Modal analysis of shear buildings

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Modal analysis of shear buildings A comprehensive modal analysis of an arbitrary multistory shear building having rigid beams and lumped masses at floor levels is obtained.

1 Modal analysis of shear buildings A comprehensive modal analysis of an arbitrary multistory shear building having rigid beams and lumped masses at floor levels is obtained. Angular frequencies (rad/sec), frequencies (Hz), periods, modal stiffness, modal damping, modal damping ratio and standard normalized amplitude ratios (for unit specific modal masses) are presented. The program is able to analyze free vibration for initial displacement and/or velocity of floors as well as excited vibration including support movement or acceleration and forced vibration. Generalized series expansion with the high accuracy can be used in the case of complicated load functions to solve ODEs of motion. The diagrams of displacements and shear forces of stories as well as base overturning moment in an arbitrary interval are plotted. Moreover, the maximum of absolute amount and extremum amount of mentioned functions are calculated. The results like the mass participation factors and spatial distribution of earthquake can be exported to Excel. Example I: The eigenvalues, natural periods of vibration and modal shapes for a four story shear building are computed. Source: Enter number of stories Enter mass, stiffness and damping of story one

and damping of story four Results show a complete agreement

2 Enter mass, stiffness and damping of story two Enter mass, stiffness and damping of story three Enter mass, stiffness and damping of story four Results show a complete agreement with those of are available in previously performed exercises.

Example II: A five story shear building with lumped mass m at each floor and same story stiffness k for all stories is considered.

3 ( ) The ratio of amplitudes, for example is calculated as follows: The results can be exported to Excel. If you don t want to export data into Excel work sheet, just close the dialogue box. The next levels of calculations will be continued by program. Example II: A five story shear building with lumped mass m at each floor and same story stiffness k for all stories is considered. Source: Dynamics of structures, Theory and applications to earthquake engineering, Anil K. Chopra, Third Edition, Chapter 12, Page 483. Mass, stiffness and damping of all stories are equal to 1, 1 and 0, respectively. Top story number is equal to 5, whereas program takes number of top story equal to 1.

The ratio of amplitudes, for example (the ratio of 1 st story amplitude

Example III: The modal expansion of the effective earthquake force

4 The ratio of amplitudes, for example (the ratio of 1 st story amplitude to 2 nd story amplitude in 4 th mode) is calculated. Example III: The modal expansion of the effective earthquake force distribution associated with horizontal ground acceleration for a two story shear building is calculated. Source: Dynamics of structures, Theory and applications to earthquake engineering, Anil K. Chopra, Third Edition, Chapter 13, Page 515.

Select No Select Yes Select Quake Program calculates amplitude ratios for 1 st mode, 0.

The reason is that the program computes amplitude ratios in standard form to have unit

It is noteworthy to mention that top story number is equal to 2, whereas program takes number

The computed modal expansion of masses matrix, SP, is same as the corresponding matrix in

The difference between mass participation factors comes from the difference between

5 Select No Select Yes Select Quake Program calculates amplitude ratios for 1 st mode, times of those are calculated in textbook. The reason is that the program computes amplitude ratios in standard form to have unit specific modal masses. The amplitude ratios coefficient for 2 nd mode is It is noteworthy to mention that top story number is equal to 2, whereas program takes number of top story equal to 1. The computed modal expansion of masses matrix, SP, is same as the corresponding matrix in textbook. The difference between mass participation factors comes from the difference between normalized amplitude ratios. Example IV: The floor displacements and story shears of Ex. III are derived. It is assumed that the ground motion function is with 10 second duration. Source: Dynamics of structures, Theory and applications to earthquake engineering, Anil K. Chopra, Third Edition, Chapter 13, Page 519.

