Optimal shakedown design of beam structures

Abstract

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.

An extended shakedown theory on an elastic–plastic spherical shell

“…Mathematical model (22)-(24) is suited for the structures with holonomic behavior because of complementary slackness conditions for mathematical programming (25): the unloading phenomenon of the cross-section is ignored in the shakedown process [36][37][38], which should always be kept in mind when formulating mathematical models for shakedown optimization problems. For this reason, in a precise mathematical model for optimization, one has to consider the lower u r;inf and upper u r;sub variation bounds of residual displacements.…”

Prisitaikančių konstrukcijų optimizavimas. sąsajos su projektavimo standartais

“…Optimalaus projekto paieškos (dažniausiai minimalaus svorio arba tūrio) problemos tinkamas formulavimas labiausiai priklauso nuo konkretaus apribojimo kriterijaus pasirinkimo, o tai priklauso nuo pasirinkto konstrukcijos ribinio būvio. Jei ribiniu įvedamas tamprusis būvis, tuomet formuluojamas tamprus optimizavimo uždavinys, pavyzdžiui, Cinquini et al (1980); jei pasirenkamas prisitaikymo kriterijus -formuluojamas prisitaikomumo optimalus projektas (Cohn ir Parimi 1973;König 1975;Giambanco et al 1994aGiambanco et al , 1994bGiambanco et al , 1998bGiambanco ir Palizzolo 1995); pagaliau, jei taikomas suirimo ribinis būvis, sudaromas standartinis plastinis optimizavimo uždavinys (Rozvany 1976(Rozvany , 1989Save ir Prager 1985). Kadangi bet kurioje iš trijų aukščiau minimų formuluočių pasirenkamas tik vienas apribojimo kriterijus, tuo užtikrinamas konstrukcijos optimalus projektas, atitinkantis būtent šį kriterijų, tačiau tokiu būdu ignoruojamos visos kitos galimos ribinės būklės.…”

Optimization of structures with unrestricted dynamic shakedown constraints

“…In such a topic it is a wellestablished practice to consider the ductility features of the structure and, as a consequence, to take into account its residual resistance capacity beyond the elastic limit, which is often very high (see, e.g., Atkočiūnas 2011;Giambanco et al 1994;Lógó 2002, 2006;König 1975;Massonet and Save 1965;Palizzolo 2004;Se-Hyu and Seung-Eock 2002;Tin-Loi 2000).…”