mechanics of maters(1)

baconossifiedMechanics

Oct 29, 2013 (3 years and 7 months ago)

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WE HAVE DISCUSSED A LOT ABOUT
MATERIALS. THERE ARE DIFFERENT
TYPES OF MATERIALS SUCH AS





METALS


CERAMICS


POLYMERS


COMPOSITES ……………………………….!



MACHINE



MECHANIC



MECHANICS



MECHANICAL ENGINEERING



MECHANISM






IN VERY SIMPLE WORDS




MECHANICS

IS THE SCIENCE OF FORCE
AND MOTION AND HENCE SCIENCE OF
MACHINERY.



IF IT IS FURTHER ELABORATED, THEN



MECHANICS

IS THE PHYSICAL SCIENCE
THAT IS CONCERNED WITH THE
CONDITION OF REST OR MOTION OF
BODIES ACTED ON BY FORCES OR BY
THERMAL DISTURBANCES.


IT CAN FURTHER BE EXPLAINED AS




MECHANICS

IS THE BRANCH OF PHYSICS
CONCERNED WITH THE BEHAVIOR OF
BODIES WHEN SUBJECTED TO FORCES OR
DISPLACEMENTS, AND THE SUBSEQUENT
EFFECTS OF THE BODIES ON THEIR
ENVIRONMENT.



THE DISCIPLINE HAS ITS ROOTS IN SEVERAL
ANCIENT CIVILIZATIONS. DURING THE
EARLY MODERN PERIOD, SCIENTISTS SUCH
AS GALILEO, KEPLER, AND ESPECIALLY
NEWTON, LAID THE FOUNDATION OF
MECHANICS AND WHAT IS NOW KNOWN AS
CLASSICAL MECHANICS.


THE STUDY OF BODIES AT REST IS KNOWN AS
STATICS



THE STUDY OF BODIES WHILE THESE ARE
MOTION IS CALLED
DYNAMICS



THE PRACTICAL APPLICATION OF SCIENCE OF
MECHANICS IS KNOWN AS
APPLIED MECHANICS.

APPLIED MECHANICS EXAMINES THE RESPONSE
OF BODIES (SOLIDS AND FLUIDS) OR SYSTEMS
OF BODIES TO EXTERNAL FORCES.



SOME EXAMPLES OF MECHANICAL SYSTEMS
INCLUDE THE FLOW OF A LIQUID UNDER
PRESSURE, AND THE FRACTURE OF A SOLID
FROM AN APPLIED FORCE.


THEREFORE, THERE ARE TWO MAJOR TYPES
OF APPLIED MECHANICS,
MECHANICS OF
SOLIDS

AND
MECHANICS OF FLUIDS

RESPECTIVELY.



THE BASIC PRINCIPLES OF STATICS AND
DYNAMICS ARE FUNDAMENTAL TO
MECHANICS OF SOLIDS AND MECHANICS OF
FLUIDS.



MECHANICS OF SOLIDS
IS THE SUBJECT TO
BE DISCUSSED IN THIS SEMESTER. THIS IS
ALSO KNOWN AS
MECHANICS OF
MATERIALS
,
STRENGTH OF MATERIALS

AND
ALSO
MECHANICS OF DEFORMABLE BODIES
.


CLASSICAL MECHANICS IS A BRANCH OF
CLASSICAL PHYSICS WHICH IS CONCERNED
WITH THE SET OF PHYSICAL LAWS DESCRIBING
THE MOTION OF BODIES UNDER THE ACTION OF
A SYSTEM OF FORCES.



THIS IS ONE OF THE OLDEST AND LARGEST
SUBJECTS IN SCIENCE, ENGINEERING AND
TECHNOLOGY.



CLASSICAL MECHANICS DESCRIBES THE
MOTION OF MACROSCOPIC OBJECTS, FROM
PROJECTILES TO PARTS OF MACHINERY, AS
WELL AS ASTRONOMICAL OBJECTS SUCH AS
SPACECRAFT, PLANET, STARS AND GALAXIES.



BESIDES THIS, MANY SPECIALIZATIONS
WITHIN THE SUBJECT DEAL WITH GASES,
LIQUIDS, SOLIDS, AND OTHER SPECIFIC
SUB
-
TOPICS.



