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: https://www.isr.umd.edu/~austin/aladdin.d/matrix-appl-building.html

Enter number of stories

Enter mass, stiffness and damping of story one

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.

√ (

)

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 1st story amplitude to 2

nd story amplitude

in 4th 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.

Program calculates amplitude ratios for 1st mode, 0.8164965812 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 2nd

mode is 0.5773502697. 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.

Select “No” Select “Yes” Select “Quake”

Select “Ground Motion Function” Select “No”

√ √ √

√

√ √

( √ (

√ )

)

(√ (√ )

)

( √ (

√ )

)

(√ (√ )

)

piecewise(0 <= t and t < 10,1.077496048*sin(0.7071067815*t)-0.2142857145*sin(2.*t)-0.235702260

4*sin(1.414213563*t),10 <= t,0.3203027701*sin(-7.071067815+0.7071067815*t)+0.4160065756*

cos(-7.071067815+0.7071067815*t)+0.9735699069e-1*sin(-14.14213563+1.414213563*t)-

0.8354180911e-1*cos(-14.14213563+1.414213563*t))

piecewise(0 <= t and t < 10,0.5387480239*sin(0.7071067815*t)-0.3571428571*sin(2.*t)+

0.2357022601*sin(1.414213563*t),10 <= t,0.1601513851*sin(-7.071067815+0.7071067815*t)+

0.2080032878*cos(-7.071067815+0.7071067815*t)-0.9735699059e-1*sin(-14.14213563+

1.414213563*t)+0.8354180902e-1*cos(-14.14213563+1.414213563*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)+

0.2080032878*cos(-7.071067815+0.7071067815*t)+0.1947139813*sin(-14.14213563+1.414213563*

t)-0.1670836181*cos(-14.14213563+1.414213563*t))

piecewise(0 <= t and t < 10,1.077496048*sin(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+

1.414213563*t)+0.1670836180*cos(-14.14213563+1.414213563*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.

∑

∑

The first part of piecewise functions is related to forced vibration and shows a complete agreement

with textbook results.

0.223158935e-1*cos(1.682507066*t)-15.45042692*cos(0.2846296764*t)+15.00000004-

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

e-2*cos(1.918985947*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 u0 and v0 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.

1 2 3 4 5 6

h 3 3 3 3 4 5

m 2E3 3E3 3E3 4E3 4E3 4E3

k 1E6 2E6 2E6 2E6 3E6 3E6

c 100 100 150 150 200 250

u0 0.01 - -0.02 0.01 0.01 -

v0 -1.00 - - 0.50 -1.00 1.00

F 1E3sin(t0.5

) - - 1E3tanh(t1.5

) - 1E3ln(t3)/(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

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 (11.87 Ton) and 396189.38 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