11.1 AXIAL STRAIN - DEUkisi.deu.edu.tr/ozgur.ozcelik/Ekonomi/Strain_AxialLoads_HookesLaw.pdf · 11.3 HOOKE’S LAW • Shown here is a stress–strain curve for a material that follows

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11.1 AXIAL STRAIN

• When an axial load is applied to a bar, normal

stresses are produced on a cross section

perpendicular to the axis of the bar.

– In addition, the bar increases in length, as shown:

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11.1 AXIAL STRAIN

• Change in length, represented by the Greek

lowercase letter δ (delta), is called deformation.

– The change in length δ is for a bar of length L.

Strain is usually expressed dimensionally

as inches per inch or meters per meter,

though it is a dimensionless quantity.

Strain is

defined by:

This equation gives average strain over a length L.

To obtain strain at a point, let length L approach zero.

• Change in length, per unit of length, represented

by Greek lowercase letterε(epsilon), is strain.

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11.1 AXIAL STRAIN

• For an axial load, stress in the direction of the

load is called axial stress.

– Strain in the direction of the load is called axial strain.

Examples and problems start on textbook page 42.

• With axial strain, comes a smaller normal or

lateral strain, perpendicular to the load.

– When the axial stress is tensile, the axial strain is

associated with an increase in length.

– Lateral strain is associated with a decrease in width.

• Tensile strain is called positive strain, and

compressive strain is called negative strain.

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11.2 TENSION TEST ANDSTRESS–STRAIN DIAGRAM

• A common material test is the tension test.

– If a large enough piece is available, and can be

machined, a round cross section could be used.

– For thin plate, a rectangular or square section.

• Deformation or change in length of the specimen

is measured for a specified distance known as

the gauge length.

– Strain is the deformation divided by the gauge length.

The profile for

a typical round

test specimen:

Fillets reduce stress

concentration caused

by the abrupt change

in section.

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• The tension specimen

is placed in a testing

machine such as the

one shown.

Strain can be measured by sensors built

into the testing machine or by separate

gauges attached directly to the specimen.

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• The electrical resistance strain gauge consists of

a length of metallic foil or fine-diameter wire that

is generally formed into a looped configuration.

The gauge is epoxied to the surface

of a test specimen and connected

via lead wires to a sensitive electrical

circuit that measures the resistance

of the foil or wire used in the gauge.

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• With no load applied, the electrical resistance has

a certain value.

• When a load is applied, the foil or wire deforms,

producing a measurable change in resistance.

– That can be directly correlated to specimen strain.

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• Values of stress are found by dividing the load

by the original cross-sectional area.

– And the corresponding value of strain by dividing

the deformation by the gauge length.

• Values obtained can be plotted in a stress–strain

curve, the shape of which will depend on the kind

of material tested.

– Temperature and speed at which the test is performed

also affect the results.

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Stress–strain curves for

three different kinds of material.

Low-carbon steel,

a ductile material

with a yield point.

A ductile material,

such as aluminum

alloy, which doesn’t

have a yield point.

A brittle material,

such as cast iron

or concrete, in

compression.

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• Proportional limit: maximum stress for which

stress is proportional to strain.

– Stress at point P.

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• Yield point: stress for which the strain increases without an increase in stress.– Horizontal portion of the curve ab. Stress at point Y.

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• Yield strength: the stress that will cause the

material to undergo a certain specified amount

of permanent strain after unloading.

– Usual permanent strain percent. Stress at point YS.

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• Breaking strength: stress in material based on

original cross-sectional area at the time it breaks.

– Fracture or rupture strength. Stress at point B.

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• Compression tests are made in a manner similar

to tension tests.

– Specimen cross section is preferably of a uniform

circular shape, although a rectangular or square

shape is often used.

• Recommended ratio of length to major cross-

sectional specimen dimension (diameter/side

length) is 2:1.

– This ratio allows a uniform state of stress to develop

on the cross section, while reducing the tendency of

the specimen to buckle sideways.

