مدل سازي پارامتريك و تحليل گرافيك استاتيك خرپاهاي وارن قوسي

Parametric modeling and graphic statics analysis of Warren Arched trusses

گرافيگ استاتيك يك روش تحليل و محاسبه بصري براي يافتن نوع و مقادير نيروهاي داخلي در سازه‌ها است كه بدور از سختي‌هاي محاسباتي و صرفاً با رويكردي هندسي متمركز بر دو نمودار متقابل فرم و نيرو به اين مهم دست مي‌يابد. در اين مقاله با استفاده از گرافيك استاتيك به تحليل خرپا‌هاي قوسي مبتني بر نوع وارن پرداخته شده است. براي اين منظور، مدل پارامتريك نمودارهاي فرم و نيرو در افزونه پارامتريك گرس‌هاپر برنامه نويسي شدند. پارامتريك سازي قابليت فرم‌يابي و تحليل هر نوع مختلفي از خرپاهاي فرم آزاد شكل گرفته مبتني بر نوع خرپاي وارن را نيز فراهم نموده ‌است. براي سنجش ميزان اعتبار روش و دقت الگوريتم نوشته شده در افزونه‌ي گرس‌هاپر، به مقايسه‌ي نتايج عددي حاصل از چند نمونه‌ي خرپاهاي قوسي تحت بارگذاري‌ متفاوت، با روش محاسباتي المان محدود پرداخته شده است. حداكثر اختلاف ميان دو روش 1/2% مشاهده شده است. نتايج شبيه‌سازي‌هاي صحت‌سنجي حاكي از دقت و سرعت بسيار زياد الگوريتم پيشنهادي دارد.

Graphic Statics is a visual analysis and calculation method to find the type and amount of internal forces in structures, which achieves this importance away from computational difficulties and only with a geometric approach focused on two reciprocal diagrams of form and force. In this paper, arched trusses based on Warren type are analyzed using graphic statics. For this purpose, the parametric model of form and force diagrams were programmed in the Grasshopper parametric plugin. Parametrization has also provided the ability to find and analyze any different types of free-form trusses based on the type of warren truss. To measure the validity of the method and the accuracy of the algorithm written in the Grasshopper add-on, the numerical results obtained from several samples of arched trusses under different loads have been compared with the finite element computational method. The results of the validation simulations indicate the high accuracy and speed of the proposed algorithm. Bow’s notation method with peer-to-peer drawing of members from form to force diagram is a suitable method for graphical calculation of forms with discontinuous components such as truss, by which the internal force of each component can be obtained separately and through drawings. This streamlines the drawing process with rules and principles and makes it easier to continue analyzing components based on calling by name. First, an algorithm for arched truss production was written. The initial inputs in this algorithm included the coordinates of the beginning and end points of the arches and two points as the key of these two arches, whose truss height and also its rise are controlled by these two points. The lower arch was then divided into (n) equal parts by an arbitrary parameter of natural numbers. This number is one less for the upper arch, i.e. (n-1), so that the Warren Arched truss can be formed correctly. After loading, an algorithm for truss naming was created based on the Bow’s notation method. To draw the part of the external forces after receiving the starting point which is located in the force line and the direction of the line which is obtained from the slope of the truss members; The intersection of the calculated lines and the points related to the truss zones are obtained and this diagram is drawn. Also, to examine the algorithm more accurately, the abutment reaction values obtained from the finite element method and the graphic statics method were compared. The comparison of simulation results shows the maximum difference about 1.2 percentage.