4-9. A beam is subjected to a normally distributed force  kN as shown.  If the maximum allowable bending stress is  MPa, what is the probability of failure?  The beam is rectangular with  mm and  mm.  Assume  and  are independent.

Solution:

Sum the forces: To find the largest bending moment, sum the forces in the  direction and take                                         the moment about .  The

By plugging in the average of P, the maximum moment is at point A with magnitude

Moment of inertia: Calculate the moment of inertia about the z axis.

 mm⁴

Bending stress: Calculate the distribution of stress that is developed in the beam.

 MPa                                     (1)

 MPa                                     (2)

 MPa

Probability of failure:

                                                                            (3)

Since kN and MPa are independent, Y also follows a normal distribution.  .

 MPa                                                                               (4)

MPa                                                                    (5)

Equation (3) can be written as

Ans.