Topology Optimization of Structures Composed of One or Two Materials
Maximization of the integral stiffness of a structure composed of one or two isotropic materials of large stiffness is considered using the homogenization technique. Material is modelled by a second rank composite, and we use the concentrations and orient
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Series Editors: The Rectors of CISM
Sandor Kaliszky - Budapest Horst Lippmann - Munich Mahir Sayir - Zurich
The Secretary General of CISM Giovanni Bianchi- Milan
Executive Editor
Carlo Tasso - Udine
The series presents lecture notes, monographs, edited works and proceedings in the field of Mechanics, Engineering, Computer Science and Applied Mathematics. Purpose of the series in to make known in the international scientific and technical community results obtained in some of the activities organized by CISM, the International Centre for Mechanical Sciences.
INTERNATIONAL CENTRE FOR MECHANICAL SCIENCES COURSES AND LECfURES - No. 332
EVALUATION OF GLOBAL BEARING CAPACITIES OF STRUCTURES
EDITED BY G. SACCHI LANDRIANI POLYTECHNIC OF MILAN, MILAN
and J. SALEN 2L, it is sure that the structure cannot be
in equilibrium together with the strength conditions being satisfied at the same time; • if the applied load Q is such that
101
~
2L, it is possible that the structure be in
equilibrium and the strength conditions be satisfied at the same time. Briefly speaking, introducing the termes "stable" and "unstable" to characterize those circumstances :
101 > 2L :
the structure cannot be stable under Q , it is surely unstable under Q ;
101
~
(2.4)
2L : the structure may be stable under Q, it is potentially stable under Q.
The loads Q+ = 2L and
a-
= -
in the loading mode defined in Fig. 2.
2L will be called the extreme loads of the structure
6
J.
Salen~on
2.3.· Comments. Statement (2.4) deserves many comments, most of which will remain relevant in the general case.
2.3.1. The analysis has been performed on the given geometry of the structure : no geometry changes are taken into account.
2.3.2. The only data required for the analysis given in Section (2.2) are the strength conditions imposed on the bars. It follows that results (2.4) hold whatever the initial internal forces in the structure for 0 = 0, whatever the loading path, whatever the constitutive equations for the bars provided they are consistent with the strength conditions (2.1) and with the preceding comment regarding the geometry where the equations are written.
2.3.3. Statement (2.4) is but a partial answer to the original question asked in Section (2.1) since it only offers a guarantee of instability (!) when presumption of stability when
101
101
> 2L and a
s 2L.
It should be understood that such a conclusion is the maximum one can logically derive starting from the only available data (2.1) on the resistance of the constituent elements of the structure imposed as a constraint. In order to be able to assert the stability of the structure under a given load 0 it would be necessary that complementary information regarding - the mechanical behaviour of the constituent elements, through their constitutive equations for instance, - the initial state of self-equilibrated interior forces, - the loading history followed to reach the actual load 0 from the unloaded initial state be available. As a matter of fact it can be easily perceive
