cmats lect1-axial stress and strain [compatibility mode]

8/3/2019 CMats Lect1-Axial Stress and Strain [Compatibility Mode]

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Civil Engineering Materials 267Stresses in Materials

Lecture 1: Axial Stress and Strain,

Kerri Bland



Civil Engineering Materials 267 (Stresses)Civil Engineering Materials 267 (Stresses)

COMMONWEALTH OF AUSTRALIA

Copyright Regulation 1969

WARNING

This material has been copied and communicated to you by or on behalf of Curtin University of Technology pursuant to Part VB of the Copyright Act

The material in this communication may be subject to copyright under the Act.

An further co in or communication of this material b ou ma be the

subject of copyright protection under the Act.

References

“Engineering Mechanics of Solids”, 2nd Ed., E.P. Popov, 1991, Prentice Hall.

Lecture 1 2

“Mechanics of Materials”, 6t Ed., R.C. Hibbeler, 2005, Prentice Hall/Pearson.




PP

PP

Which is the best section to resist the

y

Lecture 1 3




PP

PP

=

P For Axial

(Average Normal Stress distribution)

A orma s ress

Normal to plane it

Loads

Units:2

. .

N/mm2 = MPa (Preferred) (Megapascal)

Lecture 1 4

wor n an mm – answer w come ou as a




The S.I. unit of stress is ‘Newton per square metre’2 , .

,prefixes representing only the ternary powers of 10 (103,106, 10-6, etc.). Use multiples or sub-multiples of 1000’s:

1 kPa = 1000 Pa = 1000 N/m2

a = x a = x m = mm

1 GPa = 1x109 Pa = 1x103 MPa = 1x103 N/mm2

kN; Pa, kPa, MPa and GPa etc.

Lecture 1 5




Material stren th:

express in terms of allowable loads,

or allowable stresses?

Which is best? Steel rod:10mm*10mm load capacity = 25kN

15mm*15mm load capacity = 56kN

Why? 40mm*40mm load capacity = 400 kN

Steel: allowable stress = 250MPa

Lecture 1 6




Example Material A can safely handle a stress of at least 250 MPa. A

P =σ

Material B can safely handle a stress of at least 90 MPa.

1. If a rod of each material is to be produced, and they are each required to,

each material?

2. If a rod of each material has the cross sectional dimensions of,

3. If a rod is to have cross sectional dimensions of 50mm*50mm, and is to besubjected to an axial load of 500kN, what is the best material to use?

e max mum a owa e stress o a mater a n cates amaterial property that is independent of material

The stress which occurs in a particular geometric shape

Lecture 1 7

s n epen en o ma er a ype.



Civil Engineering Materials 267 (Stresses)

called a tensile stress .

force is called a compressive stress .

ens e s ress – s re c ng e ec

σx

ompress ve s ress – con rac ngeffect

Lecture 1 8




Normal Strain

When a material is subjected to load, resulting in

stresses within the material, a related deformation of thematerial occurs.

The concept of Strain is used to describe the

deformation.

Normal strain is directly related to normal stress.

Tension results in an increase in length – positive strain –

strain

Lecture 1 9




Normal Strain

Defined as ratio between the change LΔ=ε

n eng an e or g na eng . Strain is dimensionless (but is often stated in

mm/mm, m/m or μ m/m (microstrain – see below)).

Inde endent of actual len th of member. As the number is usually very small (ie: usually

ave a very sma e ormat on per metre o mater a

usually expressed in microstrain (μ).

μ = 1*10-6

Lecture 1 10

Ci il E i i M i l (S )




As mentioned before when considering stresses, in engineeringapplications, there is preference for use of prefixes representingonly the ternary powers of 10 (103, 106, 10-6, etc.). Use multiplesor sub-multiples of 1000’s (which is easier to read and

Non-preferred

power

Strain

measured

Expressed as

ternary

Expressed in

με

representation

2.63*10-4

* -5

powers

0.000263 263*10-6 263* -6 .

1.66*10-4

1.70*10-5

.

0.000166 166*10-6 1660.000017 17*10-6 17

1.02*10-30.001018 1.02*10-3

or 1018*10-6

1018

Lecture 1 11

Ci il E i i M t i l 267 (St )




Measuring stress and strain

Stress can not be measured .calculate based on the applied actions

, , ,magnitude of action)

.Strain gauges

ens on test

Lecture 1 12





STRAIN GAUGES

Lecture 1 13(Popov, p59)

Refer to slide 2 for Copyright warning





TENSION TESTTake a specimen with an originalcross-sectional area of A in the central

portion of the specimen, and a ‘gauge’

Marks are often inscribed on test specimen.

