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Optimal design problems in mechanics of growing composite solids, Part II: shape optimization
A new class of the optimal design problems arising in mechanics of growing viscoelastic composite bodies subjected to aging is proposed. In the current paper we analyze the problem of shape optimization for growing reinforced elements of construction. It is studied on the example of a growing cantilevered reinforced beam under the action of its weight. The problem consists in determining of the optimal thickness ho(x) of the beam which minimizes the maximum deflection on account of geometrical restrictions and limitations on the total volume of the beam. We propose a semi-analytical solution to this problem, and we have obtained the optimal thickness of the beam using numerical techniques. It is shown that: the optimal thickness of the growing cantilevered beam is a piecewise continuously differentiable function of the longitudinal coordinate. In the vicinity of the clamped end, the optimal thickness is uniform, and it is equal to the maximum admissible thickness, h2; in the middle of the beam, function ho(x) decreases (practically linearly); and it is uniform and takes the minimum admissible value h1 in the vicinity of the free end of the beam; further, for a nonaging material of the beam, the optimal thickness depends very weakly on the regime of construction of the reinforcement. For a fixed cost functional, the regime when the length of the reinforcement coincides with the length of the part filled by the main material demands an additional volume of the main material with respect to other regimes of growth, when the reinforcement is established before the construction of the main material; further, decrease of the mass density of the main material leads to additional material influx in the vicinity of the clamped end of the growing cantilever. This decrease shifts the optimal thickness function closer to the free end of the beam; further, aging of the main material essentially changes the optimal thickness of the growing beam. The growth of the rate of aging leads to increase of thickness in the middle part of the beam without significant changes of the total volume.
Optimal design problems in mechanics of growing composite solids, Part II: shape optimization
A new class of the optimal design problems arising in mechanics of growing viscoelastic composite bodies subjected to aging is proposed. In the current paper we analyze the problem of shape optimization for growing reinforced elements of construction. It is studied on the example of a growing cantilevered reinforced beam under the action of its weight. The problem consists in determining of the optimal thickness ho(x) of the beam which minimizes the maximum deflection on account of geometrical restrictions and limitations on the total volume of the beam. We propose a semi-analytical solution to this problem, and we have obtained the optimal thickness of the beam using numerical techniques. It is shown that: the optimal thickness of the growing cantilevered beam is a piecewise continuously differentiable function of the longitudinal coordinate. In the vicinity of the clamped end, the optimal thickness is uniform, and it is equal to the maximum admissible thickness, h2; in the middle of the beam, function ho(x) decreases (practically linearly); and it is uniform and takes the minimum admissible value h1 in the vicinity of the free end of the beam; further, for a nonaging material of the beam, the optimal thickness depends very weakly on the regime of construction of the reinforcement. For a fixed cost functional, the regime when the length of the reinforcement coincides with the length of the part filled by the main material demands an additional volume of the main material with respect to other regimes of growth, when the reinforcement is established before the construction of the main material; further, decrease of the mass density of the main material leads to additional material influx in the vicinity of the clamped end of the growing cantilever. This decrease shifts the optimal thickness function closer to the free end of the beam; further, aging of the main material essentially changes the optimal thickness of the growing beam. The growth of the rate of aging leads to increase of thickness in the middle part of the beam without significant changes of the total volume.
Optimal design problems in mechanics of growing composite solids, Part II: shape optimization
Probleme der optimalen Auslegung infolge der Volumenzunahme von festen Verbundwerkstoffen Teil 2: Gestaltoptimierung
Drozdov, A.D. (author) / Kalamkarov, A.L. (author)
Transactions of the ASME, Journal of Applied Mechanics ; 62 ; 983-988
1995
6 Seiten, 4 Bilder, 12 Quellen
Article (Journal)
English
Balken , Optimierung , Stahlbeton , Auslegung (Dimension) , Durchbiegung , Rauminhalt , Viskoelastizität , Altern (Werkstoff) , mechanische Spannung , frei tragender Träger , Brücke (Bauwerk) , Wanddicke , Querschnitt , Biegebeanspruchung , mathematisches Modell , Differenzialgleichung , Integralgleichung , Elastizitätsmodul , elastische Verformung , Dichte (Masse)
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