تحلیل قابلیت اعتماد در سازههای یک بعدی به کمک تابع چگالی احتمال صریح پاسخ استاتیکی در اجزاء محدود تصادفی

نوع مقاله : پژوهشی اصیل (کامل)

نویسندگان
ریزان چوبداریان 1
آزاد یزدانی 2
هوشنگ دباغ 2
1 دانشگاه سنندج
2 دانشگاه کردستان - گروه مهندسی عمران
چکیده


تحلیل قابلیت اعتماد سازه­ها با در نظر گرفتن تصادفی بودن متغیرهای هندسه، مصالح و بارگذاری، راهنمای مناسبی جهت طراحی در مهندسی سازه می­باشد. از روش اجزاء محدود تصادفی به منظور تحلیل سیستم سازه­ای با در نظر گرفتن عدم قطعیت در سیستم­های تصادفی، استفاده می­شود. حل معادلات اجزاء محدود تصادفی منجر به محاسبه تمامی پاسخ­های محتمل سازه با در نظر گرفتن همه­ی متغیرهای تصادفی سیستم سازه­ای می­شود. به دلیل تاثیر عدم قطعیت در پاسخ سیستم­های سازه­ای، انجام تحلیل قابلیت اعتماد ضروری است. با اینحال به علت محدودیت روش­های کلاسیک تحلیل قابلیت اعتماد، نیاز به محاسبه شاخص قابلیت اعتماد بر اساس تابع چگالی احتمال پاسخ وجود دارد. در این پژوهش با ترکیب روش اغتشاش و تغییرمتغیر و بدون وجود محدودیت نوع توزیع آماری متغیرهای تصادفی، تابع چگالی احتمال صریح پاسخ سازه محاسبه می­شود. با محاسبه تابع چگالی احتمال پاسخ استاتیکی سازه، احتمال خرابی و شاخص قابلیت اعتماد به طور مستقیم به دست می­آید. در اجزاء محدود تصادفی، به دلیل ثابت نبودن خصوصیات متغیرهای تصادفی در سازه، نیاز به تعریف یک تابع همبستگی بین متغیر تصادفی در المان­های مختلف می­باشد. بنابراین شاخص قابلیت اعتماد به عنوان معیار همگرایی مش­بندی با در نظر گرفتن مقیاس نوسانات و تابع همبستگی متغیرهای تصادفی در المان های مجاور، در نظر گرفته می­شود. در هر تعداد المان، شاخص قابلیت اعتماد سازه با استفاده از تابع چگالی احتمال صریح پاسخ و تابع ظرفیت سازه محاسبه شده تا در تعداد مشخصی از المان­ها به همگرایی برسد. در این مطالعه تحلیل اجزاء محدود تصادفی در حالت استاتیکی خطی برای یک نمونه تیر و یک نمونه ستون طره انجام می­گیرد و متغیرهای هندسه، مصالح و بارگذاری به صورت تصادفی با توزیع­های آماری واقعی مطابق پیشینه موضوع در نظر گرفته می­شوند. همان­گونه که مطابق اجزاء محدود کلاسیک قابل انتظار است، در تعداد بیشتر المان­ها و مش­بندی ریزتر، شاخص قابلیت اعتماد مقادیر بزرگتری دارد. در نظر گرفتن مقیاس نوسانات کمتر برای متغیرهای تصادفی، موجب می­شود شاخص قابلیت اعتماد در مقدار بزرگتری همگرا شود درحالیکه در مقیاس نوسانات بزرگتر به دلیل همبستگی بیشتر متغیر تصادفی در المان­های مجاور، در تعداد کمتری از المان­ها، همگرایی اتفاق می­افتد. همچنین نتایج تابع چگالی احتمال پاسخ با روش صریح در مقایسه با روش شبیه­سازی مونت­کارلو تطابق مناسبی را نشان می­دهد. مزیت استفاده از شاخص قابلیت اعتماد به عنوان معیار همگرایی در مش­بندی اجزاء محدود تصادفی، در نظر گرفتن همه­ی پاسخ­های محتمل سازه به جای استفاده از پاسخ متوسط سازه در طراحی می­باشد.

کلیدواژه‌ها

موضوعات

عنوان مقاله English

reliability analysis of one dimentional structures using explicit probability density function of static response in stochastic finite element

نویسندگان English

rezan choobdarian 1
azad yazdani 2
Hooshang Dabbagh 2
1 student
2 Associate Professor of Kurdistan University - Department of Civil Engineering
چکیده English

Reliability analysis, with considering the randomness of geometry, materials and loading variables, is a good guide for structural design in engineering. The stochastic finite element method is used to analyzing the structural systems, with regard to uncertainty in random parameters of structures. Solving the equations of stochastic finite element leads to the calculation of all possible structural responses taking into account all the random variables of the structural system. Due to the uncertainty effect on the response of structural systems, reliability analysis is essential. However, due to the limitations of the classical methods of reliability analysis, there is a need to calculate the reliability index based on the probability density function of response in structures. In this study, by combining perturbation method and change – of – variable method and without the limitation of the statistical type of random distribution of random variables, the probability density function of response is calculated. By calculating the probability density function of the static response of the structures, the probability of failure and the reliability index are obtained directly.

It is obvious that the accuracy of the result of the stochastic finite element analysis depends on the random field element meshes. For this purpose, the distributed random field is discretized over the number of elements of equal length in structural members for each random variable.

In stochastic finite element method, due to the uncertainty of the characteristics of random variables in the structure, it is necessary to define a correlation function between a random variable in different elements. The reliability index is considered as a measure of convergence by considering the scale of fluctuations and the correlation function of random variables in adjacent elements in structure. In each number of elements, the structural reliability index is calculated using the explicit probability density function of response and the structural resistance function to converge on a certain number of elements. In this study, the stochastic finite element analysis is performed in linear static mode for a simple beam and a cantilever column and the variables of geometry, materials and loading are considered randomly with real statistical distributions according to the literature review. As can be expected from the deterministic finite element method, as the number of elements increase and the meshing is smaller, the reliability index increases.Considering the lower scale of fluctuations for random variables makes the reliability index converge to a larger value. However, on a larger scale of fluctuations convergence occurs in a smaller number of elements due to the greater correlation of random variables in adjacent elements. Also, the results of the probability density function of the response with the explicit method compared to the Monte Carlo simulation method show a good match in result. The advantage of using the reliability index as a measure of convergence in meshing, the configuration of limited random components is to consider all possible structural responses instead of using the average structural response in the design of structures.

کلیدواژه‌ها English

probability density function
Reliability Index
stochastic finite element method
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مهندسی عمران مدرس
دوره 21، شماره 2
خرداد و تیر 1400
صفحه 49-62