Cross sectional Shape Selection - Wright State Universitycecs.wright.edu/~rsrin/Courses/ME474-674/Winter 2008... · 2008-02-11 · Different shapes are more or less efficient for

ME 474-674 Winter 2008 Slides 9 -1

More info: “Materials Selection in Mechanical Design”, Chapters 11 and 12

Cross sectional Shape Selection

Materials have propertiesStrength, stiffness, electrical conductivity, etc.

A component or structure is a material made into a particular shapeDifferent shapes are more or less efficient for carrying a particular type of loading An efficient shape is one that uses the least amount of material for a given strength or stiffness


Mechanical loading and associated components

Axial LoadingTension – ties or tie rods

Compression – columns

Bending – beams

Torsion – shafts

Each type of loading has a different failure mode, and some shapes are more efficient than others for that loading


Ties or Tie rods

Tensile axial loading

The stiffness of a tie rod for a given material depends only on the cross sectional area A and not the shape

The strength of a tie rod depends only on the cross sectional area A and not the shape

Therefore in tensile loading all shapes of the same cross-sectional area are equivalent

AF

LAEFS

AEFL

=

==

=

σ

δ

δ


Elastic Bending

Appendix A-3 gives the deflection of beams as a function of the type of loading. Generally

The stiffness of a beam S is defined as the ratio of load to displacement

Using either definition, S is proportional to EI

E = elastic modulus of the materialI = moment of inertia of the cross section

EICML

orEIC

FL

1

2

1

3

=

=

δ

δ

δ

δ

MS

or

FS

=

=


Elastic Bending

I = Moment of inertia of the cross section

Table 11.2 gives the section properties of different shapesFor a circular cross section

If S is the stiffness for another shape with the same cross sectional area made of the same material and subject to the same loading, then the shape factor for elastic bending is defined as

∫−

=sectionCross

dAyI 2

ππ

π

44

24

2

ArI

rA

O ==

=

21

1 4AI

EICEIC

SS

OO

eB πφ ===


Elastic Bending

Derive shape factor for elastic bending of

Square cross-section of side a

Hollow tube of radius r and thickness t where r >> t

For a square cross section

For a hollow tube

05.1124

====πφ

O

sq

O

sqeB EI

EISS

tr

AI

EIEI

SS

OO

eB ==== 24πφ


Elastic Bending - Square cross-section beam

For a square cross-section of side a

Compare with a circle with the same area A

Shape factor during elastic bending of a square cross-section relative to a circular cross section of the same area is:

Therefore, a square cross-section is about 5% stiffer than a circular cross-section

1212

24

2

AaI

aA

sq ==

=

Osq II124π

=

( )( ) 05.1

124

4/12/

2

2

=====π

πφ

AA

II

SS

O

sq

O

sqeB


Elastic Bending – Tubular beam

For a tubular beam with radius r and wall thickness t where r >> t

Shape factor during elastic bending of a tubular beam relative to a circular cross-section of the same area is:

Therefore, a thin walled tubular beam with r = 10t is 10 times as stiff as a circular cross-section beam of the same area

trIrtA

tube3

2π

π

=

=

( ) ( ) tr

rttr

AI

II

SS tube

O

tube

O

tubeeB ===== 2

3

2 24

4/ πππ

πφ

Please note that the derivations here assume a circle as the reference shape. The text book assumes the reference shape to be a square.


Elastic Bending

The shape factor φBe is dimensionless, i.e. it is a pure number that

characterizes the cross-sectional shape relative to a circular cross-section

Increasing size with constant shape

I sections withφB

e = 10

Hollow tubes withφB

e = 10


Elastic Bending

EduPack Level 3 includes most of the commercially available structural shapes made from different materials.

Using a plot of the moment of inertia versus section area one can compare different structural shapes

Section area, A (m^2)1e-4 1e-3 0.01 0.1

Sec

ond

mom

ent o

f are

a (m

ajor

), I_

max

(m^4

)

1e-11

1e-10

1e-9

1e-8

1e-7

1e-6

1e-5

1e-4

1e-3

0.01

Phi=100

Extruded Aluminum circular hollow (Y.S. 255MPa)-(32x2.6)

Hot Rolled Steel (Y.S. 355MPa) Universal Beam-(203x133x25)

Hot Fin. Steel (Y.S. 355MPa) Rect.Hollow -(150x150x5.0)

Pultruded GFRP Vinyl Ester I-section-(102x51x2.1)

24AIe

B πφ =


Failure in Bending

Failure in bending can be defined as the initiation of plastic deformation in the beam.

The stress on the top and bottom surfaces of a symmetric beam is given by

Where c is the distance of the top or bottom surface from the neutral surface

At yield,

σ = σf = yield stress

cIZ

whereZM

IMc

=

==σ


Failure in Bending

For a circular cross-section

ππ

π

44

4

2/33

4

ArZ

Thereforerc

rI

O ==

=

=


Failure in Bending

Define the shape factor for failure in bending as

Derive

For a square of side a

For a hollow cylinder of radius r and wall thickness t, where r >> t

18.13

2===

πφO

sqfB Z

Z

tr

ZZ

O

fB 8==φ

O

fB Z

Z=φ


Failure in Bending – Square cross section beam

For a square of side a

The shape factor for a square cross section is

The square cross section almost 20% stronger than a circular cross-section

( )( )

662/

122/33

4

2

Aaa

IZ

aI

aA

sq ===

=

=

( )( )( )( ) 18.1

32

4/6/

2/3

2/3

====π

πφ

AA

ZZ

O

sqfB


Failure in Bending – Tubular beam

For a tubular beam with radius r and wall thickness t where r >> t

The shape factor for a tubular beam is

The tubular beam with r = 10t has a shape factor of 8.9, i.e., the tubular beam is almost 9 times as strong as a circular cross-section beam

trrIZ

trIrtA

tube2

3

2

π

π

π

==

=

=

( )( ) ( )( ) tr

rttr

Atr

ZZ

O

tubefB 8

24

4/ 2/3

2

2/3

2

====πππ

ππφ


Elastic Torsion

During elastic torsion, the angle of twist per unit length is

Where T is the torque, J is the polar moment of inertia, and G is the shear modulus of the material.

