A widely used model describing the flexural capacity of a reinforced concrete beam is = Mn...

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A widely used model describing the flexural capacity of a reinforced concrete beam is = Mn pfybd(10.59pfy/fe), where p is the reinforcement ratio, fy the yield strength of the reinforcing bars, for the compres- sive strength of concrete, and b and d the effective width and depth, respectively, of the beam. The table below lists the predicted and measured values of the flexural capacity for 10 beams that were tested in a laboratory. Note that 1 MPa is equivalent to 1000 kN/m. To account for the model bias and error, consider the following "corrected" model form InM = InM+E, where M is the measured (true) capacity, M, the predicted (nominal) capacity using the above idealized model, and E the model error. We have used the logarithmic transform of the capacity model to satisfy the normality and homoscedasticity requirements. Assume is a normal random variable with mean and variance , both unknown. Note that describes the bias in the model. Flexural capacity in kNm fr, MPa fy, MPa b, m d, m predicted measured 28 320 0.012 0.35 0.50 309 372 28 410 0.012 0.35 0.50 386 444 28 500 0.015 0.40 0.50 631 699 32 320 0.015 0.35 0.60 551 698 32 410 0.018 0.40 0.60 918 1109 32 500 0.018 0.35 0.60 946 1031 36 410 0.015 0.30 0.50 415 393 36 500 0.018 0.35 0.60 967 922 40 410 0.020 0.40 0.60 1038 1305 40 410 0.015 0.40 0.70 1096 905 1) Employing a non-informative prior, use the normal-inverse-gamma conjugate distribution to determine the posterior joint distribution and second moments of the unknown parameters and . = 2) Consider a beam with dimensions b 0.40 m and d = 0.60 m, and reinforcement ratio p = 0.015. Let fy be a lognormal random variable with mean 430 MPa and c.o.v. equal to 0.10, and fo be a normal random variable with mean 42 MPa and c.o.v. equal to 0.20. Accounting for the model error and parameter uncertainties, determine the first-order approximations of the mean and standard deviation of the conditional reliability index. Also determine the predictive reliability index of the beam to carry a bending moment of 600 kNm that accounts for the uncertainties in the distribution parameters. A widely used model describing the flexural capacity of a reinforced concrete beam is = Mn pfybd(10.59pfy/fe), where p is the reinforcement ratio, fy the yield strength of the reinforcing bars, for the compres- sive strength of concrete, and b and d the effective width and depth, respectively, of the beam. The table below lists the predicted and measured values of the flexural capacity for 10 beams that were tested in a laboratory. Note that 1 MPa is equivalent to 1000 kN/m. To account for the model bias and error, consider the following "corrected" model form InM = InM+E, where M is the measured (true) capacity, M, the predicted (nominal) capacity using the above idealized model, and E the model error. We have used the logarithmic transform of the capacity model to satisfy the normality and homoscedasticity requirements. Assume is a normal random variable with mean and variance , both unknown. Note that describes the bias in the model. Flexural capacity in kNm fr, MPa fy, MPa b, m d, m predicted measured 28 320 0.012 0.35 0.50 309 372 28 410 0.012 0.35 0.50 386 444 28 500 0.015 0.40 0.50 631 699 32 320 0.015 0.35 0.60 551 698 32 410 0.018 0.40 0.60 918 1109 32 500 0.018 0.35 0.60 946 1031 36 410 0.015 0.30 0.50 415 393 36 500 0.018 0.35 0.60 967 922 40 410 0.020 0.40 0.60 1038 1305 40 410 0.015 0.40 0.70 1096 905 1) Employing a non-informative prior, use the normal-inverse-gamma conjugate distribution to determine the posterior joint distribution and second moments of the unknown parameters and . = 2) Consider a beam with dimensions b 0.40 m and d = 0.60 m, and reinforcement ratio p = 0.015. Let fy be a lognormal random variable with mean 430 MPa and c.o.v. equal to 0.10, and fo be a normal random variable with mean 42 MPa and c.o.v. equal to 0.20. Accounting for the model error and parameter uncertainties, determine the first-order approximations of the mean and standard deviation of the conditional reliability index. Also determine the predictive reliability index of the beam to carry a bending moment of 600 kNm that accounts for the uncertainties in the distribution parameters.