsetyield strength. For polymers, it is the stress at which thestress-strain curve becomes markedly non-linear -typically, a strain of abut 1%. For ceramics and glasses, itis the compressive crushing strength; remember that this isroughly 15 times larger than the tensile (fracture) strength.For composites it is the tensile strength. For elastomers itis the tear-strength. The chart guides selection ofmaterials for light, strong, components. The guide linesshow the loci of points for which:

(a) σ f / ρ = C (minimum weight design of strongties; maximum rotational velocity of disks)

(b) σ f 2/3 / ρ = C (minimum weight design of strong

beams and shafts)

(c) σ f 1/2

/ ρ = C (minimum weight design of strong plates)

The value of the constantC increases as the lines aredisplaced upwards and to the left. Materials offering thegreatest strength-to-weight ratio lie towards the upper leftcorner.


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Chart 3: Young's modulus, E , against Strength,

σ f The chart for elastic design. The "strength" for metals

is the 0.2% offset yield strength. For polymers, it is the1% yield strength. For ceramics and glasses, it is thecompressive crushing strength; remember that this isroughly 15 times larger than the tensile (fracture) strength.For composites it is the tensile strength. For elastomers itis the tear-strength. The chart has numerous applicationsamong them: the selection of materials for springs, elastichinges, pivots and elastic bearings, and for yield-before- buckling design. The contours show the failure strain,

E / f σ . The guide lines show three of these; they are theloci of points for which:

(a) σ f /E = C (elastic hinges)

(b) σ f 2

/E = C (springs, elastic energystorage per unit volume)

(c) σ f 3/2 /E = C (selection for elasticconstants such as knife edges; elastic diaphragms,compression seals)

The value of the constantC increases as the lines aredisplaced downward and to the right.


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Chart 4: Specific modulus, E/ ρ , against Specific

strength, σ f / ρ

The chart for specific stiffness and strength. Thecontours show the yield strain, E / f σ . The qualificationson strength given for Charts 2 and 4 apply here also. Thechart finds application in minimum weight design of tiesand springs, and in the design of rotating components tomaximize rotational speed or energy storage, etc. Theguide lines show the loci of points for which

(a) σ f 2 /E ρ = C (ties, springs of minimumweight; maximum rotational velocity of disks)

(b) = C 2 / 13 / 2 f E / ρ σ

(c) σ f /E = C (elastic hinge design)

The value of the constantC increases as the lines aredisplaced downwards and to the right.


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Chart 5: Fracture toughness, K Ic , against

Young's modulus, E

The chart displays both the fracture toughness, ,

and (as contours) the toughness, . It allowscriteria for stress and displacement-limited failure criteria( and E / K ) to be compared. The guidelines showthe loci of points for which

c1 K

E / K G 2c1c1 ≈

c1 K c1

(a) K Ic

2 /E = C (lines of constant toughness,Gc;

energy-limited failure)

(b) K Ic /E = C (guideline for displacement-limited brittle failure)

The values of the constantC increases as the lines aredisplaced upwards and to the left. Tough materials lietowards the upper left corner, brittle materials towards the bottom right.


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Chart 6: Fracture toughness, K Ic , against

Strength, σ f

The chart for safe design against fracture. Thecontours show the process-zone diameter, given

approximately by K Ic2 / πσ f

2. The qualifications on"strength" given for Charts 2 and 3 apply here also. Thechart guides selection of materials to meet yield-before- break design criteria, in assessing plastic or process-zonesizes, and in designing samples for valid fracturetoughness testing. The guide lines show the loci of pointsfor which

(a) K Ic / σ f = C (yield-before-break)

(b) K Ic2 / σ f = C (leak-before-break)

The value of the constantC increases as the lines aredisplaced upward and to the left.


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Chart 7: Loss coefficient, η , against Young'smodulus, E

The chart gives guidance in selecting material for lowdamping (springs, vibrating reeds, etc) and for highdamping (vibration-mitigating systems). The guide lineshows the loci of points for which

(a) η E = C (rule-of-thumb for estimatingdamping in polymers)

The value of the constantC increases as the line is

displaced upward and to the right.