CLASSICAL MECHANICS PROVIDES
EXTREMELY ACCURATE RESULTS AS LONG
AS THE DOMAIN OF STUDY IS RESTRICTED
TO LARGE OBJECTS AND THE SPEEDS
INVOLVED DO NOT APPROACH THE SPEED
OF LIGHT.


QUANTUM MECHANICS
IS A BRANCH OF
PHYSICS PROVIDING A MATHEMATICAL
DESCRIPTION OF THE WAVE
-
PARTICLE
DUALITY OF MATTER AND ENERGY.



QUANTUM MECHANICS DESCRIBES THE
TIME EVOLUTION OF PHYSICAL SYSTEMS
VIA A MATHEMATICAL STRUCTURE CALLED
THE WAVE FUNCTION.



QUANTUM MECHANICS DIFFERS
SIGNIFICANTLY FROM CLASSICAL
MECHANICS IN ITS PREDICTIONS WHEN THE
SCALE OF OBSERVATIONS BECOMES
COMPARABLE TO THE ATOMIC AND SUB
-
ATOMIC SCALE.






HOWEVER, MANY MACROSCOPIC
PROPERTIES OF SYSTEMS CAN ONLY
BE FULLY UNDERSTOOD AND
EXPLAINED WITH THE USE OF
QUANTUM MECHANICS.



PHENOMENA SUCH AS
SUPERCONDUCTIVITY, THE
PROPERTIES OF MATERIALS SUCH AS
SEMICONDUCTORS AND NUCLEAR
REACTION MECHANISMS OBSERVED
AS MACROSCOPIC BEHAVIOUR,
CANNOT BE EXPLAINED USING
CLASSICAL MECHANICS.




MECHANICS OF MATERIALS


BY BEER & DEWOLF MCGRAW
-
HILL



MECHANICS OF MATERIALS


BY R. CRAIG JOHN WILEY



MECHANICS OF MATERIALS


BY JAMES GERE BROOKS



MECHANICS OF MATERIALS


BY R. C. HIBBLELER
PEARSON/PRENTICE HALL


APPLICATION OF MECHANICS OF MATERIALS IS
ENDLESS AND CAN BE FOUND IN EVERY
DISCIPLINE OF ENGINEERING. FOR EXAMPLE,
BUILDINGS AND BRIDGES, MACHINES AND
MOTORS, SUBMARINES AND SHIPS, OR
AEORPLANES AND ANTENNAS.


`
THE SUBJECTS OF STATICS AND DYNAMICS ARE
ALSO IMPORTANT, BUT THESE DEAL WITH
PRIMARILY WITH THE FORCES AND MOTIONS
ASSOCIATED WITH PARTICLES AND RIGID BODIES.



HOWEVER, MECHANICS OF MATERIALS GO A STEP
FURTHER TO EXAMINE THE STRESSES AND
STRAINS THAT OCCUR INSIDE REAL BODIES THAT
DEFORM UNDER LOADS.






AS WE HAVE DISCUSSED THAT MECHANICS
OF MATERIALS DEALS WITH RESPONSE OF
SOLID BODIES SUBJECTED TO DIFFERENT
TYPES OF LOADING CONDITIONS.


`
THE SOLID BODIES FROM ENGINEERING
POINT OF VIEW ARE AXIALLY LOADED
MEMBERS, SHAFTS IN TORSION, THIN
SHELLS, BEAMS AND COLUMNS,
STRUCTURES WHICH ARE SUB
-
ASSEMBLIES
AND ASSEMBLIES OF THESE COMPONENTS.








AIM OF MECHANICS OF MATERIALS WOULD,
THEREFORE, BE THE DETERMINATION OF
THE STRESSES, STRAINS, AND
DEFLECTIONS PRODUCED BY THE LOADS.


`
A COMPLETE PICTURE OF THE MECHANICAL
BEHAVIOUR OF THE BODY CAN EASILY BE
DETERMINED IF STRESS, STRAIN AND
DEFLECTIONS ARE KNOWN FOR ALL VALUES
OF LOAD UP TO THE FAILURE LOAD.






BOTH THEORETICAL ANALYSES AND EXPERIMENTAL
RESULTS HAVE EQUALLY IMPORTANT ROLES IN THE
STUDY OF MECHANICS OF MATERIALS.



