mechanics of materials ii(2)

8/14/2019 Mechanics of Materials II(2)

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Mechanics of Materials IIMechanics of Materials II

UET, TaxilaUET, Taxila

Lecture No. (2)Lecture No. (2)



Tensile behavior of Tensile behavior of

different materials:different materials:In a typical tensile test one triesIn a typical tensile test one tries

to induce uniform extension of to induce uniform extension of the gage section of a tensilethe gage section of a tensile

specimen.specimen.



The gage section of theThe gage section of the

tensile specimen istensile specimen isnormally of uniformnormally of uniform

rectangular or circular rectangular or circular cross-section.cross-section.

The following figure shows aThe following figure shows atypical dog-bone sample.typical dog-bone sample.



Gage length

P

P

P

P



The two ends are used for fixingThe two ends are used for fixing

into the grips, which apply theinto the grips, which apply the

load. As can be seen from theload. As can be seen from thefree-body diagram to the right,free-body diagram to the right,

the load in the gage section isthe load in the gage section isthe same as the load appliedthe same as the load applied

by the grips.by the grips.



An extensometers are usedAn extensometers are used

to measure the change of to measure the change of length in the gage sectionlength in the gage section

and load cells to measureand load cells to measurethe load applied by the gripsthe load applied by the grips

on the sample.on the sample.



By the means of this it isBy the means of this it is

possible to calculate thepossible to calculate the

axial strain and normalaxial strain and normal

stress (knowing the initialstress (knowing the initial

gage length and cross-gage length and cross-sectional area of the gagesectional area of the gage

section).section).



The result is a stress-strainThe result is a stress-straindiagram, a diagram of howdiagram, a diagram of how

stress is changing in thestress is changing in the

sample as a function of thesample as a function of the

strain for the given loading. Astrain for the given loading. A

typical stress-strain diagramtypical stress-strain diagramfor a mild steel is shownfor a mild steel is shown

below.below.



Mild Steel Stress-Strain CurveMild Steel Stress-Strain Curve

Yield stress, yσ

Ultimate stress, uσ

Stress, σ

Strain, ε



The different regions of the areaThe different regions of the area

response denoted by their response denoted by their

characteristics as followscharacteristics as follows

Yield stress, y

σ

Ultimate stress, uσ

Stress, σ

Strain, ε

12

34

5

1. Linear elastic: region of proportional elastic loading

2. Nonlinear elastic: up to yield3. Perfect plasticity: plastic flow at constant load

4. Strain hardening: plastic flow with the increase of stress

5. Necking: localization of deformation and rupture



Brittle versus DuctileBrittle versus Ductile

behavior behavior Brittle materialsBrittle materials fail atfail atsmall strains and insmall strains and in

tension. Examples of tension. Examples of such materials are glass,such materials are glass,

cast iron, and ceramics.cast iron, and ceramics.



Ductile materialsDuctile materials fail atfail at

large strains and inlarge strains and inshear. Examples of shear. Examples of

ductile materials are mildductile materials are mildsteel, aluminum andsteel, aluminum and

rubber.rubber.



TheThe ductilityductility of a materialof a material

isis characterized by thecharacterized by thestrain at which thestrain at which the

material failsmaterial fails..



AnAn alternate measurealternate measure isisthe percent reduction inthe percent reduction in

cross-sectional areacross-sectional area atat

failure.failure.



Isometric of tensile testIsometric of tensile test

specimenspecimen



Different types of Different types of

response:response:Elastic response:Elastic response:

If the loading and unloadingIf the loading and unloading

stress-strain plot overlap eachstress-strain plot overlap each

other the response is elastic.other the response is elastic.

The response of steel belowThe response of steel below

the yield stress is considered tothe yield stress is considered tobe elastic.be elastic.



Elastic Response (Linear &Elastic Response (Linear &

Non-linear)Non-linear)

ε

σ

Linear Elastic

Nonlinear Elastic



After loading beyond theAfter loading beyond the

yield point, the material noyield point, the material nolonger unloads along thelonger unloads along the

loading path.loading path.There is a permanentThere is a permanent

stretch in the sample after stretch in the sample after unloading.unloading.



The strain associatedThe strain associated

with this permanentwith this permanentextension is called theextension is called the

plastic strain “plastic strain “εεpp””



As shown in next figure,As shown in next figure,

the unloading path isthe unloading path isparallel to the initialparallel to the initial

linear elastic loadinglinear elastic loadingpath (and notpath (and not

overlapping).overlapping).



ε

σ

Unloading

Loading

ε p



Most plastics when loadedMost plastics when loaded

deform over time even withoutdeform over time even without

increasing the load. Theincreasing the load. The

materialmaterial continuous extensioncontinuous extension

under constantunder constant load referred toload referred toasas creepcreep. If held at constant. If held at constant

strain, the load required to holdstrain, the load required to holdthe strain decreases with time.the strain decreases with time.



