19342648 Mechanics of Materials II2

7/30/2019 19342648 Mechanics of Materials II2

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Mechanics of Materials II

UET, Taxila

Lecture No. (2)


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Tensile behavior of

different materials:In a typical tensile test one tries

to induce uniform extension ofthe gage section of a tensile

specimen.


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The gage section of the

tensile specimen isnormally of uniform

rectangular or circularcross-section.

The following figure shows atypical dog-bone sample.


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Gage length

P

P

P

P


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The two ends are used for fixing

into the grips, which apply the

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

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

by the grips.


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An extensometers are used

to measure the change oflength in the gage section

and load cells to measurethe load applied by the grips

on the sample.


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By the means of this it is

possible to calculate the

axial strain and normal

stress (knowing the initial

gage length and cross-sectional area of the gage

section).


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The result is a stress-strain

diagram, a diagram of how

stress is changing in the

sample as a function of the

strain for the given loading. A

typical stress-strain diagramfor a mild steel is shown

below.


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Mild Steel Stress-Strain Curve

Yield stress, y

Ultimate stress, u

Stress,

Strain,

ff f


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The different regions of the area

response denoted by their

characteristics as follows

Yield stress,y

Ultimate stress, u

Stress,

Strain,

12

34 5

1. Linear elastic: region of proportional elastic loading2. Nonlinear elastic: up to yield3. Perfect plasticity: plastic flow at constant load4. Strain hardening: plastic flow with the increase of stress

5. Necking: localization of deformation and rupture


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Brittle versus Ductile

behaviorBrittle materials fail atsmall strains and in

tension. Examples ofsuch materials are glass,

cast iron, and ceramics.


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Ductile materials fail at

large strains and inshear. Examples of

ductile materials are mildsteel, aluminum and

rubber.


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The ductility of a material

is characterized by thestrain at which the

material fails.


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An alternate measure isthe percent reduction in

cross-sectional area at

failure.


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Isometric of tensile test

specimen


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Different types of

response:Elastic response:

If the loading and unloading

stress-strain plot overlap each

other the response is elastic.

The response of steel below

the yield stress is considered tobe elastic.


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Elastic Response (Linear &

Non-linear)

LinearElastic

NonlinearElastic


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After loading beyond the

yield point, the material nolonger unloads along the

loading path.There is a permanent

stretch in the sample afterunloading.


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The strain associated

with this permanentextension is called the

plastic strain p


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As shown in next figure,

the unloading path isparallel to the initial

linear elastic loadingpath (and not

overlapping).


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Unloading

Loading

p


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Most plastics when loaded

deform over time even without

increasing the load. The

material continuous extension

under constant load referred toas creep. If held at constant

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


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Relaxation

The decrease in load

over time at constantstretch is referred to

as relaxation.


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Bearing Stress:

Even though bearing stress

is not a fundamental type of

stress, it is a useful concept

for the design of

connections in which onepart pushes against

another.


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The compressive load

divided by acharacteristic area

perpendicular to it yieldsthe bearing stress which

is denoted by b.


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, ,stress is no different from the

compressive axial stress and isgiven by:

A

Fb


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Where:

F: is the compressive

load andA: is a characteristic area

perpendicular to it.


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F

p

F

F F

F

F

F

F

d

t

t

t

t

Cylindrical bolt or rivet


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For example, if two

plates are connected bya bolt or rivet as shown,

each plate pushesagainst the side of the

bolt with load F.

It i t l h t th


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It is not clear what the

contact area between thebolt and the plate is since it

depends on the size of thebolt and the shape of the

deformation that results.

Al th di t ib ti f th


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Also, the distribution of the

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

approximation one can usethe shown rectangle of

areaA

=td

Thi i t ti


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This gives a representative

bearing stress for the bolt as:

td

Fb


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Response and

Factor of SafetyLinear-elastic response:

Hookes law:In the linear elastic portion of the

response of material one canmodel the response by Hookes

law as follows


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Hookes law for extension:

= E

Hookes law for shear:

= G


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Where:Eis the elastic modulus (or

Youngs modulus), and

Gis the shear modulus.

The elastic and shear moduli

are material constants

characterizing

the stiffness of the material.


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Physical Meaning of E

(Stress Strain Curve)

E

1

P i R ti


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Poissons Ratio:

Another material

parameter is Poissons

ratio that characterizes

the contraction in thelateral directions when a

material is extended Th b l ( ) i d f


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The symbol (nu) is used for

the poison ration, which is

negative the ratio of the lateralstrain to axial strain.

a

t


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lo

do

l

d


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o

o

t d

dd


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o

a l

ll


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elastic moduli:

For an isotropic elastic material(i.e., an elastic material for whic

the properties are the samealong all directions) there are

only two independent materialconstants.


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The relation between

the three moduli aregiven by the following

equation:

Equation for the relation between the


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Equation for the relation between the

elastic moduli:

)1(2

EG

Factor of safet


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Factor of safety:

The factor of safety

denoted by n

is the ratioof the load, the structure

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


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Factor of safety

strengthRequired

strengthActualn

Therefore the factor of safety


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Therefore, the factor of safety

is a number greater than unity

(n>1).The allowable stress fora given material is the

maximum stress the materialcan take (normally the ultimate

or yield stress) divided by thefactor of safety).


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or

n

or

uyallow

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19342648 Mechanics of Materials II2

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