A set gauge length such as Lo (say, 100, , .

reference / datum – taken at central portion. 9 )

y r i g h t w a r n i n g

Change in Lo is measured by a devicecalled an ‘extensometer’. ( P

o p o v , p 5

s

l i d e 2 f o r C o

Lecture 1 14

R e f e r t o





TENSION TEST (cont.)

In tension tests, either round bars or flat rectangular barscan be used.

If Lo = the initial gauge length and L = the observed,

= (L-Lo).

The tensile (or compressive) strain is the elongation perunit of the initial au e len th:-

ε = ΔL/Lo = (L-Lo)/Lo

The strain ε is called the normal strain (or direct strain)

Lecture 1 15

.





TENSION TESTSOften used to plot stress-strain diagrams

Measure strain, calculate stress

characteristics of the tested material

Yield stress

Linear elastic re ion

Ultimate stress

Lecture 1 16





Typical stress-strain curve for ductile steel

Lecture 1 17

(Popov, p61)Refer to slide 2 for Copyright warning





HOOKE’S LAW e rs por on o e σ-ε curve s s ra g . s s rue or ma er a s

such as steel, aluminium, glass, etc.

ere are a so ma er a s w ere e n a par o e σ-ε curve s nearstraight, e.g., concrete, annealed copper, cast iron, etc.

e assump on o near re a ons p e ween σ an ε s e as s orHooke’s Law:

σ = . ε or = σ ε

where, E = slope of linear part of curve

The constant of proportionality, E,is called the:

σ u

σ y

mo u us o e as c y or oung s mo u us.

S t r e s s

Lecture 1 18

Typical Stress-strain Curve for Steel





HOOKE’S LAW(cont.)

Graphically, E is the slope of a straight line fromthe origin to the proportional or elastic limit of the

.

Hooke’s Law applies only up to the proportional.

E is in Pa or N/m2

(or MPa or GPa). r e

s s

E is the stress required to create unit strain⇒ a very large number (Esteel = 200*103 MPa)

S

ε = Δ = 1

ΔL = L (ie: doubles in length) σ y

Larger E – more stress required tocreate unit strain (ie: a stiffer

StrainStrain at yield point

of 400MPa = 0.002

Lecture 1 19

material) Typical Stress-strain Curve for Steel





Two identical rods, one withtwice the tension load of theT 2T

.

compare?

a) Assuming the strain in both

T 2T

elastic range?b) ssum ng e s ra n n one case

will not still be within the elastic

Lecture 1 20





FURTHER REMARKS ON STRESS-

Th hi h in m x. r in iv h

ultimate tensile strength of a material.

ress a e y e p a eau o e σ-ε curve s ca e

the yield strength of a material.

σ u Fracture point

s s

y

σu = ultimate tensile strength=

S t r

Strain

y

Lecture 1 21

-





In ductile materials, at the ultimatepo nt, w ere t e max mum tens estress occurs, the process of

‘necking’ starts to occur.

Lecture 1 22


(Hibbler, p88)Refer to slide 2 for Copyright warning





Stress strain curves vary

Typical cup and conefailure of a ductile material

(actually a shear failure – more on that later)


Lecture 1 23





g g ( )

(Hibbler, p92)

Lecture 1 24





g g ( )



Lecture 1 25




g g ( )

We can have linear elastic orHookean σ-ε curve.

A material which has non-near σ-ε curve, w en

unloaded, and returns backa ong e oa ng pa o sinitial stress-free state ofe orma on s ca e a non-

linear elastic material.

Lecture 1 26





g g ( )

In many materials, the yielding

ield lateau henomenon isabsent.

,repeatable yield stress, theo se me o s use .

An arbitrar 0.2% offset isusually considered.


, ,

drawn..

⇒ 0.2% = 0.002 = 200με

Lecture 1 27

e y e stress s e ne as t e po nt stress w ere t eline intersects the curve.




If in stressin a non-linearmaterial, its elastic limit is

,usually responds approx.

,

permanent deformation or set eve ops at zero oa .

The area enclosed by the loop

energy released through heat.

Lecture 1 28





Summary

u u x : A =σ

Normal strain due to uniaxial load:

LΔ

=ε

Young’s modulus: ε σ =E

a measure o e s ness o alinear elastic material)

Lecture 1 29

cmats lect1-axial stress and strain [compatibility mode]

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