A
Report on
“Theories Of Failure ”
Submitted By : 8.Yogesh bagal
16.Rahul chaure
18.sachin chole
Guided By:Prof. H.D. Mhatre
Introduction:-
The theory of failure is the science of finding the conditions under which materials fail under the action of external load .
Types of theories of failure:-
- Maximum Principle Stress Theory ( Rankine’s Theory) :-Applicable to Brittle Material.
- Maximum Shear Stress Theory (Coulomb, Tresca & Guest’s theory) :-Applicable to Ductile Material.
- Distortion Energy Theory (Huber von Mises & Hencky’s theory) :-Applicable to Ductile Material.
- Maximum Strain Theory(St. Venant’s theory)
Maximum Principle Stress Theory :-
- According to this, if one of the principal stresses σ1 (maximum principal stress), σ2 (minimum principal stress) or σ3 exceeds the yield stress, yielding would occur.
- In a two dimensional loading situation for a ductile material. where tensile and compressive yield stress are nearly of same magnitude
- σ1 = ± σy
- σ2 = ±σy
- Yielding occurs when the state of stress is at the boundary of the rectangle.
- Consider, for example, the state of stress of a thin walled pressure vessel.
- Here σ1= 2σ2, σ1 being the circumferential or hoop stress and σ2 the axial stress.
- As the pressure in the vessel increases the stress follows the dotted line. At a point (say) a, the stresses are still within the elastic limit but at b, σ1 reaches σy although σ2 is still less than σy.
- Yielding will then begin at point b.
- This theory of yielding has very poor agreement with experiment. However, the theory has been used successfully for brittle materials
- Tensile failure in socket across slot:
- Resisting area :
- Strength of socket :
- From these equation outside diameter of socket is detrmine.
3) Distortion Energy Theory:
- According to this theory yielding would occur when total distortion energy absorbed per unit volume due to applied loads exceeds the distortion energy absorbed per unit volume at the tensile yield point.
- USES
- It is also called von mises stress theory used by designer to check design will withstand a given load condition.
- Using this theory engg can say design will be safe or not.
- I f value of v.m . stress induced in the material is greater than strengthe of material.
- Mostly used in ductile material.
- Now as shown in fig. in 1st part material will fail in tension testwhile in 2nd part in actual engg problem under complex laoding condition normal stress theory cant be used then in industry mostly used theory is v.m .s .theory.
- DISTORTION ENERGY:
- Material is subjected to deformation then shape of material changes but volume do not change.
- Expression for von mises stress
- Condition of failure are as fallows:
- Material will fail von mises stress > strength of material (ductile material)
- Industrial application :
- Design of cantilever beam as material m.s,with cacity 10kn,
Maximum Shear Stress Theory :-
- According to this theory, yielding would occur when the maximum shear stress just exceeds the shear stress at the tensile yield point.
- At the tensile yield point σ2= σ3 = 0 and thus maximum shear stress is σ.y/2
- Maximum shear stress theory is used in cotter because material of the cotter is ductile.
Shear failure of cotter:
- Because of the taper provided in the cotter the two areas at top and bottom ,resisting shear force not the same.
- For easy calculation of dimension of cotter , in place of taking two different width (b1) and (b2) only mean width(b) is being taken into account. hence both the areas will be the same.
- since the areas shown in fig. parallel force p acting on the cotter , the stress produced will be shear force.
- area of the cotter = 2 b × t.
- shearing strength of the cotter =2 b × t × τ.
- Now we know shear stress = P/A
- P =2 b × t × τ
From this equation, width of cotter (b) is determine.
- Conclusion :
- MPST : best Theory of failure for brittle material design.
- MSTT : Give safe design for ductile.
- MDET : best Theory of failure for ductile material design.
Reference:-Design Of Machine Elements (V B BHANDARI)