HOWEVER, IT MUST BE NOTED THAT THEORETICAL
DERIVATIONS AND FORMULAS CANNOT BE USED IN A
REALISTIC WAY UNLESS CERTAIN PROPERTIES OF
THE MATERIALS ARE NOT DETERMINED.



THE SPECIFIC PROPERTIES OF MATERIALS ARE
AVAILABLE ONLY AFTER CONDUCTING SUITABLE
EXPERIMENTS IN THE LABORATORIES AND
EXTRACTING RESULTS.


`
FURTHERMORE, MANY PRACTICAL ENGINEERING
PROBLEMS OF GREAT IMPORTANCE CANNOT BE
HANDLED EFFICIENTLY BY THEORETICAL MEANS,
AND HENCE EXPERIMENTAL RESULTS ARE QUITE
ESSENTIAL.





DISCUSSION ABOUT SYSTEMS OF UNITS IS
ALSO VERY IMPORTANT. IF IT IS NOT VERY
CLEARLY UNDERSTOOD ACCURATE RESULTS
CAN BE OBTAINED.


`
ALL SYSTEMS ARE SAME, IT ALL DEPENDS
UPON YOUR CHOICE. HOWEVER,
UNDERSTANDING MUST BE QUITE CLEAR.



MANY SYSTEMS OF MEASUREMENTS HAVE
BEEN DEVISED OVER THE CENTURIES, BUT
TODAY THERE ARE TWO BASIC SYSTEMS OF
MEASUREMENTS USE IN ENGINEERING AND
SCIENTIFIC WORK.













INTERNATIONAL SYSTEM OF UNITS
(SI)


`
U.S. CUSTOMARY SYSTEM (USCS),


IMPERIAL SYSTEM (IS) OR BRITISH
SYSTEM (BS)










THE INTERNATIONAL SYSTEM (SI) IS BASED UPON
MASS (KG), LENGTH (METER) AND TIME
(SECONDS). THESE THREE QUANTITIES ARE
INDEPENDENT OF THE LOCATION AND HENCE THIS
SYSTEM IS KNOWN AS ABSOLUTE SYSTEM.


`
THE U.S. CUSTOMARY SYSTEM (USCS) IS BASED
UPON FORCE (POUND), LENGTH (FOOT) AND TIME
(SECONDS). THIS SYSTEM IS NOT INDEPENDENT
AS UNIT OF FORCE DEPENDS UPON
GRAVITATIONAL ATTRACTION AND HENCE IS ALSO
CALLED GRAVITATIONAL SYSTEM.


`
THE SI SYSTEM IS AN IMPROVED AND
MODERNIZED VERSION OF WELL KNOWN METRIC
SYSTEM AS SI SYSTEM HAS IMPORTANT NEW
FEATURES NOT PREVIOUSLY PART OF METRIC
SYSTEM.









STRESS IS DEFINED AS FORCE PER NIT AREA. IN SI
SYSTEM THE UNIT OF FORCE IS NEWTON (N) AND
THE UNIT OF AREA IS SQUARE METERS. HENCE
UNIT OF STRESS IS NEWTONS PER SQUARE METER
(N/M²). THIS IS ALSO KNOWN AS PASCAL (PA).


`
IN USCS SYSTEM THE UNIT OF IS POUND AND
HENCE UNIT OF STRESS WOULD BE POUNDS (P) PER
SQUARE FOOT (LB/FT²). COMMONLY STRESS IS
MEASURED AS POUNDS PER SQURE INCHES (PSI).


`
AS PASCAL IS A VERY SMALL VALUE IN SI UNITS, IT
IS TAKEN AS MEGA PASCALS (MPA) OR GIGA
PASCALS (GPA). IN USCS SYSTEM LARGER VALUES
OF STRESSES ARE ALSO MEASURED AS KSI.










POUND = 4.45 NEWTONS


KIP

= 4.45 KILONEWTONS






PSI = 6890 PASCALS




KSI = 1000 PSI


KSI = 6.890 PASCALS




PSF = 47.9 PASCALS


KSF = 47.9 KILOPASCALS








MANY NAMES FOR STRESS AND STRAIN ARE
USED IN LITERATURE SUCH NORMAL
STRESS & STRAIN, TENSILE STRESS &
STRAIN, ENGINEERING STRESS & STRAIN,
TRUE STRESS & STRAIN, NOMINAL STRESS
& STRAIN, LONGITUDINAL STRESS &
STRAIN, TRANSVERSE STRESS AND STRAIN,
UNIAXIAL, BIAXIAL AND TRIAXIAL STRESS &
STRAIN.


