Question 19.2: Analysis of 100 tests on a ceramic indicated that 11 specime......
Analysis of 100 tests on a ceramic indicated that 11 specimens failed at a stress of 85 MPa or less and that 53 failed at a stress of 100 MPa or less. Estimate the stress that would cause one failure in 10,000.
P_{S}=\exp\left[-(\sigma /\sigma_{o} )^{m}\right], \ \ln P_{S} =-(\sigma /\sigma _{o})^{m}, \ \ln P_{s2}/\ln P_{s1} =(\sigma _{2}/\sigma _{1})^{m}, m =\ln \left[\ln P_{s2}/\ln P_{s1} \right]/\ln(\sigma _{2}/\sigma _{1}). Substituting P_{s2}=1-0.53=0.47 at \sigma_{2}=100 \ MPa and P_{s1}=1-0.11=0.89 at \sigma_{1}=85 \ MPa, m =\ln \left[\ln (0.47)/\ln (0.89) \right]/\ln(100/85)=11.50. \ (\sigma_{3}/\sigma_{1} ) =(\ln P_{s3}/\ln P_{s1})^{1/m}=\left[\ln (0.9999)/\ln (0.47)\right]^{1/11.5}=0.46. Thus \sigma_{3}=(0.46)(85) = 39 \ MPa.