Appl Math Optim 38:69–94 (1998)
1998 Springer-Verlag New York Inc.
An Extended Beam Theory for Smart Materials Applications
Part II: Static Formation Problems
D. Y. Gao and D. L. Russell
Department of Mathematics,
Virginia Polytechnic Institute and State University,
Blacksburg, VA 24061-123, USA
Communicated by R. Triggiani
Abstract. In this paper we further develop the theory of the extended Timoshenko
beam model, as ﬁrst introduced in Part I  of this work, with particular emphasis on
applications of the model in formation theory , . We beginwith formal devel-
opment of the equilibrium equations of static formation theory in the context of the
extended Timoshenko model, giving a rigorous discussion of existence, uniqueness,
and regularity of weak solutions with appropriate assumptions on the coefﬁcients.
We continue to obtain the fundamental duality relationship in the context of weak
solutions and indicate its usefulness in investigations of approximate formability.
Optimal formation problems and corresponding necessary conditions for optimality
are discussed. We conclude with a discussion of a particular problem of joint op-
timization of controls and actuator densities in the context of a prismatic extended
Timoshenko beam and we present the results of some computational studies.
Key Words. Elastic beam, Timoshenko beam, Formation theory.
AMS Classiﬁcation. 35J25, 35J50, 35Q72, 49L10.
The research reported in this article was supported in part by NSF Grants DMS-9402838 and DMS-
9400565. Reproduction in whole or in part is permitted for U.S. Government purposes.