Free Vibration of Tall Buildings using Energy Method and Hamilton's Principle
Vol. 6 No. 5 (2020): May
Research Articles
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In a framed-tube tall building, shear wall systems are the most efficient structural systems for increasing the lateral load resistance. A novel and simple mathematical model is developed herein which calculates the natural frequencies of such tall buildings. The analyses are based on a continuous model, in which a tall building structure is replaced by an idealized cantilever beam that embodies all relevant structural characteristics. Governing equations and the corresponding eigen-problem are derived based on the energy method and Hamilton's principle. Solutions are obtained for three examples; using the separation of variables technique implemented in MATLAB. The results are compared to SAP2000 full model analysis; and they indicate reasonable accuracy. The computed natural frequencies for structures 50, 60 and 70 storey buildings were over-estimate 7, 11 and 14 percent respectively. The computed errors indicate that the proposed method has acceptable accuracy; and can be used during the initial stages of designing of tall buildings; it is fast and low cost computational process.
Rahgozar, P. (2020). Free Vibration of Tall Buildings using Energy Method and Hamilton’s Principle. Civil Engineering Journal, 6(5), 945–953. https://doi.org/10.28991/cej-2020-03091519
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[1] Alavi, Arsalan, Reza Rahgozar, Peyman Torkzadeh, and Mohamad Ali Hajabasi. "Optimal Design of High-Rise Buildings with Respect to Fundamental Eigenfrequency.” International Journal of Advanced Structural Engineering 9, no. 4 (September 25, 2017): 365–374. doi:10.1007/s40091-017-0172-y.
[2] Alavi, Arsalan, Peyman Rahgozar, and Reza Rahgozar, "Minimum"Weight Design of High"Rise Structures Subjected to Flexural Vibration at a Desired Natural Frequency.” The Structural Design of Tall and Special Buildings 27, No. 15 (October 2018): e1515. doi:10.1002/tal.1515.
[3] Davari, Seyed Mozafar, Mohsen Malekinejad, and Reza Rahgozar, "Static Analysis of Tall Buildings based on Timoshenko Beam Theory.” International Journal of Advanced Structural Engineering 11 (September 2019): 455–461.
[4] Kwan, Albert K.H., "Simple Method for Approximate Analysis of Framed-Tube Structures.” Journal of Structural Engineering, ASCE 120, No. 4 (April 1994): 1221-1239. doi:10.1061/(ASCE)0733-9445.
[5] Kwan, Albert K.H., "Shear Lag in Shear/Core Walls.” Journal of Structural Engineering, ASCE 122, No. 9 (1996): 1097-1104. doi:10.1061/(ASCE)0733-9445.
[6] Tavakoli, Reihaneh, Reza Kamgar, and Reza Rahgozar. "Seismic Performance of Outrigger-Braced System Based on Finite Element and Component-Mode Synthesis Methods.” Iranian Journal of Science and Technology, Transactions of Civil Engineering (July 31, 2019). doi:10.1007/s40996-019-00299-3.
[7] Tavakoli, Reihaneh, Reza Kamgar, and Reza Rahgozar. "Seismic Performance of Outrigger–belt Truss System Considering Soil–structure Interaction.” International Journal of Advanced Structural Engineering 11, no. 1 (February 1, 2019): 45–54. doi:10.1007/s40091-019-0215-7.
[8] Coull, Alexander, and Bishwanath Bose, "Simplified Analysis of Frame Tube Structures.” Journal of Structural Engineering, ASCE 101, No. 11 (1975): 2223-2240.
[9] Coull, Alexander, and Abdulla K. Ahmed, "Deflections of Framed-Tube Structures.” Journal of Structural Engineering, ASCE 104, No. 5 (1978): 857-862.
[10] Alavi, Arsalan, and Reza Rahgozar, "Optimal Stiffness Distribution in Preliminary Design of Tubed-System Tall Buildings.” Structural Engineering and Mechanics 65, No. 6 (March 2018): 731-739. doi:10.12989/sem.2018.65.6.731.
[11] Alavi, Arsalan, and Reza Rahgozar, "A Simple Mathematical Method for Optimal Preliminary Design of Tall Buildings with Peak Lateral Deflection Constraint.” International Journal of Civil Engineering 17 (July 2018): 999–1006. doi:10.1007/s40999-018-0349-1.
[12] Poon, Dennis, Show-Song Shieh, Leonard Joseph, and Ching-Chang Chang, "Structural Design of Taipei 101, the World's Tallest Building.” Proceedings of the CTBUH 2004, Seoul Conference, Seoul, Korea, 271-278.