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• The right-hand sample

(a nominal 4 x 4) is 8”

long & exhibits a typical

compressive failure.

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• The left-hand specimen,

12” in length, failed by a

combination of compression

and buckling.

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• For ductile materials, values of yield-point stress

are commonly used as the allowable stress for in-

service applications.

A good example of this

is metal used in structural

members, most of which

are made of A36 steel, an

industry designation based

on the material’s yield point

strength of 36,000 psi (~250

MPa).

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• The steel tensile specimens, below left, exhibit

the typical elongation and necking that precedes

an unmistakable point of fracture.

The copper (bottom right)

and aluminum (top right)

compression specimens

deformed under load, with

no clear signs of failure.

Due to this behavior, ductile

materials are not generally

tested in compression.

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TENSION AND COMPRESSION TEST

Copyright © 2011 Pearson Education South Asia Pte Ltd

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APPLICATIONS


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APPLICATIONS (cont)


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STRESS STRAIN DIAGRAM


• Note the critical status for strength specification

� proportional limit

� elastic limit

� yield stress

� ultimate stress

� fracture stress

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• Brittle materials, such as cast iron or concrete,

often have little or no strength in tension.

– They are used primarily for compressive loads

– Allowable stresses for these materials are generally

set at some percentage of the material’s ultimate

strength.

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11.3 HOOKE’S LAW

Figs. (a) & (b) and to a lesser degree, (c), show stress is directly

proportional to strain (the curve is a straight line) on the lower end

of the stress–strain curve.

• Based on tests of various materials and on the

idealized behavior of those materials…

– Hooke’s law states that stress is proportional to strain.

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11.3 HOOKE’S LAW

• Shown here is a stress–strain curve for a material

that follows Hooke’s law.

The slope of the stress–strain curve

is the elastic modulus or modulus of

elasticity, E.

The elastic modulus, E, is equal to

the slope of the stress–strain curve.

Hooke’s law only applies

up to the proportional

limit of the material.

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11.3 HOOKE’S LAW

• Because strain is dimensionless, the elastic

modulus, E, has the same units as stress.

– The modulus is a measure of the stiffness or

resistance of a material to loads.

• Except for brittle materials, high values of E

generally correspond to stiffer materials

– Low values are consistent with more elastic materials.

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11.4 AXIALLY LOADED MEMBERS

• From Hooke’s law:

• When the stress and strain are caused by axial

loads, we have:

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11.5 STATICALLY INDETERMINATEAXIALLY LOADED MEMBERS

• If a machine or structure is made up of one or

more axially loaded members, the equations of

statics may not be sufficient to find the internal

reactions in the members.

• The problem is said to be statically indeterminate,

and equations for the geometric fit of the

members are required.

To write the equations:

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11.6 POISSON’S RATIO

• When a load is applied along the axis of a bar,

axial strain is produced.

– At the same time, a lateral strain, perpendicular to

the axis, is also produced.

• If the axial force is in tension, the length of the

bar increases.

– The cross section contracts or decreases.

• A positive axial stress produces a positive axial

strain and a negative lateral strain.

– For negative axial stress, axial strain is negative,

and the lateral strain is positive.

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• The ratio of lateral strain to axial strain is called

Poisson’s ratio.

• It is constant for a given material provided:

– The material is not stressed above the proportional limit.

– The material is homogeneous.

– The material has the same physical properties in all

directions.

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• Poisson’s ratio, represented by Greek lowercase

letter ν(nu), is defined by the equation:

The negative sign ensures that

Poisson’s ratio is a positive number.

• The value of Poisson’s ratio,ν, varies from 0.25 to

0.35 for different metals.

– For concrete, it may be as low as ν= 0.1, and for

rubber, as high asν= 0.5.

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11.8 ADDITIONAL MECHANICALPROPERTIES OF MATERIALS

• Elastic Limit: The highest stress that can be

applied without permanent strain when the

stress is removed.

