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

The component shown below is made of AISI 4340 Oil Quenched at 800^oC and Temper at 625^oC 1hour (σ_yield=90ksi; σ_TS=110ksi) tubular steel. The component, a non-rotating closed end tube with an OD=1.25in and ID=1.0in, is subjected to a fluctuating load (see arrow) that varies between 5F and F. 

Data: The component has been cold rolled, operates at room temperature and needs a reliability of 97.5%. The design has a safety factor of 1.25. The static stress concentrations at the section HKLM are as follows: for axial stress (1.7), bending stress (1.9), torsional stress (1.68) and shear (transverse) stress (1.5). For the determination of the dynamic stress concentrators consider a fillet radius of 0.05in. Use Modified Goodman criterium.

For point L (located opposite to point H as shown)  determine the following:

Use at least four decimal places in your calculations:

The area of the section HKLM (in in^2)

The Neubers constant (in √in \sqrt{in}  

The dynamic stress concentration for axial loading 

The Marin correcting factor for surface finish (k_surf) 

The corrected endurance limit (in psi) 

The mean axial stress (in psi) 

The required force for infinite life (in lb-f) 

A hollow rectangular aluminum bar (E = 71 GPa, Poisson ratio = 0.32 and thickness 2.5mm)  is loaded by a force P and moment M at the free end as shown in the figure. Two strain gages at 60^o to the longitudinal axis recorded the following strains

ε_a=-120μ and ε_b=1000μ

A hollow rectangular aluminum bar (E = 71 GPa, Poisson ratio = 0.32 and thickness 2.5mm)  is loaded by a force P and moment M at the free end as shown in the figure. Two strain gages at 60^o to the longitudinal axis recorded the following strains

ε_a=-120μ and ε_b=1000μ

Find the strains produced by the force F=100lb, in the rosette located at Point A, if the material has the following properties: E=30,000Ksi and Poisson's ratio=0.30. The static stress concentration at point A are 1.8, 1.6 and 1.4 for axial, bending and torsional stresses respectively.

The section AB is a tube of thickness t=0.075in. The material is isotropic