[13] Bozdogan, Kanat Burak, "A Method for Free Vibration Analysis of Stiffened Multi-Bay Coupled Shear Walls.” Asian Journal of Civil Engineering (Building and Housing) 7, No. 6 (November 2006): 639-649.
[14] Bozdogan, Kanat Burak, "An Approximate Method for Static and Dynamic Analysis of Symmetric Wall-Frame Buildings.” The Structural Design of Tall and Special Buildings 18, No. 3 (September 2007): 279-290. doi:10.1002/tal.409.
[15] Dym, Clive L., Harry E. Williams, "Estimating Fundamental Frequencies of Tall Buildings.” Journal of Structural Engineering, ASCE 133, No. 10 (October 2007): 1-5. doi:10.1061/(ASCE)0733-9445(2007)133:10(1479).
[16] Kaviani, Peyman, Reza Rahgozar, and Hamed Saffari, "Approximate Analysis of Tall Buildings using Sandwich Beam Models with Variable Cross-Section.” The Structural Design of Tall and Special Buildings 17, No. 2 (August 2007): 401-418. doi:10.1002/tal.360.
[17] Lee, Jaehong, Minsik Bang, and Jae"Yeol Kim, "An Analytical Model for High-Rise Wall-Frame Structures with Outriggers.” The Structural Design of Tall and Special Buildings 17, No. 4 (October 2007): 839-851. doi:10.1002/tal.406.
[18] Mohammadnejad, Mehrdad, Hasan Haji Kazemi, "Dynamic Response Analysis of Tall Buildings under Axial Force Effects.” Journal of Civil Engineering (Journal of School of Engineering) 31, No. 2 (2018): 39-53.
[19] Mohammadnejad, Mehrdad, and Hasan Haji Kazemi, "A New and Simple Analytical Approach to Determining the Natural Frequencies of Framed Tube Structures.” Structural Engineering and Mechanics 65, No. 1 (January 2018): 111-120. doi:10.12989/sem.2018.65.1.111.
[20] Mohammadnejad, Mehrdad, and Hasan Haji Kazemi, "Dynamic Response Analysis of a Combined System of Framed Tubed, Shear Core and Outrigger-Belt Truss.” Asian Journal of Civil Engineering (BHRC) 18, No. 8 (December 2017): 1211-1228.
[21] Kamgar, Reza, and Reza Rahgozar, "Determination of Optimum Location for Flexible Outrigger System in Tall Buildings with Constant Cross-Section Consisting of Framed Tube, Shear Core, Belt Truss and Outrigger System using Energy Method.” International Journal of Steel Structures 17, No. 1 (2017): 1-8. doi:10.1007/s13296-014-0172-8.
[22] Kamgar, Reza, and Reza Rahgozar, "Critical Excitation Method for Determining the Best Location of Belt Truss System in Tall Buildings.” Iranian Journal of Structures Engineering 4, No. 2 (2017): 76-88.
[23]Tavakoli, Reihaneh, Reza Kamgar, and Reza Rahgozar. "The Best Location of Belt Truss System in Tall Buildings Using Multiple Criteria Subjected to Blast Loading.” Civil Engineering Journal 4, no. 6 (July 4, 2018): 1338. doi:10.28991/cej-0309177.
[24] Kuang, Junshang, and S.C. Ng, "Dynamic Coupling of Asymmetric Shear Wall Structures: An Analytical Solution.” International Journal of Solids and Structures 38, No. 48-49 (November–December 2001): 8723-8733. doi:10.1016/S0020-7683(01)00052-X.
[25] Wang, Quanfeng, L.Y. Wang, "Estimating Periods of Vibration of Buildings with Coupled Shear Walls.” Journal of Structural Engineering, ASCE 131, No. 12 (December 2005): 1931-1935. doi:10.1061/(ASCE)0733-9445(2005)131:12(1931).
[26] Bozdogan, Kanat Burak, Duygu Öztürk, "An Approximate Method for Free Vibration Analysis of Multi-bay Coupled Shear Walls.” Mathematical and Computational Applications 12, No. 1 (2007): 41-50. doi:10.3390/mca12010041.
[27] Kamgar, Reza, and Peyman Rahgozar. "Reducing Static Roof Displacement and Axial Forces of Columns in Tall Buildings Based on Obtaining the Best Locations for Multi-Rigid Belt Truss Outrigger Systems.” Asian Journal of Civil Engineering 20, no. 6 (April 30, 2019): 759–768. doi:10.1007/s42107-019-00142-0.
