Global vibration analysis of symmetric and asymmetric high rise buildings
This thesis presents two global analysis approaches to the calculation of the natural frequencies of high rise buildings. The structures are proportional and their component members are repeated at each storey level unless there is a step change of properties. Within this scope many geometric configurations can be encompassed, ranging from uniform structures with doubly symmetric floor plans to doubly asymmetric ones comprising plane frame and wall structures running in two orthogonal directions. The first method utilises a continuum element approach in which the structure is divided into segments by cutting through the structure horizontally at those storey levels corresponding to changes in storey properties. A typical segment is then replaced by an appropriate substitute beam that has uniformly distributed mass and stiffness. Subsequently, the governing differential equations of the substitute beam are formulated using the continuum approach and posed in the form of a dynamic member stiffness matrix that is exact to small deflection theory. Since the formulation allows for the distributed mass and stiffness of the member, it necessitates the solution of a transcendental eigenvalue problem. The required natural frequencies are thus determined using a cantilever model in conjunction with the Wittrick-Williams algorithm, which ensures that no natural frequencies can be missed. In addition, a two step process has been developed for certain asymmetric structures in which the natural frequencies corresponding to coupled motion between the planes of vibration can be obtained from the equivalent uncoupled ones through a simple cubic relationship. This enables coupled, three-dimensional vibration problems to be solved very efficiently using a two dimensional approach. The second method utilises the Principle of Multiples which, when applicable, enables any frame, regardless of the number of storeys or bays, to be simplified to an equivalent one bay frame, that has precisely the same natural frequencies. If the original frame does not fully satisfy the Principle, the same process can still be utilised, but the resulting substitute frame will yield approximate frequencies, although they will normally be acceptable to engineering accuracy. Like the first method, it can also be used for the vibration analysis of asymmetric, three-dimensional frame and wall-frame structures in a two-step procedure. First the analogous uncoupled system is analysed using substitute frames, then the relationship between the uncoupled and coupled responses is imposed through a cubic equation. Both of the above methods assume rigid floor diaphragms and require a knowledge of the building's static eccentricity at each storey level. The current methods of calculating this are cumbersome and even the definitions are open to dispute. A practical method of calculation is therefore presented and a small parametric study enables recommendations to be made. Overall, the proposed methods require little effort, offer clear and concise output and can sometimes yield solutions of sufficient accuracy for definitive checks, but more usually provide engineering accuracy for intermediate checks during tasks such as scheme development or remedial work. This claim is supported by the results of extensive parametric studies undertaken for this thesis. In all examples, the results from the proposed methods have been compared with the results of a full finite element analysis of the original structure obtained using the vibration programme ETABS. The exercise confirms that the proposed methods can yield results of sufficient accuracy for engineering calculations.
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