– To determine elastic limit would require application

of larger & larger loadings and unloadings of the

material until permanent strain is detected.

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• Elastic Range: Response of the material as

shown on the stress–strain curve from the origin

up to the proportional limit P.

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• Plastic Range: Response of the material as

shown on the stress–strain curve from the

proportional limit P to the breaking strength B.

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• Necking Range: Response of the material

as shown on the stress–strain curve from the

ultimate strength U to the breaking strength B.

Beyond the ultimate strength,

the cross-sectional area of a

localized part of the specimen

decreases rapidly until rupture

occurs.

Referred to as necking, it is a

characteristic of low-carbon

steel—brittle materials do not

exhibit it at usual temperatures.

It is part of the plastic range.

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• Percentage Reduction in Area: When a ductile

material is stretched beyond its ultimate strength,

the cross section “necks down,” and the area

reduces appreciably.

Defined by:

…where Ao is the original, and Af the

final minimum cross-sectional area.

It is a measure of ductility.

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• Percentage Elongation: a comparison of the

increase in the length of the gauge length to the

original gauge length.

It is also a measure of ductility.

Defined by:

…where Lo is the original, and Lf

the final gauge length..

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• Modulus of Resilience: The work done on a unit

volume of material from a zero force up to the

force at the proportional limit.

This is equal to the area under

the stress–strain curve from

zero to the proportional limit.

At right, Area A1.

Units of in.-lb/in.3 or N ( m/m3.)

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• Modulus of Toughness: The work done on a

unit volume of material from a zero force up to

the force at the breaking point.

This is equal to the area under

the stress–strain curve from

zero to the breaking strength.

At right, Areas A1 and A2.

Units of in.-lb/in.3 or N ( m/m3.)

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11.8 ADDITIONAL MECHANICAL PROPERTIESRecent Developments in Materials Technology

• Research in development &applications of new

materials is a continual process.

– Today, much work takes place at the microscopic

level and involves nanotechnology.

• The strongest material ever tested is a carbon

material called graphene.

– Said to be 200 times stronger than structural steel,

it consists of a single layer of graphite atoms that

may be rolled into tiny tubes (called nanotubes).

– The tubes can be used as the basis of graphite fibers

found in products requiring high strength & light weight.

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• A process has been developed in which ceramic

particles are added to molten aluminum, and a

gas is blown into the mixture.

– When solidified, a honeycombed cellular aluminum

material is formed that is lightweight yet strong.

• The use of nano-sized additives has been found

to double the effective life of concrete.

– By preventing penetration of chloride and sulfate ions

from sources such as road salt, seawater, and soils.

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• A new structural material similar to concrete has

been developed that uses a mixture of fly ash

and organic materials.

– This new material has good insulating properties &

fire resistance, as well as high & light weight.

• Several forms of enhanced steels have recently

been developed.

– Sometimes referred to as “super steels,” they are

stronger than traditional counterparts, can withstand

extreme levels of heat and radiation, and generally

have much higher resistance to corrosion.

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• A structural coating has been developed using

a combination of carbon nanotubes & polymers.

– When applied to a bridge, the coating can detect

internal cracks long before they become visible.

• The U.S. Environmental Protection Agency

recently approved public health claims that

copper, brass, and bronze are capable of killing

potentially deadly bacteria, including methicillin-

resistant Staphylococcus aureus (MRSA).

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11.9 STRAIN AND STRESS DISTRIBUTIONS: SAINT-VENANT’S PRINCIPLE

• In our discussion of the uniform axially loaded

bar, we have assumed a uniform distribution of

normal stress on any plane section near the

middle of the bar away from the load.

– We ask here what effect a concentrated compressive

load has on the stresses and strains near the load.

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Consider a short rubber bar of

rectangular cross section.

A grid or network of uniformly spaced

horizontal and perpendicular lines is

drawn on the side of the bar, as shown,

to form square elements.

(Half of the bar is shown in the figure.)

