Abstract
In this paper, closed form approximate solutions to the equations of motion for coupled shear walls are developed using Ritz-Galerkin method. Hamilton’s principle is used to derive the equations of motion. These equations and solutions were developed for two different cases, one for the coupled shear wall on fixed foundation and the other for coupled shear wall on flexible foundation. Through literature review, it is identified that previous studies addressed only the free vibration of coupled shear wall system without considering external load. The main focus of this paper is to develop equations of motion with external load applied to coupled shear walls on both fixed and flexible base using variational approach. Then cast equations of motion and corresponding boundary conditions into non-dimensional form. The solution of equations of motion is developed through the use of the Ritz-Galerkin technique. Thus attempts were made to develop equations of motion considering a driving force, p(x,t) on the structure. By using selected shape functions for the longitudinal and lateral defelections, a matrix eigenvalue equation is derived for both cases yielding closed form approximate solution.
Highlights
In tall building systems, reinforced concrete coupled shear walls are widely used to provide lateral resistance against the dynamic loads arising from earthquakes, wind and blast loads
Results of Ritz-Galerkin method for case II given by the set of matrix equations
It is shown that the equations of motion developed for fixed base using Hamilton’s principle is consistent with the sixth order differential equation developed by Mukherjee and Coull
Summary
In tall building systems, reinforced concrete coupled shear walls are widely used to provide lateral resistance against the dynamic loads arising from earthquakes, wind and blast loads. Now-a-days, it has become a standard in the design and analysis of coupled shear walls or shear wall systems to use the continuum medium technique One benefit of this approach is that the differential equation developed, under the continuum approximation, will have a closed form solution. This approach is similar to that developed by Skattum (1971), except that he did not consider the case of coupled shear walls on flexible base. It will be shown that the Hamilton’s principle, or the variational method, applied to the first case will exactly reproduce the dynamical equation developed by Mukherjee and Coull (1973), which used a different approach to compute the free vibration state of coupled shear walls on a fixed foundation. The equation of the strain energy due to flexible foundation is provided by Mukherjee and Coull (1974)
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