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Chart 8: Thermal conductivity, λ , against

Electrical conductivity, ρ e

This is the chart for exploring thermal and electricalconductivies (the electrical conductivityκ is thereciprocal of the resistivity e ρ ). For metals the two are proportional (the Wiedemann-Franz law):

e

1 ρ

κ λ =≈

because electronic contributions dominate both. But forother classes of solid thermal and electrical conductionarise from different sources and the correlation is lost.


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Chart 9: Thermal conductivity, λ , against

Thermal diffusivity, a The chart guides in selecting materials for thermal

insulation, for use as heat sinks and such like, both whenheat flow is steady, (λ ) and when it is transient (thermaldiffusivity a = λ/ρ C p where ρ is the density and C p the specific heat). Contours show values of the volumetricspecific heat, ρ C p = λ /a (J/m3K). The guidelines showthe loci of points for which

(a) λ /a = C (constant volumetric specific heat)(b) λ /a 1/2 = C (efficient insulation; thermal

energy storage)

The value of constantC increases towards the upper left.


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Chart 10: Thermal expansion coefficient, ,

against Thermal conductivity, λ The chart for assessing thermal distortion. The

contours show value of the ratioλ/α (W/m). Materialswith a large value of this design index show small thermaldistortion. They define the guide line

(a) λ/α = C (minimization of thermal distortion)

The value of the constantC increases towards the bottomright.


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Chart 11: Linear thermal expansion, , against

Young's modulus, EThe chart guides in selecting materials when thermal

stress is important. The contours show the thermal stress

generated, peroC temperature change, in a constrainedsample. They define the guide line

α E = C MPa/K (constant thermal stress peroK)

The value of the constantC increases towards the upper

right.


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Chart 12: Strength, σ f , against Maximum

service temperature T max

Temperature affects material performance in manyways. As the temperature is raised the material may creep,limiting its ability to carry loads. It may degrade ordecompose, changing its chemical structure in ways thatmake it unusable. And it may oxidise or interact in otherways with the environment in which it is used, leaving itunable to perform its function. The approximatetemperature at which, for any one of these reasons, it is

unsafe to use a material is called itsmaximum servicetemperature . Here it is plotted against strengthmaxT

f σ .

The chart gives a birds-eye view of the regimes ofstress and temperature in which each material class, andmaterial, is usable. Note that even the best polymers havelittle strength above 200oC; most metals become very soft by 800oC; and only ceramics offer strength above

1500oC.


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Chart 13: Coefficient of friction

When two surfaces are placed in contact under anormal load and one is made to slide over the other, aforce opposes the motion. This force is proportionalto but does not depend on the area of the surface –and this is the single most significant result of studies offriction, since it implies that surfaces do not contactcompletely, but only touch over small patches, the area ofwhich is independent of the apparent, nominal area of

contact . Thecoefficient friction

n F

s F

n F

n A µ is defined by

n

s F F =µ

Approximate values for µ for dry – that is, unlubricated – sliding of materials on a steel couterface are shown here.Typically, 5.0≈µ . Certain materials show much highervalues, either because they seize when rubbed together (asoft metal rubbed on itself with no lubrication, forinstance) or because one surface has a sufficiently lowmodulus that it conforms to the other (rubber on roughconcrete). At the other extreme are a sliding combinationswith exceptionally low coefficients of friction, such asPTFE, or bronze bearings loaded graphite, sliding on polished steel. Here the coefficient of friction falls as lowas 0.04, though this is still high compared with friction forlubricated surfaces, as noted at the bottom of the diagram.


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Chart 14: Wear rate constant, k a, againstHardness, H

When surfaces slide, they wear. Material is lost from both surfaces, even when one is much harder than theother. Thewear-rate , W, is conventionally defined as

slid cetan Disremoved material of Volume

W =

and thus has units of m2. A more useful quantity, for our

purposes, is the specific wear-rate

n AW =Ω

which is dimensionless. It increases with bearing pressure P (the normal force divided by the nominalarea ), such that the ratio

n F

n A

P F

W k

na

Ω ==

is roughly constant. The quantity (with units of

(MPa)-1) is a measure of the propensity of a sliding couplefor wear: high means rapid wear at a given bearing pressure. Here it is plotted against hardness, H .

ak

ak


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Chart 15 a and b: Approximate material prices,C m and ρ C m

Properties like modulus, strength or conductivity donot change with time. Cost is bothersome because it does.Supply, scarcity, speculation and inflation contribute tothe considerable fluctuations in the cost-per-kilogram of acommodity like copper or silver. Data for cost-per-kg aretabulated for some materials in daily papers and trade journals; those for others are harder to come by.Approximate values for the cost of materials per kg, and

their cost per m3, are plotted in these two charts. Mostcommodity materials (glass, steel, aluminum, and the

common polymers) cost between 0.5 and 2 $/kg. Becausethey have low densities, the cost/m3 of commodity polymers is less than that of metals.


