  Constructed facilities should be robust with respect to the loss of
load-bearing elements due to abnormal events. Yet, strengthening structures to
withstand such damage has a significant impact on construction costs.
Strengthening costs should be justified by the threat and should result in
smaller expected costs of progressive collapse. In regular frame structures,
beams and columns compete for the strengthening budget. In this paper, we
present a risk-based formulation to address the optimal design of regular plane
frames under element loss conditions. We address the threat probabilities for
which strengthening has better cost-benefit than usual design, for different
frame configurations, and study the impacts of strengthening extent and cost.
The risk-based optimization reveals optimum points of compromise between
competing failure modes: local bending of beams, local crushing of columns, and
global pancake collapse, for frames of different aspect ratios. The conceptual
study is based on a simple analytical model for progressive collapse, but it
provides relevant insight for the design and strengthening of real structures.
