MAXIMUM SHEAR STRESS THEORY OF FAILURE A THEORY OF FAILURE APPLICABLE TO DUCTILE MATERIALS Statement of the theory
When Yielin! occurs in any material, the ma"im#m $hear $tre$$ at the point of failure equals or exceeds the ma"im#m $hear $tre$$ when yielin! occurs in the ten$ion te$t $%e&imen . The theory applies to ductile materials only, be cause it is based on yielin! . The three'imen$ional (tria"ial) $tre$$ $it#ation*
In the three-dimensional stress situation, the state of stress at a particular location is fully defined by three principal stresses σ 1 , σ 2 , σ 3 . Ma"im#m $hear $tre$$ at a lo&ation of the element
The extreme alues of shear stresses then !ien by the expressions" τ 12
=
σ 1
−
2
σ 2
,
τ 13
=
σ 1
−
2
σ 3
,
σ τ 23
=
τ12 , τ13 , τ23 , in each of the three principal planes
2
−
σ
3
2
#xpressin! the principal stresses in the order of ma!nitude and si!n σ 1 σ 2 Then the maximum shear stress is !ien by
σ τ 13
=
1
−
σ 3
σ
3
2
up to vote on this title THE CASE OF SIMPLE TE+SIO+ TEST ,HE+Sign YIELDI+OCCURS
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$or the simple tension test specimen, the three principal stresses when yieldin! occurs ar S y , σ 3 %& σ σ 2 %&, 1%
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1-Maximum Shear Stress TheoryDerivation
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University of Nairobi
Engineering Desig
The aboe equation implies that the shear yield stren!th of the material S sy
=
S y 2
'ut from analysis of plane stress situation, the maximum shear stress in plane stress is al !ien in terms of plane stress elements τ max
σ x − σ y 2 = ± + τ xy 2 2
DESI-+ E.UATIO+ BASED O+ THE MAXIMUM SHEAR STRESS THEORY
This is deried by ad(ustin! the shear yield stren!th of the material with an appropriate f of safety f . s. . The desi!n equation then becomes" τ max
=
σ 1
−σ 3 2
=
S sy f . s.
=
S y 2 ) f . s.
OR
S sy S y σ x −σ y 2 2 for plane stress situation + = = τ max = ± τ xy f . s. 2 ) f . s. 2
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The principal stresses σ 1 / σ 2 / σ 3 are first determined by stress analysis. *uch analysis describes the principal stresses as a function of the loa carried, and the !eometry and Download With Free Trial element. imen$ion$ of the machine or structural
The maximum shear stress in the desi!n equation is then expressed in terms of the imen$ion$ of the machine or structural element, while the ri!ht hand side is a function to vote onofthis the ten$ile yiel $tren!th of the material. The tensile Sign yieldupstren!th thetitle material beco useful because it more easily determined from laboratory exp eriments. Useful Not useful
The fa&tor of $afety is simply a number chosen by the desi!ner. The factor of safety to!