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Chart 16: Young's modulus, E , against Relativecost, C Rρ

In design for minimum cost, material selection isguided by indices that involve modulus, strength and cost per unit volume. To make some correction for theinfluence of inflation and the units of currency in whichcost is measured, we define arelative cost per unit volume

R ,vC

rod steel mild of Density xkg / Cost material of Density xkg / Cost

C R ,v =

At the time of writing, steel reinforcing rod costs aboutUS$ 0.3/kg.

The chart shows the modulus E plotted againstrelative cost per unit volume ρ R ,vC where ρ is thedensity. Cheap stiff materials lie towards the top left.Guide lines for selection materials that are stiff and cheapare plotted on the figure.

The guide lines show the loci of points for which

(a) C C / E R ,v = ρ (minimum cost design ofstiff ties, etc)

(b) (minimum costdesign of stiff beams and columns)

C C / E R ,v2 / 1 = ρ

(c) (minimum costdesign of stiff plates)

C C / E R ,v3 / 1 = ρ

The value of the constantC increases as the lines aredisplayed upwards and to the left. Materials offering thegreatest stiffness per unit cost lie towards the upper leftcorner.


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Chart 17: Strength, σ f , against Relative cost,

C Rρ

Cheap strong materials are selected using this chart.It shows strength, defined as before, plotted againstrelative cost per unit volume, defined on chart 16. Thequalifications on the definition of strength, given earlier,apply here also.

It must be emphasised that the data plotted here andon the chart 16 are less reliable than those of other charts,

and subject to unpredictable change. Despite this direwarning, the two charts are genuinely useful. They allowselection of materials, using the criterion of "function perunit cost".

The guide lines show the loci of points for which

(a) C C / R ,v f = ρ σ (minimum cost design ofstrong ties, rotating disks, etc)

(b) (minimum cost design of

strong beams and shafts)

C C / R ,v3 / 2

f = ρ σ

(c) (minimum cost design of

strong plates)

C C / R ,v2 / 1

f = ρ σ

The value of the constantsC increase as the lines aredisplaced upwards and to the left. Materials offering thegreatest strength per unit cost lie towards the upper leftcorner.


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Charts 18 a and b: Approximate energy content

per unit mass and per unit volumeThe energy associated with the production of one kilogramof a material is , that per unit volume is ρ where p H p H

ρ is the density of the material. These two bar-chartsshow these quantities for ceramics, metals, polymers andcomposites. On a “per kg” basis (upper chart) glass, thematerial of the first container, carries the lowest penalty.Steel is higher. Polymer production carries a much higher burden than does steel. Aluminum and the other lightalloys carry the highest penalty of all. But if these samematerials are compared on a “per m3” basis (lower chart)the conclusions change: glass is still the lowest, but nowcommodity polymers such as PE and PP carry alower burden than steel; the composite GFRP is only a littlehigher.


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Chart 19: Young's modulus, E , against Energy

content, H pρ The chart guides selection of materials for stiff,

energy-economic components. The energy content perm3, ρ p H is the energy content per kg, , multiplied by the density ρ . The guide-lines show the loci of pointsfor which

p H

(a) C H / E p = ρ (minimum energy designof stiff ties; minimum deflection in centrifugal loadingetc)

(b) (minimum energy designof stiff beams, shafts and columns)

C H / E p2 / 1 = ρ

(c) (minimum energy designof stiff plates)

C H / E p3 / 1 = ρ

The value of the constantC increases as the lines aredisplaced upwards and to the left. Materials offering thegreatest stiffness per energy content lie towards the upperleft corner.

Other moduli are obtained approximately from Eusing

• ν = 1/3 ; G = 3/8E; (metals, ceramics,glasses and glassy polymers)

E K ≈

• or 5.0≈ν ; 3 / E G ≈ ; (elastomers,rubbery polymers)

E 10 K ≈

where ν is Poisson's ratio, G the shear modulus and K the bulk modulus.


