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Naeemi S R, Saleh Jalali R. Studying of some limitations of modal analysis versus D’Alembert solution in analysis of shear beam under impulsive loads. MCEJ 2022; 22 (4) :95-110
URL: http://mcej.modares.ac.ir/article-16-56370-en.html

Using modal analysis is a lot easier and more widespread among structures, but the important question is about the number of modes should be considered in the modal analysis method to reach an answer with an inevitable error but in logical tolerance. In this regard, the ratio of the dominant period of the earthquake to the main period of the structure is used as a criterion for selecting the number of modes in the modal analysis method. On the other hand, although the maximum displacement of the structure occurs above it, but when the period of the pulse is less than the main period of the structure, due to wave motion along the structure, the maximum shear strain can occur not only at the base but also in other places along the structure. In this paper, some limitations of modal analysis versus Dchr('39')Alembert solution have been studied in analysis of shear beam under impulsive loads. For this purpose, the structure is modeled with a shear beam with linear material and zero damping, and it is analyzed by discrete (modal analysis) and continuous (Dchr('39')Alembert solution) methods. The time response of modal analysis has been done by the fourth-order Runge-Kutta method. The shear beam is subjected to short, medium, and long period half-sine pulses, relative to the main period of the structure, as well as two near-field earthquakes with distinct pulse. The envelope of maximum induced displacement and shear strain (drift) along the beam have been selected to compare the two methods. The necessary number of modes in modal analysis are determined in such a way that its difference with the exact method (Dchr('39')Alembert solution) would be in acceptable range. For shear beam with linear material and zero damping, as it is expected, the results indicate that for convergence of shear strain (drift) response to the exact solution more number of modes are needed than convergence of displacement response in the modal analysis. Under short period pulse , when the ratio of the period of the pulse or the predominant period of earthquake to the main period of the beam is less than , if the minimum number of modes in modal analysis would be 20 and 50 modes for displacement and shear strain, respectively, then the percentage of error of envelope of maximum induced displacement and shear strain (drift) in beam, calculated by modal analysis, would be less than 10 percent, respect to Dchr('39')Alembert solution. Under medium period pulse , when the ratio of the period of the pulse or the predominant period of earthquake to the main period of the beam is greater than  and less than , for having ten percent difference between two methods of analyses, the necessary number of modes in modal analysis of beam would be  and  modes for displacement and shear strain, respectively. For the beam under long period pulse , when the ratio of the period of the pulse or the predominant period of earthquake to the main period of the beam is greater than , the necessary number of modes in modal analysis would be 1 and 5 modes for displacement and shear strain, respectively.