Problem 1 An unnotched solid circular shaft is made of AISI

Problem 1) An unnotched solid circular shaft is made of AISI 4142 steel heat treated to a hardness of 450 BHN. It has a diameter of 50 mm and is loaded in cyclic torsion between zero and a torque of 20 KN.m.

a) How many torsion cycles are expected to results in fatigue failure?

b) What is the safety factor according to vonMises stress theory?

c) What shaft diameter is required if 2,000,000 cycles are expected in service?

d) What shaft diameter is required if the safety factor according to vonMises stress theory must be

1.5?

e) What diameter is required if requirements (c) and (d) must be met?

A=1837 MPa, b=-.0762, yield stress = 1584 MPa, Ultimate stress = 1757 MPa.

Solution

a) Su (ultimate strength) = 1757 MPa

    Sf (fatigue strength) = 0.5 * 1757 = 878.5 Mpa

    Sf = A * N^-b

    N   = ( 1837/878.5 ) ^ (1/0.0762)   = 16006.5 cycles

b) shear stress = 16 * T / pi d^3    =   yield stress / (F.S * sqrt(3) )     (von mises)

     16 * 20000 / pi * 0.05^3   = 1584 * 10^6 / (1.732 * FS)

      FS   = 1.122

c) Stress = 1837 * (2000000)^(-0.0762)    = 608.1 MPa

     shear stress   = 16 * 20000 / pi * d^3 = 608.1 * 10^6     = > d   = 5.512 cm

d) shear stress = 16 * T / pi d^3    =   yield stress / (F.S * sqrt(3) )     (von mises)

     16 * 20000 / pi * d^3   = 1584 * 10^6 / (1.732 * 1.5 )

      d = 5.51 cm