Optimization and Multihazard Structural Design Florian A. Potra 1 and Emil Simiu, F.ASCE 2 Abstract: There is a growing interest in the development of procedures for the design of structures exposed to multiple hazards. The goal is to achieve safer and/or more economical designs than would be the case if the structures were analyzed independently for each hazard and an envelope of the demands induced by each of the hazards were used for member sizing. We describe an optimization approach to multihazard design that achieves the greatest possible economy while satisfying speciﬁed safety-related and other constraints. We then present an application to illustrate our approach. DOI: 10.1061/ASCEEM.1943-7889.0000057 CE Database subject headings: Earthquake engineering; Hazards; Optimization; Solar power; Structural engineering; Seismic design; Wind loads. Introduction There is a growing interest in the development of procedures for the design of structures exposed to multiple hazards. The goal is to achieve safer and/or more economical designs than would be the case if the structures were designed independently for each of the hazards and an envelope of the demands induced by each hazard were used for member sizing. Useful, if mostly ad hoc, approaches to multihazard design have recently been proposed Bruneau 2007. However, a broad, multidisciplinary foundation for multihazard design remains to be developed. Such a foundation should include a probabilistic com- ponent. Duthinh and Simiu 2008 have found that the current ASCE 7 Standard ASCE 2005 does not take into account the fact that failure probabilities of structures in regions exposed to both strong wind hazard and strong earthquake hazard may ex- ceed their counterparts in regions exposed to only one of the hazards, and that for this reason consideration should be given to augmenting wind and seismic load factors speciﬁed in the ASCE 7 Standard so that risk-consistency be achieved within the Standard. The mathematically and physically rigorous rationale advanced by Duthinh and Simiu 2008 in support of these state- ments can be conveyed intuitively by noting that health and life insurance premiums would likely be higher for a professional motorcycle racer if he/she would also be active as a stunt artist. The purpose of this note is to submit that a multidisciplinary foundation of multihazard design theory would beneﬁt from the inclusion of appropriate optimization approaches. For simplicity we address in this note the case in which the loads corresponding to a nominal probability of exceedance of the failure limit state are speciﬁed. We explore the potential of optimization under mul- tiple hazards as a means of integrating the design so that the greatest possible economy is achieved while satisfying speciﬁed safety-related and other constraints. Our formulation the multihazard design problem rests on the fact that optimization under N hazards N  1 imposes m i i =1,2,..., N sets of constraints, all of which are applied simul- taneously to the nonlinear programming problem NLP associ- ated with the design. Following a description of our approach we present a simple illustrative application. Finally, we present a set of conclusions and suggestions for future research. Multihazard Design as a Nonlinear Programming Problem We consider a set of n variables i.e., a vector d with n compo- nents d 1 , d 2 ,..., d n  characterizing the structure. In a structural engineering context we refer to that vector as a design. Given a single hazard, we subject those variables to a set of m constraints g 1 d 1 , d 2 ,..., d n   0 g 2 d 1 , d 2 ,..., d n   0,..., g m d 1 , d 2 ,..., d n   0 1 Examples of constraints are minimum or maximum member di- mensions, allowable stresses or design strengths, allowable drift, allowable accelerations, and so forth. A design d 1 , d 2 ,..., d n  that satisﬁes the set of m constraints is called feasible. Optimization of the structure consists of selecting, from the set of all feasible designs, the design denoted by d ¯ 1 , d ¯ 2 ,..., d ¯ n  that minimizes a speciﬁed objective function f d 1 , d 2 ,..., d n . The objective func- tion may represent, for example, the weight or cost of the struc- ture. The selection of the optimal design is a NLP for the solution of which a variety of techniques are available. We emphasize that in this note we do not consider topological optimization. Rather, we limit ourselves to structures whose conﬁguration is speciﬁed, and whose design variables consist of member sizes. As noted earlier, in multihazard design each hazard i i =1,2,..., N imposes a set of m i constraints. Typically the opti- mal design under hazard i is not feasible under i.e., does not satisfy the constraints imposed by hazard j  i. For example, 1 Mathematician, Information Technology Laboratory, Mathematical and Computational Sciences Div., National Institute of Standards and Technology, Gaithersburg, MD 200899-8911. E-mail: ﬂorian.potra@ nist.gov 2 NIST Fellow, Building and Fire Research Laboratory, National Insti- tute of Standards and Technology, Gaithersburg, MD 20899-8611 corre- sponding author. E-mail: emil.simiu@nist.gov Note. This manuscript was submitted on September 10, 2008; ap- proved on May 1, 2009; published online on May 4, 2009. Discussion period open until May 1, 2010; separate discussions must be submitted for individual papers. This technical note is part of the Journal of Engi- neering Mechanics, Vol. 135, No. 12, December 1, 2009. ©ASCE, ISSN 0733-9399/2009/12-1472–1475/$25.00. 1472 / JOURNAL OF ENGINEERING MECHANICS © ASCE / DECEMBER 2009 Downloaded 08 Dec 2009 to 129.6.88.32. Redistribution subject to ASCE license or copyright; see http://pubs.asce.org/copyright