تعیین میدان تنش در اطراف مغارهای ذخیره سازی گاز با استفاده از توابع پتانسیل مختلط

نوع مقاله: مقاله پژوهشی

نویسندگان

1 دانشجوی دکترای تخصصی معدن؛ دانشکده فنی؛ دانشگاه شهید باهنر کرمان

2 استاد؛ گروه مهندسی معدن؛ دانشکده فنی؛ دانشگاه شهید باهنر کرمان

3 استادیار؛ گروه مهندسی مکانیک؛ دانشکده فنی؛ دانشگاه شهید باهنر کرمان

چکیده

در این مقاله راه حلی تحلیلی برای تعیین میدان تنش در اطراف مغارهای گازی، تحت فشار داخلی ثابت، ارائه می شود. توده سنگ بصورت الاستیک و ایزوتروپ در نظر گرفته شده است. بعلت پیچیدگی در حل مساله برای شکل غیردایروی مغار، توابع پتانسل مختلط موسخلیشویلی همراه با نگاشت همدیس مورد استفاده قرار گرفته اند. در نهایت جوابهای تحلیلی با مقادیر بدست آمده از نرم افزار اجزاء محدود Phase 2 برای دو حالت مدل وزن دار و بدون وزن مقایسه شده اند که همگرایی خوبی را بین جوابها نشان می دهد. تنها در نقاطی از دیواره که انحناء و درنتیجه تمرکز تنش بیشتر است اختلاف میان جوابها افزایش می یابد. از این روش می توان بعنوان روشی با فرایند حل سریع و دقتی بمراتب بیشتر از روشهای عددی در تحلیل تنش اطراف حفریات غیردایروی و یا در مسائل آنالیز برگشتی بمنظور بدست آوردن پارامترهای توده سنگ استفاده نمود.

کلیدواژه‌ها

موضوعات


عنوان مقاله [English]

Determination of stress field around gas storage caverns using complex potential functions and conformal mapping

نویسندگان [English]

  • Ali Reza Kargar 1
  • Reza Rahmannejad 2
  • Mohamad Ali Hajabasi 3
1 Ph.D. Candidate; Department of Mining Eng., Faculty of Eng., Shahid Bahonar University of Kerman
2 Professor; Department of Mining Eng., Faculty of Eng., Shahid Bahonar University of Kerman
3 Associate Professor; Department of Mechanical Eng., Faculty of Eng., Shahid Bahonar University of Kerman
چکیده [English]

A new analytical solution for stress state around gas storage cavities, under constant internal pressure, is presented in this study. Rock mass is assumed to behave as isotropic, linear elastic materials. Due to complexity of the problem for noncircular cross-section configuration of cavities, Muskhelishvili complex potential functions combined with conformal mapping has been used. Finally, the results obtained from analytical solution are compared with those obtained by phase 2 finite element software for both weighted and weightless models, and as a result, good agreement between the analytical solution and modeling results has been obtained except in some points of the cavity wall where large curvature and high stress concentration exist.
 
Introduction
One of the important problems in rock mechanics is to investigate stress and displacement around excavations. Although numerical methods are widely used in determining stress and displacement components, they have some disadvantages such as they are highly dependent on the dimensions of the model, boundary conditions, mesh grid size and approximation functions. In contrast to numerical methods, analytical methods presents closed-form solutions describing the general trend of the input parameters.
One of the analytical methods used in obtaining stress field and displacement in the theory of elasticity is Muskhelishvili's complex variable method. In this approach, using conformal mapping and Muskhelishvili's complex potential functions, characteristic equations are determined, and displacement and stress fields are obtained.
In this study, using complex variables in plane elasticity theory, stress is determined around gas storage caverns under constant internal pressure.
 
Methodology and Approaches
After expressing the complex potential functions in terms of series, the problem is solved through mapping the cavern shape into a unite-radii circle, and implementing Couchy integral. By making the system of equations and determining the series coefficients, the stress and displacement components are obtained through Muskhelishvili's complex variable method. The results are then compared with those obtained by phase 2 software for two modes of weighted and weightless models and consequently, good agreement between the analytical solution and modeling results is achieved.
 
Results and Conclusions
In this paper, an elastic solution has been presented based on Muskhelishvili's potential functions for determining stress field around gas storage caverns. Considering in-situ stress and internal gas pressure, this method can determine stress field around the cavern faster and more accurately than numerical methods.
A comparison between the analytical solution and Phase 2 finite element software results has indicated good agreement between the results from these two approaches. Only for the roof and floor of caverns in the weightless model and cavern's floor in weighted model, the difference is marginally increased.

کلیدواژه‌ها [English]

  • Cavities
  • Linear elasticity
  • Complex potential functions
  • conformal mapping
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