An Approach for Free Vibration Analysis of Framed Buildings with and without Infill Walls

Abstract

In the dynamic analysis of buildings, mass can be modeled either discrete or continuous. Although the discrete mass model is generally preferred in practice, the continuous mass calculation model can be used especially in preliminary dimensioning and in checking the results of the discrete mass calculation model. In the analysis of buildings according to the continuous system calculation model, structures are modeled as equivalent beams. The simplest of the equivalent beam models is the shear beam model, which is based on shear behavior. This model is particularly suitable for frames and masonry buildings where shear effects are dominant. In this study, a method based on the shear beam model is proposed for estimating the fundamental periods of bare and infilled frames. In the study, the continuous system calculation model was originally used to determine the periods of infilled frames. Furthermore, unlike similar studies previously conducted in the literature, a correction factor suitable for the shear beam model has been defined in order to adapt the continuous system calculation model to the discrete system calculation model. To evaluate the compatibility of the proposed approach with the finite element method, two case studies were conducted. A total of twelve frame structures, both with and without infill walls, were designed: 6-, 5-, and 4-story frames for Example 1, and 8-, 7-, and 6-story frames for Example 2. These frames were analyzed using both the proposed method and SAP2000, and the resulting periods were compared. The comparison revealed that, for bare frames, the accuracy of the proposed method improves with increasing number of stories. Additionally, it was observed that infilled frames yield better agreement with SAP2000 results compared to their bare counterparts.

Keywords

• Frame
• Infill wall
• Shear beam
• Free Vibration

Dolgu Duvarsız ve Dolgu Duvarlı Çerçeveli Binaların Serbest Titreşim Analizi İçin Bir Yaklaşım

Abstract

Binaların dinamik analizinde kütle, ayrık veya sürekli olarak modellenir. Uygulamada genellikle ayrık kütle modeli tercih edilmekle birlikte, sürekli kütle hesap modeli özellikle ön boyutlandırmada ve ayrık kütle hesap modelinin sonuçlarının kontrolünde kullanılabilmektedir. Binaların sürekli sistem hesap modeline göre analizinde yapılar eşdeğer bir kiriş olarak modellenmektedir. Eşdeğer kiriş modellerinden en basiti kayma davranışını esas alan kayma kirişi modelidir. Bu model, özellikle kayma davranışı baskın çerçeveler ve yığma binalar için uygundur. Bu çalışmada boş çerçevelerin ve dolgu duvarlı çerçevelerin periyotlarının belirlenmesi için kayma kiriş modelini esas alan bir yaklaşım önerilmiştir. Çalışmada özgün olarak sürekli sistem hesap modeli dolgu duvarlı çerçevelerin periyotlarının tayini için kullanılmıştır. Ayrıca çalışmada literatürde daha önce yapılmış benzer çalışmalardan farklı olarak sürekli sistem hesap modelini ayrık sistem hesap modeline uygun hale getirmek üzere kayma kiriş modeline uygun bir düzeltme katsayısı tanımlanmıştır. Çalışmanın son bölümünde sunulan yaklaşımın sonlu elemanlar yöntemine uygunluğunu değerlendirmek üzere iki örnek çözülmüştür. Örnek 1’de 6, 5 ve 4 katlı, örnek 2’de ise 8, 7 ve 6 katlı dolgu duvarlı ve dolgu duvarsız toplam 12 adet çerçeve yapısı tasarlanmıştır. Tasarlanan örnekler hem sunulan yaklaşım ile hem de SAP2000 ile çözülerek periyotlar karşılaştırılmıştır. Yapılan karşılaştırma sonucunda dolgu duvarsız çerçeve yapılarında kat sayısı arttıkça önerilen yöntemin SAP2000’den elde edilen sonuçlara yaklaştığı gözlemlenmiştir. Ayrıca sonuçlardan dolgu duvarlı çerçeve yapılarının dolgu duvarsız çerçeve yapılarına kıyasla daha iyi sonuçlar verdiği görülmüştür.

Keywords

• Çerçeve
• Dolgu duvarı
• Kayma kirişi
• Serbest titreşim

References

1. Thomson, W. T. (1972). Theory of vibration with applications, Chap. 8, Prentice-Hall, Englewoods Cliff, N.J., 275–276.
2. Blevins, R. D. (1979). Formulas for natural frequency and mode shape, Chap. 8, Van Nostrand-Reinhold, New York, 171–175.
3. Weaver, W., Timoshenko, S. P., & Young, D. H. (1990). Vibration problems in engineering, 5th Ed., Chap. 5, Wiley Interscience, New York, 433–460.
4. Clough, R. W., & Penzien, J. (1993). Dynamics of structures, 2nd Ed., Chap. 26, McGraw-Hill, New York, 628–628.
5. Euler, L. (1750). Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici lattissimo sensu accepti. Opera Omnia, Series 1, Vol. 13.
6. Rayleigh, J. W. S. (1877). The Theory of Sound (Vol. 1). Macmillan.
7. Timoshenko, S. (1921). On the Correction of the Formula for the Flexural Rigidity of Beams. Philosophical Magazine, 42(251), 769-779.
8. Cheng, F. Y., & Tseng, W-H. (1973). Dynamic matrix of Timoshenko beam columns. Journal of the Structural Division, 99(3), 527–549.

