ارزیابی عددی روش انرژی اصلاح‌شده در تحلیل مسائل سازه‌ای با هندسه غیرخطی

نوع مقاله : علمی - پژوهشی

نویسندگان

1 دانشجوی دکتری سازه، گروه عمران، واحد تهران مرکزی، دانشگاه آزاد اسلامی، تهران، ایران

2 استادیار، گروه عمران، واحد تهران مرکزی، دانشگاه آزاد اسلامی، تهران، ایران

چکیده

در بسیاری از مسائل سازه‌ای مانند تعیین بار خرابی و همچنین تحلیل مکانیزم کمانش باید از تحلیلی‌های غیرخطی هندسی بهره گرفت. بااین‌وجود، به علت ماهیت پیچیده این نوع از مسائل و عدم وجود یک راه‌حل تحلیلی جامع برای آن‌ها در عمل از روش‌های عددی برای تقریب پاسخ دقیق مسئله استفاده می‌گردد. از مشکلات روش‌های عددی نیز عدم همگرایی و یا یافتن مسیر صحیح تعادل خصوصاً در مواجه با نقاط انشعابی است. ازاین‌رو هدف از این تحقیق، به‌کارگیری روش انرژی اصلاح‌شده (که در دینامیک سازه‌ها معرفی شده است) در مسائل شبه-استاتیک با هندسه غیرخطی و دارای نقاط انشعابی است تا بتوان کارایی این روش را در مقایسه با سایر روش‌های موجود نظیر تکنیک‌های عددی نیوتنی و همچنین نیرو-تغییرمکان-قید موردبررسی قرار داد. برای رسیدن به اهداف این تحقیق، پس از مرور کوتاه بر روی روش‌های تحلیلی نیرو-مبنا موجود در عمل، روش انرژی برای این نوع از مسائل تشریح شده و در ادامه مراحل اجرای کامپیوتری آن در غالب یک روند گام‌به‌گام تشریح می‌گردد. سپس با کدنویسی در نرم‌افزار متلب و به‌کارگیری روش مذکور بر روی مثال‌های عددی به کار گرفته‌شده توسط دیگر محققین نظیر سازه‌های خرپایی و قابی، پاسخ‌های به‌دست‌آمده با حل تحلیلی و همچنین نتایج دیگر روش‌ها مانند نیوتن-رافسون و طول قوس مورد صحت سنجی واقع شده‌اند. درمجموع، تفسیر نتایج به‌دست‌آمده از شبیه‌سازی‌های عملی صورت گرفته بیانگر این موضوع بوده‌اند که روش عددی معرفی‌شده در تحلیل مسائل با غیرخطی هندسی دارای دقت بهتری در مقایسه با روش طول قوس بوده و همچنین نسبت روش نیوتن-رافسون نیز به‌خوبی می‌تواند از نقاط انشعابی در منحنی بار-تغییرمکان بدون واگرایی عبور کند.

کلیدواژه‌ها


عنوان مقاله [English]

Developing and Investigation the Numerical Efficiency of Modified Energy Method in Solid Mechanics With Geometric Nonlinearity and Bifurcation Points

نویسندگان [English]

  • Ahmad Razaghi 1
  • Jafar Asgari Marnani 2
  • Mohammad Sadegh Rohanimanesh 2
1 Ph.D. student, Department of Civil Engineering, Central Tehran Branch, Islamic Azad University, Tehran, Iran
2 Assistant professor, Department of Civil Engineering, Central Tehran Branch, Islamic Azad University, Tehran, Iran
چکیده [English]

Geometric nonlinear analyses are used in many structural problems, such as the determination of failure load, as well as the study of buckling mechanism. Nevertheless, due to the complex nature of this type of problems and the absence of a comprehensive analytical solution for them, numerical methods are utilized in practice to approximate the exact response of these systems. In the application of numerical methods, there are also some difficulties such as divergence or finding the correct path of equilibrium, especially in the case of bifurcation points. Hence, the main purpose of this research is to apply the modified energy method (introduced in the dynamics of structures) in quasi-static problems with geometric nonlinearity and bifurcation points so that the efficiency of this method can be compared to others, such as Newtonian numerical techniques and force-displacement-constraint approaches. To achieve the objectives of this research, after briefly reviewing the current force-based computational methods in practice, the energy method is described for such problems, and then the step-by-step process of its computer implementation will be presented. Afterward, by coding in MATLAB software and applying the method to numerical examples employed by other researchers such as truss and frame structures, the numerical results are verified by analytical solution as well as those obtained by other methods, such as Newton-Raphson and Arc Length techniques. Generally, the interpretation of the results obtained from performed simulations has shown that the presented numerical method in analyzing nonlinear geometric problems has better accuracy compared to the Arc Length method; moreover, it can well pass through bifurcation points in the force-displacement curve without divergence in comparison with the Newton-Raphson method.

