Linear Modal Time History Analysis V1.1

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Linear Time History Analysis of MDOF Structure by Mode Superposition Methodusing Newmarks MethodCarleton University Mostafa Tazarv Graduate Student Version 1.1 (April 2011)

Cite as: Tazarv, M., Linear Time History Analysis of MDOF Structures by Mode Superposition Method, available at: http://alum.sharif.edu/~tazarv/

Modal Time History Analysis

Mostafa Tazarv

Introduction Structural dynamic is a mandatory graduate level course for structural/earthquake engineering student all around the world. One of the most important topics of this course is to solve modal equations of motion of a Multi Degree of Freedom (MDOF) structure by a numerical method such as Nemarks method and Wilson- method. Modal response should be assembled to obtain each DOF response. Applied load can be a base excitation earthquake or time-dependent loads on stories. Here, I will introduce a MATLAB function which can do a time history analysis of an n-DOF structure with a certain Number of Modes (nom). Then, I will show how to use this function with two examples, one excited by half-cycle harmonic on two stories and another excited by Elcentro earthquake. This function will also be verified by an example detailed in Dynamic of Structure by Chopra.

Time History Response by Newmark Method: NM I tried to write a self-explanatory m-file. Therefore, Ill explain a bit more the important part of the function called NM in this section. A new feature is to give you the option to specify the number of modes you want to consider in mode superposition analysis. For example, there is a 100-DOF structure (the size of mass and stiffness matrixes are 100 100). However, you want to do the modal analysis only for first 10 modes not all the modes which is very common in real situation. In this function to solve equation of motions for different modes, Newmark Linear Method has been used. We can decompose time-dependent applied force, , into two components where F is spatial distribution of load on DOFs and f(t) is time-variant component of load. By modifying inputs F (a vector) and f(t), you can analyze the structure for either earthquake (base-seismic-excitation) or time-dependent load applied to different stories. I will show it in examples later. That's your responsibility to organize eigenvectors () and eigenvaluses ( in which frequencies are sorted from smallest to greatest ( f1< f2< f3