A simply supported beam of rectangular section is loaded with two concentrated loads equidistance from each of the beam ends that act in synchronicity. The concentrated load (F) changes with time according to the equation F = F₀ (A + B sin ωt), where “t” is time and “w” is the frequency and their inclination is given by the distances u and v in the x- and y-direction respectively (see figure). Inspection by ultrasonic techniques carried out before loading detected an edge crack located at mid-span, top part of a beam of a_i in length. Determine the cyclic life of the beam (assume that no other cracks are present). 

Experimental observations on the steel used for the beam construction indicate that the relationship between the crack growth rate (in m/cycle) and the stress intensity factor range (in MPa.m^0.5) is as follows:

Table 1  

Hint: as the geometric factor Y depends on the crack length… then use the initial crack a_i to calculate the Y₁ for the different loads. Use the Y₁’s to determine the final crack size a_f1.

With a_f1  calculate again the geometric factors Y₂’s.. Take the average (Y₁+Y₂)/2 as the final value of the geometric factor Y.  calculate the final crack length  a_f