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Unformatted text preview: also heavier, when we increase xi ; there is a tradeoﬀ between
stiﬀness and weight.
Our goal is to design the stiﬀest truss, subject to bounds on the bar crosssectional areas and total
truss weight:
l ≤ xi ≤ u, i = 1, . . . , m,
Wtot (x) ≤ W,
where l, u, and W are given. You may assume that K (x) ≻ 0 for all feasible vectors x. To obtain
a speciﬁc optimization problem, we must say how we will measure the stiﬀness, and what model of
the loads we will use.
(a) There are several ways to form a scalar measure of how stiﬀ a truss is, for a given load f . In
this problem we will use the elastic stored energy
1
E (x, f ) = f T K (x)−1 f
2
to measure the stiﬀness. Maximizing stiﬀness corresponds to minimizing E (x, f ).
Show that E (x, f ) is a convex function of x on {x  K (x) ≻ 0}.
Hint. Use Schur complements to prove that the epigraph is a convex set.
(b) We can consider several diﬀerent scenarios that reﬂect our knowledge about the possible
loadings f that can occur. T...
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 Fall '13
 F.Borrelli
 The Aeneid

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