9. Cheng, F. Y., & Pantelides, C. P. (1988). Dynamic Timoshenko beam– columns on elastic media. Journal of Structural Engineering, 114(7),1524–1550.
10. Geist, B., & McLauglin, J. R. (1997). Double eigenvalues for the uniform Timoshenko beam. Applied Mathematics Letters, 10(3), 129–134.
11. Abbas, B. A. H. (1984). Vibration of Timoshenko beams with elastically restrained ends. Journal of Sound and Vibration, 97, 541–548.
12. Xie, J., & Wen, Z. (2008). Measure of drift demand for earthquake ground motions based on timoshenko beam model. The 14th World Conference on Earthquake Engineering, Beijing, China.
13. Miranda, E., & Akkar, S. (2006). Generalized interstory drift spectrum. Journal of Structural Engineering, 132(6), 840–852.
14. Khaloo, A. R., & Khosravi, H. (2008). Multi-mode response of shear and flexural buildings to pulse-type ground motions in near-field earthquakes. Journal of Earthquake Engineering, 12(4), 616–630.
15. Yang, D., Pan, J., & Li, G. (2010). Interstory drift ratio of building structures subjected to near-fault ground motions based on generalized drift spectral analysis. Soil Dynamics and Earthquake Engineering, 30(11), 1182– 1197.
16. Goodman, L. E., & Sutherland, J. G. (1951). Discussion of natural frequencies of continuous beams of uniform span length. Journal of Applied Mechanics, 18(2), 217–218.
17. Hurty, W. C., & Rubenstein, J. C. (1964). On the effect of rotatory inertia and shear in beam vibration. Journal of the Franklin Institute, 278, 124–132.
18. Heidebrecht, A. C., & Naumoski, N. D. (1997). Development and application of a displacement-based design approach for momentresisting frame structures. Seismic design methodologies for the next generation of codes, P. Fajfar and H. Krawinkler, eds., Balkema, Rotterdam, 217–228.
19. Tekeli, H., Atimtay, E., & Turkmen, M. (2015). An approximation method for design applications related to sway in RC framed buildings. International, Journal of Civil Engineering, 13(3), 321-330.
20. Heidebrecht, A. C., & Stafford Smith, B. (1973). Approximate analysis of tall wall-frame structures. Journal of the Structural Division, 99(2), 199–221.
21. Miranda, E. (1999). Approximate lateral deformation demands in multistory buildings. Journal of Structural Engineering, 125(4), 417-425.
22. Miranda, E., & Reyes, C. J. (2002). Approximate lateral drift demands in multistory buildings with nonuniform stiffness. Journal of Structural Engineering, 128(7), 840–849.
23. Westergaard, H. M. (1933). Earthquake-shock transmission in tall buildings, Engineering News-Record, 111(22), 654-656.
24. Rosenblueth, E., Elorduy, J., & Mendoza, E. (1968). Shears and over turning moments in shear buildings during earthquakes. Proc., 2nd Mexican Conference on Earthquake Engineering, Mexican Society of Earthquake Engineering, Veracruz, Mexico.
25. Montes, R., & Rosenblueth, E. (1968). Shears and over turning moments in chimneys. Proc., 2nd Mexican Conf. on Earthquake Engineering, Mexican Society of Earthquake Engineering, Veracruz, Mexico.
26. Chitty, L. (1947). On the cantilever composed of a number of parallel beams interconnected by cross bars. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 38(285), 685–699.
27. Stafford Smith, B. S., Kuster, M., & Hoenderkainp, C. D., (1984). Generalized Method for Estimating Drift in High-Rise Structures. Journal of Structural Engineering, 110(7), 1549-1562.
28. Chopra, A. K., & Goel, R. K. (2000). Building period formulas for estimating seismic displacements. Earthquake Spectra, 16(2), 533–536. https://doi.org/10.1193/1. 1586125
29. Gioncu, V. & Mazzolani, F. (2013). Seismic design of steel structures, CRC Press.
30. Landolfo, R. (2018). Seismic design of steel structures: New trends of research and updates of Eurocode 8, European Conference on Earthquake Engineering, Thessaloniki, Greece, 413–438.
