# 8 Physical reliability models Submission Date March 24, 2017 1. The time to failure of a drill bit is lognormal with s = 1.43. From field data the median life of the bit has been found to depend on the Brinell hardness and the density of the material being drilled. The following function was empirically derived: u x 12.31 0.0157 x1 0.35 x 2 where x1 is the Brinell hardness and x2 is the density (mg/m3). If a steel plate has been heat-treated to a Brinell hardness of 200 and has a density of 7.3 mg/m 3, determine its reliability over 20 cutting hours. 2. Consider the following load distribution: 3 x 2 / 10 9 f x x 0 0 x 1000 kg otherwise If the strength is a constant 950 kg, determine the static reliability. 3. The strength of a concrete structure designed to support a fixed load of 464 lb has the following probability density function: f y y 3y 2 10 9 0 y 1000 lb (a) Compute the static reliability. (b) If the load is also random having the probability density function below, find the static reliability. f x ( x) 2(0.001 0.000001x) 0 x 1000 lb (c) If the load in part (a) is applied at random according to a Poisson process with a mean occurrence rate of 0.01 per year, compute the design life for a reliability of 0.99. 4. The buckling force under an axial compressive load for a column made from timber and hinged at both ends is given by Euler's formula: Fc 2 EI L2 where Fc E I = = = critical load in pounds modulus of elasticity = 1.6 106 pounds per square inch for timber moment of inertia of the cross section of the column = 5.359 1 inches4 for a 2-inch by 4-inch column L = the length of the column in inches A 10-foot column is used to support a load that has a Weibull distribution with a shape parameter of 1.76 and a scale parameter of 2,500 lb. (a) What is the static reliability? (b) If a material stronger than timber is used that has a safety factor of 3 (i.e., F c is three times the mean load), what is the static reliability? 2