6 Select Ground Motion Function Select No

7 ( ( ) ) ( ( ) ) ( ( ) ) ( ( ) ) piecewise(0 <= t and t < 10, *sin( *t) *sin(2.*t) *sin( *t),10 <= t, *sin( *t) * cos( *t) e-1*sin( *t) e-1*cos( *t)) piecewise(0 <= t and t < 10, *sin( *t) *sin(2.*t) *sin( *t),10 <= t, *sin( *t) *cos( *t) e-1*sin( *t) e-1*cos( *t)) Top story number is equal to 2, whereas program takes number of top story equal to 1.The first part of piecewise functions is related to forced vibration and shows a complete agreement with textbook results. ( ( ) ) ( ( ) )

( ( ) ) ( ( ) ) piecewise(0 <= t and t < 10,0.5387480241*sin(0.7071067815*t)+0.1428571426*sin(2.*t)- 0.4714045205*sin(1.414213563*t),10 <= t,0.1601513850*sin(-7.071067815+0.7071067815*t)+ 0.

7142857142*sin(2.*t)+ 0.4714045202*sin(1.414213563*t),10<= t,0.3203027702*sin(-7.071067815+0.7071067815*t)+ 0.4160065756*cos(-7.071067815+0.7071067815*t)-0.1947139812*sin(-14.14213563+ 1.

8 ( ( ) ) ( ( ) ) piecewise(0 <= t and t < 10, *sin( *t) *sin(2.*t) *sin( *t),10 <= t, *sin( *t) *cos( *t) *sin( * t) *cos( *t)) piecewise(0 <= t and t < 10, *sin( *t) *sin(2.*t) *sin( *t),10<= t, *sin( *t) *cos( *t) *sin( *t) *cos( *t)) Example V: The modal expansion of the effective earthquake force distribution and base overturning moment associated with horizontal ground acceleration for Ex. II are derived. Source: Dynamics of structures, Theory and applications to earthquake engineering, Anil K. Chopra, Third Edition, Chapter 13, Page 526.

Select Ground Acceleration Function Select Yes The unit ground acceleration is imposed on frame. The height of all stories are assumed equal to 1.

00000004-0.924456097e-1*cos(1.309721468*t)+0.5246409809*cos(0.8308300260*t)-0.408437860 e-2*cos(1.

9 Select Ground Acceleration Function Select Yes The unit ground acceleration is imposed on frame. The height of all stories are assumed equal to 1. The first part of piecewise functions is related to forced vibration and shows a complete agreement with textbook results e-1*cos( *t) *cos( *t) e-1*cos( *t) *cos( *t) e-2*cos( *t) Example VI: Determine the maximum amounts of base shear and base overturning moment for a six story shear building. The geometrical and mechanical properties as well as initial conditions and external concentrated forces at roofs are presented in the table below. All parameters have SI units. The parameters u 0 and v 0 are initial displacement and initial velocity, respectively. The imposed forces at roof levels and the corresponding time intervals are presented by F and T, respectively.

10 h m 2E3 3E3 3E3 4E3 4E3 4E3 k 1E6 2E6 2E6 2E6 3E6 3E6 c u v F 1E3sin(t 0.5 ) - - 1E3tanh(t 1.5 ) - 1E3ln(t 3 )/(1+t -2 ) T [2,5] - - [1,4] - [2,4] If you don t want to change preferences, let checkboxes be unmarked The default name is Modal Analysis For stories without initial conditions, just click the enter button

11 Select appropriate amounts to accommodate diagrams. For stories without force, just click the exit button.

The maximum amount of base shear force and maximum amount of base overturning moment are 116371.85 N (11.

It is possible to calculate the maximum amount of displacements and shear forces for other stories in an arbitrary

12 The maximum amount of base shear force and maximum amount of base overturning moment are N (11.87 Ton) and N.m (40.4 T.m), respectively. It is possible to calculate the maximum amount of displacements and shear forces for other stories in an arbitrary time interval. If you have any suggestions to make, please let me know. a_heydari@alum.sharif.edu Abbas Heydari (PhD) Department of Civil and Structural Engineering, Sharif University of Technology

3. MDOF Systems: Modal Spectral Analysis

3. MDOF Systems: Modal Spectral Analysis Lesson Objectives: 1) Construct response spectra for an arbitrarily varying excitation. 2) Compute the equivalent lateral force, base shear, and overturning moment