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Strength of Materials Basics and Equations | Mechanics of Materials Equations

Strength / Mechanics of Material Menu

Strength of materials, also called mechanics of materials, is a subject which deals with the behavior of solid objects subject to stressesand strains .

In materials science, the strength of a material is its ability to withstand an applied load without failure. A load applied to a mechanicalmember will induce internal forces within the member called stresses when those forces are expressed on a unit basis. The stressesacting on the material cause deformation of the material in various manner. Deformation of the material is called strain when thosedeformations too are placed on a unit basis. The applied loads may be axial tensile or compressive, or shear . The stresses and strainsthat develop within a mechanical member must be calculated in order to assess the load capacity of that member. This requires acomplete description of the geometry of the member, its constraints, the loads applied to the member and the properties of thematerial of which the member is composed. With a complete description of the loading and the geometry of the member, the state ofstress and of state of strain at any point within the member can be calculated. Once the state of stress and strain within the member isknown, the strength load carrying capacity of that member, its deformations stiffness qualities, and its stability ability to maintain itsoriginal configuration can be calculated. The calculated stresses may then be compared to some measure of the strength of themember such as its material yield or ultimate strength. The calculated deflection of the member may be compared to a deflectioncriteria that is based on the member's use. The calculated buckling load of the member may be compared to the applied load. Thecalculated stiffness and mass distribution of the member may be used to calculate the member's dynamic response and then comparedto the acoustic environment in which it will be used.

Material strength refers to the point on the engineering stressstrain curve yield stress beyond which the material experiencesdeformations that will not be completely reversed upon removal of the loading and as a result the member will have a permanentdeflection. The ultimate strength refers to the point on the engineering stressstrain curve corresponding to the stress that produces

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fracture.

The following are basic definitions and equations used to calculate the strength of materials.

Stress (normal)

Stress is the ratio of applied load to the crosssectional area of an element in tension and isexpressed in pounds per square inch psi or

kg/mm2.

Load LStress, = =Area A

Strain(normal)

Ameasureofthedeformationofthematerialthatisdimensionless.

changeinlength LStrain, = =

originallength L

Modulusofelasticity

Metaldeformationisproportionaltotheimposedloadsoverarangeofloads.

Sincestressisproportionaltoloadandstrainisproportionaltodeformation,thisimpliesthatstressisproportionaltostrain.Hooke'sLawisthestatementofthatproportionality.

Stress= =E

Strain

Theconstant,E,isthemodulusofelasticity,Young'smodulusorthetensilemodulusandisthematerial'sstiffness.Young'smodulusisintermsof

106psior103kg/mm2.IfamaterialobeysHooke'sLawitiselastic.Themodulusisinsensitivetoamaterial'stemper.Normalforceisdirectlydependentupontheelasticmodulus.

Proportionallimit

Thegreateststressatwhichamaterialiscapableofsustainingtheappliedloadwithoutdeviatingfromtheproportionalityofstresstostrain.

Expressedinpsi(kg/mm2).

Ultimatestrength(tensile)

Themaximumstressamaterialwithstandswhensubjectedtoanappliedload.Dividingtheloadatfailurebytheoriginalcrosssectionalareadeterminesthevalue.

Elasticlimit

Thepointonthestressstraincurvebeyondwhichthematerialpermanentlydeformsafterremovingtheload.

Yieldstrength

Pointatwhichmaterialexceedstheelasticlimitandwillnotreturntoitsoriginshapeorlengthifthestressisremoved.Thisvalueisdeterminedbyevaluatingastressstraindiagramproducedduringatensiletest.

Poisson'sratio

TheratioofthelateraltolongitudinalstrainisPoisson'sratio.

lateralstrainV=

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longitudinalstrain

Poisson'sratioisadimensionlessconstantusedforstressanddeflectionanalysisofstructuressuchasbeams,plates,shellsandrotatingdiscs.

Bendingstress

Whenbendingapieceofmetal,onesurfaceofthematerialstretchesintensionwhiletheoppositesurfacecompresses.Itfollowsthatthereisalineorregionofzerostressbetweenthetwosurfaces,calledtheneutralaxis.Makethefollowingassumptionsinsimplebendingtheory:

1. Thebeamisinitiallystraight,unstressedandsymmetric2. Thematerialofthebeamislinearlyelastic,homogeneousandisotropic.3. Theproportionallimitisnotexceeded.4. Young'smodulusforthematerialisthesameintensionandcompression5. Alldeflectionsaresmall,sothatplanarcrosssectionsremainplanarbeforeandafterbending.

Usingclassicalbeamformulasandsectionproperties,thefollowingrelationshipcanbederived:

3PLBendingstress, b=

2wt2

PL3

Bendingorflexuralmodulus,Eb=

4wt3y

Where: P = normalforce

l = beamlength

w = beamwidth

t = beamthickness

y = deflectionatloadpoint

Thereportedflexuralmodulusisusuallytheinitialmodulusfromthestressstraincurveintension.

Themaximumstressoccursatthesurfaceofthebeamfarthestfromtheneutralsurface(axis)andis:

Mc MMaxsurfacestress, max= =

I Z

Where: M = bendingmoment

c = distancefromneutralaxistooutersurfacewheremaxstressoccurs

I = momentofinertia

Z = I/c=sectionmodulus

Forarectangularcantileverbeamwithaconcentratedloadatoneend,themaximumsurfacestressisgivenby:

3dEt

max=

2l2

themethodstoreducemaximumstressistokeepthestrainenergyinthebeamconstantwhilechangingthebeamprofile.Additionalbeamprofilesaretrapezoidal,taperedandtorsion.

Where: d = deflectionofthebeamattheload

E = ModulusofElasticity

t = beamthickness

l = beamlength

Yielding

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Yieldingoccurswhenthedesignstressexceedsthematerialyieldstrength.Designstressistypicallymaximumsurfacestress(simpleloading)orVonMisesstress(complexloadingconditions).TheVonMisesyieldcriterionstatesthatyieldingoccurswhentheVonMisesstress, exceedstheyieldstrengthintension.Often,FiniteElementAnalysisstressresultsuseVonMisesstresses.VonMisesstressis:

( 1 2)2+( 2 3)2+( 1 3)2

=2

where 1, 2, 3areprincipalstresses.

Safetyfactorisafunctionofdesignstressandyieldstrength.Thefollowingequationdenotessafetyfactor,fs.

YSfs=

DS

WhereYSistheYieldStrengthandDSistheDesignStress

SeeourMaterialTermsandLinkspageforadditionalinformation.

Related:

BeamStressDeflectionandStructuralAnalysisSectionAreamomentInertiaEquationsCalculatorsTolerances,EngineeringDesignLimitsandFits

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