Materials Selectioms Process 9

7/24/2019 Materials Selectioms Process 9

1/15

Section shape Is important for certain modes of loading. When shape is avariable a new

term, the shape factor appears in some of the macenaJ Indkes: they then allow

optimum selection of material and shape.

A material can be thought of as having properties hut no shape.

A compownr or structure is a material made into a shape

he shape factor is a measure of the efficiency material usage.

!aterial !acro"shape

Shaped material


2/15

!echanical efficiency is ol#ained by combinir maeriaJ with macroscopic shape.he

shape is characteri$ed by a dimensionless shape factor, %. he schematic issuggested by

&arkhouse '()*+.

-eniuon of symbols

Symbol -efinition

! !oment 'm

/ /orce '

0 1oung s modulus of the nutenal 2 tfle section +3a

tu 1ield or failure strerth of the material of the section'!&a

p -ensity cf the material of che section '!g4rn5

in, !ass per urr length of the section 'kg6m

A 7ross"sectional area of section 'm8

I Second moment of area of che section 'm+

Second moment of area of the s9uare reference section

'm+

Section modulus of che section 'm5

+ Section modulus of the siare reference section 'm5

; !acro shape factor for elastic bendmg deflection '


3/15

'oo 0ssential term in bendir strength '.m

t Web and f=ange thickness 'm

c Web hein'm

d Section heIght 'it > c of sandwich 'm

b Section 'flange width 'm

= Section ler'th 'm

SHAPE FACTORS

0lastic bending of beams and twisting of shafts

7ommon modes of Ioadng and the secuon.shapes chat are chosen to support them:'a a#ial

tvnion 'b bending 'c torsion nd 'd ial corrre?sion. which can ;eid tob@ckling.

he bending stiffness S of a beam is proportional to the product 0J


4/15

ere 0 is 1oung5s modulus and J is the second moment of area of the beam

about the a#is of bending 'the # a#is:

he second moment of area, (g,. for a reference beam of s9uare

section with edge"length bB and section"area A C b is simply

he bending stiffness of the shaped section differs from that of a s9uare onewith

the same area A by the factor D where

We call the shape factor for elastic bending. ote that it is dimensionless


5/15


6/15

'a A set of rectangular sections with C 8E 'b a set oil"sections with C (Band

'c a set ci wbes with + C (8. !embers ola sec differ In si$e but not inshape.

he second moment of areaJ. plotted against section area A. 0fficenc strucwreshave


7/15

h, values of the ratio JA8E inefficient structures 'ones that bend easilyhave low

values. Feal structural sections have values of I and A that he in che shaded$ones. ote

chat there are limits on A and on che ma#amum shape efficiency 6 that depend

on materia=

Shapes that resist bending well may not be so good when twisted.

he stiffness of a shaft< the tor9ue divided by the angle of twist, G

< is proportional to H, where ' is its shear modulus and its

torsional moment of area. /or circular sections is identical with the polar

moment of area, J:


8/15

where dA is the differential element of area at the radial distance r,measured

from the center of the section. /or non"circular sections, is less than JEit

is defined such that the angle of twist G is related to the tor9ue by

where = is length of the shaft and H the shear modulus of the material ofwhich

it is made.

he shape factor for elastic twisting is dened, as before, by the ratio of the

torsional stiffness of the shaped section, S, to that, ?. of a solid s9uare

shaft

of the same length = and cross"section A,

he torsional constant B for a solid s9uare section

with bCk is


9/15

Gnset of failure in bending and twisting

&lasticity starts when the stress, somewhere, first reaches the yieldstrength,

uE fracture occurs when this stress first e#ceeds the fracture strength, aIrE

fatigue failure if it e#ceeds the endurance limit a.

In bending, the stress a is largest at the point tm in the surface of thebeam

that lies furthest from the neutral a#isE it is:

where ! is the bending moment. /ailure occurs when this stress first e#ceeds

hus, in problems of failure of beams, shape enters through the section

iriodulus,' I4ym

he strength"efficiency of the shaped beam, D, is mcasurcd by the ratio 4B,

where B is the section modulus of a reference bcam of s9uare scction with

the same cross"sectional area, A:


10/15

A beam with an failure shape"efficiency factor of (B is (B times

stronger in bending than a solid s9uare section of the same weight.

/igure is a plot of against A for values of +

As before, the contour for D < ( describes the s9uare section reference beam.

he other contours describe shapes that are more or less efficient, assuggested by the icons.

he section modukis, plotted against section area A. 0Wcient structures havegh

values of the ratio IAKE Inefficier0 structures 'ones that bend easily havelow values.

Feal structural sections have values ol and A tnt he in the shaded $ones.ote that

there are limits on A and on the ma#mum shape efficiency D that depend onmaterial.


11/15

In torsion the problem is more complicated. /or circular rods or tuhes

subLected to a tor9ue the shear stress r is a ma#imum at the outer surface,

at the radial distance rm from the a#is of bending:

he 9uantity in i4m twisting has the same character as uym in bending.

/or non"circular sections with ends that are free to warp. the ma#imum surface

stress is given instead by

where M. with units of mK now plays the role of i4m or . his allows the

definition of a shape factor, for failure in torsion, following the same

pattern as before:

!icroscopic or micro"structural shape factors

N!icroscopic shapeO

!aterial !acro"shape


12/15

Shaped matenal

!echanical efficiency can be obtained by combinir material with microscopic.or internal.

shape. which repeats itself to give an e#tensive structure. he shape ischaracteri$ed by

microscopic shape factors. P.

!icro"structural shape can be combined with macroscopic shape to giveefflcQent

structures. he overall shape factor is the product of the microscopic andmacroscopic

shape factors.


13/15

awral materials with internal, or microscopic. shape. Feading from the topleft cock.

balsa wood. sponge. cancdous bocee. oeral. cuttie"bone, and pabi pbnc scR.


14/15

/our e#tensive micro"structured materials that are mechanically efficient 'aprismatic

cells. 'b fibers embedded In a foamed matri#. 'c concentric cylindricalthells with

foam between, and 'd parallel plates separated b5 foamed spacers.

/igure shows four e#tensive structures with microscopic shape, all of

them found in nature.

!icroscopic shape factors. 7onsider the gain in bending stiffness when a solid

s9uare beam like that shown as a black s9uare of side bB in /igure .

is e#panded, at constant mass, to a larger s9uare section with any one of the

structures that surround it in the figure. he bending stiffness S of theoriginal

solid beam is propruonal to the pr'#luc' of its modulus 0 and its second

moment of area I:


15/15

where the subscript sT means a property of the solid beamT and I b4 (8.

When the beam is e#panded at constant mass its density falls from to p and

its edge"length increases from b, to b where

with the result that its second moment of area increases from i to

If the cells, fibers, rings or plates in /igure e#tend parallel to the a#is of

the beam, the modulus parallel to this a#is falls from that of the solid, 0,to

he bending stiffness of the e#panded beam scales as 0l, so thai it is stifferthan

the original solid one by the factor

We refer ro as the microscopic shape faaor for elastic bending. hat for

prismatic structures like those of /igure is simply the reciprocal of the

relative density, 4p?. ote that, in the limit of a solid 'when p4ps i5 takesthe

value (, as it obviously should. A similar analysis for failure in bendinggives

the shape factor

Materials Selectioms Process 9

Documents

Transcript of Materials Selectioms Process 9