Kuhn-Tucker conditions in shakedown problems - Atkočiūnas, Juozas (Journal of Civil Engineering and Management, 1996)
An elastic perfectly plastic structure at shakedown to given cyclić loading is under consideration. The stress-strain field of dissipative system in general is related to the history of loading. And only in a particular case, i.e. at the moment prior to the failure of an elastic perfectly plastic structure the distribution of the actual residual forces is unique for each prescribed history of loading (the safety factor of shakedown approaches unity). Nevertheless, there exist some domains where the plastic strains are equal to zero. The residual forces in the statically indeterminate parts of the structure may be non-unique: the stress field is only determined by the equilibrium equations. The extremum energy principle of minimum complementary energy allows to derive the actual residual forces out of all statically admissible residual forces at the moment prior to cyclic plastic failure. Then the stress-strain field analysis problem at the moment prior to the cyclic plastic failure is formulated as a problem of non-linear mathematical programming. Formulating the dual pair of non-linear programming problem (statical and kinematic formulation of analysis problem) the differential constraints are neglected or replaced by algebraic conditions. When the safety factor is approching a unity, the degeneracy of the statical formulation of the analysis problem often can occur. In this case a mathematical model is proposed for obtaining an upper bounds for the displacement at shakedown. It is pointed out that the known Kuhn-Tucker conditions of mathematical programming theory (i.e. compatibility equations of residual strains) in concert with restriction, limiting the maximum value of total energy dissipation, make up the adaptation conditions of the structure to given cyclic loading. Kuhn-Tucker conditions used in above—mentioned problem allow to correctly interprete the physical aspect of the degeneracy problem at shakedown. When the safety factor is larger than unity an artificial degeneracy situation for the statical formulation of analysis problem can be created. Then the mathematical models presented can be applied to the analysis of unloading elastoplastic structures. With this aim in view a fictitious equiplastic structure the behaviour of which is holonomic is derived. The displacements of the fictitious structure enclose the displacements of the actual structure subject to cyclic loading. Kuno-Takerio sąlygos prisitaikomumo uždaviniuose Santrauka. Nagrinėjama idealiai tampriai plastinė, prisitaikanti prie kintamos-kartotinės apkrovos, konstrukcija. Konstrukcijos įtempimų-deformuoto būvio prieš pat ciklinį-plastinį suirimą (prisitaikomumo atsargos koeficientas artimas vienetui) analizė pateikiama kaip netiesinio matematinio programavimo uždavinys. Sudarant netiesinio programavimo dualių uždavinių porą (statinė ir kinematinė analizės uždavinio formuluotės) diferencialinės priklausomybės yra įgnoruojamos arba pakeičiamos algebrinėmis sąlygomis. Esant atsargos koeficientui artimam vienetui, dažnai gaunamas išsigimęs statinės formuluotės analizės uždavinys. Šiam atvejui siūlomas prisitaikymo poslinkių viršutinių ribų nustatymo uždavinio matematinis modelis. Parodyta, jog žinomos matematinio programavimo teorijoje Kuno-Takerio sąlygos (liekamųjų deformacijų darnos lygtys) kartu su energijos pilnos disipacijos maksimalią reikšmę ribojančia sąlyga formuoja konstrukcijos prisitaikomumo duotai ciklinei apkrovai sąlygas. Kuno-Takerio sąlygos, naudojamos minėtame poslinkių įvertinimo uždavinyje, įgalina korektiškai interpretuoti išsigimusio prisitaikomumo analizės uždavinio fizinę prasmę. Article in Russian. First Published Online: 26 Jul 2012