31. Mazzolani, F. M. & Piluso, V. (1996). Theory and Design of Seismic Resistant Steel Frames, CRC Press, an imprint of Chapman & Hall.
32. Asteris, P. G., Repapis, C. C., Tsaris, A. K., Trapani, F., & Cavaleri, L. (2015). Parameters affecting the fundamental period of infilled RC frame structures. Earthquakes and Structures, 9(5), 999-1028.
33. Asteris, P. G., Tsaris, A. K., Cavaleri, L., Repapis, C. C., Papalou, A., Trapani, F. & Karypidis, D. F. (2016). Prediction of the fundamental period of infilled RC frame structures using artificial neural networks, Computational Intelligence and Neuroscience, 5104907.
34. Asteris, P. G., Repapis, C. C., Repapi, E. V. & Cavaleri, L. (2017). Fundamental period of infilled reinforced concrete frame structures, Structure and Infrastructure Engineering, 13(7), 929-941.
35. Young, K. & Adeli, H. (2014). Fundamental period of irregular moment-resisting steel frame structures, The Structural Design of Tall and Special Buildings, 23, 1141-1157. https://doi.org/10.1002/tal.1112
36. Shafei, A. & Alirezaei, M. (2014). Evaluation of the fundamental period of vibration of irregular steel structures, International journal of engineering sciences and research technology, 3(4), 6083-6090.
37. Hatzigeorgiou, G. D. & Kanapitsas, G. (2013). Evaluation of fundamental period of low-rise and mid-rise reinforced concrete buildings. Earthquake Engineering and Structural Dynamics, 42(11), 1599-1616.
38. Kwon, O.-S. & Kim, E.-S. (2010). Evaluation of building period formulas for seismic design. Earthquake Engineering and Structural Dynamics, 39(14), 1569-1583. https://doi.org/10.1002/eqe.998
39. UBC (1997). Uniform Building Code, International Conference of Building Officials, California, Wilier.
40. EC 8 (1994). Design provisions for earthquake resistance of structures. Part 1. Brussels: CEN (European Committee for Standardization).
41. FEMA 356 (2000), Federal Emergency Management Agency, Prestandard and Commentary for the Seismic Rehabilitation of Buildings, Washington, DC, 2000.
42. Özkaynak, H., Sürmeli, M., & Yüksel, E. (2016). A capacity curve model for confined clay brick infills. Bulletin of Earthquake Engineering, 14, 889-918.
43. Murashev, V., Sigalov, E., & Baikov, V. N. (1976). Design of reinforced concrete structures, Mir Publishers, Moscow.
44. Bilyap S. (1987). Betonarme yüksek yapılarda burulmasız perde-çerçeve sistemlerinin yatay kuvvetlere göre yaklaşık hesap yöntemleri ve dinamik karakteristikleri, Ege Üniversitesi Yayınları.
45. Ertutar Y. (1987). Yatay Yüklerin Etkisi Altında Bulunan ve Çerçeve Kayma Rijitliği Yapı Yüksekliğince Nonlineer Değişen Yapılarda Yatay Yerdeğiştirmelerin Hesabı. Deprem Araştırma Bülteni,57,83-92.
46. Ertutar Y. (1995). Betonarme yüksek yapılarda yatay yük etkisi, Dokuz Eylül Üniversitesi Mühendislik Fakültesi Yayınları.
47. Bozdogan K.B. & Kaya E. (2003). Taşıyıcı sistemi çerçevelerden oluşan yapıların taşıma matrisi ve dinamik matris yöntemleriyle birinci ve ikini mertebe dinamik analizi. Selçuk Üniversitesi Mühendislik Mimarlık Fakültesi Dergisi,18(2),31-37.
48. Bozdogan,K.B, Ozturk,D. & Nuhoğlu, A.(2005). Çok katlı yapıların sürekli sistem hesap modeline göre dinamik analizi için pratik bir yöntem. Sigma Mühendislik ve Fen Bilimleri dergisi,2005(4),65-77.
49. Ozturk,D. & Bozdogan K.B.(2020). Determination of the dynamic characteristics of frame structures with non-uniform shear stiffness. Iranian Journal of Science and Technology -Transactions of Civil Engineering.44(1),37-47.
50. Zalka, K. A. (2001). A simplified method for calculation of the natural frequencies of wall–frame buildings, Engineering Structures, 23(12), 1544–1555. Choudhury, S. (2024). Performance-Based Seismic Design of Structures (1st ed.). CRC Press, Boca Raton. https://doi.org/10.1201/9781003441090.