Chapter 3 Simple stress and strain SOM.pdf

SOM materials chapter 4 simple stress and strain

1. Chapter 3
Simple Stress and Strain

2. Deformable bodies
●The bodies which change their shape and size
under the application of the external loads or
forces

3. Internal forces
●Forces that are developed within the body itself so as to balance
the effect of the externally applied forces
●Axial forces, shearing forces, bending and torsional moments are the types of internal forces.

4. Meaning of stress
●Whenamemberissubjectedtoloadsitdevelopsresistingforces.
●Tofind theresistingforces developedasection plane may bepassed
through the member and equilibriumofany one part may be
considered.
●Each partis inequilibrium under the actionofapplied forces and
internalresistingforces.
●Theresistingforcesmay beconvenientlysplit into normal and parallel
tothesectionplane.
The resistingforce parallelto theplaneiscalledshearing resistance.
The resistingforce normalto thesectional planeiscalledNormalresistingforce.

5. ●All structural membersundergo deformationundertheaction of loads and
at certain magnitude of loading,thefailure ofthemembermay occur.
●However, inmostofthecases,deformationsaresmallandaretherefore
neglected.
●Butitbecomes necessarytostudythedeformation in ordertodetermine
thecondition under which failuremay occur.
●During deformation,thematerial ofthebodyresiststhetendency ofthe
loadtodeformthebodyandwhentheload influenceistaken over bythe
internalresistanceofthematerial, it becomes stable.
●This internalresistancewhichthebodyoffers tocopetheload iscalled
stress.

6. Stress
●The internal force of resistance per unit area
offered by the body against deformation is known
as stress.
●The external force acting on the body is called the load. The load is applied on the body while the stress is induced in the material of the body.
●Mathematically,

7. TYPES OF STRESS
SIMPLE OR DIRECT STRESS:
●Simple stress is often called direct stress because it
develops under direct loading conditions.
a)Normal stress: tensile stress, compressive stress
b)Shearing stress
c)Bearing stress
INDIRECT STRESS
a)Bending stress
b)Torsionalstress

8. Normal stress
●Stress developed when the applied force is in line with the
axis of the member
●In case of axial loading, the resistance offered by the material per unit area along the direction of force is called normal stress.
Tensile stress Compressive stress
Ifelongationofthematerial along
the directionof theforcetakes
placethentheresistanceoffered
by thematerialperunitareais
calledtensilestress
Ifcontractionofthe material
alongthedirectionoftheforce
takesplacethen the resistance
offeredbythematerialperunit
areaiscalledcompressivestress

10. Shearing stress
●Shearstressisdeveloped when equal,paralleland
oppositeforcestendtocausea surface to slide
relativelytotheadjacentsurface.

11. Bearing stress
●Bearing stressisthecontactpressurebetweentheseparate bodies.
●Itis generally developed inthestructurefastenerssuchasrivets,bolts and
pins.

12. INDIRECT STRESS
●Stress developed in a member when it is subjected to a
load which is perpendicular to the axis of the member
i.e. transverse loading
●Bending stress
●Torsionalstress

13. STRAIN
●It is change in dimension to the original
dimension
TYPES:1.Nor: the deformation per unit
length caused by normal force in its direction is called normal strain/ linear strain.
•T Strain corresponding
to tensile stress
•Compressive strain: Strain
corresponding to compressive stress

14. 2. SHEAR STRAIN
●In case of a shearing load, a shear strain will be produced which is
measured by the angle through which the body distorts.

15. 3. VOLUMETRIC STRAIN
●When a member is subjected to stresses, it undergoes deformation in all
directions. Hence, there will be change in volume.
●The ratio of the change in volume to original volume is called volumetric
strain.

17. 4. LATERAL STRAIN

19. Hooke’s Law
●Robert Hooke, an English mathematician conducted several experiments and
concluded that
When a material is loaded within its elastic limit,
stress is proportional to strain.●This is called Hooke’s law.
whereEis theconstant of proportionality ofthematerial,
knownasmodulus ofelasticityor Young’s modulus,named
after the English scientist ThomasYoung(1773– 1829).

