Can the factors be exchanged directly? Of course not, as everyone knows. If you understand the structural reliability theory, you shall agree that the factors cannot be exchanged directly. I show you my familiar calculating approach, Monte Carlo method, to descript the reasons by the LRFD theoretical equation, and I think you should be able to understand what the problem is. Based on Bernoulli's law of large numbers, the sample mean of the Monte Carlo method converges in probability to the expectation for any ε>0. The Monte Carlo method can be expressed by the following equation:
where ψ is the resistance factor, Rn is the nominal strength, γ is the load factor, Qn is the load, n is the test times, μ_n is the occurring times in the event, p is the possibility of the event occurring, G(x) is the limit-state function, f(x) is the probability density function, I[∙] is the indicator function, N is the sample size, and N_f is the number of failure times.
If the load combination function is changed, the limit-state function G(x) shall be substituted according to the new load combination function at the same time. Because the limit-state function has been changed, the failure probability must get a new value by the Monte Carlo method. Of course, the resistance factor cannot satisfy the LRFD theoretical equation, so as to re-define this value by the simulation results of the Monte Carlo method. Hence, the factors cannot be exchanged directly, or the reliability index β cannot be controlled by engineers.
Once I try to find the highest factor of safety, the best dead load shall be zero by the calculus method. The highest factor of safety is 1.6, and the weight of structure is an impossible value. However, the structural design is controlled by the member strength, so that the correct explanation is the structure gets a minimum weight. As everyone knows, this is not a safe design, and engineers cannot control the factor of safety. Besides, the LRFD correctness depends on what the ratio of dead load and live load is, but the ratio of dead load and live load for the real structural design is unknown. Compared with the LRFD Code, the ASD Code is just to add all the loads and define the allowable stress, so that it is quite easy to control the minimum requirement for the factor of safety.
Although the LRFD is based on the probability method, its correctness depends on what the ratio is. However, once the ratio is defined, all of the possible design results shall effect by ratio actually. Besides, the ACI 318 Code adopts the Ultimate Strength Design, and it also chooses the ASCE 7 load combination equations which must meet the same problem. The former ACI 318 Code is safer, but it is also not correct to avoid the same problem. In the Civil 401-100, the Appendix C is the former load combination equations, and Taiwanese engineers can adopt these equations to replace the new version. Before the solution is proposed, this is just a safer measure for engineers.
References
where ψ is the resistance factor, Rn is the nominal strength, γ is the load factor, Qn is the load, n is the test times, μ_n is the occurring times in the event, p is the possibility of the event occurring, G(x) is the limit-state function, f(x) is the probability density function, I[∙] is the indicator function, N is the sample size, and N_f is the number of failure times.
If the load combination function is changed, the limit-state function G(x) shall be substituted according to the new load combination function at the same time. Because the limit-state function has been changed, the failure probability must get a new value by the Monte Carlo method. Of course, the resistance factor cannot satisfy the LRFD theoretical equation, so as to re-define this value by the simulation results of the Monte Carlo method. Hence, the factors cannot be exchanged directly, or the reliability index β cannot be controlled by engineers.
Once I try to find the highest factor of safety, the best dead load shall be zero by the calculus method. The highest factor of safety is 1.6, and the weight of structure is an impossible value. However, the structural design is controlled by the member strength, so that the correct explanation is the structure gets a minimum weight. As everyone knows, this is not a safe design, and engineers cannot control the factor of safety. Besides, the LRFD correctness depends on what the ratio of dead load and live load is, but the ratio of dead load and live load for the real structural design is unknown. Compared with the LRFD Code, the ASD Code is just to add all the loads and define the allowable stress, so that it is quite easy to control the minimum requirement for the factor of safety.
Although the LRFD is based on the probability method, its correctness depends on what the ratio is. However, once the ratio is defined, all of the possible design results shall effect by ratio actually. Besides, the ACI 318 Code adopts the Ultimate Strength Design, and it also chooses the ASCE 7 load combination equations which must meet the same problem. The former ACI 318 Code is safer, but it is also not correct to avoid the same problem. In the Civil 401-100, the Appendix C is the former load combination equations, and Taiwanese engineers can adopt these equations to replace the new version. Before the solution is proposed, this is just a safer measure for engineers.
References
- T. V. Galarmbos, B. Ellingwood, J. G. MacGregor, and C. A. Cornell (1982) "Probability-Based Load Criteria: Assessment of Current Design Practice," Journal of the Structural Division, ASCE, Vol.108, No.ST5, 959-977.
- R. Y. Rubinstein & D. P. Kroese (2008) Simulation and the Monte Carlo Method, 2nd edition, Wiley, New Jersey.
- 內政部營建署(2011)建築技術規則,內政部營建署。
- AISC(2012)AISC ANSI 360-10, American Institute of Steel Construction.
- 內政部營建署(2010)鋼構造建築物鋼結構設計技術規範(一)鋼結構容許應力設計法規範及解說,內政部營建署。
- 內政部營建署(2010)鋼構造建築物鋼結構設計技術規範(二)鋼結構極限設計法規範及解說,內政部營建署。
- ACI Committee 318(2008)ACI 318-08M, American Concrete Institute.
- 內政部營建署(2011)混凝土結構設計規範,內政部營建署。