RelaxationRelaxationTheThe decrease in loaddecrease in load

over time at constantover time at constantstretchstretch is referred tois referred to

asas relaxation.relaxation.



Bearing Stress:Bearing Stress:

Even though bearing stressEven though bearing stress

is not a fundamental type of is not a fundamental type of

stress, it is a useful conceptstress, it is a useful concept

for the design of for the design of

connections in which oneconnections in which onepart pushes againstpart pushes against

another.another.



The compressive loadThe compressive load

divided by adivided by acharacteristic areacharacteristic area

perpendicular to it yieldsperpendicular to it yieldsthe bearing stress whichthe bearing stress which

is denoted by “is denoted by “σσbb“.“.



, ,, ,stress is no different from thestress is no different from the

compressive axial stress and iscompressive axial stress and isgiven by:given by:

A

F b =σ



Where:Where:

F:F: is the compressiveis the compressive

load andload and A: A: is a characteristic areais a characteristic area

perpendicular to it.perpendicular to it.



F

ε p

F

F F

F

F

F

F

d

t

t

t

t

Cylindrical bolt or rivet



For example, if twoFor example, if two

plates are connected byplates are connected bya bolt or rivet as shown,a bolt or rivet as shown,

each plate pusheseach plate pushesagainst the side of theagainst the side of the

bolt with loadbolt with load F F ..



It is not clear what theIt is not clear what the

contact area between thecontact area between thebolt and the plate is since itbolt and the plate is since it

depends on the size of thedepends on the size of thebolt and the shape of thebolt and the shape of the

deformation that results.deformation that results.

Al h di ib i f h



Also, the distribution of theAlso, the distribution of the

load on the bolt varies fromload on the bolt varies frompoint to point, but as a firstpoint to point, but as a first

approximation one can useapproximation one can usethe shownthe shown rectangle of rectangle of

areaarea A A

==td td

Thi i iThi i t ti



This gives a representativeThis gives a representative

bearing stress for the bolt as:bearing stress for the bolt as:

td

F b =σ



--Response andResponse and

Factor of SafetyFactor of SafetyLinear-elastic response:Linear-elastic response:

Hooke’s law:Hooke’s law:

In the linear elastic portion of theIn the linear elastic portion of the

response of material one canresponse of material one canmodel the response by Hooke’smodel the response by Hooke’s

law as followslaw as follows



Hooke’s law for extension:Hooke’s law for extension:

σ σ = E = E ε ε

Hooke’s law for shear:Hooke’s law for shear:

τ τ = G= G



Where:Where:E E is the elastic modulus (or is the elastic modulus (or

Young’s modulus),Young’s modulus), andandGG is the shear modulus.is the shear modulus.

TheThe elastic and shear modulielastic and shear moduli

are material constantsare material constants

characterizingcharacterizing

the stiffness of the material.the stiffness of the material.



Physical Meaning of EPhysical Meaning of E

(Stress Strain Curve)(Stress Strain Curve)

ε

σ

E

1

P i ’ R ti



Poisson’s Ratio:Poisson’s Ratio:

Another materialAnother material

parameter is Poisson’sparameter is Poisson’sratio that characterizesratio that characterizes

the contraction in thethe contraction in thelateral directions when alateral directions when a

material is extendedmaterial is extended

Th b lTh b l ( ) i d f( ) i d f



The symbolThe symbol ν ν (nu) is used for (nu) is used for

the poison ration, which isthe poison ration, which is

negative the ratio of the lateralnegative the ratio of the lateral

strain to axial strain.strain to axial strain.

a

t

ε

ε ν −=



l o

d o

l

d



o

o

t d

d d −=ε


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oa l

l l −=ε



The relation betweenThe relation between

the three moduli arethe three moduli aregiven by the followinggiven by the following

equation:equation:

Equation for the relation between theEquation for the relation between the



Equation for the relation between theEquation for the relation between the

elastic moduli:elastic moduli:

)1(2 υ +

=

E G

F t f f tF t f f t



Factor of safety:Factor of safety:

The factor of safetyThe factor of safety

denoted bydenoted by ““n” n”

is the ratiois the ratioof the load, the structureof the load, the structure

can carry,can carry, divided by thedivided by theload it is required to take.load it is required to take.



Factor of safetyFactor of safety

strength Required

strength Actual n =

ThereforeTherefore the factor of safetythe factor of safety



Therefore,Therefore, the factor of safetythe factor of safety

is a number greater than unityis a number greater than unity

((nn>1).>1). The allowable stress for The allowable stress for

a given material is thea given material is the

maximum stress the materialmaximum stress the materialcan takecan take (normally the ultimate(normally the ultimate

or yield stress) divided by theor yield stress) divided by thefactor of safety).factor of safety).



or

n

or

u yallow

u yallow

τ τ τ

σ σ σ

=

=

mechanics of materials ii(2)

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