Abstract
In this article the theory of mathematical programming is used, composing improved mathematical models of nonlinear problems of frame loading optimization at shakedown and performing its numerical experiment. An elastic perfectly-plastic frame is considered. Frame geometry, material, load application places are considered known. Time independent load variation bounds are variable (history of loading is unknown). Mathematical model of load variation bounds optimization problem includes strength and stiffness constrains. The mentioned optimization load combines two problems. First problem is connected with the distribution of statically admissible moments at shakedown. This is a problem of residual bending moments analysis which is presented in two ways. In the first case it is formulated as a quadratic programming problem, where the objective function is non-linear, but the objective function of load optimization problem remains linear. The problem is solved by iterations, influential matrixes of residual displacements, and stresses are used. In next case, the equations of problem analysis and dependences are presented according to complete equation system of plasticity theory. Then the objective function of optimization problem becomes non-linear and it is solved in single stage. Solving the second problem, we check if it is possible to satisfy frame rigidity constrains, which are inferior or superior limits of residual displacement. This is considered as a linear programming problem. Mathematical model of frame load optimization problem at shakedown was made with the help of non-linear mathematical programming theory. Numerical experiment was realized with Rozen's gradients projecting method and using the penalty function techniques. Mathematical programming complementarity conditions prohibit taking into account the dechargable phenomena in some cross-sections, therefore analysis of residual deformation compatibility equations are performed, using linear mathematical programming.
Highlights
First problem is connected with the distribution of statically admissible moments
This is a problem of residual bending moments analysis
which is presented in two ways
Summary
Disipacinese sistemose (tokia yra konstrukcija, patirianti plastini deformavimll) ir!lZOS ir poslinkiai priklauso nuo apkrovimo istorijos [1]. Matematika ir mechanika visada turejo itakos viena kitos raidai, keliant mokslines idejas bei ieskant bendf\! Mechanikoje efektyviai taikyti kompiuterines technologijas, svarbu remtis ir matematinio programavimo teorija, taikyti jos metodus sudetingl! OptimaHems sprendiniams rasti, ypac pasinaudoti mechanine sill metod\! Mechanikos principl!, matematinio programavimo teorijos taikymas formuluojant ir sprendziant tampri~l! Konstrukcijl!, taip pat ir reml!, prisitaikomumo uzdavinius tampa logiskas, zvelgiant is abiejl! Straipsnyje matematinio programavimo teorija taikoma, sudarant patobulintus prisitaikiusio remo apkrovos optimizacijos netiesinil! Pasiiilytas naujas optimizacijos u:Zdavinio sprendimo algoritmas, sudarytas remiantis Rozeno optimalumo kriterijaus mechanines prasmes interpretacija [5,6,7], kuris is esmes skiriasi nuo sio straipsnio autoril! Taip gaunamas uzdavinio matematines formuluotes ir skaitinio sprendimo metoda loginis rysys
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