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Chart 20: Strength, f , against Energy content, H pρ

The chart guides selection of materials for strong,energy-economic components. The "strength" for metalsis the 0.2% offset yield strength. For polymers, it is thestress at which the stress-strain curve becomes markedlynon-linear - typically, a strain of about 1%. For ceramicsand glasses, it is the compressive crushing strength;remember that this is roughly 15 times larger than thetensile (fracture) strength. For composites it is the tensilestrength. For elastomers it is the tear-strength. The energycontent per m3, ρ p H is the energy content per kg, ,multiplied by the density ρ . The guide lines show theloci of points for which

p H

(a) C H / p f = ρ σ (minimum energy design ofstrong ties; maximum rotational velocity of disks)

(b) (minimum energy design of

strong beams and shafts)

C H / p3 / 2

f = ρ σ

(c) (minimum energy design of

strong plates)

C H / p2 / 1

f = ρ σ

The value of the constantC increases as the lines aredisplaced upwards and to the left. Materials offering thegreatest strength per unit energy content lie towards theupper left corner.


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Process attribute charts

Process classes and class members

A process is a method of shaping, finishing or joining a material.Sand casting ,injection molding , fusion welding and polishing are all processes. The choice, for agiven component, depends on the material of which it is to be made, on its size, shapeand precision, and on how many are required

The manufacturing processes of engineering fall into nine broad classes:

Process classes Casting (sand, gravity, pressure, die, etc)Pressure molding (direct, transfer, injection, etc)Deformation processes (rolling, forging, drawing, etc)Powder methods (slip cast, sinter, hot press, hip)Special methods (CVD, electroform, lay up, etc)Machining (cut, turn, drill, mill, grind, etc)Heat treatment (quench, temper, solution treat, age, etc)Joining (bolt, rivet, weld, braze, adhesives)Surface finish (polish, plate, anodise, paint)

Each process is characterised by a set ofattributes: the materials it can handle, theshapes it can make and their precision, complexity and size and so forth. ProcessSelection Charts map the attributes, showing the ranges of size, shape, material, precision and surface finish of which each class of process is capable. They are used inthe way described in "Materials Selection in Mechanical Design". The procedure doesnot lead to a final choice of process. Instead, it identifies a subset of processes whichhave the potential to meet the design requirements. More specialised sources must then be consulted to determine which of these is the most economical.

The hard-copy versions, shown here, are necessarily simplified, showing only a limitednumber of processes and attributes. Computer implementation, as in the CES Edusoftware, allows exploration of a much larger number of both.


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Chart P1 The Process – Material matrix .A given process can shape, or join, or finish some materials but not others. The matrix shows the links between materialand process classes. A red dot indicates that the pair arecompatible. Processes that cannot shape the material of choiceare non-starters. The upper section of the matrix describesshaping processes. The two sections at the bottom cover joining and finishing.


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Chart P2 The Process – Shape matrix. Shape is the most difficult attribute to characterize. Many processes involve rotation or translation of a tool or of theworkpiece, directing our thinking towards axial symmetry,translational symmetry, uniformity of section and such

like. Turning creates axisymmetric (or circular) shapes;extrusion, drawing and rolling make prismatic shapes, both circular and non-circular.Sheet-forming processesmake flat shapes (stamping) or dished shapes (drawing).Certain processes can make 3-dimensional shapes, andamong these some can make hollow shapes whereas otherscannot.

The process-shape matrix displays the links betweenthe two. If the process cannot make the desired shape, itmay be possible to combine it with a secondary process togive a process-chain that adds the additional features:casting followed by machining is an obvious example.

Information about material compatibility is included atthe extreme left.


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Chart P3 The Process – Mass-range chart.

The bar-chart shows the typical mass-range ofcomponents that each processes can make. It is one offour, allowing application of constraints on size (measured by mass), section thickness, tolerance and surfaceroughness. Large components can be built up by joiningsmaller ones. For this reason the ranges associated with joining are shown in the lower part of the figure. Inapplying a constraint on mass, we seek single shaping- processes or shaping-joining combinations capable ofmaking it, rejecting those that cannot.