[28] Malekinejad, Mohsen, and Reza Rahgozar, "Free Vibration Analysis of Tall Buildings with Outrigger-Belt Truss System.” International Journal of Earthquakes and Structures 2, No. 1 (March 2011): 89-107. doi:10.12989/eas.2011.2.1.089.
[29] Meirovitch, Leonard, "Computational Methods in Structural Dynamics.” The Netherland Rockville, Maryland, USA.
[30] Rahgozar, Reza, Zahra Mahmoudzadeh, Mohsen Malekinejad, and Peyman Rahgozar. "Dynamic Analysis of Combined System of Framed Tube and Shear Walls by Galerkin Method Using B-Spline Functions.” The Structural Design of Tall and Special Buildings 24, no. 8 (November 21, 2014): 591–606. doi:10.1002/tal.1201.
[2] Alavi, Arsalan, Peyman Rahgozar, and Reza Rahgozar, "Minimum"Weight Design of High"Rise Structures Subjected to Flexural Vibration at a Desired Natural Frequency.” The Structural Design of Tall and Special Buildings 27, No. 15 (October 2018): e1515. doi:10.1002/tal.1515.
[3] Davari, Seyed Mozafar, Mohsen Malekinejad, and Reza Rahgozar, "Static Analysis of Tall Buildings based on Timoshenko Beam Theory.” International Journal of Advanced Structural Engineering 11 (September 2019): 455–461.
[4] Kwan, Albert K.H., "Simple Method for Approximate Analysis of Framed-Tube Structures.” Journal of Structural Engineering, ASCE 120, No. 4 (April 1994): 1221-1239. doi:10.1061/(ASCE)0733-9445.
[5] Kwan, Albert K.H., "Shear Lag in Shear/Core Walls.” Journal of Structural Engineering, ASCE 122, No. 9 (1996): 1097-1104. doi:10.1061/(ASCE)0733-9445.
[6] Tavakoli, Reihaneh, Reza Kamgar, and Reza Rahgozar. "Seismic Performance of Outrigger-Braced System Based on Finite Element and Component-Mode Synthesis Methods.” Iranian Journal of Science and Technology, Transactions of Civil Engineering (July 31, 2019). doi:10.1007/s40996-019-00299-3.
[7] Tavakoli, Reihaneh, Reza Kamgar, and Reza Rahgozar. "Seismic Performance of Outrigger–belt Truss System Considering Soil–structure Interaction.” International Journal of Advanced Structural Engineering 11, no. 1 (February 1, 2019): 45–54. doi:10.1007/s40091-019-0215-7.
[8] Coull, Alexander, and Bishwanath Bose, "Simplified Analysis of Frame Tube Structures.” Journal of Structural Engineering, ASCE 101, No. 11 (1975): 2223-2240.
[9] Coull, Alexander, and Abdulla K. Ahmed, "Deflections of Framed-Tube Structures.” Journal of Structural Engineering, ASCE 104, No. 5 (1978): 857-862.
[10] Alavi, Arsalan, and Reza Rahgozar, "Optimal Stiffness Distribution in Preliminary Design of Tubed-System Tall Buildings.” Structural Engineering and Mechanics 65, No. 6 (March 2018): 731-739. doi:10.12989/sem.2018.65.6.731.
[11] Alavi, Arsalan, and Reza Rahgozar, "A Simple Mathematical Method for Optimal Preliminary Design of Tall Buildings with Peak Lateral Deflection Constraint.” International Journal of Civil Engineering 17 (July 2018): 999–1006. doi:10.1007/s40999-018-0349-1.
[12] Poon, Dennis, Show-Song Shieh, Leonard Joseph, and Ching-Chang Chang, "Structural Design of Taipei 101, the World's Tallest Building.” Proceedings of the CTBUH 2004, Seoul Conference, Seoul, Korea, 271-278.
[13] Bozdogan, Kanat Burak, "A Method for Free Vibration Analysis of Stiffened Multi-Bay Coupled Shear Walls.” Asian Journal of Civil Engineering (Building and Housing) 7, No. 6 (November 2006): 639-649.
[14] Bozdogan, Kanat Burak, "An Approximate Method for Static and Dynamic Analysis of Symmetric Wall-Frame Buildings.” The Structural Design of Tall and Special Buildings 18, No. 3 (September 2007): 279-290. doi:10.1002/tal.409.
[15] Dym, Clive L., Harry E. Williams, "Estimating Fundamental Frequencies of Tall Buildings.” Journal of Structural Engineering, ASCE 133, No. 10 (October 2007): 1-5. doi:10.1061/(ASCE)0733-9445(2007)133:10(1479).
