A machine element is subjected to the biaxial state of stress σ_x = 80 MPa, σ_y = 20 MPa and τ_xy = 40 MPa. If the shear strength of the material is 100 MPa then factor of safety as per maximum shear stress theory is.

Solution:

Tresca's theory: 

• Tresca's criterion states that a material point yields when the maximum shear stress at that point reaches the maximum shear stress in a uniaxial tension specimen at yield. 
• This theory is suitable for ductile materials.

​In a uniaxial tensile test, the principal stress at the yield point will be σ1=σy, σ2=σ0, σ3=σ0, where σy= yield stress

• So, maximum shear stress at yielding for the uniaxial test,

• Then according to maximum shear stress theory, for no failure: 

• For design, the maximum shear stress formula is given as : 

where, F.O.S = Factor of safety

Calculation:

Given:

σx = 80MPa ,  σY= 20 MPa , Τxy = 40Mpa

Principal stresses:

So, 

σ1 = 100 MPa,  σ2 = 0 MPa

Shear stress, 

As per Tresca's theory,