`
IN FACT ALL THESE TERMINOLOGIES ARE
SAME; DIFFERENCE IS THAT HOW YOU ARE
MANIPULATING AND INTERPRETING THE
CONDITION OF APPLICATION OF LOAD.










VERY SIMPLY, STRESS IS DEFINED AS THE
INTENSITY OF INTERNAL RESISTING FORCE
THAT IS, THE INTERNAL RESISTING FORCE
PER UNIT AREA .


`
SIMILARLY IN GENERAL TERMS
DEFORMATION PER UNIT DIMENSION OF
THE MATERIAL OR SIMPLY STRAIN IS IS THE
ELONGATION PER UNIT LENGTH OF
MATERIAL.


`
STRESS HAS DIMENSION WHILE STRAIN IS
DIMENSIONLESS PARAMETER.








IN ORDER TO INVESTIGATE THE INTERNAL STRESSES
PRODUCED IN AN AXIALLY LOADED BAR, A SECTION
OF PRISMATIC BAR IS CUT PERPENDICULAR TO THE
LONGITUDINAL AXIS OF THIS BAR. THE TENSILE LOAD
“P” ACTS AT THE RIGHT
-
HAND END OF FREE BODY.



AT THE OTHER END ARE FORCES REPRESENTING THE
ACTION OF THE REMOVED PART OF THE BAR UPON
THE PART THAT REMAINS. THESE FORCES ARE
UNIFORMLY DISTRIBUTED OVER THE CROSS SECTION.



FROM THE EQUILIBRIUM OF THE BODY RESULTANT
THESE FORCES MUST BE EQUAL IN MAGNITUDE AND
OPPOSITE IN DIRECTION TO THE APPLIED LOAD “P”.
HENCE STRESS ARE






σ

= P/A








THIS EQUATION (
σ

= P/A) HOLDS FOR THE UNIFORM
STRESS IN AN AXIALLY LOADED, PRISMATIC BAR OF
ARBITRARY CROSS
-
SECTIONAL SHAPE.


`
IF THE BAR IS STRETCHED BY THE FORCES, THE
RESULTING STRESSES ARE KNOWN AS
TENSILE
STRESSES

AND THESE BEAR POSITIVE SIGN, AND
IF THE BAR IS COMPRESSED THE RESULTING
STRESSES ARE KNOWN AS
COMPRESSIVE
STRESSES
AND BEAR NEGATIVE SIGN.



IF THE STRESS ACTS IN A DIRECTION
PERPENDICULAR TO THE CUT SURFACE, IT IS ALSO
KNOW AS
NORMAL STRESS
. NORMAL STRESS MAY
BE BOTH TENSILE OR COMPRESSIVE STRESS. IN
CONTRAST TO NORMAL STRESS THERE IS
ANOTHER TYPE OF STRESS KNOWN AS
SHEAR
STRESS.









IN ORDER FOR THE EQUATION TO BE VALID, THE
STRESS MUST BE UNIFORMLY DISTRIBUTED
OVER THE CROSS
-
SECTION OF THE BAR. IN CASE
STRESS IS NOT UNIFORM, THE EQUATION FOR
STRESS WOULD YIELD THE
AVERAGE NORMAL
STRESS.



UNIFORMITY OF STRESSES OVER THE CROSS
-
SECTION WOULD ONLY EXIST IF THE AXIAL
FORCE ACTS THROUGH THE CENTROID OF THE
CROSS
-
SECTIONAL AREA.



WHEN THE LOAD DOES NOT ACT AT THE
CENTROID, A MORE COMPLICATED SITUATION
WILL ARISE AS BENDING WOULD OCCUR.



`










IF THE INITIAL AREA OF THE BAR IS USED TO
CALCULATE STRESS, THE RESULTING
STRESS IS CALLED
NOMINAL STRESS
,
CONVENTIONAL STRESS

AND
ENGINEERING
STRESS.


