Theories of failure or strength under combined stresses

Failure by yielding under combined stresses occur when a stress, strain or strain energy in case of combined stresses equals the value of the corresponding stress, strain or strain energy in simple tension or compression

The combined stresses in bi-axial case: S1, S2 and Ss

The strength theories are

Maximum stress theory (Rankin theory)Maximum shear theory (Coulomb theory)Maximum strain theory ( St Venant theory)Maximum strain energy theoryDistortion energy theory (Von Mises- Hencky)

S1S2Ss

Maximum stress theory:Yielding in the element subjected to combined stresses occurs when one of the principle stresses becomes equal to yield stress in simple tension or compression

S1 = Sy or S1 = -SyS2 = Sy or S2 = -Sy

This theory is applicable to brittle materials SySy+S1+S2SySy* P-S2-S1

Maximum shear theory

Assumes that yielding is produced when the maximum shear stress reaches the value of maximum shear stress at yield under uni-axial stressReasonably applicable to ductile materials. Yield stress in tension and compression to be nearly equalConsider principle stresses S1, S2 and S3 on an element of the material

Max shear stresses are (S1-S2)/2 (S2-S3)/2 and (S3-S1)/2

Equating to shear stress at yield point, (S1-S2)/2 = Sy /2 ; (S2-S3)/2 = Sy/2 and (S3-S1)/2 = Sy/2 When S3 = 0, (S1-S2) = +/- Sy S2 = +/- Sy S1 = +/- Sy

It requires that yield stress in tension and compression are equal

S1S2SySySySyMaximum shear theory -S1-S2*

Maximum strain theory

Failure by yielding in case of combined stresses occurs when the max value of principle strains equal the value of strain at yielding in simple tension and compression

S1 u S2 = +/- Sy S2 u S1 = +/- Sy

This theory is in agreement with tests for thick wall cylinders such as gun barrels

Maximum strain energy theory

Yielding occurs when the elastic strain energy under combined stresses equals that at yield in simple tension or compression It is assumed that yield stress in tension and compression are equal

Principal strains 1= S1/E S2/E 2= S2/E S1/EWhere S1 and S2 are principal stresses

Strain energy U = S1x e1/2 + S2e2/2 = (1/2E)[ S1**2 2 S1S2 S2**2] to be = (1/2E) Sy**2

Failure is defined by S1**2 2 S1S2 + S2**2 = Sy**2

Distortion energy theory

It assumes that yielding begins when distortion energy produced in a unit element subjected to combined stresses becomes equal to the distortion energy at yield when subjected to simple tension

Total energy U = Energy causing volume change + energy causing distortion = Uv + Ud Uv = (1-2 )( S1 + S2 )**2 / 6E Ud = (1+ )[ S1**2 + S2**2 S1S2 ] / 3E At failure Ud = (1+ ) Sy **2 / 3E which can be expressed as [ S1**2 + S2**2 S1S2 ] = Sy**2

It is assumed that Sy is equal in tension and compression. Experimental results for ductile materials show the best agreement with distortion energy theory

Distortion energy theory : envelop of yielding S1-S1-S2S2SySySySy

Ultimate strength theories for combined stressesThe theories are based on maximum stress, maximum shear and internal friction which do not need the assumption of Hooks law (linearity of stress strain curve)

SuSuSuSuS1S2-S2-S1SuSuMax stress theory: S1 = Su or S1= -Su S2 = Su or S2= -Su

Su = ultimate tensile strength Su = ultimate compressive strengthMax shear theory: S1-S2 = SuS1= +/- SuS2= +/- SuInternal friction theory