تحليل تيرها و قاب‌هاي يك و دو طبقه با طول دهانه‌ي برابر با استفاده از دنباله هاي بازگشتي

Analyzing Beams and One-and Two-Story Frames with Equal Span Lengths, using Recurrence Sequences

قرن‌هاست كه رياضي‌دانان از دنباله هاي بازگشتي و توابع مولد آنها براي حل برخي از مسايل در رياضيات استفاده مي كنند، ولي استفاده از اين بخش زيباي رياضيات در تحليل سازه ها مهجور باقي مانده است. در اين نوشتار، در ابتدا براي يك تير سراسري چنددهانه با طول برابر در حالت‌هاي انتها مفصل، انتها گيردار و انتها گيردار غلتكي، كه تحت اثر لنگر خمشي متمركز در ابتداي تير قرار گرفته اند، ضابطه‌ي لنگر تكيه گاه ها به‌دست آمده و لنگر آن محاسبه شده است. سپس به منظور نشان‌دادن كاربرد روش ارايه‌شده ، قاب‌هاي يك و دو طبقه با طول دهانه‌ي برابر در نظر گرفته شده و با صرف‌نظركردن از سختي محوري ستون‌ها ، از فنرهاي پيچشي به جاي ستون‌ها استفاده و مجدداً با استفاده از دنباله‌هاي بازگشتي لنگرهاي محل اتصال تير و ستون محاسبه شده است.

For centuries, mathematicians have used recurrence relations, such as the famous Fibonacci recurrence sequence, and their produced functions for solving mathematics problems. The Fibonacci sequence is named after Leonardo of Pisa, who was also known as Fibonacci. The Fibonacci sequence is a set of numbers that start with a one or a zero, followed by a one, and proceeds based on the rule that each number is equal to the sum of the preceding two numbers. Unfortunately, this amazing aspect of mathematics has not been considered in the analysis of structures. This article is, therefore, particularly focused on using recurrence relations as an innovative method in the analysis of structures. In this research, first, a continuous multi span beam with equal lengths under end, fixed end and sliding end conditions is considered subjected to a concentrated bending moment at the beginning of the beam. Then, the function of the support moments is calculated and moments are obtained. In this regard, one - and two- story frames with equal span lengths have been considered in order to present the applied procedure. Accordingly, torsional springs have been used instead of columns, ignoring their axial stiffness. The bending moments are then computed again at the connection of the beam and column using recurrence relations. Analyzing the sub-frame by this method is one of the products of using recurrence sequences for approximate analysis of the frames subjected to gravity loads. Finally, the effect of shear deformations has been considered for continuous multi span beams with equal span lengths and the structure is analyzed again. In all above mentioned cases, the structures have been analyzed by software as well. The results obtained from analyzing the structures using recurrence sequences are strongly in accordance with those using software. This article shows that the distribution of internal forces follow recurrence sequences in structures, like many other proportions seen in natural phenomena, such as the arrangement of leaves in plants.