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Constructed facilities should be robust with respect to the loss ofload-bearing elements due to abnormal events. Yet, strengthening structures towithstand such damage has a significant impact on construction costs.Strengthening costs should be justified by the threat and should result insmaller expected costs of progressive collapse. In regular frame structures,beams and columns compete for the strengthening budget. In this paper, wepresent a risk-based formulation to address the optimal design of regular planeframes under element loss conditions. We address the threat probabilities forwhich strengthening has better cost-benefit than usual design, for differentframe configurations, and study the impacts of strengthening extent and cost.The risk-based optimization reveals optimum points of compromise betweencompeting failure modes: local bending of beams, local crushing of columns, andglobal pancake collapse, for frames of different aspect ratios. The conceptualstudy is based on a simple analytical model for progressive collapse, but itprovides relevant insight for the design and strengthening of real structures.

Risk-based Design of Regular Plane Frames Subject to Damage by Abnormal Events: a Conceptual Study