The optimal design of plane beam structures made of elastic perfectly plastic material is studied according to the shakedown criterion. The design problem is formulated by means of a statical approach on the grounds of the shakedown lower bound theorem, and by means of a kinematical approach on the grounds of the shakedown upper bound theorem. In both cases two different types of design problems are formulated: one searches for the minimum volume design whose shakedown limit load is assigned; the other searches for the design of the assigned volume whose shakedown limit load is maximum. The optimality conditions of the four problems above are found by the use of a variational approach; such conditions prove the equivalence of the two types of design problems, provide useful information on the structural behaviour in optimality conditions, and constitute a fifth possible way to determine the optimal design. Whatever approach is used, the strong non-linearity of the corresponding problem does not allow the finding of the analytical solution. Consequently, in the application stage suitable numerical procedures must be employed. Two numerical examples are given.
The minimum volume design problem of elastic
perfectly plastic finite element structures subjected to a combination of fixed and perfect cyclic loads is studied.
The design problem is formulated in such a way that incremental collapse is certainly prevented. The search for the structural design with the required limit behaviour is effected following two different formulations, both developed on the grounds of a statical approach: the first one operates below the elastic shakedown limit and is able to provide a suboptimal design; the second one operates
above the elastic shakedown limit and is able to provide the/an optimal design. The Kuhn–Tucker conditions of the two problems provide useful information about the different behaviour of the obtained structures.
An application concludes the paper; the comparison among the designs is effected, pointing out the different behaviour of the obtained structures as well as the required computational effort related to the numerical
solutions
A class of thin vaults, the so-called "bòvedas tabicadas", which represent one of the most common Spanish traditional building techniques at the end of XIX century are studied here, treating the relevant analysis problem through a numerical, as well as an experimental, approach. At first the problem is studied by searching for the behaviour of the material effecting suitable experiments. Once the constitutive behaviour of the materials and the structural elements are experimentally characterized, a semi inverse method for the identification of the optimum mechanical parameters to assign to an equivalent homogeneous ideal material through analysis reproducing the executed experimental tests is adopted on the grounds of finite element analysis. Moreover, other tests are processed on a real structure in order to determine the mechanical and geometric response to assigned loads. These results, compared to the related numerical analyses, can be utilized in order to propose some useful simple criteria in the reinforcing design of the referenced structures.
A formulation of a special design problem devoted to elastic perfectly plastic steel frame structures subjected to different combinations of static and dynamic loads is presented. In particular, a minimum volume design problem formulation is presented and the structure is designed to be able to elastically behave for the assigned fixed loads, to elastically shakedown in presence of serviceability load conditions and to prevent the instantaneous collapse for suitably chosen combinations of fixed and ultimate seismic loadings as well as of fixed and wind actions. The actions that the structure must suffer are evaluated by making reference to the actual Italian seismic code. The dynamic response of the structure is performed by utilizing a modal technique. In order to prevent other undesired collapse modes further constraints are introduced in the relevant optimization problem taking into account the risk of element buckling. Different applications devoted to flexural frames conclude the paper. The sensitivity of the structural response has been investigated on the grounds of the determination and interpretation of the Bree diagrams of the obtained optimal structures
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