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• Compressive loads are applied to the bar.

Close to the load, the elements are

subjected to large deformations or

strain, while other elements on the

end of the bar, away from the load,

remain virtually free of deformations.

Moving axially away from the load,

we see a gradual smoothing of the

deformations of the elements, and

thus a uniform distribution of strain

and the resulting stress along a

cross section of the bar.

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• Observations are verified by results from the

theory of elasticity.

Showing distribution

of stress for various

cross sections of an

axially loaded short

compression member.

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• Sections are taken at distances of b/4, b/2, and b

from the load where b is the width of the member.

It appears the

concentrated load

produces a highly

nonuniform stress

distribution & large

local stresses near

the load.

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• Sections are taken at distances of b/4, b/2, and b

from the load where b is the width of the member.

Notice how quickly the

stress smoothes out to

a nearly uniform

distribution.

Away from the load at

a distance equal to the

width of the bar, the

maximum stress differs

from the average stress

by only 2.7 percent.

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• The smoothing out of the stress distribution is

an illustration of Saint-Venant’s principle.

Barre de Saint-Venant, a French engineer and

mathematician, observed that near loads, high

localized stresses may occur, but away from the

load at a distance equal to the width or depth of

the member, the localized effect disappears and

the value of the stress can be determined from

an elementary formula.

Such as:

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• This principle applies to almost every other type

of member and load as well.

– It permits us to develop simple relationships between

loads and stresses and loads and deformations.

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11.10 STRESS CONCENTRATIONS

• Stress near a concentrated load is several times

larger than the average stress in the member.

– Similar conditions exist at discontinuities in a member.

Shown here is stress distribution in flat bars under an axial

load, with a circular hole, semicircular notches, and quarter-

circular fillets.

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• Whether stress concentrations are important in

design depends on the nature of the loads and

the material used for the member.

– If the loads are applied statically on a ductile material,

stress concentrations are usually not significant.

– For impact or repeated loads on ductile material, or

static loading on brittle material, it cannot be ignored.

• Stress concentrations are usually not important in

conventional building design.

– They may be important in the design of supports for

machinery and equipment and for crane runways.

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• Stress concentrations should always be

considered in the design of machines.

– In most machine part failures, cracks form at points

of high stress.

– The cracks continue to grow under repeated loading

until the section can no longer support the loads.

• The failure is usually sudden and dangerous.

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11.11 REPEATED LOADING, FATIGUE

• Many structural and most machine members are

subjected to repeated loading and the resulting

variations of the stresses in the members.

– These stresses may be significantly less than

the static breaking strength of the member, but

if repeated enough times, failure due to fatigue

can occur.

• The mechanism of a fatigue failure is progressive

cracking that leads to fracture.

– If a crack forms from repeated loads, the crack

usually forms at a point of maximum stress.

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• In a fatigue test, a specimen of the material is

loaded and unloaded until failure occurs.

– The repeated loading produces stress reversals or

large stress changes in either tension or compression.

• The lower the stress level the greater the number

of cycles before failure.

– In the test, the stress level is lowered in steps until

a level is reached where failure does not occur.

– That stress level is the fatigue strength/endurance limit.

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• Nonferrous metals such as aluminum and

magnesium do not exhibit a fatigue strength.

– They must be tested until the number of cycles

(service life) the metal will be subjected to is reached.

For example, if an aluminum

member has a service life of

107 cycles, the maximum

stress would be 17.5 ksi

(121 MPa) without stress

concentrations.

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• If handbook data are not available and stress

concentrations are not present, the fatigue

strength for ferrous materials may be taken

at approximately 50% of the tensile strength.

– For nonferrous metals, it must be lowered to

approximately 35%

Documents

11.1 AXIAL STRAIN - DEUkisi.deu.edu.tr/ozgur.ozcelik/Ekonomi/Strain_AxialLoads_HookesLaw.pdf · 11.3 HOOKE’S LAW • Shown here is a stress–strain curve for a material that follows