The stiffness of a solid circular shaft in torsion ST is defined as the ratio of load to angle of twist per unit length

The shape factor for a different cross section is defined as

JGT

=θ

ππ

π

22

24

2

ArJ

rA

==

=

GJTS OTO==

θ

( )πφ2/2A

JGJGJ

SS

OT

TeT

O

===


Elastic Torsion

For a hollow shaft with radius r and wall thickness t where r >> t

Shape factor for elastic torsion is

Therefore, a thin walled shaft with r = 10t is 10 times as stiff as a circular cross-section shaft of the same area

trJrtA

322π

π

≈

=

( ) tr

trtr

Atre

T === 222

32

2

3

44

2/2

ππ

ππφ


Failure by plastic deformation during Torsion

The shear stress at the surface of a cylindrical shaft subject to a torque T is

Failure occurs when the stress reaches the shear yield stress, or one-half of the tensile yield stress

The shape factor for a shaft of a different cross-section can be defined as

( )

ππ

π

τ

22

2

/

2/33

4

ArQ

and

rJ

whereQT

rJT

JTr

O

O

OOO

==

=

===

O

ff Q

T==

2σ

τ

2/32AQ

QQ

O

fT πφ ==


Shape factor for failure in Torsion of a Hollow Shaft

For a hollow shaft with radius r and wall thickness t where r >> t

Shape factor is

Therefore, a thin walled shaft with r = 10t is 4.5 times as strong as a circular cross-section shaft of the same area

trrJQ

trJrtA

2

3

2

22

π

π

π

==

≈

=

( ) tr

rttr

AQf

T2

2222 2/3

2

2/3 ===ππππφ


Homework Assignment

Show that the shape factor of elastic buckling is the same as that for elastic bending


Empirical upper limits for the different shape factors

The limits to the different shape factors derived above based on manufacturing considerations, as well as competing failure mechanisms is given in Table 11.4

3<6Elastomers1315Wood (solid section)45812Polymers (nylon)792639GFRP and CFRP8103144AA 60617132565Structural Steel

Material ( )maxeBφ ( )max

eTφ ( )max

fBφ ( )max

fTφ


Co-selecting shape and material for stiff beams

Suppose it is desired to make a beam with a stiffness of SB and length L with a minimum mass.

This problem can be translated as

MaterialSize and Shape of cross section

Free variables:

•Length L is specified•Stiffness SB is specified

ConstraintsMinimize mass m = ρLA is ObjectiveBeamFunction



The stiffness in bending is given by

where C1 depends upon exactly how the load is distributed

If we replace the moment of inertia I by

Then

31 LEICSB =

π

πφφ

4

4

2

2

AI

since

AII

O

EB

OEB

=

==

23

1

4A

LECS e

BB φπ

=



Eliminating A from the equation for mass m we get

The material index to be maximized is therefore

So if we want to co-select both shape and material for a stiff beam, the basis for comparison is the material index M, above.

Graphically, we can co-select materials and shape by assuming new material properties of E* and ρ*

( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛= 2/1

2/52/1

1

4E

LCSm

eB

B

φρπ

( ) ( )( )

( )*

*/

/ 2/12/12/1

ρφρφ

ρφ EEEM e

B

eB

eB ===


Co-selecting shape and material for other loading

By a similar analysis, for elastic torsion, the material index is

For failure in bending it is

And for failure in torsion it is

( )ρ

φ2/2EM

fB=

( )ρ

φ2/3EM

fT=

( )ρ

φ2/1EM

eT=


Example: The wing-spar of a human powered plane

See example 12.1 in the book

The human powered plane is basically a large model airplane capable of flying under the power of a human being

The design requirement is that the weight (or mass) of the plane be minimized

MaterialSize and Shape of cross section

Free variables:

•Length L is specified•Stiffness SB is specified

ConstraintsMinimize massObjectiveWing sparFunction


Example: The wing-spar of a human powered plane

231071.5 – 1.6100 – 160CFRP142032.8 – 2.8271 – 73AA-7075-T69251.87.82 – 7.84200 – 210Steel12280.36 – 0.449.8 – 11.9Spruce152100.17 – 0.244.2 – 5.2Balsa Wood

Modified Material Index (EφB

e)1/2/ ρ

Shape factorφB

e

Material Index(E1/2/ρ)

Densityρ (Mg/m3)

ModulusE (GPa)

Material

Without taking shape into account, Balsa wood appears to have the best properties, and aluminum has a relatively poor performance

If we take shape into account, using typical values of the shape factor for beams of different materials, CFRP is best, while AA-7075 performs as well as the woods

Early planes were made of balsa, but later designs used aluminum or CFRP (if cost was not an issue).

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Cross sectional Shape Selection - Wright State Universitycecs.wright.edu/~rsrin/Courses/ME474-674/Winter 2008... · 2008-02-11 · Different shapes are more or less efficient for