کلیدواژه‌ها [English]

  • Numerical Study
  • modified energy method
  • solid mechanics
  • Geometric nonlinearity
  • bifurcation points
[1] Crisfield M. A. Non-linear finite element analysis of solids and structures. Wiley; 1993.
[2] M. J. Tuner, E. H. Drill, and H. C. M. R. J. Melosh, “Large deflection of structures subject to heating and external load, J,” Areo Sci, vol. 27, pp. 97–106, 1960.
[3] R. H. Gallagher, R. A. Gellatly, R. H. Mallett, and J. Padlog, “A discrete element procedure for thin-shell instability analysis.,” AIAA J., vol. 5, no. 1, pp. 138–145, 1967.
[4] R. H. Gallagher and J. Padlog, “Discrete element approach to structural instability analysis,” AIAA J., vol. 1, no. 6, pp. 1437–1439, 1963.
[5] K. K. Kapur and B. J. Hartz, “Stability of plates using the finite element method,” J. Eng. Mech. Div., vol. 92, no. 2, pp. 177–196, 1966.
[6] I. Holand and T. Moan, “The fi finite element in plate bukling,” Finite Elem. Method. Stress Anal. ed. I. Hol. al., Tapir, 1969.
[7] J. H. Argyris, Recent Advances in Matrix Methods of Structural Analysis/progress in Aeronautical Sciences. Pergamon Press, 1964.
[8] J. H. Argyris, “Continua and Discontinua, opening address to the 1-st Conf. Matrix Methods in Structural Mechanics.” Wright-Patterson AFB, Dayton, Ohio, 1965.
[9] P. V Marcal, “Finite Element Analysis with Material Nonlinearities: Theory and Practice,” 1969.
[10] O. C. Zienkiewicz, “The Finite Element Method in Engineering Science McGrawHill Book Company,” Inc, London, 1971.
[11] Y. Yamada, N. Yoshimura, and T. Sakurai, “Plastic stress-strain matrix and its application for the solution of elastic-plastic problems by the finite element method,” Int. J. Mech. Sci., vol. 10, no. 5, pp. 343–354, 1968.
[12] O. C. Zienkiewicz, S. Valliappan, and I. P. King, “Elasto‐plastic solutions of engineering problems ‘initial stress’, finite element approach,” Int. J. Numer. Methods Eng., vol. 1, no. 1, pp. 75–100, 1969.
[13] J. T. Oden, “Numerical Formulations of Nonlinear Elasticity Problems,” J. Struct. Div., vol. 93, no. 3, pp. 235–356, 1967.
[14] R. H. Mallett and L. A. Schmit, “Nonlinear structural analysis by energy search,” J. Struct. Div., vol. 93, no. 3, pp. 221–234, 1967.
[15] F. K. Bogner, R. L. Fox, and L. A. Schmit, “Finite deflection structural analysis using plate and shell discreteelements.,” AIAA J., vol. 6, no. 5, pp. 781–791, 1968.
[16] G. C. Nayak and O. C. Zienkiewicz, “Elasto-plastic stress analysis. A generalization for various contitutive relations including strain softening,” Int. J. Numer. Methods Eng., vol. 5, no. 1, pp. 113–135, 1972.
[17] M. F. and B.MASSICOTTE, “GEOMETRICAL INTERPRETATION OF THE ARC-LENGTH METHOD,” Comput. Struct., vol. 46, no. 4, pp. 603–615, 1993.
[18] H. B. Coda and M. Greco, “A simple FEM formulation for large deflection 2D frame analysis based on position description,” Comput. Methods Appl. Mech. Eng., vol. 193, no. 33–35, pp. 3541–3557, 2004.
[19] P. M. A. A. and T. B. T. Rabczuk, “A simplified mesh-free method for shear bands with cohesive surfaces,” pp. 993–1021, 2007.
