You are doing a dynamic analysis of the frame below for motion in the x-directio

Question

You are doing a dynamic analysis of the frame below for motion in the x-direction. In order to have more open space on the first story, larger columns were used as indicated in the figure. You may assume that the first-story columns are fixed at the base and that the floor systems are rigid enough, relative to the columns, that there is no rotation at the ends of the columns. (a) When performing an eigenvalue analysis for the building, Mathcad gives the following results (lambda are the eigenvalues, v are the eigenvectors): lambda = (2474.5 546.34)^1/s^2 v = (0.6446 0.5101 -0.7646 0.8601) Find the frequency and period of the building for the second mode. (b) On the exam handout, the RISA mode shape for the second mode is shown. Verify that the RISA mode shape is the same as the mode shape found with Mathcad. (c) Using the response spectrum on the exam handout, find the maximum displacement of the roof using the second mode. The second floor has a mass of 4.0 kip middot sec^2/ft. The modal masses are m_1 = 2.52 kip middot sec^2/ft (first mode) and m_2 = 2.83 kip middot sec^2/ft (second mode).

Explanation / Answer

a) Let w be the angular velocity of second mode

w2 = 546.34 (which is the second eigen value)

w = sqrt(546.34) = 23.374 rad/s

frequency of oscilattion = 23.374/(2*pi) = 3.72 Hz

Period of oscillation = 1/3.72 = 0.269 second

for parts b and c, RISA mode shape and reponse spectrum are needed

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