`
IN CASE ORIGINAL AREA OF BAR IS USED A
MORE ACCURATE VALUE OF STRESS IS
OBTAINED KNOWN AS
TRUE STRESS.


`









IF A PRISMATIC BAR IS AXIALLY LOADED IT
WOULD EXTEND IN LENGTH IN CASE OF
TENSION AND WOULD SHORTEN IN CASE OF
COMPRESSION.



IN FACT CHANGE IN LENGTH IS THE
CUMULATIVE RESULT OF THE STRETCHING
AND COMPRESSION OF ALL ELEMENTS OF
THE MATERIAL.


`
IN GENERAL THE ELONGATION OR
CONTRACTION OF A SEGMENT WOULD BE
EQUAL TO ITS LENGTH DIVIDED BY THE
TOTAL LENGTH AND MULTIPLIED BY THE
TOTAL ELONGATION.









IN THIS WAY WE CAN CONCEIVE THE
CONCEPT OF ELONGATION PER UNIT
LENGTH, OR STRAIN. THIS IS CALCULATED
AS FOLLOWED:







ε

=
δ
/L



SIMILAR TO STRESS IF BAR IS IN TENSION,
THE STRAIN IS CALLED
TENSILE STRAIN

AND IF IN COMPRESSION IT IS CALLED
COMPRESSIVE STRAIN.









THE DEFINITIONS OF
NORMAL STRESS AND
STRAIN

ARE BASED UPON PURELY STATICAL
AND GEOMETRICAL CONSIDERATIONS AND
HENCE THESE TWO EQUATIONS BE USED FOR
ANY MAGNITUDE OF LOAD AND FOR ANY
MATERIAL.



HOWEVER, THE PRINCIPAL REQUIREMENT IS
THAT THE DEFORMATION OF THE BAR MUST BE
UNIFORM (BE PRISMATIC), THE LOAD ACTS
THROUGH THE CENTROID OF THE CROSS
-
SECTION AND THE MATERIAL MUST BE
HOMOGENOUS.




IF ALL THESE CONDITIONS ARE FULFILLED THE
RESULTING STATE OF STRESS AND STRAIN IS
CALLED UNIAXIAL STRESS & STRAIN.







THE REQUIRED MECHANICAL PROPERTIES OF
MATERIALS ARE DETERMINED BY TESTS
PERFORMED ON SMALL SPECIMENS WITH
STANDARD DIMENSIONS IN LABORATORIES.


`
THE DIMENSIONS OF TEST SPECIMENS AND
METHODS OF APPLYING LOADS HAVE
STANDARIZED BY SOME ORGANIZATIONS SUCH AS
ASTM, ASA, ANSI, AND BSI, PSI.


`
THE MOST COMMON TEST IS THE TENSION TEST
IN WHICH TENSILE LOADS ARE APPLIED TO A
CYLINDRICAL SPECIMEN. IN A STATIC TEST THE
LOAD IS APPLIED VERY SLOWLY, AND IN DYNAMIC
TEST THE RATE OF LOADING MAY VERY HIGH.







AFTER PERFORMING TENSILE TEST AND DETERMINING
THE STRESS AND STRAIN AT VARIOUS LOADS STRESS
-
STRAIN DIAGRAM CAN BE DEVELOPED. A TYPICAL
DIAGRAM IS SHOWN HERE.






THIS STRESS

STRAIN CURVE PROVIDE
FOLLOWING PROPERTIES OF MATERIAL:


1: TRUE ELASTIC LIMIT

2: PROPORTIONALITY LIMIT

3: ELASTIC LIMIT

4: OFFSET YIELD STRENGTH



DURING TENSILE TESTING OF A MATERIAL
SAMPLE, THE
STRESS

協RAIN=CURVE
=
I匠A=
GRA偈ICAL=R䕐R䕓䕎TA呉ON=OF=呈䔠
RELATIONSHIP BETWEEN STRESS AND STRAIN,
I.E. ELONGATION, COMPRESSION, OR
DISTORTION.







THE SLOPE OF STRESS
-
STRAIN CURVE AT ANY
POINT IS CALLED THE TANGENT MODULUS; THE
SLOPE OF THE ELASTIC (LINEAR) PORTION OF
THE CURVE IS A PROPERTY USED TO
CHARACTERIZE MATERIALS AND IS KNOWN AS
THE YOUNG’S MODULUS OR MODULUS OF
ELASTICITY, E.





