آنالیز کمانش و ارتعاشات تیر غیر منشوری بر بستر وینکلر – پاسترناک با دو روش اجزاء محدود و رایلی ریتز

در مقاله‌ی حاضر، کمانش و ارتعاشات آزاد تیری اولر – برنولی غیر منشوری بر بستر وینکلر – پاسترناک با دو روش اجزاء محدود و رایلی ریتز بررسی می‌شود. برای مدلسازی، تغییرات ممان اینرسی و تغییرات مساحت تیر در سه حالت، به‌صورت تابعی بر حسب ممان اینرسی ابتدایی و مساحت ابتدایی در معادله اعمال می‌شود. برای حل معادله، در گام اول با استفاده از روش همیلتون، معادله دیفرانسیل حاکم استخراج می‌گردد. در گام بعدی، شکل ضعیف معادله دیفرانسیل محاسبه شده و از توابع میان‌یابی هرمیتی (اجزاء محدود) و سری توانی (رایلی ریتز) به‌عنوان توابع تغییر مکان عرضی و توابع وزن استفاده می‌گردد. در گام آخر، پس از استخراج ماتریس‌های سختی مصالح، سختی هندسی، ماتریس سختی بستر وینکلر و پاسترناک و ماتریس جرم در نهایت مقادیر ویژه معادله بررسی می‌شوند. نتایج نشان می‌دهد که افزایش هم‌زمان شیب مقطع β و ثابت فنری وینکلر (k_w ) ̅ و ثابت برشی پاسترناک (k_p ) ̅ تاثیر قابل‌توجهی در افزایش ضریب طول موثر k و کاهش ظرفیت بارکمانشی p_cr در تمامی شرایط مرزی تکیه گاهی مختلف دارد. هم چنین افزایش همزمان شیب مقطع β و ثابت فنری وینلکر (k_w ) ̅ و ثابت برشی پاسترناک (k_p ) ̅ بسته به نوع شرایط مرزی تکیه گاهی سبب افزایش یا کاهش فرکانس طبیعی بی بعد ω ̅ می گردد. برای کاربردی بودن نتایج در محاسبات مهندسی، از منحنی های هم تراز برای ارائه ی نتایج و نمایش نمودارها استفاده می شود. نتایج مقاله با دو روش اجزاء محدود و رایلی ریتز صحت سنجی و بررسی گردید. تطابق قابل قبولی بین نتایج مقاله‌ی‌ حاضر و پژوهش پیشین برقرار است.

analysis of buckling and vibrations of non-prismatic beam on winkler-pasternak foundation with finite element and riley-ritz methods

in the analysis of structures such as building foundations, railway rails, reservoirs, airport runways and docks, it is necessary to consider the effect of the elastic foundation in modeling. for this reason, different theories such as winkler, pasternak, and reisner have been introduced. on the other hand, a member with a variable cross-section has a higher load-bearing capacity than a prismatic member with a larger cross-section. nowadays, engineers use members with variable cross-sections in the design of structural members to minimize the consumption of materials, to reduce the weight of the structure, and to increase the critical buckling load capacity. numerous studies have been conducted to investigate the critical load capacity of compressive members, particularly focusing on the stability of non-prismatic columns on elastic foundations. for the first time, timoshenko, wang, bazzant, morley, and dinnik studied the critical buckling load of elastic columns based on closed-form solutions or numerical approximations of the governing differential equation. in these studies, researchers aimed to provide simplified relations for practical use by designers. in the present paper, the vibrations and stability of a non-prismatic beam on a winkler-pasternak foundation are examined. the variations in the moment of inertia and cross-sectional area of the beam are incorporated into the equation in three cases, as functions of the initial moment of inertia and initial cross-sectional area. the effects of displacement and vertical pressure of the soil are modeled through elastic springs based on the winkler model, and shear deformations are taken into account according to the pasternak model. to solve the governing differential equation, both the finite element method and the rayleigh-ritz method are employed. in the finite element method, third-order hermite interpolation functions are used, while in the rayleigh-ritz method, a power series expansion is employed as the shape function. finally, eigenvalue-solving techniques are applied to determine the response parameters, including dimensionless natural frequency and effective length factor. all calculations, including the computation of material stiffness matrices, geometric stiffness, mass, and winkler-pasternak foundation stiffness, were carried out using matlab coding.the results indicate that the simultaneous increase in the section slope coefficient, winkler spring constant, and pasternak shear constant under various support conditions leads to an increase in the effective length and a decrease in the beam’s buckling load capacity. additionally, the simultaneous increase in the section slope coefficient, winkler spring constant, and pasternak shear constant, depending on the type of variations in the moment of inertia along the member, results in either an increase or decrease in the dimensionless natural frequency.both the finite element method and rayleigh-ritz method were used to solve the governing equation. the rayleigh-ritz method, compared to the finite element method, is more efficient. with fewer terms, the rayleigh-ritz method provides a more accurate calculation of the target parameters in comparison to the finite element method. all parameters involved in this study, including the section slope coefficient (β), winkler spring constant (k_w), pasternak shear constant (k_p), effective length factor (k), and natural frequency (ω), are dimensionless. the ninth topic (year 2020) and the tenth topic (year 2022) of iran’s national building regulations do not provide relations for analyzing the stability and vibrations of non-prismatic beams on the winkler-pasternak foundation. in this paper, aligned curves are used to present the results. the findings of this study serve as practical research that can be used by engineers and designers for the design of non-prismatic beams on winkler-pasternak foundations.furthermore, a practical example demonstrating aligned curves to calculate the critical buckling load capacity and natural frequency of the system is provided. the results of previous research are used for verification. there is an acceptable agreement between the present results and previous research.