[16] Kaviani, Peyman, Reza Rahgozar, and Hamed Saffari, "Approximate Analysis of Tall Buildings using Sandwich Beam Models with Variable Cross-Section.” The Structural Design of Tall and Special Buildings 17, No. 2 (August 2007): 401-418. doi:10.1002/tal.360.
[17] Lee, Jaehong, Minsik Bang, and Jae"Yeol Kim, "An Analytical Model for High-Rise Wall-Frame Structures with Outriggers.” The Structural Design of Tall and Special Buildings 17, No. 4 (October 2007): 839-851. doi:10.1002/tal.406.
[18] Mohammadnejad, Mehrdad, Hasan Haji Kazemi, "Dynamic Response Analysis of Tall Buildings under Axial Force Effects.” Journal of Civil Engineering (Journal of School of Engineering) 31, No. 2 (2018): 39-53.
[19] Mohammadnejad, Mehrdad, and Hasan Haji Kazemi, "A New and Simple Analytical Approach to Determining the Natural Frequencies of Framed Tube Structures.” Structural Engineering and Mechanics 65, No. 1 (January 2018): 111-120. doi:10.12989/sem.2018.65.1.111.
[20] Mohammadnejad, Mehrdad, and Hasan Haji Kazemi, "Dynamic Response Analysis of a Combined System of Framed Tubed, Shear Core and Outrigger-Belt Truss.” Asian Journal of Civil Engineering (BHRC) 18, No. 8 (December 2017): 1211-1228.
[21] Kamgar, Reza, and Reza Rahgozar, "Determination of Optimum Location for Flexible Outrigger System in Tall Buildings with Constant Cross-Section Consisting of Framed Tube, Shear Core, Belt Truss and Outrigger System using Energy Method.” International Journal of Steel Structures 17, No. 1 (2017): 1-8. doi:10.1007/s13296-014-0172-8.
[22] Kamgar, Reza, and Reza Rahgozar, "Critical Excitation Method for Determining the Best Location of Belt Truss System in Tall Buildings.” Iranian Journal of Structures Engineering 4, No. 2 (2017): 76-88.
[23]Tavakoli, Reihaneh, Reza Kamgar, and Reza Rahgozar. "The Best Location of Belt Truss System in Tall Buildings Using Multiple Criteria Subjected to Blast Loading.” Civil Engineering Journal 4, no. 6 (July 4, 2018): 1338. doi:10.28991/cej-0309177.
[24] Kuang, Junshang, and S.C. Ng, "Dynamic Coupling of Asymmetric Shear Wall Structures: An Analytical Solution.” International Journal of Solids and Structures 38, No. 48-49 (November–December 2001): 8723-8733. doi:10.1016/S0020-7683(01)00052-X.
[25] Wang, Quanfeng, L.Y. Wang, "Estimating Periods of Vibration of Buildings with Coupled Shear Walls.” Journal of Structural Engineering, ASCE 131, No. 12 (December 2005): 1931-1935. doi:10.1061/(ASCE)0733-9445(2005)131:12(1931).
[26] Bozdogan, Kanat Burak, Duygu Öztürk, "An Approximate Method for Free Vibration Analysis of Multi-bay Coupled Shear Walls.” Mathematical and Computational Applications 12, No. 1 (2007): 41-50. doi:10.3390/mca12010041.
[27] Kamgar, Reza, and Peyman Rahgozar. "Reducing Static Roof Displacement and Axial Forces of Columns in Tall Buildings Based on Obtaining the Best Locations for Multi-Rigid Belt Truss Outrigger Systems.” Asian Journal of Civil Engineering 20, no. 6 (April 30, 2019): 759–768. doi:10.1007/s42107-019-00142-0.
[28] Malekinejad, Mohsen, and Reza Rahgozar, "Free Vibration Analysis of Tall Buildings with Outrigger-Belt Truss System.” International Journal of Earthquakes and Structures 2, No. 1 (March 2011): 89-107. doi:10.12989/eas.2011.2.1.089.
[29] Meirovitch, Leonard, "Computational Methods in Structural Dynamics.” The Netherland Rockville, Maryland, USA.
[30] Rahgozar, Reza, Zahra Mahmoudzadeh, Mohsen Malekinejad, and Peyman Rahgozar. "Dynamic Analysis of Combined System of Framed Tube and Shear Walls by Galerkin Method Using B-Spline Functions.” The Structural Design of Tall and Special Buildings 24, no. 8 (November 21, 2014): 591–606. doi:10.1002/tal.1201.
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