Force, stress and strain

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Force, stress and strain Mass in kilograms, kg
often called weight Produces, in gravity, Force
in Newtons. 1kg produces a force of 9.8
Newtons A force pulling (or pushing) on a rod of
a cross-sectional area A Produces a stress,
measured in Pascals (Newtons per square
meter) Stress Force/(Area of cross
section) For a weight of 10 kg hanging from a
rod that is 1cm (0.01meters) in diameter Force
98 N Area pr2 78.5x10-6 m2 Stress 1.2x106
Pa, 1.2 MegaPascals This will cause a strain
(change in length/original length) in rod.
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Mechanical Properties Response to Stress
Elastic reversible strain (e.g a
spring) Plastic permanent strain (e.g. a bent
paperclip) Fracture propagation of a crack
(e.g. breaking glass) Viscous flow of a
liquid Visco-elastic slow elastic strain and
recovery (e.g. toffee) Fatigue slowly growing
crack in cyclic loading Creep slow permanent
strain due to vacancy motion (e.g. old lead pipe)
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Elastic modulus in tension Linear
elasticity Hookes Law, 1660 Deformation ?
Load Youngs modulus 1807 Stress Modulus x
Strain, ? E . ? Engineering Stress
Load/Original Area True Stress Load/ Actual
Area Engineering Strain Extension/Length True
strain log (L/Lo)
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Stiffness is good for Stiff low elastic
deflection under load Aircraft stiff light
aluminum alloy, carbon fiber composite Columns
prone to elastic compressive buckling
pillars, legs, bicycle frames Radio
masts Stiffness is bad for Trees Buildings
in earthquakes Bungee cords Structures subject to
impacts bicycle frames, auto suspensions, bodies
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A column under a centric axial load exhibiting
the characteristic deformation of buckling.
K column effective length factor, whose value
depends upon the conditions of end support of the
column, as follows. For both ends pinned
(hinged, free to rotate), K 1.0. For both ends
fixed, K 0.50. For one end fixed and the other
end pinned, K 0.70. For one end fixed and the
other end free to move laterally, K 2.0. l
unsupported length of column, I Moment of inertia
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Other moduli Shear tg.e g1/3E Bulk
PK.DV/V Poissons ratio, n Dw/DL
1/2 And many more.
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Complex loading In bending, one surface is in
tension, one is in compression and the center is
neutral. Shear stress is highest at the center.
For a load P, length L, width b, thickness d, the
stress at the top surface is
For our weight lifter P 3000 Newtons, L 1
meter (2x bone length) b 2 cm, d2cm Stress
562 MPa, enough to break the bone easily.
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Failure by plastic deformation or
fracture Strength is the stress at failure.
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Yield is characteristic failure of tough
materials This one is being stretched. More
often metals yield in bending and become
permanently deformed.
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Stronger new materials
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High strength is good for Heavy tensile
loadings Cables Heavy plastic buckling loads
auto wheels, connecting rods High yield
strength in compression Hardness Good for
resistance to scratching, abrasion and wear Low
strength may be used to assure safe failure
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Strength brittle fracture Crack starts and
grows across a material Strength depends on
size of surface or internal cracks Ductile
failure is usually localized and provides a
warning. Brittle fracture is frequently
disastrous. Many alloys go brittle at low
temperatures, high speed or when pre-cracked.
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Stress concentration
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Fracture depends on existing cracks These may
grow from defects or form as a result of fatigue
after cyclic loading (vibration) Occurs when the
biggest crack in a part reaches the critical
length. Often the crack first grows slowly by
fatigue or corrosion. Griffith Theory of
fracture Strength ? E Elastic Modulus, c
crack length, ? is surface energy
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In a brittle material, cracks may grow slowly in
cyclic fatigue and then lead to sudden failure
once they are long enough. If the same material
were ductile, the metal at the crack tip would
yield and blunt the crack so it stops
growing. Fatigue cracks at stress concentrations
are a big worry because they can be
unpredictable.
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Liberty ships, WWII Welded rather than riveted,
which led to higher stress concentrations. Strong
er steels have a ductile-brittle transition at
low temperatures, met in the North
Atlantic. About 300 out of 3,000 cracked.
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The Comet The first commercial jet
airliner. Several disappeared mysteriously
before it was discovered that fatigue at a window
corner led to cracking in the Al-4Cu skin alloy
that reached a critical size and then zipped off
the front of the aircraft.
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Aloha Airlines flight 243 The roof of this 737
unzipped at 24,000 feet after fatigue cracks
grew, due to the many take-offs and landings on
short haul routes in Hawaii and salt aggravation.
A flight attendant was sucked out but everyone
else lived.
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The Alexander Kielland drilling rig resembled a
five-legged stool and it was to travel from place
to place in the North Sea in attempt to locate
oil beneath the sea floor. Each leg was
constructed of steel members that were each
supported on a pontoon. However even before it
left the fabrication site in 1976 the
French-built drilling rig was considered
obsolete, since more advanced rigs were already
operating in the North Sea. Therefore the
oil-company decided to convert the drilling rig
into a floating hotel so that it could offer
proper rest to the drilling crews in their
wandering for new oil sites. Probably due to
bad welding a crack of 80 mm of size was already
present in one of the struts that connected the
five legs together. As the drilling rig was put
in service the crack expanded each time a wave
hit the strut. The crack propagated undetected to
2/3 of the circumference of the strut before it
caused the failure. When the strut failed the
fifth leg got loose and turned the drilling rig
into an unstable structure that flipped upside
killing two thirds of the occupants.
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Hardness Is related to yield strength
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Toughness Energy absorbed during fracture,
crucial for impact resistance Most materials can
be hardened only at the cost of reduced
toughness
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