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Chart P4 The Process – Section thickness chart.The bar-chart on the right allows selection to meetconstraints on section thickness. Surface tension and heat-flow limit the minimum section of gravity cast shapes.The range can be extended by applying a pressure or by pre-heating the mold, but there remain definite lowerlimits for the section thickness. Limits on rolling andforging-pressures set a lower limit on thickness achievable by deformation processing. Powder-forming methods are

more limited in the section thicknesses they can create, butthey can be used for ceramics and very hard metals thatcannot be shaped in other ways. The section thicknessesobtained by polymer-forming methods – injectionmolding, pressing, blow-molding, etc – depend on theviscosity of the polymer; fillers increase viscosity, furtherlimiting the thinness of sections. Special techniques,which include electro-forming, plasma-spraying andvarious vapour – deposition methods, allow very slendershapes.


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Chart P5 The Process – Tolerance chart. No process can shape a partexactly to a specified dimension.Some deviation ∆ x from a desired dimension x is permitted;it is referred to as thetolerance, T , and is specified as

mm, or as 01.0 mm. This bar chartallows selection to achieve a given tolerance.

1.0100 x ±= 001.050 x+−=

The inclusion of finishing processes allows simple processchains to be explored


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Chart P6 The Process – Surface roughness chart. The surface roughness R, is measured by the root-mean-square amplitude of the irregularities on the surface. It isspecified as 100 R < µm (the rough surface of a sandcasting) or 01.0 R < µm (a highly polished surface).The bar chart on the right allows selection to achieve agiven surface roughness.

The inclusion of finishing processes allows simple

process chains to be explored.


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Chart P7 The Process – Economic batch-size chart.Process cost depends on a large number of independentvariables. The influence of many of the inputs to the costof a process are captured by a single attribute: theeconomic batch size . A process with an economic batchsize with the range B1 – B2 is one that is found byexperience to be competitive in cost when the output liesin that range.


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Table A2 Strength-limited design at minimum mass (cost, energy, eco-impact)

FUNCTION and CONSTRAINTS Maximize

TIE (tensile strut) stiffness, length specified; section area free ρ σ / f

SHAFT (loaded in torsion)

load, length, shape specified, section area free

load, length, outer radius specified; wall thickness free

load, length, wall-thickness specified; outer radius free

ρ σ / 3 / 2 f

ρ σ / f

ρ σ / 2 / 1 f

BEAM (loaded in bending)

load, length, shape specified; section area free

load length, height specified; width free

load, length, width specified; height free

ρ σ / 3 / 2 f

ρ σ / f

ρ σ / 2 / 1 f

COLUMN (compression strut)load, length, shape specified; section area free ρ σ / f

PANEL (flat plate, loaded in bending)stiffness, length, width specified, thickness free

ρ σ / 2 / 1 f

PLATE (flat plate, compressed in-plane, buckling failure)collapse load, length and width specified, thickness free

ρ σ / 2 / 1 f

CYLINDER WITH INTERNAL PRESSUREelastic distortion, pressure and radius specified; wall

thickness free

ρ σ / f

SPHERICAL SHELL WITH INTERNAL PRESSUREelastic distortion, pressure and radius specified, wallthickness free

ρ σ / f

FLYWHEELS, ROTATING DISKSmaximum energy storage per unit volume; given velocitymaximum energy storage per unit mass; no failure

ρ

ρ σ / f

Table A1 Stiffness-limited design at minimum mass (cost, energy, eco-impact)

FUNCTION and CONSTRAINTS Maximize

TIE (tensile strut)stiffness, length specified; section area free ρ / E

SHAFT (loaded in torsion)

stiffness, length, shape specified, section area freestiffness, length, outer radius specified; wall thickness freestiffness, length, wall-thickness specified; outer radius free

ρ / G 2 / 1

ρ / G

ρ / G 3 / 1

BEAM (loaded in bending)

stiffness, length, shape specified; section area freestiffness, length, height specified; width freestiffness, length, width specified; height free

ρ / E 2 / 1

ρ / E

ρ / E 3 / 1

COLUMN (compression strut, failure by elastic buckling) buckling load, length, shape specified; section area free ρ / E 2 / 1

PANEL (flat plate, loaded in bending)stiffness, length, width specified, thickness free ρ / E 3 / 1

PLATE (flat plate, compressed in-plane, buckling failure)collapse load, length and width specified, thickness free ρ / E 3 / 1

