mechanics of materials ii(1)

8/14/2019 Mechanics of Materials II(1)

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Mechanics of Materials II

UET, Taxila


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References:

- Strength and fracture of engineering

solids. David K. Felbeck & A. G. Atkins,Prentice Hall (1995)

ISBN: 9780138561130

- Mechanics of materials (An Introductioto the Mechanics of Elastic

nd Plastic Deformation of Solids andtructural Materials, E. J. Hearn,

utterworth (1997) ISBN 0750632658


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How Materials Carry Load Basic modes of loading amaterial are:

Tension,

compression

and shear


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Tension Compression Shear


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Definition of Stress:

Loads applied on amaterial which are

distributed over asurface.


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For example, the pointload shown in the

following figure mightactually be a uniformlydistributed load that hasbeen replaced by itsequivalent point load.


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P

A


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Another Definition of stress

Stress is the loadapplied per unit area

of the surface it isapplied on.


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Normal stress

Normal stress is the stressnormal to a surface and isdenoted by the symbol

"" (sigma). In the abovefigure the normal stress is

uniform over the surfaceof the bar and is givenby:


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Normal Stress Equation

A

P=

Where:

P is the normal load &

A is the area


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Shear Stress

Shear stress is the

stress tangent to asurface.


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If in the following figurethe shear stress (tau)that results in the shear

load V is uniformlydistributed over thesurface, then the shear

stress can be calculatedby dividing the shear forceby the area it is appliedon.


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Shear Stress Equation

AV=

Where:

V is the shear load &

Ais the area


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V

A


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Units of Stresses

The units of stress arethe units of load

divided by the units ofarea.


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In the SI system the unitof stress is "Pa"

and in the U.S. system it is"Psi".

Pa and Psi are related tothe basic units throughfollowing relations:


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Pa & Psi Equations

KsiPsi

in

lbPsi

MPaPam

NPa

110

1

11

1101

11

3

2

6

2

==

==


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Pressure gauge (Same units as stress)
http://www.answers.com/main/Record2?a=NR&url=http%3A%2F%2Fcommons.wikimedia.org%2Fwiki%2FImage%3APsidial.jpg


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Conversion

1 Pa = 145.04106

psi


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Basic modes of deformation

Basic modes of deformation of a materialare:

Extension,Contraction &

shearing


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Material element can beextended, compressed, orsheared. The following

figure shows how thesquare section to the left

changes its shape duringextension, contraction andshearing.


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Extension Contraction Shearing


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Definition of Strain:

Strain is the wayengineers represent

the distortion of abody.


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Another definition

strain is thegeometrical expression

of deformation causedby the action of stress

on a physical body.


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Axial strain

Axial strain (normalstrain) in a bar is a

measure of theextension of a bar per

unit length of the barbefore deformation.


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The following figureshows a bar of initial

length lo that isextended by theapplication of a load

to the length l


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Representation of strain

lo

l


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Axial StrainThe axial strain, denoted by

(epsilon), in a

homogeneously deformingbar is calculated by dividingthe amount the bar extendsby its initial length.

Strain Equation


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Strain Equation

This yield the equation:

lll

=


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Positive and negative strains

A positive axial strainrepresents extension and

a negative axial strainrepresents a contraction.Strain has no units since itis one length divided byanother length.


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Shear Strain

Shear strain, denoted by (gamma), is a measure of howthe angle between orthogonallines drawn on an undeformedbody changes with

deformation. In the followingfigure the square has beensheared into a parallelogram.


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Shear Strain

1h

u


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Equation of Shear Strain

The shear strain is calculated from theequation:

hu

=


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As can be seen from the followingfigure, the shear strain is equal to thetangent of the change in angle or thetwo orthogonal sides.

h

u


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Another Equation of shear strain

h

u== )tan(

Th diff b t


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The difference between and becomes less and

less as the angle (inradians) becomes small.

This is since the tangent ofan angle, given in radians,

can be approximated bythe angle for small values

of the angle.


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In most structuralmaterials, the shearing is

small and we can use theapproximation

1


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Tensile behavior different

materials:In a typical tensile test one

tries to induce uniformextension of the gage

section of a tensilespecimen.

Th ti f th


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The gage section of the

tensile specimen isnormally of uniform

rectangular or circularcross-section.

The following figureshows a typical dog-

bone sample P


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Gage length

P

P

P

P


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The two ends are used forfixing into the grips, whichapply the load. As can be

seen from the free-bodydiagram to the right, the

load in the gage section isthe same as the load appliedby the grips.


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Using extensometers to measure the

change of length in the gage sectionand a load cells to measure the loadapplied by the grips on the sample one

calculates the axial strain and normalstress (knowing the initial gage lengthand 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 inthe sample as a function of the strainfor the given loading. A typical stress-

strain diagram for a mild steel is shownbelow.


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Mild Steel Stress-Strain Curve

Yield stress, y

Ultimate stress, u

Stress,

Strain,

The different regions of the area response


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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

B i l D il b h i


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Brittle versus Ductile behavior

Brittle materials fail at small strains and

in tension. Examples of such materialsare glass, cast iron, and ceramics.Ductile materials fail at large strains

and in shear. Examples of ductilematerials are mild steel, aluminum andrubber.


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The ductility of a material is

characterized by the strain at which thematerial fails. An alternate measure isthe percent reduction in cross-sectional

area at failure.

Different types of response:


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Different types of response:

Elastic response:

If the loading and unloading stress-strain plot overlap each other theresponse is elastic. The response of

steel below the yield stress isconsidered to be elastic.

El ti R (Li & N li )


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Elastic Response (Linear & Non-linear)

LinearElastic

NonlinearElastic


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After loading beyond the yield point,

the material no longer unloads alongthe loading path. There is a permanentstretch in the sample after unloading.


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The strain associated with this

permanent extension is called the

plastic strain p (on the figure).


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As shown in the figure, the unloading

path is parallel to the initial linearelastic loading path.


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Unloading

Loading

Most plastics when loaded continue to


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Most plastics when loaded continue todeform over time even without increasingthe load. This continues extension under

constant referred to as creep. If held atconstant strain, the load required to holdthe strain decreases with time.

R l ti


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Relaxation

The decrease in load over time at

constant stretch is referred to asrelaxation.


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Bearing Stress:

Even though bearing stress is not afundamental type of stress, it is auseful concept for the design ofconnections in which one part pushesagainst another. The compressive load

divided by a characteristic areaperpendicular to it yields the bearing

stress which is denoted by b.


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Therefore, in form, the bearing stress is

no different from the compressive axialstress and is given by

A

Fb =


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Where:

F is the compressive load and

A is a characteristic area perpendicularto 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 by a bolt or rivet as shown,each plate pushes against the side ofthe bolt with load F. It is not clear what

the contact area between the bolt andthe plate is since it depends on the sizeof the bolt and the shape of the

deformation that results.

Also, the distribution of the load on the


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,bolt varies from point to point, but as afirst approximation one can use the

shown rectangle of area A=td to get arepresentative bearing stress for thebolt as

Fb =

mechanics of materials ii(1)

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