Selection of Materials and Shape

… when choosing among different materials with different shapes and when the design is one in which shape matters…

Introduction The concept of performance

indices extended to include SHAPE.

Maximize the performance by choosing the materials and shape in which the materials are available or feasible

The best material-shape combination depends on the mode of loading.

Mode of Loading and Shape

In axial tension, shape not important

In bending, I-beam or hollow box are better than solid sections

In torsion, circular tube is the best

A shape factor () is defined for each mode of loading

Shape Factor A shape factor is a dimensionless number

which characterizes the efficiency of a section shape in a given mode of loading

Shape factor does not depend on scale B

e is the shape factor for elastic bending; Tf is

for failure due to torsion. Two other shape factors T

e and Bf are similarly defined.

The factors reflect the load bearing capacity of the shaped material compared with a solid cylindrical bar with the same cross-sectional area

They all equal to 1 for solid circular section

Derivation of Shape Factor for Elastic Bending

Elastic bending

Displacement for 3-point bending

Stiffness, for shaped

section and for solid

circular section

Shape factor

44

24

0sec

2 ArIdAyI

tionxx

EIC

Fl

1

3

31

sec l

EICFS tion

301

l

EICScircular

20sec 4

A

I

I

I

S

S

circular

tioneB

Failure load for bending of beam with shaped section

Failure load for bending of beam with circular section

Ratio of the two strengths:

Shape factor

Derivation of Shape Factor for Bending Strength

ly

ICF f

mtionf

2sec

ly

ICF f

mcircularf

0

02

2

1

23

21sec 4 f

Bmcircularf

tionf

yA

I

F

F

21

230

34

0

0

44

4

Ar

r

r

y

I

m

23

216

m

fB yA

I

Other Shape Factors For elastic torsion where K is a

modified polar moment of area (J) For torsional strength where Q =

J/rm

For buckling of column,the Euler loadThe shape factor is, again, the ratio between the moments of area, i.e. equal to B

e

2

2

A

KeT

3

24

A

QfT

2

22

l

EInFc

Performance Indices which include Shape The procedure is similar to that

without consideration of shape, with one extra step to bring in the shape.

Shape factor has no effect on the choice of materials for Axial tension loaded components

Aim: Stiff light beam m = Al & Bending stiffness Eliminate A, gives

Comparing beams with different sections and materials

Performance Indices which include Shape - Elastic Bending

3

21

31

4 l

AE

C

l

EICS e

BB

2

4

A

IeB

21221

1

54

EC

Slm

eB

B

21

1

EM

eB

Aim: light stiff shaft m = Al Torsional stiffness Eliminate A gives

Performance Indices which include Shape - Elastic Twisting

l

AG

l

KGS e

TT

2

2

212

2132

GSlm

eT

T

21

2

GM

eT

Aim: light strong beam Failure load & Replace (I/ym) by B

f

Eliminate A gives

Performance Indices which include Shape - Failure of Beam

ly

ICF f

mf

2

23

216

m

fB yA

I

ffBf A

l

CF

213

212

4

32

21

2332

2

214

ffB

f lC

lFm

3231

3f

fBM

Aim: light strong shaft

Failure load

Eliminate A gives

Performance Indices which include Shape - Failure of Shaft

f

fT

ff

AQT

21

2321

4

3

24

A

QfT

2f

f

32

21

2332214

ffT

f lTm

3231

4f

fTM

Shaped Materials on Materials Property Charts

By rearranging, The equation means the material, when

structured, behaves like a material with modulus E* and density *

Introducing shape moves the material along a line of slope 1 on the chart

eB

eB

eB EE

M

2121

1

eB

EE

*

eB *

Shaped Materials on Materials Property Charts - Elastic Bending

• The material move from below the selection criteria line to above

• The effective performance of the shaped materials is improved

Similarly, by rearranging

The shaped behaves like a material with

Shaped Materials on Materials Property Charts - Bending Failure

fB

fBff

fBM

323231

3

fB *

fB

ff

*

Shaped Materials on Materials Property Charts - Bending Failure