Applications in Elasticity

• Published in: Introduction to Shape Optimization p. 2
• Main Authors: Mäkinen R. A. E, Haslinger J
• Format: Book Chapter
• Language: English
• Published: Society for Industrial and Applied Mathematics (SIAM) 2003; Society for Industrial and Applied Mathematics
• Series: Advances in Design and Control
• Subjects: General Engineering & Project Administration; Manufacture & Processing; Structural optimization—Mathematics
• ISBN: 0898715369; 9780898715361
• DOI: 10.1137/1.9780898718690.ch7

7.1 Multicriteria optimization of a beam In Section 2.1 the thickness optimization of an elastic beam loaded by a vertical force was analyzed. We looked for a thickness distribution minimizing the compliance of the beam. In practice, however, it is usually important to optimize structures subject to different types of loads. This section deals with a simple prototype of such problems. We shall study multiobjective thickness optimization of an elastic beam. In addition to the compliance cost functional we introduce another two functionals involving the smallest eigenvalues of two generalized eigenvalue problems. Eigenvalues represent natural frequencies of free oscillations and buckling loads of the beam and depend on the thickness distribution e. Our goal is to find a thickness minimizing the compliance of the perpendicularly loaded beam, maximizing the minimal natural frequency (i.e., the beam is stiffer under slowly varying dynamic forces), and maximizing the minimal buckling load (i.e., the beam does not lose its stability easily under the compressive load). 7.1.1 Setting of the problem Let the beam of varying thickness e and represented by an interval I = [0, ℓ], ℓ > 0, be clamped at x = 0 and simply supported at x = ℓ, yielding the following boundary conditions (b.c.): u ( 0 ) = u ′ ( 0 ) = u ( ℓ ) = u ″ ( ℓ ) = 0 . 7.1 The deflection u of the beam under a vertical force f solves the following boundary value problem: A ( e ) u ≔ ( β e 3 u ″ ) ″ = ƒ in ] 0 , ℓ [ with b.c. ( 7.1 ) . 7.2 The meaning of all symbols is the same as in Section 2.1 and e ∈ Uad, where Uad is the class of admissible thicknesses defined by (2.3).