9. A large steel gusset plate (KIC = 50 MPa-m^0.5 ) in a truss bridge undergoes cyclic tensile (300 MPa) stresses during use. Prior to use, it was inspected using ultrasonic techniques, from which the largest surface crack found was 2.5 millimeters in length (minimum crack length once loading begins). For the steel in question, the Paris law constants are

C = 1.5 × 10^-12  m/(MPa-m^0.5) per cycle

m = 2.5

Calculate the number of cycles to failure, Nf. Take the stress intensity factor, K=σ √πa. Note: Integrating Paris’s law yields

Hint: First determine the maximum crack length from the fracture strength, then use the change in actual stresses to determine DK as a function of crack length and solve for the critical number of cycles based on the minimum and maximum crack length.