STATIC STABILITY ANALYSIS OF A CANTILEVER TIMOSHENKO BEAM WITH VARYING CROSS-SECTION

Yıl 2000, Cilt: 2 Sayı: 2, 75 - 86, 01.05.2000

Binnur Gören Seçil Erim

Öz

In this study, effects of variation of various cross-sections on the static stability of
cantilever beams subjected to axial loading have been investigated. The beam is the
Timoshenko type where the effect of shear force upon the elastic curve is included to the
elastic curve. Ten different types of continuous and stepped beams have been used in the
analyses. In order to determine the critical buckling loads and buckling factors, the Finite
Element and Finite Element-Transfer Matrix Methods have been employed. In both methods,
the beam has been modeled as four degrees of freedom finite elements. Therefore, it has been
possible to compare the advantages and disadvantages of both methods. Results obtained
from both methods are similar to each other. Buckling parameters are determined from the
solution of the eigenvalue problem obtained by the application of the variational principle to
the potential energy terms of the integral form system. MATLAB 5.1 computer software has
been used in numerical calculations. Results obtained for various kinds of beams have been
presented in tables and graphics.

Anahtar Kelimeler

Static stability, Cantilever Timoshenko Beam, Finite Element Method

DEĞİŞKEN KESİTLİ ANKASTRE TIMOSHENKO KİRİŞİN STATİK STABİLİTE ANALİZİ

Öz

Bu çalışmada, eksenel yüklemeye maruz ankastre kirişlerin statik stabilitesine, çeşitli kesit değişimlerinin etkileri incelenmiştir. Kiriş, elastik eğriye kesme kuvveti etkisinin de katıldığı Timoshenko kirişidir. Analizlerde kademesiz ve kademeli olmak üzere on çeşit kiriş kullanılmıştır. Kritik burkulma yükü ve burkulma faktörlerini tespit etmek için Sonlu Eleman ve Sonlu Eleman-Transfer Matris Metotları kullanılmıştır. Her iki yöntemde de kiriş dört serbestlik derecesine sahip sonlu eleman ile modellenmiştir. Bu şekilde, iki metodun birbirlerine olan üstünlüklerini görmek mümkün olmuştur. Her iki metottan elde edilen sonuçlar birbirine çok yakındır. Burkulma parametreleri, integral formdaki sistemin potansiyel enerji ifadelerine varyasyonel prensibi uygulanmasıyla elde edilen özdeğer probleminin çözümüyle saptanmıştır. Nümerik hesaplamalarda MATLAB 5.1 bilgisayar programlama dili kullanılmıştır. Çeşitli tipteki kirişler için elde edilen sonuçlar çizelge ve grafikler halinde sunulmuştur

Anahtar Kelimeler

Statik stabilite, Ankastre Timoshenko Kirişi, Sonlu Elemanlar Metodu

Kaynakça

• Bathe K. J., (1982): “Finite Element Procedures in Engineering Analysis”, New Jersey, Prentice Hall Inc.
• Belek H.T., (1984): “Kritik Hızların Sonlu Elemanlar-Transfer Matris Yöntemiyle Hesaplanması”, 1.Ulusal Makina Teorisi Sempozyumu, Ankara. 349-358.
• Brown J.E., Hutt J.M., Salama A.E., (1968): “FE Solution to Dynamic Stability of Bars” AIAA Journal, 7, 1423-1425.
• Burnett D.S., (1987): “FE Analysis from Concepts to Applications”, New York: Addison- Wesley Publishing Company.
• Cook R.D., Malkus D.S., Plesha M.E., (1989): “Concepts and Applications of Finite Element Analysis”, (3rd edition) London: John Wiley & Sons.
• Desai C.S., Abel J.F., (1972): “Introduction to the FEM, A Numerical Method for Engineering Analysis”, Boston: Von Nostrand Reinhold Company. Flügge W., (1982): “Handbook of Engineering Mechanics”, New York: McGraw-Hill.
• Gere J.M., Timoshenko S.P., (1990): “Mechanics of Materials”, (2nd SI edition) Boston: Von Nostrand Reinhold Company.
• Gere J.M., Carter W.O., (1963): “Critical Buckling Loads for Tapered Beam-Columns”, Trans. Am. Soc. Civil Eng. 128.
• Güneş A., Yıldız K., (1997): “MATLAB for Windows”, İstanbul: Türkmen Kitabevi.
• Huebner K.H., (1975): “The FEM for Engineers”, New York: John Wiley & Sons.
• Kitipornchai, S., Trahair, N.S., (1972): “Elastic Stability of Tapered I-Beams”, Am. Soc. Civil Eng., (98).
• Krishnamoorthy C.S., (1994): “Finite Element Analysis, Theory of Programming”, (2nd edition), New Delhi: Tata McGraw-Hill Publishing Company Limited.
• Kwon Y. W., Bang H., (1997): “The Finite Element Method Using MATLAB”, New York: CRC Press.
• Massey C., Mcguire P.J., (1971): “Lateral Stability of Nonuniform Cantilevers”, Proc. Am. Soc. Civil Eng. (97).
• Morrison T.G., (1972): “Lateral Buckling of Constrained Beams”, Proc. Am. Soc. Civil Eng. (97).
• Naranaysnaswami R., Adelman H.M., (1974): “Inclusion of Transverse Shear Deformation in FE Displacement Formulations”, American Institute of Aeronautics and Astronautics Journal, 12, 1613-1614.
• Petyt M., (1990): “Introduction to Finite Element Vibration Analysis”, Cambridge: Cambridge University Press.
• Popov E.P., (1968): “Introduction to Mechanics of Solids”, New York: Macdonald.
• Reddy J.N., (1993): “An Introduction to the FEM”, (2nd edition). New York: McGraw-Hill Inc.
• Rossi R.E., Laura P.A.A., (1993): “Numerical Experiments on Vibrating, Linearly Tapered Timoshenko Beams”, Journal of Sound and Vibration, 168(1), 179-183.
• Simitses G.J., (1983): “Effect on Static Stability Preloading on the Dynamic Stability of Structures”, AIAA Journal, 21 (8), 1174-1180.
• Thomas J., Abbas B.A., (1975): “FE Model for Dynamic Analysis of Timoshenko Beam”, Journal of Sound and Vibration, 41 (3), 291-299.
• Thomas J., Abbas B.A., (1976): “Dynamic Stability of Timoshenko Beams by Finite Element Method”, Journal of Engineering for Industry, 98 (1), 1145-1151.
• Timoshenko S.P., Gere J.M., (1961): “Theory of Elastic Stability”, New York: McGraw- Hill Book Company.
• Yardımoğlu B., Sabuncu M., (1995): “Yay Sabiti ve Yay Konumunun Ankastre Bir Çubuğun Dinamik Stabilitesine Etkisi”,7. Ulusal Makine Teorisi Sempozyumu, İstanbul, 242-250.
• Young W.C., (1989): “Roark’ s Formulas for Stress and Strain”, (6th edition). New York: McGrapw-Hill Inc.