THE LINEAR RELATIONSHIP BETWEEN STRESS
AND STRAIN IS DEFINED BY HOOKE’S LAW BY
THE FORMULA




σ = E
ε



THIS EQUATION APPLIES ONLY TO ORDINARY
TENSION COMPRESSION; AND FOR MORE
COMPLICATED STATE OF STRESS SUCH AS
BIAXIAL STRESS AND PLANE STRESS, A MORE
GENERALIZED HOOKE’S LAW IS REQUIRED.




ε

= 1/E(
σ

-

ν

σ
)







THE MODULUS ELASTICITY HAS RELATIVELY
LARGE VALUES OF MATERIALS THAT ARE
VERY STIFF SUCH AS STEEL. MORE FLEXIBLE
MATERIALS HAVE A LOWER VALUES.



¾
FOR MOST MATERIALS THE VALUES OF E IS
SAME IN TENSION AND COMPRESSION.





IF A PRISMATIC BAR IS LOADED IN TENSION, THE
AXIAL ELONGATION WOULD BE ACCOMPANIED BY
LATERAL CONTRACTION, AND IT WOULD BE
NORMAL TO THE DIRECTION OF THE APPLIED
LOAD.


`
LATERAL CHANGES CAN EASILY BE OBSERVED IN
POLYMERS AND THESE NOT EASILY OBSERVABLE
IN METALS. HOWEVER, THESE CAN EASILY BE
DETECTED WITH MEASURING DEVICES.



LATERAL STRAIN IS PROPORTIONAL TO THE
AXIAL STRAIN IN THE LINEAR ELASTIC RANGE IF
THE MATERIAL IS BOTH HOMOGENOUS AND
ISOTROPIC. MOST OF THE MATERIALS MEET
THESE REQUIREMENTS.






THE RATIO OF THE STRAIN IN THE LATERAL
DIRECTION TO THE STRAIN IN THE AXIAL
DIRECTION IS CALLED THE POISSON’S RATIO,
AND IS GIVEN BY THE FOLLOWING FORMULA:





ν

=
-

Lateral Strain/Axial Strain



FOR ISOTROPIC MATERIALS POISSON’S RATIO
IS 1/4, MORE RECENT CALCULATION GIVES THIS
VALUE AS 1/3, AND BOTH VALUES ARE IN THE
RANGE OF CALCULATED VALUES.


¾
THE LATERAL CONTRACTION OF A BAR IN
TENSION , OR THE EXPANSION OF A BAR IN
COMPRESSION IS AN EXAMPLE OF STRAIN
WITHOUT ANY CORRESPONDING STRESS.







THE UNIT VOLUME CHANGE IN THE MATERIALS IS
KNOWN AS THE “DIALATATION”, AND IS
CALCULATED AS THE CHANGE IN VOLUME
DIVIDED BY THE ORIGINAL VOLUME. THIS IS
CALCULATED AS



℮ = ∆V/V =
ε
(1
-

2

ν
) =
σ
/E(1
-

2
ν
)



THIS EQUATION CAN BE USED TO CALCULATE THE
INCREASE IN VOLUME IN A BAR IN TENSION IF
THE VALUES OF AXIAL STRAIN OR STRESS AND
POISSON’S RATIO ARE KNOW.



IN THE ELASTIC REGION THE UNIT VOLUME
CHANGE WOULD BE IN THE RANGE OF 0.3
Ε

TO

0.5
Ε
.










SHEAR STRESS IS DIFFERENT FROM TENSILE
STRAIN IN THE SENSE THAT SHEAR STRESS IS
ALWAYS PARALLEL OR TANGENTIAL TO SURFACE.
THESE STRESSES ARE HIGHER NEAR THE MIDDLE
AND BECOME ZERO AT CERTAIN LOCATIONS ON
THE EDGES.



THE AVERAGE SHEAR STRESS ON A X
-
SECTIONAL
AREA IS OBTAINED BY DIVIDING THE SHEAR FORCE
V BY THE AREA A.


