بررسی برخی محدودیت های روش مودال در مقابل حل دالامبر در تحلیل تیر برشی تحت بارهای ضربه ای

نوع مقاله : پژوهشی اصیل (کامل)

نویسندگان
سید رسول نعیمی 1
رضا صالح جلالی 2
1 دانشجوی کارشناسی ارشد سازه، گروه مهندسی عمران، دانشکده فنی، دانشگاه گیلان
2 استادیار گروه مهندسی عمران، دانشکده فنی، دانشگاه گیلان
10.22034/22.4.7
چکیده
روش تحلیل مودال روشی ساده و پرکاربرد در بین مهندسین می باشد لذا پاسخ به این سوال که حداقل تعداد مودهای لازم در این روش چند باید باشد تا پاسخهای حاصل از آن دارای دقت کافی نسبت به روشهای دقیق همچون روش انتشار موج یا حل دالامبر باشد دارای اهمیت است. در این راستا از نسبت زمان تناوب غالب زلزله به زمان تناوب اصلی سازه بعنوان معیاری جهت انتخاب تعداد مودهای لازم در تحلیل مودال استفاده گردیده است. از طرفی اگر چه حداکثر جابجایی سازه همواره در بالای آن رخ می دهد لیکن وقتی زمان تناوب پالس جابجایی ورودی به سازه کوچکتر از زمان تناوب اصلی سازه باشد آنگاه بدلیل انتشار موج در طول سازه، حداکثر تغییر شکل نسبی آن می تواند نه فقط در پای سازه بلکه در نقاط دیگری نیز اتفاق افتد. در این مقاله به بررسی برخی محدودیت های روش مودال در مقابل حل دالامبر در تحلیل تیرهای برشی تحت بارهای ضربه ای پرداخته شده است. بدین منظور یک تیر برشی با فرض رفتار خطی و میرایی صفر به روش مودال و حل دالامبر تحلیل گردیده است. روش مودال بر اساس جداسازی متغیرها و پاسخ زمانی آن نیز به روش رانگ-کوتای مرتبه چهار حل گردیده است. حل دالامبر نیز بر اساس انتشار پالس ورودی در تیر برشی و تداخل امواج رفت و برگشتی در طول آن انجام گردیده است. در این تحقیق با در نظر گرفتن پالسهای جابجایی نیم سینوسی با زمان تناوب های کوتاه، متوسط و بلند، نسبت به زمان تناوب اصلی تیر برشی، و نیز دو زلزله حوزه نزدیک دارای پالس مشخص، و مقایسه پوش حداکثر جابجایی و تغییر شکل نسبی ایجاد شده در طول تیر به روش تحلیل مودال و حل دالامبر و محاسبه میزان خطا، تعداد مودهای لازم در روش مودال برای دست یابی به پاسخ قابل قبول ارائه گردیده است. برای تیر برشی با رفتار خطی و میرایی صفر نتایج نشان می دهد که درصد خطای حداکثر جابجایی و حداکثر کرنش برشی ایجاد شده در طول تیر حاصل از روش مودال نسبت به حل دالامبر تحت پالس کوتاه بترتیب با انتخاب 20 و 50 مود، تحت پالس متوسط بترتیب با انتخاب و 20 مود، و نیز تحت پالس بلند بترتیب با انتخاب و مود کمتر از 10 درصد خواهد بود.

کلیدواژه‌ها

موضوعات

عنوان مقاله English

Studying of some limitations of modal analysis versus D’Alembert solution in analysis of shear beam under impulsive loads

نویسندگان English

Seyyed Rasoul Naeemi 1
REZA Saleh Jalali 2
1 MS Student in Structural Engineering, Department of Civil Engineering, Faculty of Engineering, University of Guilan, P.O.Box 3756, Rasht, Iran
2 Assistant Professor, Department of Civil Engineering, Faculty of Engineering, University of Guilan, P.O.Box 3756, Rasht, Iran
چکیده English

Using modal analysis is a lot easier and more widespread among structures, but the important question is about the number of modes should be considered in the modal analysis method to reach an answer with an inevitable error but in logical tolerance. In this regard, the ratio of the dominant period of the earthquake to the main period of the structure is used as a criterion for selecting the number of modes in the modal analysis method. On the other hand, although the maximum displacement of the structure occurs above it, but when the period of the pulse is less than the main period of the structure, due to wave motion along the structure, the maximum shear strain can occur not only at the base but also in other places along the structure. In this paper, some limitations of modal analysis versus Dchr('39')Alembert solution have been studied in analysis of shear beam under impulsive loads. For this purpose, the structure is modeled with a shear beam with linear material and zero damping, and it is analyzed by discrete (modal analysis) and continuous (Dchr('39')Alembert solution) methods. The time response of modal analysis has been done by the fourth-order Runge-Kutta method. The shear beam is subjected to short, medium, and long period half-sine pulses, relative to the main period of the structure, as well as two near-field earthquakes with distinct pulse. The envelope of maximum induced displacement and shear strain (drift) along the beam have been selected to compare the two methods. The necessary number of modes in modal analysis are determined in such a way that its difference with the exact method (Dchr('39')Alembert solution) would be in acceptable range. For shear beam with linear material and zero damping, as it is expected, the results indicate that for convergence of shear strain (drift) response to the exact solution more number of modes are needed than convergence of displacement response in the modal analysis. Under short period pulse , when the ratio of the period of the pulse or the predominant period of earthquake to the main period of the beam is less than , if the minimum number of modes in modal analysis would be 20 and 50 modes for displacement and shear strain, respectively, then the percentage of error of envelope of maximum induced displacement and shear strain (drift) in beam, calculated by modal analysis, would be less than 10 percent, respect to Dchr('39')Alembert solution. Under medium period pulse , when the ratio of the period of the pulse or the predominant period of earthquake to the main period of the beam is greater than and less than , for having ten percent difference between two methods of analyses, the necessary number of modes in modal analysis of beam would be and modes for displacement and shear strain, respectively. For the beam under long period pulse , when the ratio of the period of the pulse or the predominant period of earthquake to the main period of the beam is greater than , the necessary number of modes in modal analysis would be 1 and 5 modes for displacement and shear strain, respectively.

کلیدواژه‌ها English

Modal Analysis
Wave propagation
D'Alembert solution
Impulsive load
Shear beam
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مهندسی عمران مدرس
دوره 22، شماره 4
خرداد و تیر 1401
صفحه 95-110