[20] A. Lorenzana, P. M. López-reyes, E. Chica, and J. M. . Teran, “A NONLINEAR MODEL FOR THE ELASTOPLASTIC ANALYSIS OF 2D FRAMES ACCOUNTING FOR DAMAGE Antol´ ın Lorenzana,” Theor. Appl. Mech., vol. 49, no. 2, pp. 515–529, 2011.
[21] I. Mansouri and H. Saffari, “An efficient nonlinear analysis of 2D frames using a Newton-like technique,” Arch. Civ. Mech. Eng., vol. 12, no. 4, pp. 485–492, 2012.
[22] S. Mamouri, E. Mourid, and a. Ibrahimbegovic, “Study of geometric non-linear instability of 2D frame structures,” Eur. J. Comput. Mech., vol. 24, no. 6, pp. 1–23, 2015.
[23] J. Radnic, R. Markic, M. Glibic, N. Grgić, and I. Banović, “Comparison of numerical models for nonlinear static analysis of planar concrete frames based on 1D and 2D finite elements,” Materwiss. Werksttech., vol. 47, no. 5–6, pp. 472–482, 2016.
[24] J. S. Moita, A. L. Ara??jo, C. M. Mota Soares, C. a. Mota Soares, and J. Herskovits, “Geometrically nonlinear analysis of sandwich structures,” Compos. Struct., pp. 1–10, 2016.
[25] M. Crusells-Girona, F. C. Filippou, and R. L. Taylor, “A mixed formulation for nonlinear analysis of cable structures,” Comput. Struct., vol. 186, pp. 50–61, 2017.
[26] D. C. Feng, G. Wu, Z. Y. Sun, and J. G. Xu, “A flexure-shear Timoshenko fiber beam element based on softened damage-plasticity model,” Eng. Struct., vol. 140, pp. 483–497, 2017.
[27] M. Jalili Sadr Abad, M. Mahmoudi, M. Mollapour Asl, R. “Application of modified energy method in the nonlinear cyclic behavior of structures,” J. Struct. Constr. Eng., 2018.
[28] O. C. Zienkiewicz, R. L. Taylor, and D. Fox, The Finite Element Method for Solid and Structural Mechanics. 2014.
[29] R. De Borst, M. a. Crisfiel, J. J. C. Remmers, and C. V. Verhoosel, Non-Linear Finite Element Analysis of Solids and Structures. 2012.
[30] E. Riks, “An incremental approach to the solution of snapping and buckling problems,” Int. J. Solids Struct., vol. 15, no. 7, pp. 529–551, 1979.
[31] M. Crisfield, “A fast incremental/iterative solution procedure that handles ‘snap-through,’” in Computational Methods in Nonlinear Structural and Solid Mechanics, Elsevier, 1981, pp. 55–62.
[32] M. Jalili Sadr Abad, M. Mahmoudi, and E. H. Dowell, “Dynamic Analysis of SDOF Systems Using Modified Energy Method,” ASIAN J. Civ. Eng., vol. 18, no. 7, pp. 1125–1146, 2017.
[33] M. Jalili Sadr Abad, M. Mahmoudi, and E. Dowell, “Novel Technique for Dynamic Analysis of Shear-Frames Based on Energy Balance Equations,” Sci. Iran., pp. 1–31, 2018.
[34] K. Kondoh and S. N. Atluri, “Influence of local buckling on global instability: Simplified, large deformation, post-buckling analyses of plane trusses,” Comput. Struct., vol. 21, no. 4, pp. 613–627, 1985.
[35] M. a. M. Torkamani and J.-H. Shieh, “Higher-order stiffness matrices in nonlinear finite element analysis of plane truss structures,” Eng. Struct., vol. 33, no. 12, pp. 3516–3526, 2011.
[36] F. W. Williams, “An approach to the non-linear behaviour of the members of a rigid jointed plane framework with finite deflections,” Q. J. Mech. Appl. Math., vol. 17, no. 4, pp. 451–469, 1964.