Strength Theories

The majority of material strength data is based on uniaxial tensile test results. Usually, all that you have to work with is the yield strength Sy and/or the ultimate tensile strength Su.

This is fine if you only have the one normal stress component present : this is true for simple tension or compression members and for parts loaded only in bending.

σ 1		σ 1	= σ x		

In this case, failure (defined as the onset of plastic deformation) occurs when σ x = σ 1 =Sy/n

‘n’ is the factor of safety.

In many loading cases, we have more than just one normal stress component.

E.g. in torsion, we have a single shear stress component:

τ xy		τ xy	

Or, combined bending and torsion in a shaft:

		τ xy			
σ	x		σ	x		

τ xy

These cases can all be reduced to a simple biaxial case by finding the principal stresses,
σ 1  and σ 2	σ 2		

σ	1		σ	1		

σ 2

Now when does failure occur? For ductile materials there are two commonly used strength theories - the Maximum Shear Stress (MSS) or Tresca theory and the von Mises or Distortion Energy theory.

Strength Theories

1.	Maximum Shear Stress: 

This states that failure occurs when the maximum shear stress in the component being designed equals the maximum shear stress in a uniaxial tensile test at the yield stress: 

This gives	τ max	=	Sy/2n	or			
				| σ 1	– σ 2 |	= Sy/n	
			or	| σ 2	– σ 3 |	= Sy/n	
			or	| σ 3	– σ 1 |	= Sy/n	

whichever of the last three leads to the safest result. The latter usually involves σ 3 being zero, i.e. plane stress, and both σ 1 and σ 2 having the same sign. Note that the yield strength is reduced by the factor of safety ‘n’.

2.	von Mises or Distortion Energy Theory: 

This states that failure occurs when the von Mises stress σ e in the component being designed equals the von Mises stress σ e in a uniaxial tensile test at the yield stress: 

This gives: σ e   =	√2/2 [( σ 1  – σ 2  )2    +   ( σ 2  – σ 3  )2    +	( σ 3  – σ 1  )2 ]0.5	=	Sy/n	
In the plane stress case we have σ 3  = 0 and hence:			
	σ e   =  [σ 12  – σ 1σ 2  + σ 22]0.5	=  Sy/n			

This is the most commonly used of the strength equations.

A third theory, the Maximum Normal Stress theory is similarly defined. It must NEVER be used for design with ductile materials. A modified version of this theory is sometimes used with brittle materials.

All three of these theories are shown on a plot the σ 1  versus σ 2  below:

Strength Theories

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