بررسي روش‌هاي تعيين علامت راه‌حل پيش‌بيني در روش طول كمان در عضوهاي خرپايي دو بُعدي

Investigating the Sign of Predictor Solution Methods in the Arc-Length Method in 2-D Truss Structures

تحليل پايداري سازه‌هاي لاغر نيازمند انجام‌دادن تحليل غيرخطي هندسي است. با دنبال‌كردن مسير غيرخطي تعادل، تشخيص پديده‌ي فروريختگي و بار كمانش و در نتيجه، ظرفيت باربري سازه ممكن مي‌شود. روش طول كمان يكي از قوي‌ترين روش‌ها در ترسيم مسير غيرخطي است. در اين پژوهش، روش‌هاي تعيين علامت راه‌حل پيش‌بيني و تاثير اين روش‌ها در عملكرد روش طول كمان مورد بررسي قرار گرفته است. در مثال‌هاي انجام‌يافته،‌ عملكرد هر يك از روش‌ها در عبور از نقاط بحراني مورد بررسي قرار گرفته است. علاوه بر روش‌هاي موجود، يك روش پيشنهادي و جديدي نيز معرفي شده است كه صحت عملكرد اين روش نيز در مثال‌ها بررسي شود. از ديگر متغيرهاي تاثيرگذار در عملكرد روش طول كمان، متغير مقياسي است. در تحليل سازه‌هاي خرپايي، روش كنترل طول كمان با متغير مقياسي پيشنهادشده، قادر به تحليل و ترسيم نمودارهايي با نقاط ويژه است.

The stability analysis of slender structures requires carrying out geometrically nonlinear analysis. Following the nonlinear equilibrium path, it is possible to understand the phenomenon of collapse or buckling, or the total bearing capacity of structures. Nonlinear equilibrium equations in analysis structures are often solved using the Newton-Raphson method, which is an incremental / iterative procedure. However, the method diverges when it reaches a limit point; therefore, only a part of the curve is obtained. To overcome the difficulties with limit points, displacement control techniques were introduced, and the arc-length method is among those displacement control methods developed in an effort to enable solution algorithms to pass critical points. The arc-length control method often fails to draw the equilibrium path after passing bifurcation and turning points. This is mostly due to the fact that the corrector steps are not carried out in the proper direction around such points. To overcome this issue, several criteria have been presented to predict the correct direction of the predictor step. Some of these methods have been investigated and discussed by means of numerical experiments; in this research, simple two-dimensional truss structures were chosen in order to keep analysis time short. The criterion recently introduced by Feng, Owen and Peric, showed itself to be insensible to bifurcation and turning points, and, therefore, has been successfully applied to draw entire equilibrium paths containing such points. In this research, the criteria of the sign of predictor solution methods and the effects of these methods for the arc-length control method were studied and implemented using Matlab software. In addition to existing methods, a new method was proposed, whose verification by these examples is investigated. The choice of a proper scaling parameter, ? , was demonstrated to have a great influence on the performance of the arc-length method. In the analyzed problems, the use of ? =0 produced the best performance of the method, regarding convergence and computing time.