CYLINDER WITH INTERNAL PRESSUREelastic distortion, pressure and radius specified; wall thickness free ρ / E

SPHERICAL SHELL WITH INTERNAL PRESSUREelastic distortion, pressure and radius specified, wall thickness free ρ ν )1 /( E −


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Table A4 Vibration-limited design

FUNCTION and CONSTRAINTS MaximizeTIES, COLUMNS maximum longitudinal vibration frequencies ρ / E

BEAMS, all dimensions prescribedmaximum flexural vibration frequenciesBEAMS, length and stiffness prescribedmaximum flexural vibration frequencies

ρ / E 2 ρ / E / 1

PANELS, all dimensions prescribedmaximum flexural vibration frequenciesPANELS, length, width and stiffness prescribedmaximum flexural vibration frequencies

ρ / E

3 ρ / E / 1

TIES, COLUMNS, BEAMS, PANELS, stiffness prescribed minimum longitudinal excitation from external drivers, tiesminimum flexural excitation from external drivers, beamsminimum flexural excitation from external drivers, panels

ρ η / E

ρ η / E 2 / 1

ρ η / E 3 / 1

Table A3 Strength-limited design: springs, hinges etc for maximum performance

FUNCTION and CONSTRAINTS MaximizeSPRINGS maximum stored elastic energy per unit volume; no failuremaximum stored elastic energy per unit mass; no failure

E / 2 f σ

ρ σ E / 2 f

ELASTIC HINGESradius of bend to be minimized (max flexibility without failure) E / f σ

KNIFE EDGES, PIVOTSminimum contact area, maximum bearing load 23 f E / σ and H

COMPRESSION SEALS AND GASKETSmaximum conformability; limit on contact pressure E / 2 / 3 f σ and E / 1

DIAPHRAGMSmaximum deflection under specified pressure or force E / 2 / 3 f σ

ROTATING DRUMS AND CENTRIFUGESmaximum angular velocity; radius fixed; wall thickness free ρ σ / f


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FUNCTION and CONSTRAINTS MaximizeTHERMAL INSULATION MATERIALSminimum heat flux at steady state; thickness specifiedminimum temp rise in specified time; thickness specifiedminimize total energy consumed in thermal cycle (kilns, etc)

λ / 1 λ ρ / C a / 1 p=

pC / 1 / a ρ λ λ =

THERMAL STORAGE MATERIALSmaximum energy stored / unit material cost (storage heaters)maximize energy stored for given temperature rise and time

m p C / C

pC a / ρ λ λ =

PRECISION DEVICESminimize thermal distortion for given heat flux / aλ

THERMAL SHOCK RESISTANCEmaximum change in surface temperature; no failure α σ E / f

HEAT SINKSmaximum heat flux per unit volume; expansion limitedmaximum heat flux per unit mass; expansion limited

∆λ / ∆ ρ λ /

HEAT EXCHANGERS (pressure-limited)

maximum heat flux per unit area; no failure under p∆

maximum heat flux per unit mass; no failure under p∆

f σ λ

ρ σ λ / f

Table A6 Thermal and thermo-mechanical designTable A5 Damage-tolerant design

FUNCTION and CONSTRAINTS MaximizeTIES (tensile member) maximum flaw tolerance and strength, load-controlled designmaximum flaw tolerance and strength, displacement-controlmaximum flaw tolerance and strength, energy-control

c1 K and f σ E / K c1 and f σ

E / K 2c1 and f σ

SHAFTS (loaded in torsion) maximum flaw tolerance and strength, load-controlled design

maximum flaw tolerance and strength, displacement-controlmaximum flaw tolerance and strength, energy-control

c1 K and f σ E / K

c1and

f σ

E / K 2c1 and f σ

BEAMS (loaded in bending) maximum flaw tolerance and strength, load-controlled designmaximum flaw tolerance and strength, displacement-controlmaximum flaw tolerance and strength, energy-control

c1 K and f σ E / K c1 and f σ

E / K 2c1 and f σ

PRESSURE VESSEL

yield-before-breakleak-before-break

f c1 / K

f 2c1 / K σ


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Table A7 Electro-mechanical design

FUNCTION and CONSTRAINTS MaximizeBUS BARSminimum life-cost; high current conductor me C / 1 ρ ρ

