Vibration of High-Rise Buildings (A Case Study of Unity Hall, KNUST)

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Very often, building contractors and civil engineers design and build high-rise buildings oblivious of the maximum possible displacement that each floor can displace due to vibrations. Apparently, building contractors and engineers lay much emphasis on the foundation to the neglect of slabs of each floor of the high-rise building. If the maximum displacement of a high-rise building is known before hand, building experts can run simulations on the building to know the safest amount of load to use on each' slab construction that can minimize cost, and yet provide the needed resistance it is designed for. The structural and sectional drawings of Unity Hall, my case study, were analyzed and the volume of each floor slab computed. The result was multiplied by the density of reinforced concrete to obtain the mass of each slab of the building. The column constants were also computed for use in our problem solving process. The results were then formulated as a symmetric positive generalized eigenvalue problem in terms of mass and stiffness matrices under free vibration and a tridiagonal system under forced vibration. The problem was solved to obtain the maximum possible displacement of each floor of the building. A second solution is obtained by including live load to each floor of the building. The Numerical stability theorem was used to confirm the stability or otherwise of the algorithms employed in this project. Under free vibration and live load absent, all displacement values fall below 0.5m which shows that the model maintains its linearity under free vibration. However, when live load is present, all displacement values fall below 1.6m, showing that the presence of live load reduces linearity of the model under free vibration. This goes to suggest that the linearity of the model is compromised when live load is included under free vibration. Under forced vibration, our model maintains its linearity up to an earthquake of magnitude 1.0 when live load is absent and 1.25 when live load is present This also confirms that, the presence of students has some effect on the linearity of the model under forced vibration. Above the stipulated magnitudes of 1.0 and 1.25 when live load is absent and present respectively, displacements exceeds 0.5m and 1.6rt) meaning the linearity assumption becomes no longer tenable and failure of the building becomes eminent. In spite of the stated magnitude thresholds, it must be noted that if the frequency of the incoming wave coincides with one of the natural frequencies of the building, the amplitude of displacement becomes large, signaling the occurrence of resonance. Where as displacements below 0.5m when live load absent, and 1.6m when live load present, will cause the building to crack, the resonance scenario will cause the buildings to fail woefully, even to the extent of collapse. 
A thesis submitted to the Department of Mathematics, Kwame Nkrumah University of Science and Technology in partial fulfilment of the requirements for the degree of Master of Science.