20. Stress-strain characteristics of mild steel
OAB—represents a straight line,
(obeys Hooke’s law.)
A — limit of proportionality
B — elastic limit
C’/C— represents the
upper/lower yield point
CD — plastic yielding
DE —strain hardening
E—ultimate stress
EF — necking
F — breaking stress

21. A—representsthelimitofproportionality—the stress beyond which
linear variation ceases.
B—representstheelasticlimit—the maximum stress uptowhicha
specimen regainsitsoriginal length on removalof theapplied
load.Thisstress,ingeneral,isnotmeasured andB isassumedto
coincidewithA.Actually,itis difficulttodeterminetheactual
stress–strain relationinthe elastic range neartheyieldpoint;
proportional and elasticlimitaretherefore consideredtobe
coincident.

22. ●C’/C—represents the upper/lower yield point—the magnitude of
stress corresponding to the upper yield point C’ depends on the
cross sectional shape of the specimen and the type of equipment
used to perform the test. The upper yield point is observed if load is
applied rapidly whereas the lower yield point is observed if the rate
of loading is slow. In many of the structural steel hot-rolled
sections, the upper yield point is not obtained due to residual
stresses from the hot rolling process. Hence, it has no practical
significance. Point C represents the yield point at which there is a
definite increase in strain without any further increase in stress. The
stress corresponding to the lower yield point C is thus yield stress
with a typical magnitude of 250 N/mm2 for mild steel.

23. ●For theregionOC,the material is elastic, andtheslopeE,istheYoung’s
modulus.OnaverageEis2 ×105N/mm2.Thestrainattheyieldstressis
about0.00125.
●CD—represents plastic yielding— it isthestrainwhichoccursafter theyield
point, with no increase instress.
●DE—representsstrainhardening—it is arange where additionalstress
produces additional strain. Strain increases fastwithstresstill ultimate loadis
reached. Theslopeofthestress–strain curveinthestrainhardening range,is
typically5to15%oftheYoung’s modulusforstructuralsteels.
●E—representstheultimatestress—the stresscorrespondingto theultimate
load.Theinitial slope of this region is about4per cent of Young’s modulus. At
a strainof at least0.2,thestressreachesitsmaximumvalue,theultimate
strength of steel. Itis also known as minimum ultimate tensile strength.

24. Br do not showyieldpoint.
Insuchcaseswheretheyieldpointisnot
clearlyshown,it is taken asthepoint of
somedefiniteamount ofpermanent strain.
Thepointwherethestress-straincurveis
cutbyastraight line parallel to its initial
straightportiondrawnatapoint ofspecified
strain gives the yieldpointforit.
Incaseofmetals,stressinducinga
permanentstrainof0.2 %defines theproof
stress.

25. Change in length of a body due to application of load

28. Acircularmetalrodof1cmdiameterisloaded intension.Wherethetensile
loadis8kN, theextensioninthe20 cmlongrodismeasuredtobe 0.029 cm.
DetermineYoung’smodulusEofthemetal.

29. A30cmlongbar is5 cmsquare insectionfor12cmofitslengthof2.5 cmdiameterfor8 cmandof4
cmdiameterfor theremaininglength.If tensileforceof100KNisappliedto thebar,calculatethe
maximum andminimumstressesproducedinit andthe totalelongation. Assumeuniform distribution
ofloadoverthecross-section. Take E =200GN/m2

51. Generalized Hooke’s law

52. 1.24A bar of steel is 60 mm x 60mm in
section and180mm long. It is
subjectedtoa tensile load of 300 KN
along the longitudinal axis and
tensile loads of750KN and 600 KN
on the lateral faces. Find the change
in dimensions of the bar and change
in volume. Take E = 200GN/m2
and Poisson’s ratio (1/m) = 0.3

56. 1.23A C.I. flat,300 mmlong and of 30mm
x 50mm uniform sectionisactedupon
by the following forces uniformly
distributed over the respective cross-
section;25KN in the direction of
length (tensile),350KN in directionof
width (compressive) and 200KNin
directionofthickness (tensile) .
Determine the change in volume of the
flat. Take E = 140 GN/m2 and m = 4

64. Thecompressiveload 20KN
issharedbyeachrods.
Thedecreaseoflengthof
eachrodwillbesame.

68. Example1.32Two copper rodsandonesteelrodtogethersupportaloadP.
if thestressesin thecopperandsteelarenottoexceed60x10^6N/m2
and120x10^6 N/m2respectively,findthesafeloadthat canbe
supported. Young’s modulus ofelasticityforsteelis twicethatof copper.