ELECTRO-MAGNET WINDINGSmaximum short-pulse field; no mechanical failuremaximize field and pulse-length, limit on temperature rise

f σ

e p / C ρ ρ

WINDINGS, HIGH-SPEED ELECTRIC MOTORSmaximum rotational speed; no fatigue failureminimum ohmic losses; no fatigue failure

ee / ρ σ e / 1 ρ

RELAY ARMSminimum response time; no fatigue failureminimum ohmic losses; no fatigue failure

ee E / ρ σ

e2e E / ρ σ


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Physical constants and conversion of units Absolute zero temperatureAcceleration due to gravity, g

Avogadro’s number, NABase of natural logarithms, eBoltsmann’s constant, kFaraday’s constant kGas constant, R Planck’s constant, hVelocity of light in vacuum, cVolume of perfect gas at STP

-273.2oC9.807m/s2

6.022 x 10232.7181.381 x 10-23J/K9.648 x 104 C/mol8.314 J/mol/K6.626 x 10-34 J/s2.998 x 108 m/s22.41 x 10-3 m3/mol

Angle,θ 1 rad 57.30oDensity, ρ 1 lb/ft3 16.03 kg/m3Diffusion Coefficient, D 1cm3/s 1.0 x 10-4m2/sEnergy, U See oppositeForce, F 1 kgf

1 lbf1 dyne

9.807 N4.448 N1.0 x 10-5 N

Length, l 1 ft1 inch1 Å

304.8 mm25.40 mm0.1 nm

Mass, M 1 tonne1 short ton1 long ton1 lb mass

1000 kg908 kg1107 kg0.454 kg

Power, P See oppositeStress,σ See oppositeSpecific Heat, Cp 1 cal/gal.oC

Btu/lb.oF4.188 kJ/kg.oC4.187 kg/kg.oC

Stress Intensity, K 1c 1 ksi √in 1.10 MN/m3/2Surface Energyγ 1 erg/cm2 1 mJ/m2Temperature, T 1oF 0.556oKThermal Conductivityλ 1 cal/s.cm.oC

1 Btu/h.ft.oF418.8 W/m.oC1.731 W/m.oC

Volume, V 1 Imperial gall1 US gall

4.546 x 10-3m33.785 x 10-3m3

Viscosity, η 1 poise1 lb ft.s

0.1 N.s/m20.1517 N.s/m2

Conversion of units – stress and pressure*

MPa dyn/cm 2 lb.in 2 kgf/mm 2 bar long ton/in 2

MPa 1 107 1.45 x 102 0.102 10 6.48 x 10-2

dyn/cm 2 10-7 1 1.45 x 10-5 1.02 x 10-8 10-6 6.48 x 10-9

lb/in 2 6.89 x 10-3 6.89 x 104 1 703 x10-4 6.89 x 10-2 4.46 x 10-4

kgf/mm 2 9.81 9.81 x 107 1.42 x 103 1 98.1 63.5 x 10-2

bar 0.10 106 14.48 1.02 x 10-2 1 6.48 x 10-3

long ton/ in 2 15.44 1.54 x 108 2.24 x 103 1.54 1.54 x 102 1

Conversion of units – energy*

J erg cal eV Btu ft lbf

J 1 107 0.239 6.24 x 1018 9.48 x 10-4 0.738

erg 10-7 1 2.39 x 10-8 6.24 x 1011 9.48 x 10-11 7.38 x 10-8

cal 4.19 4.19 x 107 1 2.61 x 1019 3.97 x 10-3 3.09

eV 1.60 x 10-19 1.60 x 10-12 3.38 x 10-20 1 1.52 x 10-22 1.18 x 10-19

Btu 1.06 x 103 1.06 x 1010 2.52 x 102 6.59 x 1021 1 7.78 x 102

ft lbf 1.36 1.36 x 107 0.324 8.46 x 1018 1.29 x 10-3 1

Conversion of units – power*

kW (kJ/s) erg/s hp ft lbf/s

kW (kJ/s) 1 10-10 1.34 7.38 x 102

erg/s 10-10 1 1.34 x 10-10 7.38 x 10-8

hp 7.46 x 10-1 7.46 x 109 1 15.50 X 102

Ft lbf/s 1.36 X 10-3 1.36 X 107 1.82 X 10-3 1


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