τ

= V/A



LIKE NORMAL STRESSES SHEAR STRESSES ALSO
REPRESENT INTENSITY OF FORCE, FORCE PER
UNIT AREA HENCE IT HAS THE SAME UNITS AS OF
TENSILE STRESS.






SIMPLE SHEAR OR DIRECT SHEAR ARISES IN
THE DESIGN OF BOLTS, PINS, RIVETS, KEYS,
WELDS, AND GLUED JOINTS.


¾
SHEAR STRESSES ALSO ARISE IN INDIRECT
MANNER WHEN STRUCTURES ARE SUBJECTED
TO TENSION, TORSION, AND BENDING.




SHEAR STRESSES ON OPPOSITE FACES OF AN
ELEMENT ARE EQUAL IN MAGNITUDE AND OPPOSITE
IN DIRECTION.


`
THESE STRESSES ON PERPENDICULAR FACES OF AN
ELEMENT ARE EQUAL IN MAGNITUDE AND HAVE
DIRECTIONS SUCH THAT BOTH STRESSES POINT
TOWARD, OR POINT AWAY FROM THE LINE OF
INTERSECTION OF THE FACES.


`
UNDER THE ACTION OF SHEAR STRESSES, SHEAR
STRAIN, SIMILAR TO TENSILE STRAIN, IS PRODUCED.
IT IS TO BE NOTED THAT SHEAR STRESSES HAVE NO
TENDENCY TO ELONGATE OR SHORTEN THE MEMBER,
RATHER THESE STRESSES PRODUCE A CHANGE IN
THE ELEMENT CHANGING IT FROM SQUARE SHAPE TO
AN OBLIQUE FORM.





THE CHANGE IN ANGLE,
ϒ
, LESS THAN
Π
/2 AT
ONE SIDE AND GREATER THAN
Π
/2 ON THE
OTHER SIDE, IS A MEASURE OF THE
DISTORTION, OR CHANGE IN SHAPE OF THE
ELEMENT AND IS KNOWN AS SHEAR STRAIN.



SHEAR STRESS ACTING ON A POSITIVE FACE
OF AN ELEMENT IS POSITIVE IF IT ACTS IN THE
POSITIVE DIRECTION OF ONE OF THE
COORDINATE AXES; AND IS NEGATIVE IF IT
ACTS IN THE NEGATIVE DIRECTION OF THE
AXIS.





SIMILARLY SHEAR STRESS ACTING ON A NEGATIVE
FACE OF AN ELEMENT IS POSITIVE IF IT ACTS IN
THE NEGATIVE DIRECTION OF AN AXIS AND
NEGATIVE IF IT ACTS IN THE POSITIVE DIRECTION.



IN THE SAME WAY SHEAR STRAIN IN AN ELEMENT
IS POSITIVE WHEN THE ANGLE BETWEEN TWO
POSITIVE OR TWO NEGATIVE FACES IS REDUCED. IT
IS NEGATIVE WHEN THE ANGLE BETWEEN TWO
POSITIVE OR TWO NEGATIVE FACES IS INCREASED.



AS A RULE OF THUMB, IT MUST BE REMEMBERED
THAT POSITIVE SHEAR STRESSES ALWAYS
PRODUCE POSITIVE SHEAR STRAINS.


SHEAR STRESS
-
STRAIN DIAGRAM CAN BE PLOTTED
IN A SIMILAR WAY TESTS. IN THESE TESTS
HOLLOW, CIRCULAR TUBES ARE TWISTING IN
ORDER TO PRODUCE A STATE OF PURE SHEAR.


`
THE MATERIAL PROPERTIES LIKE PROPORTIONAL
LIMIT, THE YIELD STRESS AND THE ULTIMATE
STRESS UNDER SHEAR ARE USUALLY ABOUT HALF
AS LARGE AS IN TENSION.


`
SIMILAR TO TENSILE STRESS, SHEAR STRESS IS
DIRECTLY PROPORTIONAL TO SHEAR STRAIN IN
THE ELASTIC REGION AND HOOKE’S LAW HOLD
TRUE AS






Τ

= G
ϒ



G IS SHEAR MODULUS OF ELASTICITY OR MODULUS
OF RIGIDITY EXACTLY SIMILAR TO LINEAR
MODULUS OF ELASTICITY, E.