81. SHEAR STRAIN
●In case of a shearing load, a shear strain will be produced which is
measured by the angle through which the body distorts.

82. Complementary Shear Stress

93. For a given material, Young's modulusis110GN/m2 and shear modulusis42GN/m2.
Find thebulk modulus and lateral contractionofa round bar of 37.5mmdiameter and 2.4
m length when stretched 2.5mm.

100. Inpracticeit isnotpossible todesignamechanical component or structural componentpermitting
stressingupto ultimatestress forthe followingreasons:
Reliabilityof materialmaynot be 100%. Theremaybesmallspots offlaws.
Resultingdeformationmayobstructthefunctional performance ofthecomponent.
Loadstakenby designerareonlyestimatedloads.Occasionally therecan beoverloading.
unexpected impact and temperature loadingsmayactin the lifetimeofthemember.
Certainideal conditions assumedin theanalysis(likeboundaryconditions);ideal conditionswill
not be available and, therefore,thecalculatedstresseswillnot be 100%realstresses.
Hence,themaximumstresstowhichanymemberisdesignedismuchlessthanthe ultimatestress,
andthisstressiscalled Working Stress.Theratioofultimatestresstoworkingstressiscalled factor
ofsafety.
WORKING STRESS AND FACTOR OF SAFETY

101. T rateofchangeofstresswithrespect tostrainisknownastangentmodulus,asshown
inFig.The valueofthe tangentmodulusisusefulin thecaseofmaterialswithouta
straight- lineportion inthestress-straindiagram.Itisaninstantaneous value of the modulus
ofelasticity atagivenstress.
TANGENT MODULUS AND SECANT MODULUS
T secantmodulusistheslope
ofthe linejoiningtheorigin toa
specifiedpointon thestress-strain
diagram.

102. •Co ofabar loaded asshowninFig.
•Whilethestressatasectionsufficientlyawayfromthe load isuniform,thestressatthe
sectionwherethe loadis appliedornear that end isnot uniform.This principle isknownas
St Venant’sprinciple.
•Thelocaleffectoftheconcentrated loadis to increasethestressaroundthe loadpoint.This
effect is calledstressconcentration.
STRESS CONCENTRATION

103. St V principleinsimpleterms states that thestressatadistance awayfromthe load
pointis equal to the averagestress.
Itis generallyassumedthat the loadPis uniformly distributedoverthe areaofthemember.
But in practice,the loadisapointload, concentratedatapoint.Stressintensity isthus
theoretically infinite at thepoint ofthe load.Theload infact cannot beapointload as all
pointloadsaredistributedovera smallarea.
Thestressdistribution atorneartheloadedpointissuchthat thereisahigherstressnear the
load.
As wem ove awayfromtheendsections, thishigherstressgoesondecreasingandat
somedistance awayfromtheend section, thestressoverthewholesectionbecomes the
averagestressP/A.This distance is generallyassumedtobethewidthofthemember.

104. Discontinuities In The Section
A discontinuity inthematerial likeaholein the section causesstressconcentration.
Asimilar effect is seen inthecaseofstepped bars having suddenchange ofsection
Thestressnearthehole ornotch,orthediscontinuity ismuchhigher thantheaverage
stressnormally calculated as load/area.
Stressconcentration factor is the ratioofthemaximum stressto the averagestress.
Stressconcentration factor= maximum stress/averagestress

105. F .(b):Effectofaholeinthemember(arivetor
bolthole)is depicted.Thestressismuchhigher ator
near the edgeoftheholethan the averagestress.The
stressconcentrationfactorcanbeashighas3in
certain cases.
Fig.(c):Effectofanotch on thesideofthemember
isshown. Hereagainthestressnear thenotchcanbe
veryhighcompared to the averagestress.
Fig.(d):This isacase ofabruptchangein section.
Thereisstressconcentrationnear the change of
section.This is takencare of byprovidingfillets,
whichchangethe section gradually.
Stressconcentration factorsareworkedoutusing the
theoryofelasticityorexperimentallyby
photoelasticity.
Thepowerfulnumericaltool, finiteelementanalysis,
allowsusto accurately determine the stress
concentration at anyformofdiscontinuity.