`
BOTH E AND G CAN BE RELATED BY THE
FOLLOWING RELATIONSHIP:






G = E/2(1+
ν
)


`
THIS RELATIONSHIP SHOWS THAT E, G AND

ν

ARE
NOT INDEPENDENT LINEAR PROPERTIES.


`
AS THE VALUE OF POISSON’S RATIO IS BETWEEN
ZERO AND ONE
-
HALF FOR ORDINARY MATERIALS,
THE RELATIONSHIP ALSO SHOWS THAT VALUE OF G
MUST BE FROM ONE
-
THIRD TO ONE
-
HALF OF E.



STRUCTURES SUCH AS BUILDING, BRIDGES,
MACHINES, AIRCRAFT, VEHICLES , SHIPS AND
MANY MORE ARE DESIGNED TO SUPPORT OR
TRANSMIT LOADS.


`
IN ORDER TO AVOID THE FAILURE OF STRUCTURES
THE LOADS THAT A STRUCTURE ACTUALLY CAN
SUPPORT MUST BE GREATER THAN THE LOADS IT
WILL BE REQUIRED TO SUSTAIN DURING SERVICE.
FACTOR OF SAFETY MUST ALWAYS BE GREATER
THAN ONE.


`
IN MORE SIMPLE WORDS THE ACTUAL STRENGTH
OF A STRUCTURE MUST EXCEED THE REQUIRED
STRENGTH; AND THE RATIO OF ACTUAL STRENGTH
TO REQUIRED STRENGTH IS KNOWN AS FACTOR OF
SAFETY N.



FOR MANY STRUCTURES, IT IS QUITE
IMPORTANT THAT THE MATERIALS REMAIN
WITHIN ELASTIC REGION TO AVOID ANY TYPE
OF PERMANENT DEFORMATION.



IN SUCH CASES FACTOR OF SAFETY TO
STRUCTURES IS APPLIED WITH RESPECT TO
YIELD STRESS TO GET ALLOWABLE OR
WORKING STRESS.


IN SOME CASES ULTIMATE STRENGTH IS ALSO
TAKEN IN TO ACCOUNT AND IS CONSIDERED IN THE
SAME WAY AS CONSIDERED FOR YIELD STRESS.



THE DETERMINATION OF A FACTOR OF SAFETY
MUST ALSO BE CONSIDERED FOR FOLLOWING
MATTERS:



PROBABILITY OF ACCIDENTAL OVERLOADING

?C
TYPES OF LOADS

?C
ACCURACY OF DETERMINATION OF LOADS


POSSIBILITIES FOR FATIGUE FAILURES


INACCURACIES IN CONSTRUCTION


VARIABILITY IN THE QUALITY OF WORKMANSHIP


VARIATIONS IN THE PROPERTIES OF MATERIALS




A steel rod 1m long and 13 mm in diameter
carries a tensile load of 13.5kN. The bar
increases in length by 0.05 cm when load is
applied. Determine the stress and strain in
the rod.





A tensile test is performed on a brass
specimen 1 cm in diameter using a gauge
length of 50mm. When applying a load of
25kN we observe that the distance between
the gauge marks is increased by 0.0152cm.
Calculate the modulus of elasticity.













A round rod of diameter 3.8cm is loaded in
tension by a force P. The change in diameter
is measured 0.08mm. Assuming E = 2.8 GPa
and Poisson’s ratio 0.4, calculate the axial
force in the rod.




A steel bar of length of 2.5m with a square X
-
section 100 mm on each side sis subjected to
an axial tensile force of 1300kN. Assuming
E=200 GPa and Poisson’s ratio 0.3, find the
elongation of the bar, the change in the X
-
sectional dimensions and the change in
volum.














Drive the formula for the increase of ∆V in the volume
of a prismatic bar of length L hanging vertically under
its own weight (W = total weight of the bar)






A solid cylinder of some with E=86GPa,
ν

= 0.30 of
diameter 60mm and length 37cm is compressed by an
axial force P = 200kN. Find (a) increase in the diameter
of the bar, (b) find the decrease ∆V in the volume of
the bar.




QUESTIONS AND QUERIES

IF ANY!


IF NOT THEN

GOOD BYE


SEE ALL OF YOU IN NEXT LECTURE

ON shear STRESS AND STRAIN,
allowable stresses &
axially loaded members